Course Notes for OSU ME 8260 Advanced Engineering Acoustics

Transcription

1 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Course Notes for OSU ME 86 Advanced Engineering Acoustics Prof. Ryan L. Harne* Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43, USA * harne.3@osu.edu Last modified: :44

2 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Table of contents Course introduction 6. Scope of acoustics 6. Mathematical notation 6 Mathematics review 7. The harmonic oscillator 7. Initial conditions 8.3 Energy of vibration 9.4 Complex exponential method of solution to ODEs.5 Damped oscillations.6 Forced oscillations 4.7 Mechanical power 9.8 Linear combinations of simple harmonic oscillations.9 Fourier's theorem, series, and transform 3 3 Wave equation analysis 7 3. Transverse waves on a string General solution to the one-dimensional wave equation Wave propagation at string boundaries Forced vibration of a semi-infinite string Forced vibration of a finite length string Normal modes of the fixed-fixed string Longitudinal waves along a one-dimensional rod Transverse waves along a beam Dispersion, phase velocity, group velocity Forced excitation of beams Point force excitation of infinite beam Arbitrary force excitation of finite beam Waves in membranes Free vibration of a rectangular membrane with fixed edges Free vibration of a circular membrane 6

3 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University 3.7 Vibration of thin plates Free vibration of thin plates Forced vibration of thin plates Impedance and mobility functions of structures 66 4 Acoustic wave equation Consolidating the components to derive the one-dimensional acoustic wave equation 7 4. Harmonic, plane acoustic waves Impedance terminology clarification Plane progressive shock waves Acoustic intensity Harmonic, spherical acoustic waves Spherical wave acoustic intensity and acoustic power Comparison between plane and spherical waves Decibels and sound levels Point source and source strength Acoustic reciprocity 85 5 Reflection and transmission Normal incidence wave propagation at a fluid interface Normal incidence wave propagation through a fluid layer Oblique incidence wave propagation at a fluid interface Oblique incidence wave propagation at a thin partition interface: the mass law Method of images Acoustic metamaterials 5.6. One-dimensional monatomic lattice 5.6. One dimensional diatomic lattice Metamaterial continua Computational methods of analysis 8 6 Sound radiation from structures 6. Monopole and point source 6. Two point sources 3

4 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University 6.. Dipole Line source in the free field Integral methods of analysis for radiating acoustic sources Directivity Rigid, circular, baffled piston source On-axis acoustic pressure from baffled piston source Far field acoustic pressure from baffled piston source Generalization of the Rayleigh's integral for planar, baffled acoustic sources in the far field Radiation impedance Acoustic arrays Line array Arrays of directional sources Spatial Fourier transform Line source far field acoustic pressure response by spatial Fourier transform method Forced, baffled rectangular plate far field acoustic pressure response by spatial Fourier transform method Arrays and far field acoustic pressure design by spatial Fourier transform method 46 7 Enclosures and waveguides Normal modes in acoustic enclosures Rectangular enclosure Cylindrical enclosure Acoustics waveguides of constant cross-section Rectangular waveguide Pipes Rigid termination of pipes Short, closed volume Open-ended pipes Helmholtz resonator Filtering properties of lumped acoustic impedances Low pass filter 65 4

5 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University 7.4. High pass filter Band stop filter 67 5

6 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Course introduction Acoustics is defined as "a science that deals with the production, control, transmission, reception, and effects of sound", while sounds are defined as "mechanical radiant energy that are transmitted by longitudinal pressure waves in a material medium (as air) and are the objective cause of hearing" [Merriam-Webster dictionary]. Alternative definitions recognize that the context of acoustics is broader than merely "sound". Contemporary use of the term acoustics now often follows the definition denoting "a science [dealing with] the generation, transmission, and reception of energy as vibrational waves in matter" [] where matter may be solid, gas, or fluid. This course will adopt the contemporary definition to undertake rigorous treatments of theories pertaining to wave propagation in matter. Yet, a greater balance of attention will be given to problems of fluid-borne waves, although they may be generated by waves or vibrations of structures.. Scope of acoustics Acoustics is an encompassing science. Lindsay's wheel of acoustics is shown in Figure, and it is an exemplary picture of how acoustical subjects impact myriad scientific disciplines and even the liberal arts. The shared core principles, at the center of the wheel, is what unites the diverse disciplines. This course will dive deep into these fundamental principles of physical acoustics and will consider many of the extensions of such foundations as applied throughout engineering practices.. Mathematical notation In the course notes, the following mathematical notations will be used consistently. Figure. Lindsay's wheel of acoustics, adapted from R. B. Lindsay, J. Acoust. Soc. Am. 36, 4 (964). We use j to denote the imaginary number, j =. j t Bold mathematics denote complex numbers, for example k = 8. + j.3, d = De ω. If written out by hand, we use an underline to denote a complex number, for example k = 8. + j.3, d j t = De ω. Mathematics with overbar denote vectors, for example v =.i +. j +.3k, u= ue x x + ue y y. u = ui+ u j e ω. Boldandbarred mathematics denote complex vectors, for example ( ) j t x y 6

7 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Mathematics review. The harmonic oscillator We first review the foundations of mechanical vibrations analysis. Consider Figure which shows the schematic of a mass-spring oscillator having one dimension of motion, x (SI units [m]), with respect to the fixed ground. There are no gravitational influences to account for in this context. The spring exerts a force on the mass m (SI units [kg]) according to the deformation of the spring with respect to the ground. Assuming that the spring has no undeformed length, the spring force is therefore f = sx where the spring stiffness s has units [N/m]. Applying Newton's second law of motion, we determine the governing equation of motion for the mass d x m = f = sx ( ) dt Rearranging terms and using the notation that that overdot indicates the d / dt operator, we have that mx + sx = ( ) Based on the fact that stability conditions require m and s to be >, we can define a new term ω = s m such that the governing equation () becomes x / x+ ω = ( 3 ) Figure. Mass-spring oscillator. Solution to the second-order ordinary differential equation (ODE) (3) determines the displacement x of the mass for all times. Solving ODEs is commonly accomplished by assuming trial solutions and verifying their correctness. By engineering intuition, such as one's visual conception of a mass at the end of a Slinky and how that object may move, we hypothesize that a suitable trial solution to (3) is x= A cosγ t ( 4 ) By substitution, we find that (4) is a solution to (3) when γ = ω. Likewise, we also discover that an alternative trial solution to (3) is x= A sinω t ( 5 ) 7

8 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University When an ODE has multiple solutions, the total solution is the superposition of the individual solutions. Thus, the general solution to (3) is ( ) cosω sinω xt = A t+ A t ( 6 ) The term ω is the natural angular frequency and it has units [rad/s]. Thus, the mass will exhibit oscillatory motion at ω. There are π radians in a cycle, which gives that the frequency in cycles per second is ω /π. We refer to this as the natural frequency f / = ω π which has units [Hz = cycles/s]. In practice, we observe the mass oscillation over a duration of time. The period T of one oscillation is therefore T = / f [s].. Initial conditions To determine the unknown constants A and A, two initial conditions are required because this is a secondorder ODE. Thus, if the mass displacement at an initial time velocity is xt ( = ) = x = u t = is xt ( ) x we can determine the constants by substitution. = = and the initial mass x = A cosω + A sinω = A ( 7 ) u = ω A sinω + ω A cosω = ω A ( 8 ) Using this knowledge, the general solution to (3), and the mass displacement described for all time, is u xt = x t+ sinω t ( 9 ) ( ) cosω ω Alternatively, we express these two sinusoidal functions using an amplitude and phase ( ) cos[ ω φ] xt = A t+ ( ) where A= x + ( u / ω ) / tan φ = u / ω x. and The time derivative of displacement is velocity ( ) = ( ) = ω sin[ ω + φ] = sin[ ω + φ] xt ut A t U t ( ) where we define the speed amplitude U = ω A. Likewise, the acceleration is ( ) ω cos[ ω φ] at = U t+ ( ) From these results, we see that the velocity leads the displacement by 9, and that the acceleration is 8 out-of-phase with the displacement, Figure 3. This response is fully dependent upon the initial conditions of displacement and velocity. Such response is referred to as the free response or free vibration. 8

9 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University x displacement [m] velocity [m/s] acceleration [m/s ] response time [s] Figure 3. Free vibration response of mass-spring oscillator..3 Energy of vibration From physics, the potential energy associated with deforming a spring is the integration of the spring force across the path of deformation Ep = sx x = sx x d ( 3 ) Here, we assume our integration constant is zero, which means that our minimum of potential energy is set to zero. If we substitute () into (3), we find that = cos [ ω + φ] ( 4 ) E p sa t By definition from physics, the kinetic energy of the mass is Ek = mx = mu ( 5 ) Likewise, by substitution of () into (5) we have = sin [ ω + φ] ( 6 ) E k mu t The total energy of this dynamic system is the summation of potential and kinetic energies E = Ep + Ek = mω A = mu = sa ( 7 ) E p where we have used the definition ω = s m and the identity / sin α cos α + =. The total energy (7) is independent of time, which is a statement of the conservation of energy. The total energy is equal to both the peak elastic potential energy (when kinetic energy is zero) and the peak kinetic energy (when potential energy is zero). Figure 4 illustrates the instantaneous exchange between potential and kinetic energies over the course of the mass-spring system oscillation. 9

10 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University energy [J] total energy potential energy kinetic energy time [s] Figure 4. Exchange of energy between potential and kinetic forms as mass-spring oscillates..4 Complex exponential method of solution to ODEs A useful technique in harmonic analysis of engineering problems is the complex exponential form of j t solution. In this course, we will use the engineering notation e ω where the imaginary number is j =. This notation is in contrast to the notation used in physics and mathematics j t e ω. When confronted with a derivation performed according to the physics notation, to convert to the engineering notation requires taking the complex conjugate. In other words, one replaces the j by j everywhere to recover the engineering notation. Table consolidates the primary acoustics textbook references which use the complex exponential notations. One observes a similar number of books in the engineering "camp" as in the "physics/math" camp. Note that the underlying principles of acoustics are derived from physics, so this result should not be surprising. Table. Primary acoustics textbook usage of the complex exponential notation. engineering physics/math j t e ω notation [] [] [3] [4] [5] j t e ω notation [6] [7] [8] [9] [] Recalling the equation (3), the more general solution method is to assume γt ( t) = e x A ( 8 ) where we use the boldface type to represent complex numbers, with the exception of the explicit description of the imaginary number j =. By substituting (8) into (3), one finds that γ = ω and thus γ = ± jω. Since two solutions are obtained, the general solution to (3) by the complex exponential notation is the superposition of both terms associated with γ = ± jω jωt jωt ( t) = e + e x A A ( 9 ) Recall the initial conditions, x ( ) = x and ( ) = x = u x. By substitution we find x = A + A ( )

11 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University u = jω A jω A ( ) from which we find A = x u j ω, and u A = x + j ( ) ω We see that the unknown constants A and A are complex conjugates. Substituting () into (9) and ± using Euler's identity e θ = cosθ ± sinθ, we find u jωt u jωt u x = x j e + x + j e = xcosωt+ sinωt ( 3 ) ω ω ω The result in (3) is the same as (9). Thus, although we used a complex number representation of the assumed solution (8), satisfying the initial conditions which are both real entailed that the complex components of the response were eliminated. In general, there is no need to perform this elimination process, because the real part of the complex solution is itself the complete general solution of the real differential equation. For example, we could have alternatively assumed only j t ( t) = e ω x A ( 4 ) Given that, in general, A = a + jb, we have that [ ] = ω ω Re x acos t bsin t ( 5 ) Then we satisfy the initial conditions x u = a ( 6 ) = ω b ( 7 ) Thus, a= x and b= u / ω. Then by substitution of a and b into (5), we again arrive at (3) confirming the conclusion that the real part of the complex solution is the complete general solution. We use complex exponential assumed solution forms similar to (4) throughout this course. According to the relation between displacement, velocity, and acceleration, we summarize u= jω Ae = jω x ( 8 ) jωt a= ω Ae ω = ω x ( 9 ) j t j t The term e ω is a unit phasor that rotates in the complex plane, while A is a complex function that modifies the amplitude of the rotating amplitude and shifts it in phase according to the complex component of A.

12 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Thus, A = a + b and tan / φ = b a such that we have A j( t ) cos[ ωt φ] Re e ω + + = φ A. Figure 5 illustrates how the complex exponential form representation may be considered in the complex plane. The magnitude-scaled phasor rotates in the plane with angular rate ω as time changes, while the Real contribution of the phasor oscillates between positive to negative values. By considering (8), the velocity phasor leads the phasor rotation of displacement by 9 while the acceleration phasor is out-of-phase with the displacement phasor by 8. The phasor rotate in the counterclockwise direction, for ω > and positive increase in time t. Figure 5. Physical representation of a phasor..5 Damped oscillations All real systems are subjected to phenomena that dissipate kinetic energy as time increases. In general, damping phenomena decrease the instantaneous amplitude of free oscillations as time increases. Unlike forces characterized with potential energy and inertial forces associated with kinetic energy, damping forces are often identified empirically, meaning that sequences of tests are conducted to study the rate at which energy decays in the system according to changes in the damping element's parameters. A common form of damping observed in mechanical systems is viscous damping, which exerts a force proportional to the velocity of the mass away from its equilibrium, Figure 6. dx = ( 3 ) fr Rm dt The damping constant R m has a positive value and SI unit [N.s/m]. We refer to this damping constant as the mechanical resistance. Typical dampers use turbulent phenomena of fluids or gases being forced through orifices and channels to induce dissipative effects proportional to the damper extensional velocity.

13 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 6. Damped harmonic oscillator schematic. Considering the damped mass-spring oscillator schematic of Figure 6, we again apply Newton's laws to yield mx + R x + sx = ( 3 ) m By incorporating our definition of the natural angular frequency, we express R m x+ x + ω x= ( 3 ) m To solve this equation, we resort to the complex exponential solution method assuming that γt ( t) = e x A ( 33 ) By substituting (36) into (35), we find R m jt γ + + ω e = γ γ m A ( 34 ) The only non-trivial solution to (37) requires that the terms in brackets be equal to zero. Thus ( ) / γ = β ± β ω ( 35 ) where we have introduced β = R /m which has units of [/s]. Often, the damping is small such that ω m >> β. We can then consider rearranging the radicand by defining the damped natural angular frequency ω = ω β ( 36 ) d such that the term γ is given by γ = β ± jω ( 37 ) d Considering the other common conventions of denoting damping, we recognize that R m m = ζω ( 38 ) where ζ is termed the damping ratio. From this, we recognize that β = ζω, and ωd = ω ζ. 3

14 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Substituting (4) into (36), the general solution to the damping mass-spring oscillator free vibration problem is x βt jωdt jωdt = e A e + A e ( 39 ) As described and demonstrated previously, only the real part of (4) is the complete, general solution to (36). Thus, we can express (4) as βt ( ) cos[ ω φ] x t = Ae t + ( 4 ) d As before, the constants A and φ are determined by applying initial conditions to (43) and its first derivative. Unlike the undamped harmonic oscillator considered in., the vibrations of the damped oscillator decay in amplitude in time by virtue of the term e βt. To assess the rate at which this decay occurs, we recognize that the term β has units of [/s]. Thus, we can therefore define a relaxation time τ = / β which characterizes the decay of the oscillation amplitude as time increases..5 displacement [m] time [s] Figure 7. Damped free oscillation. β / ω =., and the same additional parameters as those to generate Figure 3..6 Forced oscillations In contrast with free vibration, forced oscillations are those induced by externally applied forces f ( t ), Figure 8. This modifies the governing equation (34) to yield m ( ) mx + R x + sx = f t ( 4 ) For linear response by the harmonic oscillator, the total response is the summation of the individual responses. Thus, by recalling Fourier's theorem that any periodic function may be described using an infinite series of sinusoids (even when the period extends for an infinitely great duration), we consider that the force f ( t ) is composed according to f ( t) = f ( t) where each f ( ) i i i t accounts for one of the harmonic, sinusoidal components of force. Based on this, by linear superposition we only need to solve for the response of the oscillator when subjected to harmonic forces occurring at single frequencies. Once we solve f t, for the individual displacement responses ( ) i x t corresponding to their respective harmonic forces ( ) we then obtain the total response by the superposition of the individual displacement responses. i 4

15 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 8. Schematic of forced damped harmonic oscillator. Therefore, in this course we will often consider harmonic driving inputs at single frequencies, such as ( ) cos f t = F ωt. When excited by such a force from a state accounting for unique initial conditions of displacement and velocity, the oscillator will undergo two responses: transient response associated with the initial conditions and a steady-state response associated with the periodic forcing function. The steady-state response is the solution to the ODE (44) and it considers the initial conditions to be zerovalued during the solution process. The transient response is the solution to the ODE (34) accounting for the initial conditions and zero-valued forcing function. For systems with even an infinitesimal amount of damping, the transient response decays as time increases. At times sufficiently greater than the relaxation time τ = / β, the transient response is insignificantly small when compared to the steady-state response induced by the forcing function. In many engineering contexts, and often in acoustical engineering, we will be concerned with the steadystate response. In these cases, the complex exponential solution approach to the ODE (44) will be favorable. j t Consider that the real driving force f ( t) = Fcosωt is replaced with the complex driving force f ( t) = Fe ω. Then, the equation (44) becomes j t x+ mx + x= ( 4 ) m R s Fe ω Since the real part of the driving force represents the actual driving force, similarly, the real part of the complex displacement Re[ x ] that solves the equation (45) represents the actual displacement in resulting from the force. To proceed with the solution, we assume j t x= A e ω and by substitution into (45) we have jωt jωt ω m + jωrm + s A e = Fe ( 43 ) In general, the application of the above approach refers to the assumption of steady-state, time-harmonic response. We then solve for the complex displacement coefficient A and substitute that back into the assumed solution form to determine the complex displacement 5

16 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University jωt jωt x Fe Fe = = ( 44 ) ( s ω m) + jωr jω Rm + j( ωm s/ ω m ) Similar to the case of free vibration for (8), the complex steady-state velocity of the mass is u= jωx jωt Fe u = ( 45 ) R + j m s m ( ω / ω) We introduce the complex mechanical input impedance Z = R + jx ( 46 ) m m m where we define the mechanical reactance X = ωm s/ ω. The magnitude of the mechanical impedance Z is j m = Zme Θ m Z m ( ω / ω) Zm = Rm + m s / ( 47 ) while the corresponding phase angle is determined from X m ωm s/ ω tan Θ= = ( 48 ) R R m m Considering (49), the dimensions of the mechanical impedance are the same as the mechanical resistance, [N.s/m]. Considering (48), we see that the mechanical impedance is equal to the ratio of the driving force to the harmonic response velocity Z m f = u ( 49 ) From (5), we see that the complex mechanical impedance Z m is the ratio of the complex driving force f to the complex velocity u of the system. The interpretation of (5) is important. First, note that (5) is a transfer function between f and u in the frequency domain. Second, for large mechanical input impedance magnitudes, (5) indicates that large force is required to achieve a given system velocity, all other factors remaining the same. Whereas in contrast, for small mechanical impedance magnitudes it is relatively easy to apply the harmonic driving force to obtain considerable system velocity in oscillation. It is important to recognize that the mechanical input impedance can be measured by collocated force transducer and velocity transducer on the mechanical system. Because the transfer function of mechanical input impedance is in the frequency domain, it can be computed by taking the ratio of the Fast Fourier transforms of the force and velocity measurements. 6

17 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Thus knowing the mechanical input impedance, one may compute the complex velocity for a different j t harmonic forcing function according to u= f / Z m. By virtue of the assumed solution form x= A e ω, the complex displacement becomes x= f / jωz m. In other words, determining Z m is analogous to solving the differential equation of motion for the linear system. As a result, the actual displacement is the real part [ ] = [ jω ] = x= ( F ωz ) [ ωt Θ] Re x Re f / Z / sin ( 5 ) whereas the actual speed is [ ] = u = ( F Z ) [ ωt Θ] m m Re u / cos ( 5 ) m From (54), for a constant amplitude of the harmonic force, it is seen that the speed of the mass is maximized when the impedance magnitude is minimized. This occurs when ωm s/ ω = ω = s/ m = ω ( 5 ) In other words, when the harmonic excitation frequency corresponds to the natural frequency, the speed of the mass oscillation will be maximized. This phenomenon is called resonance. Thus, at resonance, the impedance is minimized and purely real, such that the speed is F e jωt u and u Re[ ] Rm F = u = cosωt ( 53 ) R m As seen in (56), for a given harmonic force amplitude, when driven at resonance the system response is damping- or resistance-controlled. In other words, the damping is the principal determinant for the amplitude of the system velocity at frequencies close to resonance. The (56) also indicates that when driven at resonance, the oscillator is perfectly in phase with the harmonic driving force, which is otherwise not generally true. When the excitation frequency is significantly less than the natural angular frequency ω << ω, we write ωzm / m= ωrm / m+ j ω ω ωzm / m ωrm / m+ j ω Z R + j s / Z js / ω [ ω] m m m ( 54 ) Thus, when the excitation frequency is much less than the natural frequency, the mass velocity is jωf e jωt u ( 55 ) s from which we see that the complex displacement is F e jωt x ( 56 ) s 7

18 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University yielding that the actual mass displacement is F x= Re[ x ] = cosωt ( 57 ) s This shows that at excitation frequencies considerably less than the natural frequency, the system response is stiffness-controlled. Similarly, for high frequencies, ω >> ω, this routine will yield that the complex acceleration is F e jωt a ( 58 ) m given a real acceleration response of F a= Re[ a ] = cosωt ( 59 ) m which shows that at high excitation frequency the system response is mass-controlled. Summarizing the findings from the above derivations: Around resonance ω ω the system is damping- or resistance-controlled and the velocity is independent of frequency, although the band of frequencies around which this occurs is typically narrow for lightly damped structures in many engineering contexts For harmonic excitation frequencies significantly below the natural angular frequency ω << ω, the system is stiffness-controlled and the displacement is independent of the excitation frequency For excitation frequencies much greater than the natural angular frequency ω >> ω, the system is mass-controlled and the acceleration is independent of the excitation frequency These results are summarized by the example plot shown in Figure 9. amplitude - velocity [m/s] displacement [m] acceleration [m/s] frequency [Hz] Figure 9. Harmonic force excitation of damped mass-spring oscillator. [N]. R m =.377 [N.s/m], m = [kg], s =39.5 [N/m], F = 8

19 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University.7 Mechanical power In acoustical engineering applications, the power and energy associated with sound sources and radiation are critical factors to assess in the determination of acoustic performance and quality. To first introduce these concepts, we consider the power relations for the mechanical oscillator. The instantaneous mechanical power Π i delivered to the harmonic oscillator is determined by the product of the instantaneous driving force and the corresponding instantaneous speed F Π i = cosωtcos[ ωt Θ ] = Re[ f] Re[ u ] ( 6 ) Z m In general, the average power Π delivered to the system is the more relevant engineering quantity to consider. The average power is the instantaneous power averaged over one cycle of oscillation T ω F π/ ω Π= { [ ]} idt cos tcos t dt T Π = ω ω π Z Θ m ω F π/ ω = { cos ωtcos cosωtsinωtsin } dt π Z Θ+ Θ ( 6 ) m F = cosθ Z m Recalling (5), X m ωm s/ ω sin Θ tan Θ= = = R R cosθ m m ( 6 ) which shows that cos Θ= R / Z and leads to the result that the average power delivered to the mechanical oscillator is R F Re m * Π= = Zm m m fu ( 63 ) The units of power, whether instantaneous or average, are Watts, [W]. When the oscillator is driven at resonance ω = ω, the average power becomes Π= F R. This shows / m that the peak average power delivered to the oscillator is also damping-controlled. Based on the in-phase relationship between the input force and oscillator speed at resonance, we find that peak average power delivery to the oscillator occurs when speed and force are perfectly in phase. This conclusion is also intuitively arrived at by considering the procedures of (64). In addition, the power delivered to the oscillator is proportional to the resistance. Also, the presence of reactance works only to diminish the power. As will become clearer through additional impedance relations in the study of wave propagation, the real part of the impedance, the resistance, is associated with energy 9

20 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University transfer or power delivery, while the imaginary part of the impedance, the reactance, is associated with energy exchange (alternatively termed reciprocating energy) which does not deliver power..8 Linear combinations of simple harmonic oscillations In numerous vibration and acoustics contexts, it is needed to determine the total response associated with a combination of individual responses. We consider only linear systems here, so the principle of linear superposition applies. Consider two harmonic oscillations at the same angular frequency ω. These two harmonic oscillations are termed coherent or correlated due to the sharing of frequency. j( t + φ ) x = Ae ω and j( t + φ ) x = Ae ω ( 64 ) The linear combination x= x + x is therefore ( + ) ( ) j ω t φ j φ Ae A e A e j φ e j ω = + t ( 65 ) Determining the magnitude A and the total phase φ is accomplished trigonometrically from considering representative phasors in the complex number coordinate plane, Figure ( cosφ cosφ ) ( sinφ sinφ ) A= A + A + A + A / ( 66 ) tanφ = Asinφ + Asinφ A cosφ + A cosφ ( 67 ) Figure. Phasor combination of x= x + x.

21 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University The real response displacement is [ ] [ ] [ ω φ] x= Re x + Re x = Acos t+ ( 68 ) As will be seen throughout later portions of this course, many response metrics of interest in acoustics involve mean-square and root-mean-square quantities. The mean-square is computed by π/ ω ω = π + π/ ω ( cos[ ω φ ]) x A t dt ω = A cos [ ωt+ φ] dt π ( 69 ) Keep in mind that the name "mean-square" indicates the operators occur to the expression from the rightto-left: thus, square the expression first and secondly take the mean. Using trigonometric identities on (7) yields π/ ω ω = cos ω d π + = x A t t A ( 7 ) The root-mean-square (RMS) is the square root of the mean-square quantity, and is typically denoted by x rms. Likewise, operate on the expression from the right-to-left: square the expression, take the mean, and then take the square root of the mean result. Thus, continuing the derivation from (73), it is seen that the RMS oscillation displacement is A x, rms = ( 7 ) so as to relate the response amplitude A to the RMS value of the harmonic component x,rms via x, rms A =. From the perspective of complex exponential representations (67), the RMS of the oscillation is likewise the amplitude of the response divided by the square-root of. For the summation of two harmonic oscillations having the same frequency (7), the mean-square is found Re x = x is by steps. First, the square [ ] [ ω φ ] [ ω φ ] [ ω φ ] [ ω φ ] x = A cos t+ + A cos t+ + AA cos t+ cos t+ = A + t+ + A + t+ + AA t+ + + ( cos[ ω φ ]) ( cos[ ω φ ]) ( cos[ ω φ φ ] cos[ φ φ ]) ( 7 ) where trigonometric identities are used to transition from the first to second line of (75). Then, we compute the mean-square x and find

22 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University x = A + A + AA cos [ φ φ ] [ φ φ ] = x + x + xx cos ( 73 ) Thus, for harmonic oscillations occurring at the same frequency ω, a significant variation in the total response may occur as relates to the mean-square output. For instance, if A = A = A and if φ = φ =, then the mean-square displacement is result with respect to an individual harmonic oscillation: φ φ π = + =, then we find [ π ] x = A + A + A = A so as to quadruple the mean-square A. On the contrary, if the A = A = A and x = A + A + A cos = A A = which shows that out-of-phase oscillations destructively interfere so as to eliminate the mean-square measure. Of course, this occurs for direct oscillation summation, as well, but this also confirms that the mean-square quantity likewise is eliminated. Again adopting the perspective of complex exponential representations (67), the RMS is computed from comparable steps. The results of (69) and (7) have indeed already yielded the bulk of the derivation: ( ω + φ ) ( ω + φ ) jφ jφ jωt ( ω φ) ( ) + x = Ae + Ae = Ae + Ae e = Ae ( 74 ) j t j t j t where A and φ are defined in (69) and (7), respectively. Considering what was shown in (7-74), the RMS of x is A x rms = ( 75 ) One may perceive a contradiction between (78) and the example cases above that demonstrated that the RMS of a summation of harmonic oscillations at the same frequency can potentially constructively or destructively interfere. Yet, recalling (69), the amplitude in (78) varies according to such phase differences between the oscillations. Therefore, for instance in the event of perfect destructive interference by waves of the same amplitude but out-of-phase, we find A, which was shown above. Note that in general, amplitude A. Thus, j t x= A e ω where the contribution of the phase is included within the complex x rms = A. Two oscillations that occur at different frequencies are termed incoherent or uncorrelated. When the oscillations do not occur at the same frequency, there is no further simplification to adopt for [ ω φ ] cos[ ω φ ] x= x + x = A cos t+ + A t+ ( 76 ) For the mean-square quantity, the computation shows

23 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University x = x + x ( 77 ) Also, the RMS quantities are the summation of the individual terms x = x + x ( 78 ) rms, rms, rms The same results for the mean-square and RMS quantities would be obtained by applying the complex exponential form of the response. Finally, by linear superposition, the procedures outlined above for two harmonic oscillations extend to any number of oscillations. In particular, for coherent or correlated sinusoid summation, it is found that / ( ncosφn) ( nsinφn) ( 79 ) A= A + A tanφ = A sinφ n n n A cosφ n ( 8 ) Thereafter, following the computation of (8), the mean-square and RMS values follow naturally from computations of (7) and (78), respectively. For incoherent or uncorrelated sinusoid summation, there are no general simplifications available to employ for the resulting oscillation itself, although the mean-square and the RMS quantities are respectively computed from x = x n ( 8 ) x = x ( 8 ) rms n, rms.9 Fourier's theorem, series, and transform Linear combinations of oscillations is a foundational means of predicting wave propagation phenomena throughout many studies in acoustics. Conversely, in the acquisition of acoustic data sets, it is regularly needed to decompose a convoluted time series measurement into frequency components that exert greater significance in contributing to the whole. To make this leap from a combination response to components, we draw from the Fourier's theorem. Fourier's theorem is a statement that any single-valued, periodic function may be expressed as a summation of simple harmonic components whose frequencies are integral multiples of the periodicity of the whole function. For a function f ( t ) that has a period T, Fourier's series expands this function by a πn πn f ( t) = + ancos t bnsin t + n= T T ( 83 ) where the angular frequency of repetition is ω = π /T. The coefficients of the series are determined from 3

24 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University t+ T π n an = f ( t) cos t dt; n,,,... T = t T ( 84 ) t+ T π n bn = f ( t) sin t dt; n,,3,... T = t T ( 85 ) In this way, the function f ( t ) with fundamental periodicity rate ω is represented by sine and cosine components of rates nω, n =,,3,..., and a constant term a that is independent of frequency. The specific constitution of the time series function ( ) f t determines the number N, n=,,..., N, required in the superposition in order to faithfully reproduce the original function, arbitrary though it may be. Of course, odd functions, f ( t) = f ( t), are expressed using only sines. In a similar way, even functions, f ( t) = f ( t), require only cosine functions for reproduction. The constant term a vanishes only when the function is symmetric about f =. Looking further, functions that are mostly sinusoidal in effect at a handful of lower-order harmonics, such as that observed in the generation of "notes" by certain musical instruments, require only a few terms, e.g. N 6, for accurate reproduction. On the other hand, the presence of sharp changes in a time series response demand a large number of harmonic terms in order to reconstruct the sudden change in the function. This is representative of significant high frequency contributions, where high is relative to the fundamental frequency of the periodic function. Example: Determine the number of terms in a Fourier series required such that a pulse train f ( t ) may be reproduced with 5% or less RMS error. The pulse train period is T = π. ( ) f t Answer: ; < t < π = ; π < t < π The function is odd, such that cosine terms are anticipated to be absent from the series. Similarly, the pulse train function is symmetric about f =, such that the constant term is anticipated to be absent. Indeed, π π an = { cos[ nt] } dt + { cos[ nt] } dt = sin[ nt] + sin[ nt] = n n π π π π π for all n =,,,..., thus confirming the anticipation. For the sine terms of the series, π π π 4

25 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University π π bn = { sin[ nt] } dt { sin[ nt] } dt π + π π π = cos[ nt] cos[ nt] π n + π n =,,,,,... π 3π 5π In other words, for ( ) n odd but b n = for n even. π π a πn πn f t = + ancos t bnsin t + n= T T, a n = for all n, while bn = 4/ nπ for.5 function f(t) time [s] The plot at left in Figure shows a reconstruction of the pulse train using 7 terms in the Fourier series. This number of terms is determined from the plot at right in Figure as approximately the point at which the RMS error is 5%. The code used to generate these results is given in Table. The overshoot observed at the discontinuity in the pulse train function at t Table. Code to generate Figure black solid curve, exact red dashed curve, Fourier series 7 terms Figure. At left, time series reconstruction of pulse train using 7 terms in the Fourier series. At right, RMS error convergence by increasing number of terms in Fourier series. = π is referred to as Gibbs phenomenon. error of Fourier series expansion to actual number of Fourier series coefficients % pulse train. f(t)=; <t<pi; f(t)=-; pi<t<*pi. T=*pi. time=linspace(,*pi,e4); f=zeros(length(time),); f(:length(time)/)=; f(length(time)/:end)=-; numcoef_all=7; clear error for ooo=:length(numcoef_all) numcoef=numcoef_all(ooo); % number of fourier series coefficients fest=zeros(length(time),); for iii=::numcoef, fest=fest+4/pi/iii*sin(iii*time)'; end; error(ooo)=rms(fest-f); disp(numstr(error,'%.5f ')); end figure(); clf; plot(time,f,'k') hold on plot(time,fest,'--r'); xlabel('time [s]'); ylabel('function f(t)'); xlim([min(time) max(time)]); 5

26 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University legend('black solid curve, exact','red dashed curve, Fourier series 7 terms') numcoef_all=logspace(,3,4); clear error for ooo=:length(numcoef_all) numcoef=numcoef_all(ooo); % number of fourier series coefficients fest=zeros(length(time),); for iii=::numcoef, fest=fest+4/pi/iii*sin(iii*time)'; end; error(ooo)=rms(fest-f); end figure(); clf; plot(numcoef_all,error); xlabel('number of Fourier series coefficients'); ylabel('error of Fourier series expansion to actual'); set(gca,'yscale','log'); set(gca,'xscale','log'); xlim([min(numcoef_all) max(numcoef_all)]); Like all integrals, Fourier's (integral) transform is the limiting case of the Fourier's series. The Fourier transform pair of the transform and inverse transform are given in (86) and (87), respectively. + jωt ( ω) ( ) F = f t e dt ( 86 ) + jωt f ( t) = F( ω) e dω π ( 87 ) The integral pair relate the time series of the function f ( t ) to its continuous representation in the frequency domain F ( ω ), in contrast to the discretized frequency domain representation given in the Fourier series. As before, ω = π /T. Note that the signs in the complex exponentials of (86) and (87) can be interchanged without loss of generality. 6

27 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University 3 Wave equation analysis 3. Transverse waves on a string The string represents one of the most elementary vibrating bodies that permits the propagation of waves according to the distributed nature of the system. This contrasts with the lumped parameter mechanical vibrations considered in Sec.. Thus, the string serves as a suitable introduction into the formation of a wave equation that governs the motion of the distributed parameter system and will consequently enable exploration of wave propagation principles. Consider the infinitesimal element of taut, stretched string shown in Figure. The string is under tension T, units [N], and possesses a uniform linear density ρ L, units [kg/m]. The stiffness of the string is presumed to be negligible and gravitational forces are omitted by virtue of considering a tension T significantly great enough such that no slack in the string occurs. The element of string is of sufficient small extent ds that the tension is nearly constant along the element. Also, the vertical displacement of the string from one end of the element to another, from an equilibrium displacement y, is also assumed to be small. Comparatively, the horizontal displacements of the string are negligibly small. Figure. Schematic of taut string element. By virtue of such assumptions, the vertical force present between the two ends of the infinitesimal string element due to the tension is ( sinθ) ( sinθ) df = T T ( 88 ) y x+ dx x where θ is the angle between the tangent to the string and the x axis. Also, ( sin ) T sinθ at the location x + dx, while ( sin ) x expansion is used to approximate ( sin ) T θ + is the value of T θ is the corresponding value at location x. A Taylor series T θ + x dx x dx ( θ) ( θ) ( T sinθ ) x T sin T sin + dx x+ dx x x ( 89 ) Consequently, the vertical force is approximately df y ( T sinθ ) x dx ( 9 ) 7

28 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University For small vertical displacements of the string, the angle tangent to the x axis is approximately sinθ tan θ dy / dx. Thus, the vertical force exerted on the string by the tension along the element length is y y df y = T dx = T dx x x x ( 9 ) The string element mass is ρ dx. Newton's nd law of motion then relates the acceleration y L of this t mass to the net vertical force according to y y T dx = ρ Ldx x t ( 9 ) which is reduced to y y = x c t where the constant is defined c T / ρl =, units [m/s]. The (93) is the one-dimensional wave equation. ( 93 ) 3.. General solution to the one-dimensional wave equation Jean-Baptiste le Rond d'alembert (77-783) derived the general solution to the one-dimensional wave equation (93). He found that a complete solution to (93) required (, ) ( ) ( ) y x t = f ct x + f ct + x ( 94 ) To show the satisfaction of (93) by (94), one applies the differential operators of (93) on (94). ( ) f ct x t ( ) f ct x x where f ( ) = c f ct x ( ) = f ct x f ( α ) =. Performing the operations also on f ( ct x) α f + f = c f + c f c +, by substitution we find ( 95 ) ( 96 ) ( 97 ) which confirms the satisfaction of (93) by (94). It is important to recognize that the arguments of functions f and f are arbitrary. We investigate this general solution (94) by considering the result of a single function f at different times, t and t, Figure 3. A triangular pulse waveform is considered. Because the wave shape remains constant in the absence of 8

29 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University dissipative or dispersive terms in the one-dimensional wave equation, we find that f ( x, t ) f ( x, t ) =. As a result, the arguments of the functions must yield the same outcome. Consider that the wave at x at time t is the same as the wave response at x at time t. Thus, the wave function arguments between these two times must yield the same result x ct x = ct x c = t x t ( 98 ) As a result, it is clear that the parameter c is the speed at which a wave propagates. It is termed the phase speed because this is the rate of travel for a point of constant phase of the wave, units [m/s]. If we repeat the analysis for f ( ct x) comparing the arguments ct +, we find that in increasing time the wave travels "to the left". Thus, x and ct + x, we see that waves travel towards increasing x values for increasing time according to the argument ( ct x ), while waves travel towards decreasing x values for increasing time according to the argument ( ct + x ). Figure 3. Wave nature of the general solutions of the one-dimensional wave equation. For a string of significantly great length, we may consider the one-dimensional wave equation as an initialvalue problem (IVP) rather than as an initial-boundary-value problem (IBVP). In the former case, we have the initial conditions, y( x,) = U( x) and y ( x,) = V( x). We then evaluate how our two functions for the general solution (94) accommodate the initial conditions by direct substitution. To this end, it may be shown that we can rearrange the order of the functions without loss of generality. Thus, in the following, we use ( ) and f ( x ct ) f x ct (,) ( ) ( ) ( ) + having the same meanings as before [3]. Thus, applying the initial conditions y x = f x + f x = U x ( 99 ) (,) = ( ) + ( ) = ( ) y x c f x f x V x ( ) We then integrate the equation for the initial condition on velocity. x ( ) ( ) ( ) f x f x = V S ds c ( ) b 9

30 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Using (98) and (), we solve to find f ( x ) and ( ) f x. x f ( x) = U ( x) V ( S ) ds c ( ) b x f ( x) = U ( x) + V ( S ) ds c ( 3 ) b Then, rather than only the argument x, the full arguments x ± ct are inserted into the functions and they are summed together to yield the general solution. y ( x, t x ct x ct ) ( ) ( ) ( ) ( ) U x ct U x ct + V S ds = c V S ds b b ( 4 ) x+ ct y( x, t) = U ( x ct ) U ( x ct ) V ( S ) ds c x ct ( 5 ) The implications of (4) are that an initial displacement of the string results in one-half of the displacement shape propagating towards increasing x values along the string length, while the other half of the initial displacement shape propagates towards decreasing x values along the string length. The rate of propagation is c. Figure 4 shows a representative case of initial-value wave propagation when the initial velocity of the string is zero. The initial displacement profile is shown in the top frame of the figure. As time elapses, the wave breaks into forward and backward traveling wave components. By the time t = a/ c, it is clear that the initial displacement shape is equally propagating forward and backward at one-half of the original shape amplitude. Figure 4. Snap-shots of one-dimensional wave propagation considering the initial displacement condition shown in the top-most panel. 3

31 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University 3.. Wave propagation at string boundaries Two basic string boundaries are worth considering since they, and their analogs in fluid-borne wave propagation, are representative of the two key limiting cases of boundary interactions involving waves. Consider a string rigidly supported at x =. For all time, the rightward and leftward x propagating waves must always sum to zero value at this boundary constraint. In other words, (, ) ( ) ( ) y t = y ct + y ct + ( 6 ) One way in which this may be satisfied is if the rightward propagating wave is always negated by its leftward propagating wave which is mirrored over y =. Thus, (, ) ( ) ( ) y x t = y ct x y ct + x ( 7 ) This is illustrated in Figure 5(a) where, for a constant phase speed c, the summation of propagating waves at y, t, leading to the summation result in the bottom figure panel. x = always yields a zero value for ( ) The second common boundary condition is a free end. In this case, the string may displace at y(, ) t but has no transverse force at its constraint, T sinθ =. As shown in the derivation of the one-dimensional y wave equation, this is comparable to T =. Since the tension must remain finite, the string slope must x be zero at the free end boundary condition. One way for the slope to the zero at the boundary condition is for rightward and leftward propagating waves to possess opposite slopes at x =, such that their summation is zero. Thus, the leftward propagating wave y ct x y ct x about x =. ( + ) can be a mirror of ( ) (, ) ( ) ( ) y x t = y ct x + y ct + x ( 8 ) This cancellation of slopes of the string vibration at the free boundary is illustrated in Figure 5(b). 3

32 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 5. Illustrations of wave reflection phenomena at (a) rigid and (b) free supports. The top row shows the two traveling wave components while the bottom row shows the actual result which is the superposition of traveling waves in the domain x. Table 3. Code used to generate Figure 5 x_l=linspace(-,,4); % create string x coordinate x_r=linspace(-,,4); % create string x coordinate y_r=(x_r-.3).^/4.*cos(4*pi*(x_r-.86)).*exp(-3.*(x_r.^-3.)); % create rightward propagating wave component y_l=-fliplr(y_r); % create leftward propagating wave component figure(); clf; plot(x_r,y_r,x_l,y_l,x_r,y_r+y_l); xlabel('x'); ylabel('y(x,t)') legend('+x propagating wave','-x propagating wave','superposition of waves'); title('string reflection at rigid support') y_l=fliplr(y_r); % create leftward propagating wave component figure(); clf plot(x_r,y_r,x_l,y_l,x_r,y_r+y_l); xlabel('x'); ylabel('y(x,t)') legend('+x propagating wave','-x propagating wave','superposition of waves'); title('string reflection at free support') 3..3 Forced vibration of a semi-infinite string Consider a string that extends infinitely from x = in the + x axis. The string is excited transversally at x = by a harmonic force, f ( t) = Fcosωt. As a result, waves only propagate in the string in the + x direction away from the excitation boundary condition at x =. This hypothetical situation could be realized simply by consideration of a string of significantly greater length than the wavelength (to be defined below) under study, such that inherent dissipation mechanisms in the string prevent x traveling waves from developing. Using the complex exponential mathematical convention, the harmonic force is denoted j t ( t) = Fe ω f ( 9 ) The string response is therefore also most generally complex, 3

33 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University ( x, t) = ( ct x) y y ( ) Application of the boundary condition at x = yields the general expression (, t) j t y = A e ω ( ) where the unknown A must be determined from the boundary condition. Prior to this, we apply () to () to relate the general result to the rightward propagating wave component, ( ct ) jk( ct) y = A e ( ) Here, we introduce the wavenumber k = ω / c ( 3 ) For all ct x, we have for () that (, ) jk( ct x) j( t kx) y xt = Ae = A e ω ( 4 ) The top of Figure 6 illustrates the waveform (4) at several points in time along the string length. It is clear that for a given time t, the elements of the string undergo harmonic oscillation along the string length, with an amplitude of A. The distance from one peak of this oscillation to the next is the wavelength λ = π /k ( 5 ) Considering (4), for every kλ = πn with n =,,... the waveform repeats in x. As shown in the inset of Figure 6, for a fixed location of the string, harmonic oscillation of the displacement ( ) y x t occurs with an amplitude of A. The time elapsed for a point of constant phase on, the wave to repeat is T, which is the period. The period is related to the frequency via T = π / ω = / f where f is in [Hz]=[cycles/s]. Figure 6. Schematic of wave propagation in string due to harmonic force excitation at x =. 33

34 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Because the waveform moves one wavelength λ in a time equal to one period T, we find that c = λ f ( 6 ) Summarizing, the wavelength λ is the spatial duration of the harmonic waveform. The wavenumber k is the amount of phase change of the waveform that occurs per unit distance. Analogously, the period T is the time duration of the harmonic waveform. The angular frequency ω is the amount of phase change of the waveform that occurs per unit time. One may also refer to the angular frequency ω as the time rate of change of the waveform for a given spatial location. Likewise, the wavenumber k is the spatial rate of change of the waveform for a given time. The complex amplitude A remains to be determined. The transverse force at the harmonically excited end results in the boundary condition y f = T ( 7 ) x x= By substitution of (9) and (4), we find ( ) jωt jωt Fe T jk e = A ( 8 ) indicating that the amplitude is F F A = = j ( 9 ) jkt kt As a result, the waveform in the string is completely described by F j( ωt kx) y ( xt, ) = j e ( ) kt The transverse speed of a string element particle is L u= y / t, so that F j( ωt kx) u ( xt, ) = e ( ) ρ c Like for mechanical oscillators described in Sec..6, impedance measures are valuable in the analysis and development of systems supporting wave propagation. Thus, we introduce the input mechanical impedance Z m and define it as the ratio of the harmonic driving force to the string element particle speed at the location of the driving force, Z m f = u (, t) ( ) For the semi-infinite string, we find that 34

35 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Z = ρ c ( 3 ) m L We find that the input mechanical impedance of the semi-infinite string is purely real. As described in Sec..6, the real part of the impedance is associated with power delivery. Likewise, the units of the real part of impedance are plainly associated with damping mechanisms (of course, the units of ρ c L are [kg/s], the same as R m ). Thus, for the harmonically driven semi-infinite string, the wave propagation away from the driving end at x = acts to remove the energy injected into the driven end in a way analogous to a damping mechanism. The instantaneous power input to the string is [ ] Re[ ] Π = Re f u ( 4 ) i x= By substitution, we have F Π i = ( Fcosωt) cosωt ( 5 ) ρ c L while the time average of the power gives the average power delivered F Π= = ρ cu ( 6 ) ρ c L L U = u, t = F / ρlc. where the input transverse speed amplitude is ( ) 3..4 Forced vibration of a finite length string When the string has finite length, rightward and leftward propagating waves must be taken into consideration and boundary conditions are of great importance. Assuming that transient responses associated with temporary disturbances or the start-up of the driving force are sufficiently decayed due to small inherent damping mechanisms and the passage of time, we conclude that two harmonic, steady-state waves must be employed to satisfy the wave equation and boundary conditions (, ) j( ωt kx) j( ωt + kx) y xt = Ae + B e ( 7 ) Forced, fixed string Consider the string to be driven by a harmonic force at x = and fixed at the opposite end at x= L. At the driven end, Fe jωt y = T x x= ( 8 ) By substitution of (7) into (8), we obtain the equation ( ) T jka jkb = F ( 9 ) 35

36 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University At the fixed end, we have the equation jkl jkl Ae + B e = ( 3 ) Solving (9) and (3) yields jkl Fe A = j ( 3 ) kt cos kl jkl Fe B = + j ( 3 ) kt cos kl Then, by substitution, we express the general solution for the string motion in two ways, (33) and (34). j ωt+ k( L x) j ωt k( L x) { } F y ( xt, ) = j e e ( 33 ) kt cos kl ( x) F sin k L jωt y ( xt, ) = e ( 34 ) kt cos kl The (33) and (34) are mathematically equivalent, and are transformed from one to another merely by use of Euler's identity. Yet, they indicate a striking phenomena associated with wave motion in systems of finite spatial extent. The (33) describes the resulting waveform as a combination of rightward and leftward traveling waves, where the waves have the same wavelength and amplitude. The (34) describes the j t resulting waveform in terms of a standing wave that oscillates harmonically by e ω ; this standing wave is determined according to the frequency of the excitation, via the wavenumber in the sine numerator term. Thus, in systems of finite spatial extent, the resulting wave motion may be conceived as either waves traveling in opposite directions superposing to a unique response, or conceived as a single, standing wave shape that oscillates harmonically. Considering the standing wave perspective in further detail, depending on the frequency ω, the numerator term may possess zeros. In other words, sin k( L x) may have zeros called nodes associated with zero displacement at all times. These nodes are found to exist according to the relation k( L x) = nπ where n =,,,... kl / π. The resulting node locations are x = L nλ /; n=,,,..., L/ λ ( 35 ) n A representative standing wave is shown in the top panel of Figure 7, with node locations shown by the locations along the string that retain zero displacement at all times, such as around x.37 [m]. Conversely, antinodes are observed around locations x. [m], x.55 [m], and x.8 [m]. Antinodes are locations of local peak displacement in a standing waveform. The standing wave wavelength spans two nodes or two antinodes. Table 4. Code used to generate Figure 7 L=; % [m] string length 36

37 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University c=e3; % [m/s] phase speed x=linspace(,l,3); % [m] spatial duration of string F=.3; % [N] harmonic driving force amplitude T=.8; % [N] string tension omega= e+3; % [rad/s] angular frequency of harmonic force k=omega/c; % [/m] wavenumber times=*pi/omega.*[/8 /7 /6 /5]; % [s] times for iii=:length(times) y(:,iii)=f/k/t/cos(k*l)*sin(k*(l-x))*exp(j*omega*times(iii)); end figure(); clf; plot(x,real(y)) xlabel('length along string [m]'); ylabel('string transverse displacement'); legend('/8 period','/7 period','/6 period','/5 period','location','best'); title(['c=' numstr(c) ' [m/s]. omega=' numstr(omega) ' [rad/s]. L=' numstr(l) ' [m].']); ylim(.*[min(min(real(y))) max(max(real(y)))]) c= [m/s]. omega= [rad/s]. L= [m]. string transverse displacement /8 period /7 period /6 period /5 period string transverse displacement length along string [m] x 4 c= [m/s]. omega= [rad/s]. L= [m]. 4 /8 period /7 period /6 period /5 period length along string [m] Figure 7. Top panel, example of off-resonance excitation of forced-fixed string. Bottom panel, example of on-resonance excitation. Note the relative amplitudes of the string transverse displacement in comparison from top to bottom panels to exemplify the resonance condition in the bottom panel example. Another feature is revealed from (34). The denominator becomes zero according to cos kl = when n kl = π ; n =,,3,... ( 36 ) 37

38 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Thus, the frequencies of the harmonic force associated with these conditions are f n n c = ( 37 ) 4 L which are referred to as resonance frequencies. When the string is driven by the force at frequencies identified by (37), the string is driven at resonance such that the string transverse displacement will exhibit infinite-valued peaks at the antinodes. Thus, for even vanishingly small input force amplitudes F, the string response at the antinodes becomes unbounded. An example of this resonant excitation of the string is shown in the bottom panel of Figure 7. Note the large difference in vertical axis range comparing the off-resonant and on-resonant excitation of the string from the top to bottom panels of the figure. When the forced-fixed string is driven at resonance, there is always an antinode at the driven end, such that the string transverse speed is maximized. Conversely, when cos kl = ±, the denominator of (34) is at its greatest value, such that string motion is minimized along its length. The frequencies at which these conditions occur are f m mc = ; m =,,3,... ( 38 ) L which are termed antiresonance frequencies. Under such conditions, the string possesses a node at the driven end, such that the string transverse speed is zero. The input mechanical impedance of the string is determined as before using (). By substitution of (34) into (), it is found that Z m = jρlccot kl ( 39 ) In this case, (39) shows that the input mechanical impedance of the forced-fixed string is purely imaginary, thus only reactance contributes to the impedance. For a purely lossless (undamped) string, there is no way for energy to leave the system and thus with the passage of time the string vibration should always become unbounded regardless of on- or off-resonance excitation. In reality, inherent damping mechanisms in real strings result in a balance of energy dissipated and energy injected by the harmonic force. Resonant excitation examples result in significantly larger amplitudes of such string motion than off-resonant excitation, but the energy balance between dissipation and energy input occurs as well. Considering (39), substitution of the values kl corresponding to resonance, (36), reveals that the input mechanical impedance vanishes. Because (39) is a pure reactance, the more general statement to conclude is that for any mechanical system, resonant excitation is associated with a vanishing of the input mechanical reactance Normal modes of the fixed-fixed string Consider a string fixed at both ends, x = and x L =. Thus, y( t) y( Lt), =, =. While we established the efficacy of the assumed solution (7) for the case of harmonic excitation of the string, in the absence 38

39 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University of the harmonic force we now take a step back and address the one-dimensional wave equation (93) by a more general approach often used to analyze PDEs. According to the method of separation of variables, we assume that a separable solution to (93) may be found whereby (, ) X( ) T( ) y xt = x t ( 4 ) where the response y is purely real and is determined as the product of two functions X( x ) and T( t ) that are dependent only on their respective space and time variables. Substitution of (4) into (93) yields X ( x) T( t) = X ( x) T ( t) ( 4 ) c The corresponding boundary and initial conditions are (4) and (43), respectively. XT ( ) ( t ) = ; ( ) ( ) X L T t = ( 4 ) ( x) ( ) = U( x) ; X( x) T( ) = V( x) X T ( 43 ) The prime operator indicates derivatives with respect to x and the overdot operator indicates derivatives with respect to t. The (4) is rearranged to be X X ( x) ( ) T = ( t) ( ) x c T t ( 44 ) It is seen that the left-hand side of (44) is independent of t while the right-hand side of (44) is independent of x. As a result, for (44) to be true for all x and t, both sides of the equation must be equal to the same constant, σ. X X ( x) ( x) = σ ( 45 ) c T T ( t) ( t) = σ ( 46 ) The (45) is then the BVP of this IVBP, while (46) is the IVP of this IVBP. Considering (45), ( x) σ ( x) X X = ( 47 ) with the boundary conditions ( ) ( L) X = X = ( 48 ) 39

40 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University it is discovered that the nature of (47) is governed by the unknown constant. Investigations of such equations shows that solutions exist only when σ < []. For later convenience, we select the constant to be of the form σ = φ. Thus, ( x) φ ( x) X + X = ( 49 ) Past experience shows that solutions to (49) are the summation of two sinusoidal functions, ( ) φ X x = A sin x+ A cosφx ( 5 ) Satisfying the fixed-fixed boundary conditions, (48), indicates that A = ; A sinφ L= ( 5 ) where non-trivial solutions to (5) occur only when sinφ L =, thus nπ φ n = ; n =,,3,... ( 5 ) L Consequently, the total solution to (49) is the superposition of an infinite number of terms X n nπ = Ansin x L ( 53 ) Since the constants φ n are now known, they may be substituted into the IVP (46). T nπ ( t) + c T( t) = ( 54 ) L Like before, the general solution to (54) is a summation of sinusoidal functions. Now, we consider the of these solutions which is th n nπc nπc Tn( t) = Ansin t + An cos t L L ( 55 ) where the superscripts in (55) are notational and do not refer to power operations. Taking the product of the the fixed-fixed string is th n solutions for the IVP and BVP, we find the general th n solution to (93) for (, ) sin n π c n n sin n π y x t a t x bncos n π c t sin n π = + x L L L L ( 56 ) where the constants a n and b n are the products of the constants AA and n n AA, respectively. n n The general response of the string is therefore 4

41 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University nπc nπ nπc nπ y( x, t) = ansin t sin x + bncos t sin x ( 57 ) n= L L L L The initial conditions (43) are applied to determine the constants a n and b n. nπ U( x) = bn sin x n= L ( 58 ) The functions sin ( n x/ L) π are orthogonal, such that we take advantage of the orthogonal property L n m ; m n sin x sin x dx mn L π π = = δ = L L ( 59 ) ; m n Thus, the constants n L b are found by multiplying (58) by sin ( m x/ L) π and integrating over the length, nπ bn = U ( x) sin x dx L L ( 6 ) A similar procedure is used in the application of the second initial condition, on the string velocity along y x, = V x, (44). its length ( ) ( ) L nπ nπ an = V ( x) sin x dx = V ( x) sin x dx nπc L ω L L n L ( 6 ) Because there is no harmonic force excitation, these results indicate that the string response only occurs at th frequencies ω nπc/ L sin nπ x/ L. The extents to which the n of these n = and in spatial distributions ( ) frequencies and spatial distributions are activated are determined by the initial conditions through the constants (6) and (6). The shapes sin ( n x/ L) π are referred to as the normal modes of vibrations. The frequencies ω = nπc/ L are referred to as the natural frequencies. Thus, when given initial disturbances and in the absence of excitation mechanisms, the string will undergo free vibration at the natural frequencies each of which is associated with a normal mode. Reflecting upon the steps used to determine the normal modes and the natural frequencies, the application of the boundary conditions yields both such information while application of the initial conditions via Fourier's theorem identifies the exact contribution of normal modes to a given free vibration response. The orthogonality property (59) indicates that the normal modes are linearly independent. Thus, if the string is given an initial condition of displacement identical to one of the normal modes, the string will respond only in that particular mode for all time. One arrives at the same results as (57), (6), and (6) if a complex exponential form of assumed solution is adopted, like (7). For instance, see [] Sec... The utilization of the method of separation of variables n 4

42 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University here demonstrates an alternative and general means by which these conclusions may be reached regarding the free vibration response of systems that permit wave propagation. 3. Longitudinal waves along a one-dimensional rod Unlike the string displacements which are transverse to the axis of wave motion, longitudinal waves involve harmonic compression and rarefaction of particles in the axis of wave propagation. Longitudinal waves occur in myriad structural and acoustic contexts and we first encounter them in the elementary structure of the one-dimensional rod. Consider a rod whose lateral dimensions are small compared to the length, such that each cross-sectional plane of the rod moves identically according to the stresses acting on the elemental face. Figure 8 provides a schematic of the rod. The cross-sectional area is S units [m ], while the rod density and Young's modulus are, respectively, ρ [kg/m 3 ] and Y [N/m ]. Figure 8. Schematic used to formulate equation of motion for the longitudinal waves propagating in a one-dimensional rod. The free-body diagram of the differential element of the rod is shown in Figure 8(b). The forces on the faces of the cross-sectional areas are determined via Hooke's law. Considering the left face of the differential element, we find (, ) σ (, ) F xt = S xt ( 6 ) where ( xt, ) σ is the stress on the face, and we adopt the convention that the stress is positive in compression and negative in tension. Stress is related to the material strain via the constitutive relation σ ( xt, ) Yε( xt, ) = ( 63 ) In (63), the positive compressive stress is a result of a negative strain; because the Young's modulus Y is positive, the right-hand side of (63) is likewise a positive value. Although this convention is the reverse of that used by materials scientists, it is convention in the context of studying acoustics because positive pressure changes correspond to decreases in the volume of a fluid. Finally, we use the strain-displacement relation for the compressive stress 4

43 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University ε ( xt, ) ( xt, ) ξ = x ( 64 ) All together, the force on the differential element face is (, ) F x t ( xt, ) ξ = SY x ( 65 ) The (65) is Hooke's law for the rod. The force on the right face of the differential element is ( x dx, t) ξ + F ( x + dx, t) = SY x ( 66 ) A Taylor series approximation for the displacement is taken to yield ( xt, ) ξ ξ( x + dx, t) ξ( x, t) + dx x ( 67 ) This approximation is illustrated in Figure 8(c). Of course, for small deformations from the original element, the approximation is reasonably accurate. For all purposes in this course, we will consider such small amplitude perturbations from an equilibrium which form the underpinnings of a significant proportion of topics in acoustics. By using (67), we find that (66) is ( xt, ) ( xt, ) ( xt, ) ξ ξ ξ F ( x + dx, t) SY ξ ( x, t) + dx = SY SY dx x x x x ( 68 ) The average acceleration of the differential element is illustrated in Figure 8(c) such that the inertial force is ρs ξ t ( xt, ) dx Using Newton's second law, we obtain the governing equation of motion for the rod ( 69 ) ( xt, ) ξ F ( x, t) F ( x + dx, t) = ρs dx t ( xt, ) ( xt, ) ( xt, ) ( xt, ) ξ ξ ξ ξ SY SY SY dx ρs dx = x x x t ξ SY x ( xt, ) ξ( xt, ) = ρs t ( 7 ) ( 7 ) ( 7 ) The (7) is then rearranged 43

44 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University c ξ ξ = x t ( 73 ) The (73) is also a one-dimensional wave equation. There is clearly a fundamental shared physics between the propagation of transverse waves in a stretched string and the propagation of longitudinal waves in rods. Indeed, further shared physics will be seen in the continued study of other elastic and acoustic systems throughout this course. Similar to the case of strings, we likewise refer to term c as the phase speed. Now the relation of the material properties of the rod to the phase speed is c= Y / ρ ( 74 ) In SI units, the phase speed is [m/s]. Because the longitudinal deformations of the rod are governed by the one-dimensional wave equation, we use the general form of solution to satisfy (73). Thus, ( x, t) ( ct x) ( ct x) ξ = ξ + ξ + ( 75 ) The complex exponential form of assumed solution is here (, ) j( ωt kx) j( ωt + kx) ξ xt = Ae + B e ( 76 ) Because (73) is the same as (93), we promptly (and rightly) conclude that a fixed-fixed rod exhibits the same normal modes and natural frequencies a a fixed-fixed string. As a result, we study a different example of boundary conditions applied to the rod. Consider an example case of a rod with free boundaries at both x = and x= L. While a finite length string with both ends free can be realized using two slider ends, such as that illustrated in Figure 5(b), it is much easier to realize a free-free rod in practice (e.g. rest a rod on a soft foam bed) and such an elementary structure may be used in the study of vibration or wave control. A free boundary on the rod must have no force acting on the cross-sectional area element. Thus, ξ F (, t) = = SY x (, t) ; F ( L t) ( Lt, ) ξ, = = SY x ( 77 ) It is observed then that ξ (, t) ξ( Lt, ) x = = x ( 78 ) Inserting the assumed solution (76) into the two components of (78) yields jk + jk = A B ; B=A; ( ) j t ξ x, t = A e ω cos kx ( 79 ) ka sin kl = ( 8 ) 44

45 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University where (8) indicates that responses are permitted only when nπ c sin kl = ωn = ; n =,,3,... ( 8 ) L The natural frequencies of the free-free rod are the same as those of the fixed-fixed rod. On the other hand, the normal modes are cosines and not sines. The normal modes of the free-free rod are nπ cos kx n = cos x L ( 8 ) Thus, the ( ) th n mode response of the free-free rod vibrates according to j nt ξ xt, = A e ω cos kx ( 83 ) n n n Also recognize that in the process of steps (79) we simplified the general expression of assumed solution (76) using Euler's identity and implicitly transitioned our perspective from referring to oppositely traveling waves to standing waves. Yet, the mathematical procedures all ultimately yield the same results once computed. Thus, the free-free rod undergoing vibration in the normal modes may be perceived as oscillating in standing waves or as permitting a pair of oppositely traveling waves along its length. 3.3 Transverse waves along a beam While longitudinal waves and vibrations are permitted in many engineering structures, a more common form of response due to arbitrary loads is bending, which is a motion transverse to the axis of the structure. The same rod that allows longitudinal waves to travel along its length is likely to oscillate in bending if an excitation acts off the center of the axis. In an opposite extreme of excitation to the rod, excitations that are normal to the axis will induce only bending motions. To differentiate bending vibrations and waves from longitudinal counterparts, we refer to the elementary structure here as the beam although, as described above, a realistic structure that permits longitudinal compressions along the primary axis will also permit bending motions transverse to its axis. Consider the schematic shown in Figure 9(a). The beam has a uniform cross-section S and is symmetric through its thickness. The x coordinate denotes positions along the beam length while y denotes the transverse displacement, similar to our convention in deriving the wave equation for the string. The ξ denotes longitudinal stretching of a filament of the beam along its length dimension in x. 45

46 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 9. Schematic of beam bending. When the beam is bent, the lower part of the beam becomes compressed while the upper part is under tension, as exemplified in the schematic of Figure 9(a). The neutral axis length remains unchanged from the undeformed state. The bending is characterized by the radius of curvature R of the neutral axis. The strip of the beam r from the neutral axis is stretched (or compressed) by an amount δx ( ξ / ) a result, a longitudinal force develops for this strip of the beam given by = x dx. As δx ξ df = YdS = YdS dx x ( 84 ) The sign convention according to Figure 9 is that the positive stretch δ x denotes a tensile force, which is shown by the negative sign of the force in (84). Geometrically, we have dx + δx dx δx r = = R + r R dx R ( 85 ) Thus, we have that the force is df Y = rds ( 86 ) R The bending moment M in the beam develops by virtue of the distortion of the infinitely many beam strips, so that Y R ( 87 ) M = rdf = r ds A constant is introduced as r ds S κ = ( 88 ) where κ is referred to as the radius of gyration. For a beam, the radius of gyration is often given according to its second moment of area I and cross-sectional area S according to κ = I / S. For beams of crosssectional area of width b and height h bent along their height dimension, the relevant moment of area is the common expression I = bh 3 /. 46

47 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Thus, the bending moment is simplified in expression to be M YSκ = ( 89 ) R When the displacements of the beam from its undeformed state are small y/ x<< such that the radius of curvature is large, we may use the approximation y + x R y x 3/ y x ( 9 ) which enables the approximation of the bending moment as y κ x y x M = YS = YI ( 9 ) The (9) relates the bending moment to the out-of-axis (transverse) displacement y. Shear forces are also developed in a beam in bending. Consider Figure 9(b) that denotes the establishment of upward shear force F y at x while a downward shear force is created at x + dx along the beam length. For a beam in equilibrium, no net moment of the beam segment must be induced. Thus, taking moments about x, we have the constraint that ( ) ( ) ( ) M x M x + dx = F x + dx dx ( 9 ) y Expanding the moments using a linearized Taylor series, assuming a small beam segment and small moments, yields the expression Fy = = YS = YI x x x 3 3 M y y κ 3 3 ( 93 ) The net force on the beam segment in the y (vertical) axis is therefore 4 4 Fy y y dfy = Fy( x) Fy( x + dx) = dx = YSκ dx = YI dx 4 4 x x x ( 94 ) Newton's nd law then equates this force (94) to the net inertial force. Considering a beam volumetric density given by ρ, we have that 4 4 y YI y y ( κc) = = 4 4 x ρs x t ( 95 ) where the constant is defined c Y / ρ =. 47

48 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Eq. (95) is not a wave equation. In particular, the general function f ( ct x) evidence that transverse waves do not travel along beams with unchanging shape. is not a solution. This is To solve (95), we apply the method of separation of variables, now assuming time-harmonic form for the time-dependent component. Thus, ( xt, ) = ( ) j t y Y x e ω ( 96 ) Then (96) is substituted into (95), yielding 4 4 d Y ω = 4 Y ( 97 ) dx v v ( c) = ωκ ( 98 ) where (98) states that the frequency ω is a function of κ c, and v has units of speed. Assuming a form of x Y ( x) = e γ, substitution into (97) reveals that this assumed solution is valid so long as = ± ( ω / v) γ = ± j ( ω / v) γ and. The four possible forms of γ are anticipated due to the fourth-order differential equation (97). Thus, we define a new term g ρsω YI = / v= 4 ( 99 ) ω so that the complete solution to (95) is gx gx jgx jgx ( x) e e = + + e + e Y A B C D ( ) gx gx jω t j( ωt+ gx) j( ωt gx) ( xt, ) = ( e + e ) e + e + e y A B C D ( ) It is apparent that the unknown g has units of inverse length, which reveals that g is the wavenumber for the transverse bending waves along the beam. The general solution () is associated with two traveling waves, associated with complex amplitudes C and D, and two standing waves with spatial attenuation coefficients, associated with complex amplitudes A and B. The real part of () is the actual beam response (, ) [ cosh sinh cos sin ] cos( ω φ) y x t = A gx + B gx + C gx + D gx t + ( ) while the unknown constants are related to the complex unknown constants in () through application of boundary and initial conditions. Application of the boundary conditions alone, at a minimum, provides definition of the wavenumber values g for free vibration. Interestingly, (), (99), and (98) show that the wave propagation in the beam is frequency dependent. This leads to a discussion of dispersion in wave propagation. 48

49 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University 3.4 Dispersion, phase velocity, group velocity Waves traveling in media may exhibit two types of speeds. The more common terminology to use here are velocities, and we temporarily employ this nomenclature. Consider a monofrequency waveform y( x, t) Y cos( ωt kx) =. The wavenumber is k = π / λ and the angular frequency is ω = π /T. The phase velocity is therefore c = ω / k = λ / T. This represents the velocity at which points of constant phase on the wave travel. Now consider a wave composed of two frequencies, ω and ω, but the same amplitude Y (, ) cos( ω ) cos( ω ) y x t = Y t k x + Y t k x ( 3 ) k = k + k; ω = ω+ ω; ω << ω ( 4 ) k = k k; ω = ω ω; k << k ( 5 ) p y(x,t) x Figure. Wave packet travel. Also see Trigonometric identities enable us to write (3) as (, ) cos( ω ) cos( ω ) y x t = Y t kx t kx ( 6 ) where the final term in (6) is the wave modulation. This phenomenon is shown in Figure. The distance between individual waveform peaks corresponds to the period T = π / ω, while the distance between the slow modulation nulls is T = π / ω. Considering the phase velocity definition, the modulation velocity is ω / k. This is referred to as the group velocity c g ω ω = cg = k k ( 7 ) 49

50 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University The group velocity is the rate at which a wave packet travels. Because a series of wave phase information is encapsulated in the packet, the group velocity may be considered as the rate at which information in the wave travels. Dispersion is the phenomenon in which phase velocity depends on frequency. A consequence of this is that the group and phase velocities are no longer the same. For non-dispersive media, cp = cg. For dispersive media, the ω is a nonlinear function of k. Consider the longitudinal wave propagation along a one-dimensional rod. We found that c ξ ξ = x t ( 8 ) With a time-harmonic assumed solution currently considering wave propagation only in the ( xt, ) j( t kx) e ω + x direction, ξ = A ( 9 ) we have that = = ( ) c k ω ω kc The angular frequency is a linear function of the wavenumber. Thus, the phase velocity and group velocity are () and (). c = ω / k = c ( ) p cg ω = = c k ( ) Thus, the longitudinal wave propagation in a rod is non-dispersive, cp = cg. Formally, the dispersion relation is the expression relating the frequency to the wavenumber, although here the relation is ( k ) which is linear such that longitudinal wave propagation in a rod is non-dispersive. Now consider the transverse bending wave propagation in a beam. We found ω = kc 4 y y ( κc) = 4 x t ( 3 ) With a time-harmonic assumed solution currently considering wave propagation only in the ( xt, ) j( t kx) e ω + x direction, y = A ( 4 ) we have that ( ) 4 κc k = ω ω = k κc ( 5 ) 5

51 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University The angular frequency ω is a nonlinear function of k. The dispersion relation for transverse bending wave propagation is ( ) ω k = k κc. Thus, the phase velocity and group velocity are (6) and (7). c = ω / k = kκc ( 6 ) p ω cg = = kκc= c k p ( 7 ) Thus, transverse bending wave propagation in a beam is dispersive, cp cg. Contextually, it is often the group velocity that is more important since the group velocity governs the transmission rate of the information or wave packet while the phase velocity is the transmission rate of points of constant phase on the wave. Such points of constant phase are not inherently associated with information since information requires a series of phase data to establish the wave packet. Considering (5) in another light, the dispersion of the waves is manifest by the fact that k = ω / κc, which reveals that higher frequency waves move at faster rates than lower frequency waves. A comparison of non-dispersive and dispersive wavenumber-frequency relationships is given in Figure..4. wavenumber [/m] k=ω/c k=[ω/c] / frequency [rad/s] Figure. Dispersive and non-dispersive wavenumber-frequency relationships. 3.5 Forced excitation of beams Examination of several cases of forced excitation of beams is useful to appreciate how the injected energy contributes to waves propagation in dispersive structures Point force excitation of infinite beam Considering a time-harmonic form of driving force ( t) j t f = Fe ω at a central point x = on a beam of infinite spatial extent in ± x directions, we use the notation of the transverse beam bending equation of motion that employs the alternative relations to the radius of gyration, thus 5

52 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University y x 4 YI ρsω = Fδ x 4 ( ) y ( 8 ) where δ ( x) is the Dirac delta function. For the unforced beam, we identify k ρsω YI 4 f = ± following substitution of the comparable terms for radius of gyration into (5). For completeness, also k f ρsω YI = ± 4. j For the forced system, we introduce the spatial Fourier transform to assess (8). The spatial Fourier transform uses a pair of integral relations to make the connection between spatial frequency (wavenumber) and displacement. This integral pair is analogous to the traditional Fourier transform that relates functions of frequency and functions of time. Thus, the spatial Fourier transform pair is ( ) ( ) jkx + Y k = y x e dx ( 9 ) + jkx y( x) = ( k ) e dx π Y ( ) Multiplication of (8) by 4 4 ( k f ) jkx e and integration from negative to positive infinity yields F Y ( k ) = ( ) YI k Analogous to Laplace transform operations, the inverse transform () is used to convert the spatial frequency domain (wavenumber domain) response () back into the response in the spatial domain. With considerable simplifications [], we obtain for the response x > jf f f (, ) = 3 ( ) jk x k x jωt y x t e je e ( ) 4YIk f A similar result is obtained for waves in the x < domain. The result () shows that two waves propagate away from the driving point force into the + x direction. One wave travels, having a complex exponential term jk f x e, while a second wave decays exponentially away from the force, having the standard k x e f exponential term. The prior is of course a traveling wave while the latter is referred to as an evanescent wave. This is evidence of the dispersion in the media: that the wave shape does not remain constant in the course of wave travel. Yet, it should be emphasized that evanescent waves are not only found in dispersive media. Indeed, evanescent waves appear prominently in the study of fluid-borne wave propagation through multiple fluid media, and fluids are non-dispersive. Finally, we note that the real part of () is the actual or measurable bending wave response in the beam. An example of the combined nature of the traveling and evanescent waves for the + x domain of the infinite beam is given in Figure. The evanescent components of the wave decay relatively quickly away from 5

53 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University the excitation point at x = as evidenced by the relatively quick convergence of the waveforms (at different time snap shots) to common waveform shapes for increasing values of x. y(x,t) 6 x /8 period /7 period /6 period /5 period x Figure. Evidence of evanescent and traveling waves at a snap shot of time along an infinite beam forced at its center point x =. F = [N], Y = 7e9 [N/m ], ρ = 3 e [kg/m 3 ], S =.9 [m ], I = e 3 [m 4 ], ω = 93 e [rad/s] Arbitrary force excitation of finite beam Consider again a harmonic force excitation of the beam, but now the force is arbitrary in spatial distribution ( xt, ) = ( ) x = and j t f F x e ω and the beam is of finite extent with simply-supported boundary conditions x = L. The governing equation of motion is ( ) 4 y F x k = 4 x YI 4 jωt f y e ( 3 ) while the associated boundary conditions are ( t) ( Lt) y, = y, = ( 4 ) y (, t) y( Lt, ) x = = x ( 5 ) Substitution of the assumed solution (, ) j( ωt kx) j( ωt + kx) y xt = Ae + B e into the homogeneous form of (3) and into the associated boundary conditions reveals the normal modes (6), and natural frequencies and wavenumbers (7) of the system. ( ) sin ( ) y x = Y k x ( 6 ) n n fn / nπ ; k fn YI ωn = ρs L nπ = ( 7 ) L 53

54 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University In this way, the total bending response of the simply-supported beam is (, ) n sin ( fn ) j t y xt Y k xe ω ( 8 ) = n= For the inhomogeneous form of (3), in other words with the harmonic force j t F( xe ) ω, we hypothesize that the response of the beam will also be a linear superposition of the normal modes where the contribution from each mode to the total response is computed in a way similar as for the initial condition contribution in the free response, such as for the string Sec Substituting (8) into (3), we find n= ( ) 4 nπ ρs nπ F x ω Yn sin x = L YI L YI 4 4 nπ F x kfn k f Yn sin x = n= L YI ( ) ( 9 ) Taking advantage of the orthogonality property of the normal modes, we multiply (9) by sin m π x L and integrate over the beam length, to find the modal amplitudes. L nπ Yn = F ( x) sin x dx 4 4 ( 3 ) YIL k L ( fn k f ) To investigate (3), a form of the force distribution F( x ) must be examined. Consider then a point force j t < x < L. The force is therefore f ( x, t) = Fδ ( x x ) e ω. acting on the beam at location x, where Evaluation of the integral (3) yields Y F nπ n = sin 4 4 x YIL k fn k f L ( 3 ) Then, the total response is determined by (8), F jωt y ( x, t) = sin 4 4 ( kfnx ) sin ( kfnx) e ( 3 ) YIL k k n= fn f 4 recalling the distinction that k ( nπ / L) 4 fn = and k = S YI. 4 f ρ ω / jk fnx jk fnx Recognizing an embodiment of Euler's identity using kfnx j( e e ) sin = /, (3) is again indicative of the duality of perceiving the oscillations of a system of finite spatial extent as being composed of standing waves or a pair of oppositely traveling waves. When ω ω, the response is more associated with a pair of traveling waves along the beam. Alternatively, when ω ω the denominator of (3) nearly n n 54

55 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University vanishes and there is strong reinforcement of the normal mode response which would commonly appear as more of a significant standing wave in practice. Examples of such off- and near-resonant point force excitation of the simply-supported beam are shown in Figure 3 using modes in the linear superposition, and a point force located at one-third the distance along the beam length. For the off-resonant excitation case shown in the top panel, the off-center point force excitation is evident by virtue of the significant asymmetry in response that is weighted towards the excitation location L /3. Yet, this feature is not readily observed when the beam is excited very near to resonance, as shown in the bottom panel since the normal mode motion dominates. Table 5. Code used to generate Figure 3 num_modes=; % number of modes to use in summation L=; % [m] beam length x=l/4; % [m] point force excitation point along beam length F=; % [N] point force amplitude Y=7e9; % [N/m^] young's modulus of beam I=e-5; % [m^4] second moment of area rho=e3; % [kg/m^3] density S=.; % [m^] cross-sectional area omega=.*sqrt(y*i/rho/s)*(3*pi/l)^; % [rad/s] excitation frequency time=*pi/omega*[/8 /7 /6 /5]; % [s] times 4 x - frequency [rad/s],.3 times the third natural frequency y(x,t), beam bending displacement [m] y(x,t), beam bending displacement [m] length along beam [m] x -5 frequency [rad/s], times the third natural frequency /8 period /7 period /6 period /5 period /8 period /7 period /6 period /5 period length along beam [m] Figure 3. Top panel, example of slightly off-resonant point force excitation of simply-supported beam. Bottom panel, example of near-resonant excitation.

56 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University x=linspace(,l,4); % [m] define beam spatial extent y=zeros(length(x),length(time)); % pre-allocate beam bending vector for ooo=:length(time) for iii=:num_modes y(:,ooo)=y(:,ooo)+/((iii*pi/l)^4-(rho*s*omega^/y/i)).*sin(iii*pi/l*x).*sin(iii*pi/l*x)'.*exp(j*omega*time(ooo)); end end figure(); clf; plot(x,*f/y/i/l*real(y)) xlim([min(x) max(x)]); xlabel('length along beam [m]'); ylabel('y(x,t), beam bending displacement [m]'); legend('/8 period','/7 period','/6 period','/5 period','location','best') title(['frequency ' numstr(omega) ' [rad/s], ' numstr(omega/(sqrt(y*i/rho/s)*(3*pi/l)^)) ' times the third natural frequency']); 3.6 Waves in membranes Having evaluated numerous examples of wave propagation in one-dimensional systems, characteristics of wave propagation in two-dimensional systems is a logical next step for study. We first consider the membrane, which is the two-dimensional analog of the string. We assume that the membrane is thin, is stretched uniformly in directions x and z, and undergoes small displacements y( xzt,, ) away from equilibrium. A schematic of the membrane element under consideration is shown in Figure 4. The membrane is stretched with tension τ per unit length [N/m] and possesses a linear density of ρ S [kg/m ]. Forces exerted by the tensions in x and z axes displace the membrane outof-the-page, respecting the view of Figure 4. Figure 4. Schematic of membrane element. Such net upward force induced by the opposing tensions τ dz is τdz = τ x x+ dx x x x y y y dxdz ( 33 ) 56

57 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University where a linearized Taylor series expansion is used for by the opposing tensions τ dx is τdx = τ z z+ dz z z z y y y dxdz y x x+ dx. Similarly, the net upward force induced ( 34 ) The summation of these forces must be equal to the net inertial force, by Newton's nd law of motion. As a result, the governing equation for the membrane is y y y + = x z c t ( 35 ) where the constant is defined c τ / ρs =. The more general expression of (35) is y = y c t ( 36 ) where is the Laplacian operator, which is dependent upon the coordinate system considered. The (36) is the two-dimensional wave equation. The Laplacian operator is ordinarily selected for convenience based on the system geometry under study. For a system that is mostly rectangular in geometry, a Cartesian coordinate system is convenient. Thus the Laplacian for Cartesian coordinates is x z = + ( 37 ) For systems that possess a circular geometry, the polar coordinate system ( r, θ ) is preferable such that the Laplacian becomes r r r r θ = + + ( 38 ) Assumption of a general time-harmonic response encourages a separable assumed solution to (36) of j t y = Y e ω ( 39 ) where the complex component Y is a function only of space. Substitution of (39) into (36) yields the Helmholtz equation + k = Y Y ( 4 ) Solution to (4) is the means to compute the normal modes of the system subject to known boundary conditions. 57

58 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University 3.6. Free vibration of a rectangular membrane with fixed edges Consider a rectangularly-shaped membrane with edges fixed at x =, x= Lx, z =, and z = Lz. ( ) ( ) ( ) ( ) y, z, t = y L, z, t = y x,, t = y x, L, t = ( 4 ) x z Following the logic of (39), we assume that the membrane responds according to ( xzt,, ) = (, ) j t y Y xze ω ( 4 ) By substitution of (4) into (36), we find Y x Y z + + k = Y ( 43 ) The method of separation of variables is then employed. We assume independence of the x and z responses in the complex function Y ( xz, ) ( xz, ) = ( x) ( z). Thus Y X Z ( 44 ) Therefore, (43) becomes d X d Z X dx Z dz + + k = ( 45 ) In order for (45) to be true, the first and second terms must be equal to constants. k k + k = ( 46 ) x z As a result, (45) becomes the set of equations d X k x dx + X = ; d Z dz + k z = Z ( 47 ) Satisfying (47) requires a pair of general sinusoidal functions, leading to (,, ) = sin ( + φ ) sin ( + φ ) j t y xzt A kx x x kz z z e ω ( 48 ) where the wavenumber components and phases are determined by the boundary conditions. For the membrane fixed according to (4), the phases φ x and φ z must be zero to ensure the sine functions are zero. Also, it is seen that k k x z mπ = ; m=,,3,... ( 49 ) L x nπ = ; n=,,3,... ( 5 ) L z Therefore the standing waves on the membrane are expressed by 58

59 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University ( ) jωt y xzt,, = A e sin kxsin kz ( 5 ) x z Considering (49), (5), and (46), the free vibration of the membrane may occur only at particular frequencies f mn ωmn c m n = = + π Lx L z / ( 5 ) where each combination (, ) mn is associated with a normal mode, sin kxsin kz. This analogous to the x z.5 mode (,) of fixed-fixed rectangular membrane.45.4 mode (,) of fixed-fixed rectangular membrane z [m].5 x [m] z [m] x [m] mode (,) of fixed-fixed rectangular membrane.4.3 mode (,) of fixed-fixed rectangular membrane z [m].5 x [m] z [m] x [m] -.4 Figure 5. Examples of normal modes of fixed-fixed rectangular membrane. one-dimensional case for the string vibration response. Figure 5 shows several examples of the lowerorder, normal modes of the fixed-fixed rectangular membrane. Clear nodal lines occur where the membrane possesses no displacement. Thus, if excited only in a given mode, additional fixed supports could be positioned along nodal lines without affecting the response of the membrane. Table 6. Code used to generate Figure m=; % mode number in x direction n=; % mode number in z direction Lx=pi/; % [m] dimension of x extent of membrane 59

60 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Lz=; % [m] dimension of z extent of membrane x=linspace(,lx,6); % [m] x extent z=linspace(,lz,5); % [m] z extent kx=m*pi/lx; % [/m] x wavenumber kz=n*pi/lz; % [/m] z wavenumber clear y y=zeros(length(x),length(z)); for iii=:length(z) y(:,iii)=.5*sin(kx*x)*sin(kz*z(iii)); end figure(); clf; surf(x,z,y','linestyle','none'); xlabel('x [m]'); ylabel('z [m]'); colorbar axis equal zlim([-.5.5]) view(-45,5) title(['mode (' numstr(m) ',' numstr(n) ') of fixed-fixed rectangular membrane']); 3.6. Free vibration of a circular membrane Circular drum heads are one of the most common surface geometries employed for percussive instruments. This is effectively a circular membrane with fixed edge at the radius, by our assumptions employed to arrive at (36). Consider a membrane of radius a. Using the appropriate Laplacian operator, the associated Helmholtz equation is Y Y Y r r r r θ k = Y ( 53 ) where k = ω / c according to the time-harmonic assumption. A separable solution is again assumed ( r, θ) = ( r) Θ( θ) Y R ( 54 ) satisfying the boundary condition R ( a ) = ( 55 ) A constraint must be imposed on the azimuthal function Θ ( θ ), namely that it is smooth and continuously a function of θ. Substituting (54) into (53) yields R Θ R R Θ Θ k R Θ= ( 56 ) r r r r θ Rearrangement and pre-multiplication by r RΘ yields r d R dr d Θ + kr + = R dr r dr Θ dθ ( 57 ) Like for prior implementations of the method of separation of variables, the independent sides of the (57) must be equal to the same constant, m. Considering the right-hand side, 6

61 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University d Θ + m Θ= ( 58 ) dθ which is solved using ( θ) cos( mθ γ ) Θ = + ( 59 ) m The terms γ m are associated with azimuthal dependence in the initial conditions. The second equation to result from an application of the method of separation of variables to (57) is Bessel's equation, (6). d R dr m + + k R = ( 6 ) dr r dr r Solutions (6) are Bessel functions of order m of the first and second kind, respectively, J ( ) Ym ( kr ). Thus, ( r) = J ( kr ) + Y ( kr ) m m m kr and R A B ( 6 ) Consider the polar coordinate system to have as its origin the center of the membrane. For a membrane with bounded displacements at its center, the terms B = for all m. Therefore, ( r) = J ( kr ) R A ( 6 ) m The boundary condition (55) requires J m ( ka ) =. We denote the arguments of the function as j mn that cause the Bessel function of the first kind to be zero. Therefore, the wavenumbers associated with these arguments are kmn = jmn / a. Appendix A5 of Ref. [] provides a list of these zeros of the Bessel function, which enable the computation of the associated wavenumber and reconstruction of the normal modes of the circular membrane that is fixed at its edge. An abbreviated version of the tabulated results is given in Table 7. Collectively, the response of the ( mn, ) mode is (, θ, ) = ( ) cos( θ + γ ) j t ymn r t A mnjm kmnr m mn e ω ( 63 ) where kmna j f mn =. The natural frequency associated with the (, ) mn mn mode is jmnc = ( 64 ) π a Table 7. Zeros of the Bessel function of the first kind, J ( j ) = m mn m n 3 j mn 6

62 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University mode (,) of fixed-rim circular membrane mode (,) of fixed-rim circular membrane y x.5. y x.5. mode (,) of fixed-rim circular membrane mode (,) of fixed-rim circular membrane y -. y x -.5 x.5. Figure 6. Examples of normal modes of fixed-edge circular membrane. Table 8. Code used to generate Figure 6 a=.; % [m] radius of membrane c=3e3; % [m/s] speed constant m=; % mode number in theta direction n=; % mode number in r direction bzeros=[ ; ; ; ]; % bessel zeros r=linspace(,a,5); % [m] create radial coordinate theta=pi*linspace(-,,4); % [rad] create azimuthal coordinate y=zeros(length(r),length(theta)); % pre-allocate for displacement 6

63 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University for iii=:length(theta) y(:,iii)=.*besselj(m,(r*bzeros(m+,n+)/a))*cos(m*theta(iii)); end for ooo=:length(r) for iii=:length(theta) x(ooo,iii)=r(ooo)*cos(theta(iii)); w(ooo,iii)=r(ooo)*sin(theta(iii)); z(ooo,iii)=y(ooo,iii); end end figure(); clf; surf(x,w,z,'linestyle','none'); axis equal xlabel('x'); ylabel('y'); zlim(.*[- ]) view(-5,5) title(['mode (' numstr(m) ',' numstr(n) ') of fixed-rim circular membrane']); 3.7 Vibration of thin plates While the restoring forces in membranes arise from the tension applied to maintain their shape, plates develop restoring forces by virtue of their inherent bending stiffness. To assess the vibrations of plates as a next stage in development of standing wave principles, we move straight to the governing equation of motion for plates leaving the detailed derivation to reference [3]. Consider a plate of small thickness with respect to the span-wise dimensions extending in the x y plane. The displacement of the plate out of this plane from an undeformed configuration is w( xyt,, ). (,, ) w xyt t 4 YI w( xyt,, ) + ρh = p xyt,, ( ) ( 65 ) where I is the moment of inertia per unit width, p( xyt,, ) is an applied external pressure or load, ρ is the volumetric density, and h is the plate thickness. The moment of inertia is thus I h 3 /( ν ) the Poisson's ratio is ν. = where 4 The biharmonic operator is and its selection depends on the coordinate system. Numerous engineering systems are composed of rectangular-shaped plates and so we consider the Cartesian form of the biharmonic operator here. x x y y = ( 66 ) Considering the homogeneous form of (65) for a plate with infinite span in the x y plane, we assume that a wave propagates with an angle α measured counterclockwise from the x axis. This wave adopts the form j( t kx x ky y ) w xyt = A e ω ( 67 ) (,, ) Substitution of (67) into the homogeneous form of (65) reveals 63

64 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University ( ) x y YI k + k ρhω = ( 68 ) Consequently, we define k + k = k ( 69 ) x y f k 4 f ρhω = ( 7 ) YI Similarly, the relations between the wavenumber components to (7) are k = k cosα ( 7 ) x f k = k sinα ( 7 ) y f The perspective of (7) and (7) is illuminating since the assumed solution alternatively adopts the form j( t kf cos x kf sin y) w xyt = A e ω α α ( 73 ) (,, ) which plainly indicates that the wave propagates as a common front in a direction α away from the x axis in the x y plane. Thus, k f is the amplitude of a wavenumber vector kf = kx x + ky y where the overbars indicate vectors, and x and y are respective unit vectors Free vibration of thin plates Consider the homogeneous form of (65) for a plate that is simply-supported at all edges. For a simplysupported plate, the boundary conditions on the edges require w( xyt,, ) = along all edges ( 74 ) (,, ) w xyt x (,, ) w xyt y = = along edges x = and x= Lx ( 75 ) along edges y = and y = Ly ( 76 ) An assumed solution to (65) for the simply-supported plate may adopt our established experience that sine functions satisfy separable equations pertaining to the normal modes. Thus, we assume (,, ) (, ) j ω = t = sin ( ) sin ( ) jωt w xyt W xye A kx m kye n ( 77 ) Substitution of (7) into (65), and satisfaction of the boundary conditions (67), (68), and (69) gives non-trivial results so long as (78) and (79) are satisfied. k m mπ = ; m=,,3,... ( 78 ) L x 64

65 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University k n nπ = ; n=,,3,... ( 79 ) L y The associated natural frequencies of the plate are therefore f mn / / YI m π nπ π YI m n = + = + π ρh L x L y ρh L x L y ( 8 ) with ω = π f. mn mn 3.7. Forced vibration of thin plates For the inhomogeneous form of (65), the pressure load on the plate is presumed to be harmonic by p ( xyt,, ) = (, ) j t Pxye ω. In the way performed for beams, we assume that the response of the thin plate is a superposition of its normal modes weighted by modal coefficients. (,, ) j ω t jωt xyt = e ( xy, ) = e sin ( kx) sin ( ky) w W A ( 8 ) mn mn m n m= n= m= n= The (8) is substituted into (65), then the normal modes are multiplied by the result and integrated over the plate domain. In this way, the modal amplitudes A mn are determined. 4 A P x y k x k y dxdy ( 8 ) x ( ) (, ) sin ( ) sin ( ) L Ly mn = m n ρhlxly ω ωmn When the plate is excited by a point force at (, ) jωt (,, ) = (, ) = δ( ) δ( ) x y, jωt p x y t P x y e P x x y y e ( 83 ) the evaluation by (8) gives the modal amplitudes ( kx) sin ( ky) 4P sin m n A mn = ( 84 ) ρhl L ω ω ( ) x y mn Summarizing, the forced vibration of the thin plate to a point force at (, ) ( ) ( kx) sin ( ky) jωt m n,, = sin m sin ρhlxly m= n= ω ωmn ( ) ( ) ( ) x y is 4P sin w xyt e k x ky ( 85 ) As emphasized in Sec. 3.5., by use of Euler's identity on the normal modes, we see the duality of (85) as perceived either as standing waves associated with modes of vibration or as oppositely traveling waves in the x and y axes. n 65

66 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University 3.8 Impedance and mobility functions of structures As described in Sec..6, determination of the impedance for the system is equivalent to solving the differential equation of motion. This referred to the case of an ordinary differential equation. The question may be asked whether this broad statement of differential equation solution applies to partial differential equations that govern the motions of numerous and diverse engineering structures. In fact, the first questions to ask are whether the impedance of a distributed parameter system is defined and whether it is useful in the analysis of that system's dynamic response. To the first part of the question, for distributed parameter systems, we must distinguish between drive point (or input) impedance and transfer impedance. Recall the infinite beam driven by a harmonic point force at the beam center. The beam response due to this force was determined to be (), which is repeated for convenience and expanded for both ± x domains. jf f f (, ) = 3 ( ) jk x k x jωt y x t e je e ; for x > ( 86 ) 4YIk f jf + f + f (, ) = 3 ( ) jk x k x jωt y x t e je e ; for x < ( 87 ) 4YIk f The drive point impedance of the beam is Z = f y ( t) (, t) ( 88 ) Given the time-harmonic assumptions, we find that the drive point impedance is 3 4YIk f Z = ( 89 ) ω ( j) Eq. (89) characterizes the resistance and reactance, i.e. respectively energy transfer and energy exchange, of the harmonic energy attempted to be injected into the infinite beam by the point force. It is clear that (89) is complex and frequency dependent. The transfer impedance from the force point x, here x =, to the spatial location x along the beam is Z x = f y ( t) ( xt, ) ( 9 ) Evaluating, we find Z 4YIk 3 f x = jk x k x ω f f ( e je ) ( 9 ) 66

67 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University The transfer impedance characterizes the ability for energy to be injected at location A and be delivered (or returned from) location B. Here, (9) shows that the impedance is complex, exhibits decaying reactance components as one considers beam locations further and further removed from the point location, and is also frequency dependent like the drive point impedance. Analysis of drive point and transfer impedances in structures is a critical part of engineering system development and evaluation throughout design processes. This analysis is useful in the determination of sensitive paths of vibration or wave energy transfer through built-up systems. As a result, impedance-based analysis of structural, mechanical, and (as we will find) acoustic systems is common in many engineering practices for its value in first approximations of key sub-systems that govern the dynamics [4]. Drive point and transfer mobilities are the inverse relationships of drive point and transfer impedances. Thus, the transfer mobility from the force point x to the spatial location x along the beam is ( ) ( ) Yx = y xt, / f t. Expressions for impedances and mobility of various infinite and finite dimensioned elementary structures are available in texts [5] [4]. 67

68 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University 4 Acoustic wave equation Towards deriving the acoustic wave equation, we first define acoustic variables and constants. In the following, we assume that the fluid is lossless and inviscid meaning that viscous forces are negligible, and that the fluid undergoes small (linear), relative displacements between adjacent particles prior to a pressure change. These fluid particles imply an infinitesimal volume of fluid large enough to contain millions of fluid molecules so that within a given small fluid element, the element may be considered as a uniform, continuous medium having constant acoustic variables (defined below) throughout the element. Of course, at smaller and smaller length scales, the individual fluid molecules that make up the fluid element are in constant random motion at velocities considerably in excess of the particle velocities associated with acoustic wave propagation. Thus within any given fluid element, these molecules may in fact leave the element in an infinitesimal duration of time. Yet, they are replaced by other molecules entering the element. The consequence is that evaluating the relatively slow motions associated with wave propagation does not need to account for the very small time-scale dynamics of the individual fluid molecules. Because we consider only linear relative displacements between fluid particles, our theoretical development is limited to accurate treatment of linear acoustic phenomena. The same applies to the consideration of pressure and density fluctuations which are very small with respect to ambient pressure and ambient density. While focused on linear contexts, these linear acoustic phenomena are the core of a significant proportion of all acoustic events in air-borne and water-borne acoustic applications. The linearity of acoustic phenomena also corresponds to nearly lossless wave propagation, which indicates that thermal losses associated with fluid particle motion are negligible. In other words, the acoustic processes are isentropic. Finally, at this stage we consider only the acoustic free field which means we examine only unbounded fluid media. Also, we consider that this acoustic field contains no sources, and instead aim to characterize the type of wave functions that satisfy the acoustic wave equation itself, irrespective of the specific source that may be present. Acoustic pressure p( xt, ) is the pressure fluctuation around the equilibrium (atmospheric) pressure P : p( xt, ) P( xt, ) P P =. The SI units of pressure are Pascals [Pa=N/m ]. The atmospheric pressure is [kpa] near sea level. Here, x is the spatial dimension, limited currently to one dimension. The fluid particle displacement is ξ ( xt, ) while the fluid particle velocity is u( xt) equilibrium (atmospheric) fluid density is ( xt, ) ξ, = = ξ, t ρ xt,. ρ, while the instantaneous fluid density is ( ) To derive the acoustic wave equation, three components are required: the equation of state the equation of continuity the Euler's equation ( xt). The 68

69 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Equation of state. The fluid must be compressible to yield changes in pressure. Therefore, the thermodynamic behavior of the fluid must be considered. All fluids may be described by an appropriate equation of state that relates pressure, density, and temperature. For isentropic acoustic processes, the pressure is a function of density and for small pressure changes this can be expressed via a Taylor series P P P= P ( ρ ρ ) ( ρ ρ ) ρ ρ ρ ρ ( 9 ) Because the pressure and density fluctuations within the fluid element associated with a significant proportion of acoustic sound pressure levels are so small, the terms in (9) of order and greater are insignificantly small, thus P P P + ( ρ ρ ) ρ ρ ( 93 ) which is the linear approximation of the pressure change from ambient condition. From (93) we rearrange to yield P P P ρ ρ ρ ρ ρ ρ ( 94 ) The term on the right-hand side of (94) in brackets [ ] is called the condensation s ρ ρ ρ = ( 95 ) The condensation is the relative deviation of fluid element density from a reference value. The term on the right-hand side of (94) in the curly brackets { } is the adiabatic bulk modulus B. P B = ρ ρ ρ ( 96 ) Finally, as already defined, the term on the left-hand side is the acoustic pressure. p = Bs ( 97 ) Equation (97) is the equation of state. It may be thought of as "Hooke's law for fluids", since it relates a fluid "stress" (pressure) to the fluid "strain" (condensation) through the fluid "stiffness" (bulk modulus). A particular mathematical difference from the solid mechanics analogy is that (97) is a scalar equation since pressure has no directionality within a differential element. 69

70 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 7. Schematics for deriving (a) equation of continuity and (b) Euler's equation, respectively, giving attention to constant volume element and constant fluid element. Equation of continuity. Within an unchanging, defined volume of space, pressure changes cause mass to flow in and out of the volume. By conservation of mass, the net rate with which mass flows into the volume through the enclosing surfaces of the volume must be equal to the rate with which the mass increases within the volume. Figure 7(a) illustrates the scenario. A constant volume element dv = dxdydz is fixed in space. The mass flow rate in the volume in the x axis from the left/rear face of area dydz is dydzρ u x ( 98 ) where the particle velocity u component in the x axis is u x. The mass leaving the volume on the front/right face is approximated by a linearized Taylor series expansion to be ( ρu ) x x dydz ρux + dx ( 99 ) The net rate of mass influx in the x axis is therefore ( ρu ) ( ρu ) x x x dydzρux dydz ρux + dx = dv x ( 3 ) Similar expressions are obtained for the net influx of mass in the y and z directions. Consequently, ( ρu ) ( ρuy ) ( ρu ) dv x z + + = x y z ( ρ ) u dv ( 3 ) The rate at which the mass in the control volume changes is equal to ρ dv t ( 3 ) Thus, by conservation of mass, we obtain the equation of continuity (33) s ρ + ( ρ u ) = t ( 33 ) 7

71 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University where the dependence of the density ρ ρ ( s ) condensation is a small value, to permit the substitution. = + is assumed to be a weak function of time and the The continuity equation states that the time rate of change of mass in the control volume is equal to the net influx of mass in the same time duration. For small (linear) changes of acoustic pressure and weak dependence of the equilibrium density ρ on space, the linearized equation of continuity is s + u = t ( 34 ) Euler's equation. Unlike for the continuity equation, we now focus on a fluid element that is deformed in consequence to the pressure differences between opposing faces, see Figure 7(a,b,c). The fluid element moves with the remaining, host fluid media in which it exists. The pressure on the left-most face is P( xyzt,,, ) while, by a linearized Taylor series approximation, the pressure on the right-most face is P( xyzt,,, ) P( x, y, z, t) + dx. Thus, the force difference between left and right faces is x (,,, ) P( xyzt,,, ) P xyzt P( x, y, z, t) P( x, y, z, t) + dx dydz = dv x x ( 35 ) Analogous expressions to (35) are found for the net forces in the y and z axes. Because the force is directional, the total force vector is therefore df = PdV ( 36 ) The total acceleration of the fluid particle is [] u u u u u a = + ux + uy + uz = + ( u ) u t x y z t ( 37 ) The net force due to the pressure differentials (36) balances the net inertial force acting on the fluid particle, which is determined by the product of the total acceleration (37) and the fluid particle mass ρdv. Thus, by Newton's nd law, we find u PdV = ρ + ( u ) u dv t ( 38 ) In the absence of acoustic excitation, consideration of small absolute condensation values, and weak spatial dependence on the particle velocity vector in comparison to the time dependence, we simplify (38) to be u p = ρ t ( 39 ) The (39) is the linear Euler's equation. 7

72 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University 4.. Consolidating the components to derive the one-dimensional acoustic wave equation To summarize, equation of state p = Bs ( 3 ) equation of continuity s + u = t u Euler's equation p = ρ t ( 3 ) ( 3 ) To derive the acoustic wave equation, we first take the gradient of (3) and assume that the equilibrium is a weak function of time and space u t p = ρ ( 33 ) Second, we take the time derivative of (3), use the method of mixed partials to change the order of derivatives on the particle velocity, and multiply by the equilibrium density s ρ t u t + ρ = ( 34 ) Then, rerranging the terms of (34), we finally add (33) and (34) to yield s p = ρ t ( 35 ) Using (3), we obtain the acoustic wave equation (36) p = p c t ( 36 ) where the sound speed is c= B/ ρ. To summarize the developments and assumptions, the (36) is the linear (small pressure changes), lossless (inviscid) acoustic wave equation. The adiabatic bulk modulus is P B= ρ = γp ρ ρ ( 37 ) where γ is the ratio of specific heats which is specific to a given gas. In general P / ρ is almost independent of pressure so that the sound speed is primarily a function of temperature. Thus, an alternative derivation of the sound speed for air yields c = γ rtk = c + TC / 73 ( 38 ) 7

73 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University where T C is the temperature in degrees [ C]. c is the sound speed in air at [ C] which is approximately c =33.5 [m/s] at [atm] pressure (sea level) and at [ C]. Appendix A of [] provides a table of material properties for common gases. Mathematically, the curl of the gradient of a function must vanish, f =, so that by Euler's equation the particle velocity is irrotational, u =. This directs us to introduce a new function, which is a scalar, Φ : u = Φ ( 39 ) The function Φ is termed the velocity potential. Substituting (39) into (3) and assuming the equilibrium pressure is a weak function of space shows that p + ρ Φ = t ( 3 ) Consequently, we find p = ρ Φ t ( 3 ) Therefore, the velocity potential Φ satisfies the acoustic wave equation by substitution of (3) into (36). Φ= c Φ t ( 3 ) Thus, solving the (3) permits the subsequent determination of the fluid particle velocity and acoustic pressure by (39) and (3), respectively. The purpose of introducing the potential is to relate the particle velocity to the acoustic pressure via a common function, here a non-physical scalar. Thus, the acoustic pressure is related to the time rate of change of this scalar quantity at a location in space (3), while the particle velocity is related to the gradient of this scalar quantity at the same location (39). This is a general set of relations and is not limited in its interpretation to the coordinate system or waveforms of interest (e.g. whether plane, cylindrical, spherical waves). 4. Harmonic, plane acoustic waves When the acoustic variables are constant in the cross section spanning dy and dz, the Laplacian operator is = / x. Thus, in one Cartesian dimension, the acoustic wave equation is p p = x c t ( 33 ) 73

74 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University A scenario that realizes this one-dimensional variation of acoustic variables would be a duct stretching along the x axis and considering low frequencies of harmonic oscillations, respecting the duct crosssectional dimensions. Of course, (33) is similar to the one-dimensional wave equation determined for transverse oscillations of the string (93) as well as the wave equation for the longitudinal harmonic deformations of the rod (73). As a result, we anticipate that the complex exponential form of harmonic solution that satisfied (93) (73), via the (7) (76), will also satisfy (33). Therefore, we assume that the time-harmonic form (, ) j( ωt kx) j( ωt + kx) p xt = Ae + B e ( 34 ) is a solution to (33). These are harmonic plane acoustic waves that occur at angular frequency ω and respectively travel in the ± x directions according to the arguments kx in the exponentials. The particle velocity is determined from Euler's equation (3) p u = ρ x t ( 35 ) p = jωρu= jkρcu ( 36 ) x jk j( ωt kx) j( ωt+ kx) j( ωt kx) j( ωt+ kx) u( xt, ) = e e e e jωρ A + B = ρ c A B ( 37 ) The (36) exemplifies the time-harmonic form of the Euler's equation for plane waves. j( t kx) Using that k = ω / c and expressing p A e ω and j( t kx) p B e ω +, the particle velocity components in + = the positive + and negative - directions are found to be u± = ± p ± / ρc ( 38 ) Breaking (38) up, we see / ρ c u+ = p + ; = u = p / ρ c ( 39 ) The specific acoustic impedance is the ratio of acoustic pressure to fluid particle velocity and, like for mechanical vibrations, is a measure of resistance and reactance of the fluid media to inhibit or assist the propagation of waves: z=p/u ( 33 ) Therefore, for plane waves, z= ± p ± p = ± ρc / ρ c ( 33 ) 74

75 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University the specific acoustic impedance is purely real and dependent upon the direction of wave travel. The interpretation of a purely real impedance for traveling waves is that energy is transferred without return from a previous source or origin. For air at [ C], the specific acoustic impedance is ρ c =45 [Pa.s/m]. Because the product of atmospheric density and sound speed constitutes the impedance, their combination is typically more important in acoustics applications than the individual values alone. Appendix A of [] reports the impedances of numerous fluid media. 4.. Impedance terminology clarification There are several variations of impedance used throughout the study of acoustics. The terminology must be clarified since each variant possesses different units and has different contexts of appropriate use. The mechanical impedance is the ratio of force to velocity, units [N.s/m]. The specific acoustic impedance is the ratio of acoustic pressure to particle velocity, units [N.s/m 3 ]. The acoustic impedance is the ratio of acoustic pressure to volume velocity of the fluid, units [N.s/m 5 ]. For plane waves, where the acoustic pressure is constant over a constant cross-section S, the acoustic impedance is the specific acoustic impedance normalized by the area S, while the mechanical impedance is the specific acoustic impedance multiplied by the area S. 4.3 Plane progressive shock waves A detailed derivation of the nonlinear acoustic wave equation is considerably beyond the scope of this course. Interested individuals should consult [3] or [5] for such derivation. It suffices us to consider the equation directly: ( ) Φ Φ Φ γ c Φ = + ( γ ) Φ + Φ Φ( Φ u ) + ( Φ) t t t ( 33 ) where c is the linear sound speed, γ is the ratio of specific heats and u is the fluid medium mean speed. If the fluid is not flowing in this way, then u =. Of course, should all of the nonlinear terms on the righthand side of (33) vanish, then (33) returns to the linear acoustic wave equation as expressed according to the velocity potential, (3). No assumptions as to the significance of the acoustic variable perturbations are made in the derivation of (33). Thus (33) holds for all acoustic variable fluctuations for lossless fluid media. The (33) is derived by use of two equations, which under the assumptions of plane wave propagation reduce to (333) and (334). c c γ + u + c u = t x x ( 333 ) 75

76 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University u + u u + c c = t x γ x ( 334 ) The (333) and (334) govern the time- and space-dependent sound speed c( xt, ) and particle velocity u( xt, ). The dependencies on the sound speed are of course unique with respect to linear acoustic wave disturbances. Integral evaluations of (333) and (334), and further manipulations, reveal γ c= c + u ( 335 ) Then by substitution of (335) into (334), it is seen that u u + ( c + βu) = t x ( 336 ) where the constant is β = ( γ + ) ( 337 ) Solving (336) is accomplished with relative ease, compared to (33). First, the linearized result when u βu is negligible is x x u = f t c ( 338 ) which is of course the general form of wave solution argument for waves propagating in the + x direction. We then consider that replacement of c with c + βu in (338) is a way to satisfy the full form of (336). By substitution, indeed this is the case. Thus, u = f t c x + βu ( 339 ) The argument of (339) has units [s]. For harmonic, lossless, progressive plane waves, the argument is assumed to be in units [rad] to accommodate a sinusoidal profile. Thus, u( xt, ) = Usin ω t c x + βu ( 34 ) The (34) is a nonlinear function of the particle velocity and its interpretation is revealing to the formation of shock waves. For great enough speed amplitude U, the propagation of the wave to increasing values x causes the crests of the wave (traveling at c + βu ) to overtake the troughs of the wave (traveling at c βu ). Figure 8 shows a representative case of this trend. For increasing x a point is reached at which the solution to the nonlinear (34) results in a multi-valued speed. In reality this is the onset of a shock wave 76

77 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University and an "N"-shaped wave profile is measured in practice. Thus, a point in the acoustic field is subjected to a near-instantaneous increase in acoustic pressure from a resting, equilibrium value. In some contexts, shock waves are referred to as sonic booms, particularly in aerospace applications. A measured shock wave from a line charge (explosion) is shown in Figure 9. The "N" shape to the waveform is very apparent. Figure 8. Example of plane progressive shock wave formation. Figure 9. Measured shock wave. Courtesy of M.C. Remillieux of Ref. [6]. 4.4 Acoustic intensity As observed in (33) by the lack of imaginary terms, acoustic pressure and the particle velocity are in phase for plane waves. As a result, acoustic power is transmitted. 77

78 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University The instantaneous intensity I ( t) = pu of an acoustic wave is the rate per unit area at which work is done by one element of the acoustic fluid on an adjacent fluid element. Note that in general, the particle velocity is a vector u and thus the intensity is also vector quantity. The units of I ( t ) are [W/m ]. The average acoustic intensity I is the time average of I ( t ) T I = I ( t ) = pu = pu d t T T T ( 34 ) where, for harmonic waves with angular frequency ω, the period is T = π / ω, and the integration considers the real components of pressure and particle velocity. In general throughout this course, unless otherwise specified, use of the term acoustic intensity refers to the time-averaged version (34) and not the instantaneous measure of intensity. For plane waves, the intensity is one-dimensional so that we drop the overbar, thus recognizing that "travel" in the opposing direction is indicated by a change of sign: I p = = ρcu ( 34 ) ρc Using the relations for mean-square, RMS, and amplitude, the intensity may be written as prms P I = prmsu = rms ρ c = ρ c ( 343 ) In general, when using the complex exponential representation of acoustic variables, thus considering timeharmonic oscillations, the time-harmonic [average] intensity is computed from I T * = Re[ ] Re[ ] dt Re T p u = pu ( 344 ) where the asterisk denotes the complex conjugate, the vector notation on particle velocity is retained, and the / multiple results from the time-harmonic integration. This time-harmonic intensity is only a vector if u spans more than one dimension. Note that for a complex function x, * Re xrms x x x = =. 4.5 Harmonic, spherical acoustic waves For spherically symmetric sound fields, use of the appropriate Laplacian operator in (36) and use of the chain rule of differentiation allows us to express the wave equation as ( rp) ( rp) = r c t ( 345 ) 78

79 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University It is apparent that this equation is similar to (36) with the variable mapping p rp. Thus, we use the same general solution to the wave equation ( ) ( ) rp = f ct r + f ct + r ( 346 ) Then, we express the acoustic pressure, without loss of generality as p( r, t) = f( ct r) f( ct r) r + r + ( 347 ) which is valid for all r >. For the first term of (347), the waves are outgoing, from the origin r =, and spread in greater and greater areas from the origin. For the second term of (347), the waves are incoming and increase in intensity approaching an origin. In this second case, there are few examples in linear acoustics that pertain to such self-focusing of sound energy by a wavefield converging to an origin. So we hereafter neglect the second term of (347). By and large, the most important spherically-spreading acoustic waves are harmonic, in which case we assume a solution to (345) of the form j( ωt kr) p ( rt, ) = A e ( 348 ) r Using the Euler's equation (3) in time-harmonic form for spherically symmetric wave propagation p = jωρu= jkρcu ( 349 ) r the particle velocity is then derived to be ( rt, ) p u ( rt, ) = j ( 35 ) kr ρ c Using (348) and (35), the specific acoustic impedance for spherical waves is ( kr ) kr ( ) ( ) p jkr z = = ρc = ρc + j u + jkr + kr + kr ( 35 ) The specific acoustic impedance of spherical waves contains resistive and reactive components. A comparison of this characteristic to plane waves is made in Sec Spherical wave acoustic intensity and acoustic power The acoustic intensity for spherical waves is computed in the same manner as for plane waves, it is the time average of work per unit area that fluid particles exert on neighboring particles. Using the time-harmonic equation form with ( rt, ) A j( ωt kr) p = e and 35, we have r 79

80 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University * A j( ωt kr) A j( ωt kr) I = Re Re e j e pu = + ( 35 ) r kr ρc r I = A I ρ cr = ( 353 ) where the overbar is dropped due to the uni-axial (radial) propagation of the waves. Expressing (353) using P= A/ r, we have P I = ρ c ( 354 ) The (354) is equivalent to (343) for plane waves, but the interpretation of P is distinct between (354) and (343). For harmonic plane waves, P is independent of the spatial coordinate x. Thus, in lossless acoustic media, energy transfer persists without diminishing in intensity during propagation. In contrast, for spherical waves the change of P is in inverse proportion to the change in radial coordinate r. Thus, spherical wave intensity changes in proportion to the square of the radial distance of travel r for the wave. This means that the acoustic energy radiated to a field point that is a distance r from the spherical wave origin (source) is reduced in proportion to the squared distance between source and point, and explains why sounds decay in amplitude as they travel from their origins. In general, the acoustic power is the sound energy per time radiated by an acoustic source, and is defined according to the intensity via Π= ns d ( 355 ) S I where S is a surface that encloses the sound source and n is the unit normal to the surface. Thus the dot product in (355) refers to the component of the intensity vector that is normal to the enclosing surface under consideration. For a given radial distance from a source of sound, if the intensity has the same magnitude, then the acoustic power is simply Π= IS. For plane and spherical waves, only one direction of wave propagation is considered, either a single axis of one-dimensional motion or a radial direction spreading, respectively. For spherical waves, the area through which the sound spreads is 4π r. Thus, the acoustic power is p Π= = ( 356 ) rms 4πrI 4πr ρc 4.6 Comparison between plane and spherical waves A comparison of the plane wave acoustic pressure and particle velocity with those corresponding values kr = π / λ r. In for spherical waves shows significant differences. It is instructive to consider the term ( ) other words, when kr < the observation radial location of the acoustic variables is approximately within one acoustic wave wavelength from the origin of radial wave propagation. 8

81 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Thus, when kr <, the imaginary component of the spherical wave particle velocity (35) is as important or more significant than the real component. This means there is a phase difference between acoustic pressure (348) and particle velocity (35) for spherical waves near the origin of the sound source from which the waves propagate. As a result, the specific acoustic impedance of spherical waves near the sound source contains real and imaginary components, Figure 3 and (35). For radial distances far from the acoustic origin, kr >>, the imaginary component of the particle velocity (35) is insignificantly small. As a result, for kr >> the spherical wave particle velocity is in phase with the acoustic pressure. Figure 3 shows this trend that the imaginary contribution of the specific acoustic impedance of spherical waves converges to zero for kr >> while the real component approaches that of plane waves. specific acoustic impedance normalized by ρ c plane wave real(spherical wave) imag(spherical wave) kr, normalized distance from acoustic source origin Figure 3. Specific acoustic impedance normalized by plane wave specific acoustic impedance. Comparing plane and spherical acoustic waves, we refer to Table 9. Table 9. Comparison between plane and spherical acoustic waves plane waves radiate in direction x spherical waves radiate in direction r pressure amplitude is constant as distance x changes pressure amplitude decreases as /r acoustic intensity is proportional to p rms acoustic intensity is proportional to /r p rms and decreases as pressure and particle velocity are in phase pressure and particle velocity are only in phase for kr >>, and in general are not in phase the specific acoustic impedance is real the specific acoustic impedance is real for kr >>, and in general is complex Thus, for point acoustic sources (infinitesimally small radiating spheres), we refer to the acoustic far field as the location in space where spherical wave characteristics approach those of plane waves. The acoustic far field satisfies kr >>. In the acoustic far field, spherical waves have the following characteristics 8

82 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University the acoustic pressure and the particle velocity are in-phase the acoustic intensity is in the radial direction r the specific acoustic impedance is purely real and equal to ρ `c In general, the acoustic far field refers to a long distance between the receiving point of acoustic pressure waves and the acoustic wavelength. This is evident by kr πr λ π( r λ) = / = / >>. 4.7 Decibels and sound levels In acoustics, we often use logarithmic scales to characterize the amplitude of acoustic pressures to which we are subjected. The sound pressure level is defined p SPL = log ( 357 ) where rms pref p ref is the reference sound pressure and is often taken to be p ref = μpa for sound in air. The units of SPL are decibels [db]. Equation (58) is also written prms SPL = log ( 358 ) p ref Recall that p = P/ where P is the amplitude of the acoustic pressure in units [Pa]. rms In addition to sound pressure level, we are also often interested in sound power level L Π Π = log Π ref ( 359 ) where Π ref = - [W]. The sound power level for spherical waves may be also expressed using p 4 r pref L = log + π Π log Π ρ c rms pref = SPL + log r + ref ( 36 ) To combine incoherent acoustic signals, the total RMS pressure of the incoherent frequencies ω i is computed from (36), after which the SPL is determined from (357). An alternative computation using the individual SPL values at the incoherent frequencies is given in (36). ( ) p = p ω ( 36 ) rms rmsi i i SPL ( ω )/ SPL = log i i ( 36 ) i 8

83 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University 4.8 Point source and source strength The point source is the fundamental acoustic source constituent employed in the study of more complex and realistic sources of acoustic waves. This is because, by summation or integral forms according to the Huygens' principle, arbitrary-shaped sources of sound may be analyzed by virtue of the appropriate combination of point sources. j t Consider that a sphere of radius a oscillates radially with a velocity of Ue ω, Figure 3, and therefore has a displacement amplitude U / ω. This acoustic source is called the monopole. To satisfy the boundary condition at the sphere surface, the particle velocity of the acoustic fluid must be equal to the surface normal velocity of the sphere, which is everywhere U. Therefore, we have that ( at, ) p jωt u( a, t) = j = Ue ka ρc A = j e = Ue ka aρ c j( ωt ka) jωt ( 363 ) jka jka A = ρcua e ( 364 ) + jka Consequently, the pressure field generated by a monopole is a jka j ω t k( r a) p = ρcu e ( 365 ) r + jka By substitution, the particle velocity is a j ω t k( r a) u ( rt, ) = U e ( 366 ) r j t Note that (366) meets the boundary condition that r = a, we have u ( a, t) = Ue ω. Note also that (366) is only valid r > a. At distances satisfying ka <<, which is equivalent to π( a ) / λ <<, the pressure is a j( ωt kr) p = jρcu ( ka) e ( 367 ) r We call such spherical acoustic sources (367) that are small relative to the acoustic wavelength as point sources. 83

84 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 3. Monopole in spherical coordinate system showing location vector r denoting point (grey) in field. The complex source strength Q is computed from the integral of the surface normal velocity over the surface. j t Qe ω = u nds ( 368 ) S Considering the development defining the monopole, the complex source strength is equal to the product of the uniform velocity amplitude and the sphere surface area. Q = Q= 4π au ( 369 ) The units of source strength are illuminating: [m 3 /s]. Thus, the source strength characterizes the rate at which fluid volume is harmonically pumped back and forth at the source surface. The expression (368) using j t Q e ω is referred to as the complex source volume velocity, and is used regularly in the analysis and evaluation of sound power delivery from acoustic sources. Substituting (369) into the expression for the acoustic pressure emitted by the point source gives p = = = Q a jρc ( ka) e 4π a r Q j( ωt kr) jρc ke 4π r Q j( ωt kr) jρc e λr j( ωt kr) ( 37 ) While (37) is valid only for field points many acoustic wavelengths from the source center, the expression for source strength (369) may be substituted into the full relation between the monopole surface normal velocity amplitude and the acoustic pressure emitted (365) since the determination of (368) is an integral evaluation over the source surface. 84

85 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University 4.9 Acoustic reciprocity A detailed establishment of the principle of acoustic reciprocity is beyond the scope of this book. Interested individuals should consult Ref. [7] for further details and the relevant integral formulations. It suffices us to understand what acoustic reciprocity means and its impacts. Acoustic reciprocity is the principle that under time-invariance in non-flowing, unbounded fluid, and regardless of the presence of scattering objects, the positions of an acoustic source and receiver may be interchanged without altering the received acoustic pressure [] [7]. There are numerous implications to this principle that will be encountered later in this course. Nevertheless, at this time, the unique symmetry of acoustics embodied in reciprocity is to be appreciated. 85

86 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University 5 Reflection and transmission Acoustic waves incident on interfaces between two media are partially reflected and partially transmitted. The theories employed in this portion of the course are applicable for media interfaces between multiple fluids under all circumstances of wave incidence while the theories are applicable for media interfaces between fluids and solids only under special cases. Such factors will be elaborated upon at the appropriate times in the following sections. Figure 3. Schematic of normally incident, reflected, and transmitted pressure waves through two fluid media. 5. Normal incidence wave propagation at a fluid interface Consider a normally-incident, plane acoustic pressure wave p i with complex amplitude P i arriving at the boundary between a source fluid (fluid ) of specific acoustic impedance r = ρ c and a second fluid (fluid ) of specific acoustic impedance r = ρ c, as shown schematically in Figure 3. A reflected, plane wave p r with amplitude P r is induced back within the source fluid while a transmitted, plane wave p t with amplitude P t is induced in the second fluid. The pressure reflection coefficient and the pressure transmission coefficient are defined, respectively, using (37) and (37). r R = P P t T = P P i i ( 37 ) ( 37 ) The pressure reflection and transmission coefficients are most generally complex. The intensity of plane waves is P /r, where P is the amplitude of the complex pressure and as above r is the specific acoustic impedance. The intensity reflection coefficient and the intensity transmission coefficient are, respectively, computed by (373) and (374). R I r I = = R ( 373 ) Ii 86

87 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University T I I I = t = T ( 374 ) i r r The power reflection coefficient and the power transmission coefficient are computed by (375) and (376), respectively. Note that the incident and reflected areas are the same of a different cross-section interface. A i while the transmitted area may be A t for instance in the event of a duct of changing cross-section at the fluid R = Π R = R ( 375 ) I T A A r = t t TI A = Π i Ar T ( 376 ) i Based on the normalization of the constants and by virtue of the conservation of energy, we have that R Π + T = ( 377 ) Π Considering the interface plane between fluids and to be given at x =, as shown in Figure 3, we define the incident, reflected, and transmitted acoustic pressures. i e j( ω t kx ) i p = P ( 378 ) j ( t kx ) r = e ω + r p P ( 379 ) t e j( ω t kx ) t p = P ( 38 ) Because the waves must occur at the same frequency to maintain contact, the wavenumbers between fluids and are related by k = ω / c and k = ω / c. Thus, the rates of which points of constant phase in the waves travel are different from fluid to fluid. Correspondingly, this indicates that the acoustic wavelength in each fluid is unique. In fluid, the wavelength is λ c / f = while in fluid the wavelength is λ = c / f. In order for the fluids to remain in contact at x =, two boundary conditions must be satisfied. Continuity of pressure keeps the interface at x =, otherwise a net force would produce a shift in the boundary in the x direction. Continuity of normal particle velocity keeps the fluid particles at the interface in contact. Applying these two boundary conditions leads to pi + pr = p t ( 38 ) ui + ur = u t ( 38 ) A third, derivative boundary condition is observed by virtue of dividing (38) by (38). The continuity of normal specific acoustic impedance is therefore generally given by 87

88 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University p + p = p u + u u i r t i r t ( 383 ) By the notations of this section, plane wave pressure and particle velocity are related by p/ u = ± r. Therefore, we make appropriate substitutions for the particle velocities in (383) to find p + p r = r i r pi pr ( 384 ) Rearrangement of (384) gives the expression for the pressure reflection coefficient between the two fluid media. r r r r R = = r r + r ( 385 ) + r Division of (38) by p i shows that + R = T. Therefore, the pressure transmission coefficient is found to be r r r T = = r r + r ( 386 ) + r The intensity reflection and transmission coefficients for normal incidence acoustic pressure between the two fluids are R T I I r r r r = = r r + r + r r 4 4rr r = = ( r + r) r + r ( 387 ) ( 388 ) Studying the pressure reflection coefficient, we first observe that R is always real. When r < r, the reflected pressure is in phase. Thus, the pressure near to the interface on the side of incidence is greater in amplitude than the amplitude of the incident wavefront itself. This is schematically shown in Figure 33(a) and would be similar to the case of a normally incident wave in air arriving at an air-water interface. 88

89 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 33. Examples of reflected and transmitted pressures based on the impedance difference between two fluids. When r > r, similar to a water-air interface, Figure 33(c), the reflected pressure is out of phase with the incident pressure. Thus, the pressure near to the interface on the side of incidence is less in amplitude than the amplitude of the incident wavefront itself. Of course, when r = r, there is no "interface" between fluids because they are the same and no reflected pressure wave is induced, Figure 33(e). The pressure transmission coefficient is always real and positive, so that transmitted waves are always in phase with the incident wavefront. Yet, the transmitted waves will either be reduced or magnified in amplitude based on the conditions r > r or r < r, respectively. The transmitted pressure trends are shown in Figure 33(b,d,f), including the case when r = r and the pressure transmission coefficient takes on a unit value. In the limit of r / r =, a rigid boundary is realized. The transmitted pressure is doubled although the power transmission coefficient is zero. In the limit of r / r =, a pressure release boundary is realized. As a result, there is neither pressure transmitted nor power transmitted. Importantly, there is no difference in the intensity reflection and transmission coefficients in the consideration of the direction of the incident wave. 5. Normal incidence wave propagation through a fluid layer Consider a fluid layer of specific acoustic impedance r = ρ c and thickness L that is bounded by a fluid with incident wavefront p i and specific acoustic impedance r = ρ c and by a final fluid layer permitting a transmitted wavefront p t and specific acoustic impedance r3= ρ3c3. The situation is shown in Figure

90 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 34. Schematic of wave transmission through a fluid layer. Incident and reflected waves in fluid are i e j( ω t kx ) i p = P ( 389 ) j ( t kx ) r = e ω + r p P ( 39 ) while the waves transmitted into fluid (the fluid layer) are a j( t kx ) e ω p = A ( 39 ) j ( t kx ) b = e ω + p B ( 39 ) while the transmitted wave into the final fluid 3, which does not have bound for continued increase in x, is t e j( ω t kx 3 ) t p = P ( 393 ) Continuity of normal specific acoustic impedance at x = and x= L yields (394) and (395), respectively. Pi + Pr r A+ B = P P r A B i r ( 394 ) Ae A + Be r jkl jkl 3 = jkl jkl e Be r ( 395 ) Following considerable manipulation of (394) and (395), one obtains the pressure reflection coefficient r r r cos kl + j sin kl r3 r3 r R = ( 396 ) r r r + cos kl + j + sin kl r3 r3 r In addition, the intensity and power transmission coefficients, which are the same presuming the crosssectional areas of the fluid interfaces are all the same, are 9

91 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University TI 4 = TΠ = r r r rr + + cos + + sin r r rr r 3 3 kl 3 3 kl ( 397 ) Evaluating special cases of the most general result of (397) provides illumination to the opportunities for wave transmission and reflection. When r = r 3, we have that T = Π 4 r r r r + sin kl ( 398 ) Continuing, when the layer has specific acoustic impedance that is significantly greater than the fluid impedances bounding it, r >> r, (398) becomes (399). T = Π + 4 r r sin kl ( 399 ) The transmission of normally incident pressure waves through such a near-rigid media is comparable to sound transmission from one room to another through most walls of moderate thickness. In fact, considering the realistic case that ( ) r/ r sin kl>> when the fluid is air and the cross-sectional dimensions of the wall layer are much greater than the thickness, we obtain a further-reduced form of (399) that approximately holds for normally incident sound transmission through a wall T r Π = rsin kl ( 4 ) Considering practical building designs, unless the frequency of consideration is extremely high, (4) reduces further by virtue of sin kl kl T r Π = klr ( 4 ) Therefore, the transmitted pressure amplitude is inversely proportional to the wall thickness while the transmitted power is inversely proportional to the square of the thickness. This is effectively a re-statement of the mass law in characterization of the sound insulation capabilities of panels, where it is seen that the transmitted pressure amplitude is inversely proportional to the mass per unit area (product of density and wall thickness). 9

92 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University A case of particular interest in the general form of the power transmission coefficient (397) occurs when kl = n π where n =,,3... As such, we find T = Π r 4rr 3 rr 3 + r ( 4 ) c As a result, when the frequency of the incident wave is near f = n L and the specific acoustic impedance of the separating fluid layer is the geometric mean of the bounding media impedances, r = rr 3, there is total transmission T Π of the acoustic energy. This corresponds to a form of acoustic resonance. For instance, by expressing these frequencies in terms of the acoustic wavelength λ, we find that the lengths of the separating fluid layer for which this phenomenon occurs are 3 5 L= ( n ) λ ; L= λ ; L= λ ; L= λ ;... ( 43 ) The use of quarter-wavelength devices to permit large wave transmission through interfacing layers is a means to permit high efficiency power transmission between media. This technique is effectively demanded in medical ultrasonics where piezoelectric-based acoustic radiators must interface with the human body, which has a specific acoustic impedance similar to water [8]. Thus, several such "matching layers" are employed whereby each layer is approximately the geometric mean of its adjacent neighbors. Note that this resonant phenomenon is only encountered for wave propagation at select frequencies. A similar resonant phenomenon occurs for kl = nπ. In such events, the power transmission coefficient becomes T 4 rr /( r r ) Π = + which takes on a unit value when the bounding fluid media are the same, 3 3 r = r3 = r. Thus, we find that half-wavelength layers, L = nλ /, permit perfect sound transmission at the frequencies associated with this through-thickness resonance, f = nc /L. 5.3 Oblique incidence wave propagation at a fluid interface Consider the incident plane wave impinging on the interface between two fluid media at an angle of incidence θ i where the interface is at x =. Figure 35 gives a schematic of the situation. The incident, reflected, and transmitted waves are expressed by j( t kx cos i ky sin i) i = e ω θ θ i p P ( 44 ) j( t kx cos r ky sin r) r = e ω + θ θ r p P ( 45 ) e j( ω t kx cosθt ky sin θt) t = t p P ( 46 ) 9

93 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University There is no fluid interface in the y = plane such that the assumed directions of wave propagation within fluid in the y axis are correct as shown in Figure 35 according to our finding of Sec. 5.. The angle of the reflected wave to the normal is θ r while a more general, complex angle form is used for the transmitted wave θ t for reasons to be discovered through derivation. Figure 35. Schematic of obliquely incident, reflected, and transmitted pressure waves through two fluid media. Continuity of pressure at x = gives Pe + Pe = P e ( 47 ) jky sinθi jk ysinθr jkysin θt i r t The result of (47) must be true for all values y which requires the exponents of (47) to all be equal. Thus, first we conclude that sinθ = sinθ ( 48 ) i r which requires θi = θr. Also, the exponents from left- to right-hand sides of (47) must be equal so that Snell's law is obtained. sinθ i sin θ = t c c ( 49 ) In addition, division of (47) by the incident acoustic pressure also relates the pressure reflection to pressure transmission coefficients. + R = T ( 4 ) Continuity of the normal component of particle velocity must be satisfied lest the two fluids lose contact at x =. Therefore, u cosθ + u cosθ = u cosθ ( 4 ) i i r r t t The (4) is written in terms of the pressure amplitudes and specific acoustic impedances of the fluids to yield r = cosθ t R T ( 4 ) r cosθi 93

94 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Together, (4) and (4) may be manipulated to give the pressure reflection coefficient R r cosθt r r r cosθ cosθ cosθ i t i = = r cosθt r r + r cosθ i cosθ t + cosθ i ( 43 ) The (43) is referred to as Rayleigh's reflection coefficient. In addition, rearrangement of Snell's law (49) gives cos c sin θ t = θi ( 44 ) c Considering the (43) and (44), the angle of wave transmission is strongly dependent on the relative difference between the sound speeds of the fluids. If c > c the angle of the transmitted wave θ t is real and less than the incident angle θ i. Figure 36(a) gives an example of this outcome which can happen at an interface separating water and fluid domains, with the incident wave coming from the water domain. In this case, the transmitted waves are bent towards the normal to the interface. If c < c and θi < θc, the transmitted angle θ t is real but greater than the angle of incidence. In this case, the transmitted waves are bent away from the normal to the interface. The angle θ c is termed the critical angle. By virtue of (44), the critical angle is computed from sin c c θ = ( 45 ) c The pressure reflection coefficient is unity when incident waves impinge on the interface at the critical angle. Thus, the reflected waveform has the same amplitude and is in phase with the incident waveform. By (4), the transmitted acoustic pressure (and power) under such circumstances is zero. Figure 36. Examples of oblique incidence acoustic pressure wave reflection and transmission phenomena between two fluids. 94

95 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University When c < c and θi > θc, the cosine of the transmitted angle cosθ t is imaginary, while by Snell's law the sine of the transmitted angle sin θ t is real. Consequently, the transmitted pressure is alternatively written γ x j( t ky ) sin i p = P ( 46 ) t t e e ω θ where the constant of exponential decay is γ k c = sin θi c ( 47 ) From (46), it is seen that the y axis component of transmitted wave propagation persists while the component normal to the interface, in the x axis, decays in amplitude exponentially away from the interface. This evanescent wave consequently does not transmit acoustic energy into the fluid (beyond a very short distance in x ). This example is shown in Figure 36(b) and could occur for such "steep" angles of incidence at an air-water interface where the incident wave arrives from the air fluid domain. In this case θ > θ, the pressure reflection coefficient (43) becomes alternatively expressed using Euler's identity and i c manipulations j R = e φ ( 48 ) ρ cosθ ρ cosθi c tan φ = ( 49 ) As clearly shown by (48), the reflected wave has the same amplitude as the incident wave. Yet, the phase of the reflected wave is shifted respecting the incident waveform. For angles of incidence slightly exceeding the critical value, the reflection coefficient is near R =+ such that the reflected wave is of the same amplitude and in phase with the incident wave, which is similar to the special case indicated above and is comparable to a rigid boundary. As the incident angle approaches grazing incidence, θ 9, the reflection coefficient approaches R =-. Therefore, the reflected wave has the same amplitude but is perfectly out of phase to the incident wave, similar to a pressure release boundary. In the general case, the power transmission coefficients are r cosθt 4 r cos i T = θ Π r cosθ t + r cosθi ; when θ t is real ( 4 ) T Π = ; when θ t is imaginary ( 4 ) which confirms the aforementioned claims that no acoustic power is transmitted when the incident angle exceeds criticality due to the decaying exponential factor that is a function of x in (46). 95

96 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University An interesting phenomenon occurs when r / r = cos θ t / cosθi. Then, T Π =. In other words, all incident acoustic power is transmitted into the second fluid. Manipulating (4) under such conditions leads to the means to compute the angle of intromission θ I : r r r r sinθi = = r c ρ r c ρ ( 4 ) This phenomenon enables the complete transfer of obliquely incident acoustic energy from one fluid media to another, regardless of the frequency. Intromission thus creates perfect acoustic transparency between two distinct media. Intromission requires either [ r > r and c < c] to occur, or [ r < r and c > c] to occur. While the combinations of these properties are not met in natural bulk materials, there are ways of developing or engineering artificial materials to achieve perfect acoustic transparency from an air-material interface. For example, see Ref. [9] and select references therein. 5.4 Oblique incidence wave propagation at a thin partition interface: the mass law In building or architectural acoustics, a formula known as the mass law serves as a ballpark estimate of the significance of sound insulation from one room to another provided by a thin partition. The mass law may be derived directly by the methods undertaken in this chapter pertaining to oblique incidence sound transmission through multiple media. Similar to Figure 34, consider an obliquely incident wave from an unbounded air domain impinging on a thin, flexible partition material at x = of thickness L, on the other side of which is another unbounded region of air. Thus, the difference between Figure 34 and this scenario is that the incident wavefront is obliquely incident to the intermediate layer. When the partition is thin with respect to the acoustic wavelength, kl<<. Therefore, an application of Snell's law reveals that the angle of transmission θ t is the same as the angle of incidence θ i. As a result, the normal component of particle velocity at the thin interface is ui + ur = u t ( 43 ) The surface density of the partition is ρs = ρl where ρ is the volumetric density. Newton's nd law normalized by area is thus an equivalence between the pressure balance from both sides of the layer to the area-normalized inertial force. p + p p = jωρ u cosθ ( 44 ) i r t S t Multiplying (43) by the specific acoustic impedance (recalling that there is only one fluid) yields 96

97 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University p p p u + u = u u u u i r t i r t i r t ( 45 ) and division by the incident acoustic pressure on (45) and (44), respectively yields (46) and (47). R = T ( 46 ) ωρs + R = T+ j T cosθ ( 47 ) r Solving for the power transmission coefficient gives Τ Π = ωρs + cosθ r ( θ ) ( 48 ) The transmission loss TL is computed from the power transmission coefficient by TL = log T Π ( 49 ) Therefore, the transmission loss of a flexible partition in the moderate frequency range, i.e. for most frequencies associated with speech and music, the mass law is TL ωρ log cosθ r S = + ( 43 ) In many cases, the denominator of (48) is overwhelmed in magnitude by the second term, so long as the angle of incidence is not significant. Thus, considering (48) and (43), each doubling of surface density reduces the power transmission coefficient by approximately four times, while the TL correspondingly log ~ 4 6. A common reality in building and architectural acoustics is that mass increases by 6 [db], ( ) between two rooms is the one assured means to inhibit sound transmission. The mass law can guide decision-making in such practices. 5.5 Method of images If the wavefront incident on a plane interface is spherical, such as at the z = plane interface in Figure 37, we can use the method of images to evaluate the waves received at locations elsewhere in the acoustic media from where the waves originated. The exact plane interface type, e.g. air-to-water or water-to-air, does not matter so long as the boundary conditions pertaining to the interface are satisfied, thus satisfying the acoustic wave equation. The method of images replaces the reflected wavefront by an imaged wavefront that hypothetically emanates from the fluid below the plane z =. The superposition of incident and imaged wavefronts are determined so as to satisfy the boundary conditions at the interface. Consequently, the acoustic field above 97

98 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University the plane surface, in the domain of the real point source, is correctly reconstructed. On the other hand, the acoustic field below the surface is not correctly reconstructed. Figure 37. Schematic of physical point source above a plane interface at z =, and an image source below the plane. Consider the example shown in Figure 37. A point source of spherical waves is at location ( xz, ) = (, d) above a perfectly rigid plane z =. The field point is at (,, ) xyz, where positive y locations are into the page. The direct pressure wave received at the field point as emitted by the point source is A j( t kr ) i e ω p = ( 43 ) r ( ) r = z d + y + x ( 43 ) We then imagine that the fluid domain from where the waves arrive continues without interupt by the plane. Then, we introduce a hypothetical, coherent image point source at a location below the plane interface that is identical to the distance above the plane of the real point source. The image point source has the same amplitude as the point source. A j( t kr ) r e ω + p = ( 433 ) r + ( ) r = z + + d + y + x ( 434 ) The rigid plane boundary condition requires that the normal component of particle velocity (in the z axis) at the z = plane must vanish. Because the sources have the same pressure amplitude and are identical distances away from the now-hypothetical plane interface, it is clear that the normal component of particle velocity vanishes at z =. Thus, instead of a rigid plane interface, the direct and reflected pressures in the fluid domain z > are recreated by an unbounded fluid domain possessing original and imaged point sources. The acoustic field 98

99 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University in z > is correctly reconstructed by this approach while the field in z < is not correctly reconstructed by this approach. Simplifying the problem and considering the schematic of Figure 37, we assume that the field point is a radial distance r from (,,) y such that r >> dcosθ. Thus, we find that the superposition of original and imaged point source spherical wavefronts at the field point gives jk r jk r A j( ωt kr) e e p( r, θ, t) e + r r r + r r ( 435 ) where r = dsinθ. When the field point is many times r away from the origin line (, y,), r >> r, the inclusion of this term into the denominators of (435) may be safely disregarded. On the other hand, the presence of r in the complex exponentials may not be disregarded since, by Euler's identity, the function arguments repeat every π. Thus, by use of Euler's identity and with the simplifications described above, we have that A j( ωt kr) p( r, θ, t) e cos[ kd sinθ] ( 436 ) r When the source is very near to the rigid plane with respect to the wavelength kd <<, the pressure received A j( ωt kr) at the field point is p( r, θ, t) e. In other words, a point source near a rigid plane projects twice r the amplitude of acoustic pressure into the field as the same point source in the free field. Simple calculations may show that the intensity in the real (not imaged) acoustic field increases 4 times in consequence to the rigid plane while the acoustic power increases times. In practice, a baffle is the term used to describe the rigid plane onto or into which an acoustic source is placed in order to enhance its projection of sound power into the field. If the interfacing surface is a pressure release boundary, rather than rigid boundary, then it can be shown that the imaged source must be of the same amplitude but of opposite phase as the original point source. The resulting pressure superposition at the plane satisfies the boundary condition of p( xyz=,, ) = while, using polar coordinates, the pressure in the field is A j( ωt kr) p( r, θ, t) j e sin[ kd sinθ] ( 437 ) r To apply the method of images, the acoustic waves need not be single frequency since the superposition of general spherical wave solutions for the direct and image sources, according to the form p = f ( ct r ), r satisfies the boundary condition where the normal component of particle velocity vanishes at the plane 99

100 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University defined by ( xy,,). Thus, the method of images is applicable for all waveforms: single frequency, transient, stochastic, etc. Should the receiver and original point source be switched, so that the receiver is now located at (, yd, ) and the point source is located at ( xyz,, ), then the application of the method of images reveals that the resulting pressure received at the field point receiver is the same. This is illustrated by the two cases shown in Figure 38. The result exemplifies that the principle of acoustic reciprocity holds regardless of the presence of boundaries. In other words, as subject to the assumptions of the linear acoustic wave equation, switching of source and receiver locations does not influence the acoustic variables at the receiving point. Figure 38. Switch of point source and receiver locations to demonstrate that the principle of acoustic reciprocity holds regardless of the presence of boundaries. Demonstrated by application of the method of images. 5.6 Acoustic metamaterials Metamaterials are engineered structures, often composed of strategically assembled or periodic constituents, that exhibit unprecedented bulk properties by virtue of the composition. Because such properties are unconventional with respect to materials, the term metamaterials is warranted, despite the fact that the engineered systems are often realized in structural forms. An illustration of the concept of metamaterial is given in Figure 39(a) where a rod-like object in the bulk (top) is actually composed of a mass-spring lattice, although such microarchitecture may not be observable from a macroscopic perspective. Studies on metamaterials were mostly concentrated in photonic and electromagnetic applications in early years [], while more recent decades have seen an acceleration of acoustic metamaterials research as it pertains to waves in fluids and solids [] []. Because the term acoustic waves is often interchangeably used to refer to any waves satisfying a linear wave equation, as we have seen in Sec. 3 there are numerous solid or structural media that have equivalent wave equations as that derived for fluid-borne wave propagation, Sec. 4. Thus, the use of the term acoustic metamaterials is used in the scientific communities to refer to metamaterials that have unusual elastic and fluid-borne wave propagation properties [3].

101 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 39. Schematics of fundamental metamaterial architectures. (a,b) show respectively atomic and diatomic lattice representations while (c) shows a fluid media representation analogous to the diatomic lattice of (b). While our studies to date have focused upon wave propagation in systems governed by PDEs, the analysis of wave propagation through periodic-types of metamaterials is best understood by first considering discrete mass-spring lattices, such as that shown in Figure 39(a) One-dimensional monatomic lattice The perfect periodicity of the composition of the lattice in Figure 39(a) warrants its name the monatomic th lattice. When the n mass m of the lattice in Figure 39(a) is displaced an amount w n from equilibrium, forces are exerted upon it by the neighboring elements of the monatomic lattice. Thus, a force acts on the th n mass to restore it to equilibrium according to ( ) ( ) F = s w w s w w ( 438 ) n n+ n n n Therefore, by Newton's nd law, the equation of motion for the ( ) n n+ n n th n mass m of the lattice is mw = s w + w w ( 439 ) Due to the periodicity of this mass-spring composition, we refer to these constituents as the unit cell of the monatomic lattice. We seek one-dimensional traveling wave solutions to (439) according to an assumed form n j( t kna) e ω + w = θ ( 44 ) where na, termed the propagation constant, is a quantity similar to the continuous spatial coordinate x in the study of plane wave propagation in an elastic or fluid medium. Due to the lumped-parameter formulation

102 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University and the spatial periodicity of (439), there is no particular need to distinguish the arguments for "forward" or "backward" propagating waves. ± jkna exponential This form of assumed solution (44) to wave propagation in a periodic media is an expression of Bloch's theorem. In other words, time-harmonic waves permitted in periodic media are modulated according to the period of the media. As a result, the study of the unit cell, respecting its location within the periodic media, enables the full characterization of the wave propagation characteristics of the whole periodic media. Thus, the use of (44) as an assumed solution, and similar forms for multi-dimensional wave propagation, to study time-harmonic wave propagation is termed Bloch wave analysis for periodic media. Substituting (44) into (439), we find jka jka ( ) ω m= s e + e ( 44 ) Using Euler's identity, we find that (44) yields non-trivial solutions for the wave frequency ω only so long as ω ( k ) 4s = ± sin ka m ( 44 ) The (44) is the dispersion relation ω ( k ) for the monatomic lattice, relating the frequency of wave propagation ω to the wavenumber k (or, phase change per unit length) characterizing the propagation. By (44), it is evident that the monatomic lattice is dispersive since the frequency is a nonlinear function of the wavenumber. The dispersion relation for the monatomic, one-dimensional mass-spring lattice is shown in Figure 4. Negative frequencies are not shown because they are non-physical, purely mathematical outcomes of (44). The dependence of ω on k is symmetric about k =, and the maximum frequency that is permitted to propagate through the lattice is ω = 4 s/ m which occurs when k = ± π / a. This indicates that the maximum frequency that may propagate corresponds to adjacent masses oscillating jka perfectly out of phase. In other words, most generally we have wn / w = n e, such that when + k = km, we find n / n w w + =. Considering small values of the wavenumber, (44) becomes ω( π / ) m m s k << a ka. m Considering the density to be mass per length to be ρ = m/ a and considering the equivalent elastic B stiffness B = sa, the aforementioned limit shows ω k = ck, defining c= B/ ρ = a s/ m. This ρ shows that the monatomic mass-spring lattice permits wave propagation in an identical manner as a continuous media satisfying the linear wave equation when the waves are at low frequencies respecting the spatial periodicity of the lattice.

103 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University ω/(s/m) (/) normalized frequency [dim] ka/π, normalized wavenumber [dim] Figure 4. Dispersion relation for monatomic mass-spring lattice Several factors about the elementary, monatomic mass-spring lattice are illuminating as they pertain to the broader class of acoustic metamaterials and periodic structures studied in recent research. These factors, described below, are largely the outcome of realizing periodic media that in some respects are analogous to continuous media from a macroscopic perspective and in some respects are different than continuous media Periodicity Values of wavenumber greater than π / a repeat the trends from π / a< k <+ π / a. This indicates that for a given frequency of wave propagation, the wavelength, by k = π / λ, is not uniquely defined. In fact, this corresponds to a spatial aliasing phenomenon in lattices that also governs the direction of the wave propagation that will occur. The wavenumber-frequency space for this lattice k [ π / a, π / a] (FBZ) while the truly unique space k [, π / a] is referred to as the first Brillouin zone, is referred to as the irreducible Brillouin zone (IBZ). These areas of the dispersion relation are named in honor of an applied mathematician Léon Brillouin whose textbook on the subject of wave propagation in periodic structures has stood the test of time and continues to be cited regularly almost 8 years after its publication [4]. Conveniently, the text is an easy and good read. The unique frequencies of wave propagation permitted in the monatomic lattice are encapsulated in the IBZ. Similar domains are identified for multi-dimensional lattices in the respective spatial extents of the lattice composition, and hence in wave propagation directions Dispersion The dispersion relation for the lattice is given in (44). The phase velocity of the monatomic lattice is sin ka ω 4s cp = = sin ka c k k m = ka ( 443 ) 3

104 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University The phase velocity of the lattice is distinctly a nonlinear function of the wavenumber, unlike the phase velocity of one-dimensional linear wave propagation in a rod or in fluids. Thus, waves at high frequencies (within the band of frequencies able to be propagated in the lattice) travel slower than lower frequencies. Yet, for long wavelengths, the limit ka << shows cp c, showing a parallel to linear wave propagation in a rod or fluids. Similarly, the group velocity of the monatomic lattice is ω s cg = = a cos ka = ccos ka k m which shows a vanishing of group velocity as the wavelength approaches the limits of the IBZ, ka / π. This indicates that information ceases to travel as the frequency of the wave increases. As shown in Sec. 5.6., the adjacent masses merely oscillate out-of-phase when excited at this frequency. This provides a physical explanation for why "information" cannot travel in such a periodic lattice at high frequency: the energy of vibration is merely exchanged between each adjacent oscillating mass, as a transition between kinetic and potential energy, so that no energy remains to flow through the system. The trends of variation in the phase and group velocity are plotted in Figure 4. red solid, normalized phase velocity c p /(a(s/m) (/) ) [dim] blue dash, normalized group velocity c g /(a(s/m) (/) ) [dim] ka/π, normalized wavenumber [dim] Figure 4. Phase and group velocity of monatomic lattice Propagating and attenuation bands The dispersion relation (44) does not permit waves at frequencies above ω m = 4 s/ m to propagate. This is evidence of a stopband or bandgap. These are frequency ranges in which waves are not permitted to travel by virtue of the fact that the dispersion relation does not have a fully real-valued relationship between wavenumber and frequency in such a parametric range. Specifically, stopbands refer to ranges of frequencies greater than some value in which waves do not propagate, while bandgaps refer to a finite 4

105 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University bandwidth of frequencies in which waves do not propagate. The monatomic lattice possesses a stopband while the diatomic lattice, as shown in Sec. 5.6., possesses a bandgap and stopband. Considering the forced excitation of the lattice at frequencies ω < ωm, waves will propagate through the lattice with phase speeds (443) and according to the spatial variation of mass displacement of jka wn / w = n e. Yet, when the forcing frequency exceeds + ω m, there are no corresponding real-valued wavenumbers obtained from the dispersion relation. In this case, the waves from the driven point of the lattice will be evanescent and will not propagate. The development and exploitation of bandgaps in metamaterials is a commonly used means to manipulate elastic and acoustic waves [] []. To characterize the significance of the wave attenuation in the stopband or bandgap, the dispersion relation m is written as the wavenumber as a function of the frequency cos ka = s ω and solved for a range of frequency ω. The resulting computation yields real k r and imaginary k i components of the wavenumber k = k + jk, as shown in Figure 4. Because the wavenumber is complex, the assumed solution is r i ( ) jkina j t krna consequently characterized by propagating and attenuating components n e θe ω + w = exemplifying the inability for the monatomic lattice to propagate energy in such frequency ranges. ω, wave propagation frequency red solid, real component of wavenumber. blue dash, imaginary component of wavenumber ka/π, normalized wavenumber [dim] Figure 4. Real and imaginary wavenumbers computed from dispersion relation One dimensional diatomic lattice Consider the diatomic lattice shown in Figure 39(b). The unit cell consists of two masses and two springs. In the diatomic lattice, masses m are located at positions with odd indices, while masses M are located at positions with even indices. Application of Newton's nd law for the th n unit cell reveals 5

106 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University ( ) Mw = s w + w w ( 444 ) n n+ n n ( ) mw = s w + w w ( 445 ) n+ n+ n n+ Similar to the methodology used to analyze the monatomic lattice, we assume that the lattice permits wave propagation according to n j( t nka) e ω + w = θ ( 446 ) = n+ j( t [ n ] ka) e ω + + w φ ( 447 ) Substituting (446) and (447) into (444) and (445) yields ( jka jka ) ( jka jka ) = + ω Mθ sφ e e sθ = + ω mφ sθ e e sφ ( 448 ) The (448) has unique solutions for θ and φ so long as the determinant of coefficients is zero. ω cos s M s ka s cos ka s mω = ( 449 ) Solving for the wave propagation frequency yields 4sin ka ω = s + ± s + M m M m mm / ( 45 ) This pair of dispersion relations is shown in Figure 43 within the irreducible Brillouin zone. Two branches in the plot are shown. Figure 43. Dispersion relations for diatomic lattice. 6

107 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University At low frequencies, it may be shown that θ φ, such that the masses oscillate in phase with nearly the same amplitude. As we know from Sec. 4, this is how waves propagate through acoustic fluids: the fluid molecules within a fluid particle are so numerous that from one molecule to the next the motions are nearly identical. Thus, in this diatomic lattice, at low frequencies waves propagate in a way similar as in acoustic fluids, warranting the term for this portion of the dispersion relation as the acoustic branch. At high frequencies, it may be shown that θ / φ m/ M. Thus, the masses oscillate perfectly out of phase with an amplitude ratio proportional to the mass ratio. Phonons exhibit this wave propagation phenomenon, which has led to the use of the term optical branch to denote the dispersion relation component corresponding to high frequency wave propagation. A bandgap is apparent from the dispersion relation. Frequencies of waves from s/ M < ω < s/ m will not propagate through the lattice, which is evidence of a bandgap. High frequencies greater than ω > s + will also not propagate, which is evidence of a stopband. M m Metamaterial continua The periodic system of continuous elastic or fluid layers, such as the two-layer media shown in Figure 39(c), is analyzed using similar methods as that utilized in Sec but now the attention is drawn to the original wave equations of motion and associated boundary conditions. Namely, (45) must be satisfied for each domain p p = ; i =, i i x ci t ( 45 ) where c = B / ρ. Assuming time-harmonic waves and by invoking Bloch's theorem [5], we presume i i i that the pressure response is periodic with spatial periodicity d, p ( x) p ( x d) assumed solutions to be ( x) ( n) jk ξ ( n) p a e a e jkξ = + ; ( x) ( n) jk ξ ( n) jkξ = + ; p b e b e i = +. Consequently, we take < ξ < d ( 45 ) d < ξ < d ( 453 ) whereξ = x + nd, k = ω / c, and k = ω / c. In (45) and (453), the shared time-harmonic dropped for brevity. i j t e ω terms are At ξ = and ξ = d, the boundary conditions between fluid layer to are applied to yield a set of four ( n) ( n) ( n) ( n) equations in terms of the amplitude coefficients a, a, b, and b. The determinant of this coefficient matrix must be zero in order for non-trivial coefficients and corresponding wave propagation amplitudes to exist. The result of this determinant is a dispersion relation given by 7

108 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University coskd = coskd coskd r r + sin kd sin kd r r ( 454 ) where d = d d. Recalling that k = ω / c and k = ω / c, (454) defines an implicit relationship of the frequency ω on the wavenumber k. Figure 44 shows representative results of the dispersion relation for a periodic fluid system composed of air and steam layers as fluids and, respectively. The corresponding parameters are c =343 [m/s], ρ =. [kg/m 3 ], c =44 [m/s], ρ =.6 [kg/m 3 ], d = [mm], and d =5 [mm]. The method of solving (454) is either to prescribe a range of frequency ω and solve for the corresponding wavenumber, or prescribe a range of wavenumber k (often normalized as kd / π ) and solve for the corresponding wave frequencies. Taking the latter approach requires determination of the non-unique responses and may be more challenging from an algorithm-coding perspective. While diatomic lattices possess only two "branches", due to the infinite-dimensional nature of the fluid continua and the partial differential governing equations of motion, an infinite number of dispersion branches occur in periodic continua. It is only a matter of how many eigenfrequencies associated with the branches, as determined by the dispersion relation (454), are ultimately examined in a plot. In Figure 44, four of the branches are shown, revealing bandgaps around. [khz],.3 [khz], and 3.7 [khz]. 5 4 frequency [khz] normalized wavenumber, kd/π Figure 44. Dispersion plot of waves through periodic two-fluid-media unit cell Computational methods of analysis While lumped parameter periodic systems are useful, simplified representations of practical periodic metamaterials, more realistic models incorporate the continua representations of the domains, as exemplified in Sec Yet, the closed-form analysis of metamaterials becomes significantly 8

109 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University cumbersome for more intricate architectures and when the metamaterials possess a greater number of physical dimensions. Consequently, computational methods are well-established means of investigating acoustic metamaterials. The method herein described is based on the finite element method, as used via COMSOL Multiphysics software. Alternative computational methods are available [3], but the ease of learning to appropriately use COMSOL and its wide implementation in the research communities warrants its consideration here. A concise introduction to the effective use of COMSOL Multiphysics is provided here. This software is best used by undertaking the following steps Define parameters Define geometry Define materials and assign domain to prescribed materials Associate physics with domains and assign boundary conditions Define mesh with suitable refinement Construct parameter study (as applicable) and other relevant study characteristics Plot and assess results The following overview uses a one-dimensional Pressure Acoustics model as an example Define parameters Inserting parameters as "hard coded" numbers into a COMSOL model is tedious when one inevitably needs to change that parameter. Thus, defining parameters using the model-wide Global Definitions is wise. A representative set of parameters for a one-dimension acoustic model is shown in Figure 45. Figure 45. Defining parameters in COMSOL. 9

110 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Define geometry The geometry is defined entirely according to the parameters set aside in the Global Definitions. Importantly, one should retain the original default units of [m] and [deg], as shown in Figure 46. Figure 47 shows an example of geometry definition for a line interval. Figure 46. COMSOL units definition. Figure 47. Definition geometry in COMSOL.

111 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Define materials and assign domains Once a geometry is defined, the various domains must be assigned material properties. Figure 48 shows an example of such definitions. Figure 48. Assigning parameters to material properties and assigning domains to the material Associate physics with domains and assign boundary conditions In models with multiple physics, the domains must be associated with the appropriate physics. For instance, in a fluid-structure interaction problem, only the fluid domain should be associated with the Pressure Acoustics physics, while the Solid Mechanics physics should only be associated with the structural domains. In our one-dimensional acoustics model, both intervals (domains in one dimension) are assigned to the Pressure Acoustics physics. The boundary conditions for determining the dispersion relation are the Periodic Condition, by Floquet periodicity. The destination boundary is arbitrary in a one-dimensional model, but is usually the "right" or "top" boundary in the model. The Bloch's theorem is alternatively termed the Floquet-Bloch theorem, which explains the term used in COMSOL. Figure 49 shows the assignment of the edge points of the domains to this boundary condition and its definition according to the wavenumber k that was previously identified as a parameter above. Figure 5 shows an example of prescribing the destination boundary.

112 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 49. Floquet periodicity boundary condition assignment and definition. Figure 5. Destination definition for Floquet-periodicity Define mesh with suitable refinement In any finite element or boundary element model of wave processes, it is the established convention that at least 6 elements are required per wavelength in order to yield reasonably accurate results [6]. This is because the spatial discretization is similar to time discretization. Thus, in data acquisition, at least 6 points in time are needed to viably reconstruct a sinusoid. Similarly, at least spatial sampling points are required

113 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University to viably reconstruct a sinusoidal wave in space. Yet, here we are determining the frequency associated with a given wavenumber. As a result, we will first need to run the model with a moderate to excessive spatial discretization (high refinement), assess the resulting frequencies respecting the associated wavelengths and mesh discretization, and then decide whether further mesh refinement is desired for greater accuracy. Figure 5 shows one example of the mesh definition. It is recommended to always define a mesh with respect to key spatial parameters instead of using the COMSOL-based mesh refinements of "Coarse", "Fine", "Finer", and so on. Figure 5. Mesh discretization definition Construct the study Here, a parameter study will be used to iterate through potential wavenumbers and solving the associated finite element-based eigenvalue problem. This will yield the dispersion relation. Figure 5 shows the definition of a parameter sweep for the normalized wavenumber k [,] that interacts with the Periodic Boundary condition. The Eigenfrequency study step here solves for the lowest 4 eigenfrequencies. This small number is used for conciseness of results presentation and speed of computation. In general, it is better to compute a greater number of eigenfrequencies so long as the mesh discretization is refined enough for accurate computations, according to the requirements described in Sec

114 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 5. Parameter sweep definition for the eigenfrequency study Plot and assess results Here, we use a Global plot to evaluate the eigenfrequency as a function of the sweep parameter, the normalized wavenumber k. The subplot shown in Figure 53 appears very similar to the analytical plot shown in Figure 44, which is to be expected because the analytical model (computed by MATLAB) employed the same parameters as the COMSOL model. The final check is the assessment of suitable mesh discretization. The highest eigenfrequency determined is near 5 [khz]. The shortest wavelength at 5 [khz] between the two fluid media included in the study is around 69 [mm]. The mesh discretization defined the largest element size to be d / =.5 [mm], see Figure 5. Thus, 6 elements would span approximately 9 [mm]. In other words, the existing mesh discretization is appropriate for the frequency range computed in this one-dimensional acoustics study. 4

115 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 53. Example Global plot for the eigenfrequencies determined according to each parameter in the sweep Nuances for two-dimensional dispersion relation COMSOL study The parameter and periodic boundary condition definitions are unique for two-dimensional models. As observed in Sec. 3.7, two-dimensional wave propagation may be defined according to a wavenumber vector that includes components in the x and y axes. This is the concept employed in the study of twodimensional acoustic metamaterials composed of continuous media, rather than lumped parameter constituents. Interested individuals should consult Refs. [3] [7] for further details. In COMSOL, the implementation of the wavenumber vector requires the explicit definition of the wavenumber components in x and y axes as parameters. For a rectangular periodic domain, the irreducible Brillouin zone (IBZ) of a two-dimensional acoustic metamaterial is described according to the region outlined in Figure 54 by the label range Γ X M. This is the conventional notation used in the literature [3]. For points evaluated at Γ, the wavenumber vector components are k = k = which is the rigid body mode. For points evaluated at X, the wavenumber x y vector components are k = π / d [/m] and k y =, which corresponds to wave propagation only in the x x axis. For points evaluated at M, the wavenumber vector components are k = π / d [/m] and k = π / d [/m], which corresponds to plane wave propagation at an angle 45 from both x and y axes. The variation of the wavenumber vector components from these limits of the IBZ yield the full range of corresponding wavenumbers and hence associated frequencies that are permitted within the periodic domain, while the symmetry of the IBZ warrants the omission of the wavenumber vector point that would correspond to k x = and ky = π / d. x y 5

116 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 54. Irreducible Brillouin zone outlined by Γ X M Y of two-dimensional acoustic metamaterial with rectangular domain. To implement this computational procedure in COMSOL, the wavenumber vector components are defined according to if/else criteria. In other words, by sweeping the wavenumber k from values arbitrarily assigned from to, the wavenumber vector components k x and k y are swept from the values corresponding to Γ to those corresponding to X. Then, the process repeats by sweeping the wavenumber k from values arbitrarily assigned from to, wherein the wavenumber vector components k x and k y are swept from the values corresponding to X to those corresponding to M. Then a sweep from M to Y is accomplished. From X to M to Y, the relative contribution of wavenumbers in x and y axes is assessed, thus characterizing wave propagation at angles through the lattice respecting the x y plane. Then the final sweep from Y to Γ is undertaken. From Y to Γ and from Γ to X, wave propagation strictly in y or x axes is evaluated, respectively. The parameter definition in COMSOL is shown in Figure 55 and Figure 56 where it is observed that one set of material properties is similar to steel, while Figure 57 shows the representative geometry where in scaled circle (ellipse) is the steel domain. The specific parameter definitions for k x and k y, written in the COMSOL if/else style, are k x, if(k<,pi/d*k,if(k<,pi/d,if(k<3,pi/d*(3-k),))) k y, if(k<,,if(k<,(k-)*pi/d,if(k<3,pi/d,(4-k)*pi/d))) The corresponding boundary condition definition is given in Figure 58. The parameter sweep, in Figure 59, now indicates the wavenumber k [,4]. The resulting plot of results is also given in Figure 59 within the subplot. Note from the results that the dispersion relation is periodic from the plot limits of k = and k = 4. Typically, the axes of the plots are not labeled numerically, but are instead labeled Γ, X, M, and Y and are correspondingly "stretched" to accommodate the different spatial lengths associated with the right triangular space of the IBZ. Based on the symmetries of the geometry, the IBZ of a given two-dimensional periodic lattice structure may be reduced or expanded, such as due to hexagonal arrangements or circular symmetries [8]. Also, the output from COMSOL orders the eigenfrequencies from low to high values 6

117 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University although the spatial motion, the eigenmode, associated with each frequency is distinct. Thus, there are "crossover" points in the plot of Figure 59 which are only due to the ordering provided by COMSOL. Thus, an eigenmode plot of points below and above such points in the wavenumber space will reveal the connections between eigenmotion and eigenfrequency. Figure 55. Two-dimensional COMSOL parameter definitions for dispersion analysis, kx. 7

118 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 56. Two-dimensional COMSOL parameter definitions for dispersion analysis, ky. Figure 57. Two-dimensional geometry definition. 8

119 RL Harne ME 86, Adv. Eng. Acoust. 7 The Ohio State University Figure 58. Two-dimensional periodic boundary condition definition. Figure 59. Two-dimensional wavenumber parameter sweep definition and representative dispersion plot results. 9