Chapter 6: Sound Wave Equation

Transcription

1 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 Chater 6: Sound Wave Equation. Sound Waves in a medium the wave equation Just like the eriodic motion of the simle harmonic oscillator, waves have certain characteristics. The ones we will concentrate on are the frequency, eriod, wave seed and the wavelength. A icture of a transverse wave in a medium at some time, maybe t=0 sec. We wrote an equation to describe this icture: s( x, t) s0 sin x where: s = article dislacement Distance that the fluid article is moved from its equilibrium osition at any time, t. so = maximum article dislacement or amlitude λ = distance over which the wave begins to reeat k = π/λ= a conversion factor that relates the change in hase (angle) to a satial dislacement. We call k the wavenumber. When we let this wave begin to move to the right with a seed, c, the osition is shifted in the governing equation from x to x-ct. s( x, t) s0 sin ( x ct) Below is a icture of the same traveling wave shown at some later time, t. Now, if instead of taking a sna shot of the wave in the medium at two different times, what if we had set a sensor somewhere in sace maybe at x = 0 m, and recorded the wave s dislacement over time. The equation governing the wave would become: Note that we have emloyed a similar strategy regarding the grou of constants in front of the time variable that we used when discussing the wavenumber, k. Since the wave reeats every π change in hase and that corresonds to a time eriod, T, angular frequency, ω = π /T, is nothing more than a conversion factor from time to hase angle. The symmetry with wavenumber is striking causing many eole to identify the wave number as the secial frequency and to secifically refer to angular frequency, ω, as the temoral frequency.

2 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 To be clear, the seed of the wave c, is not the seed of the medium. t is the seed of the wave disturbance enveloe and is often called the hase seed. t is the seed you would need to run next to the medium in order to stay in hase with a oint on the disturbance. The seed of the medium is also called the article seed and is found by taking the derivative of the dislacement with resect to time. u = s t is article seed Distance that the medium travels er unit time. Note that the average value of the article velocity over any cycle is zero. Putting these three ictures together, we have an exression for a traveling wave in a medium s( x, t) s0 sin x t) T or more comactly, s( x, t) s sin kx wt) 0 We also have a new way of defining the seed of the wave. t makes good sense that the wave seed is the distance the wave travels in one cycle (the wavelength) divided by the time it takes the wave to comlete one cycle. t is a simle matter to substitute the frequency, f, for the eriod: c = T = f The wave seed can also be calculated from the angular frequency and the wavenumber: c = π Tπ = w k We call waves modeled using this result lane waves because in three dimensions the locus of oints all having the same hase are lanes. We call these lanes wavefronts and often draw them as lines on a age searated by one wavelength. n fact, the wavefronts are actually arallel lanes. We also find it convenient to show the direction the wave is traveling using a ray which is constructed erendicular to the wavefronts.

3 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 Sound Waves: Acoustic Presssure Acoustic waves constitute one kind of ressure fluctuation that can exist in a comressible fluid. The restoring forces resonsible for roagating a wave are the ressure changes that occur when the fluid is comressed or exanded. ndividual elements of the fluid move back and forth in the direction of the forces, roducing adjacent regions of comression and rarefaction. n the case of ressure, static ressure from the height of the column of fluid above the wave are always resent. This force is constant with time. We learned how to calculate this ressure,, using the following equation: = o + ρg where ρ is the density of the fluid and h is the height of the fluid column. The acoustic ressure due to the condensations and rarefactions sits on this static ressure and oscillates around it due to the resence of the acoustic wave motion. While we could consider the entire ressure variation in describing an acoustic wave, we will, by convention, instead consider only the ressure variation from the static ressure. We saw that simle harmonic motion has a governing differential equation called the equation of motion whose solution gives the osition of a mass as a function of time. n the case of a traveling wave, there is an analogous equation whose solution describes the medium s article dislacement as a function of osition and time. This artial differential equation is known as the wave equation. n the next section we will show how the wave equation follows directly from some fundamental Physics rinciles. Sound Waves in a medium the wave equation Equations governing acoustic henomena are derived from general hydrodynamic equations and are generally quite comlex, but can be simlified due to the following. We are dealing with a continuous medium. The medium (fluid) is homogeneous, isotroic, and erfectly elastic. Dissiative effects, that is viscosity and heat conduction, are neglected. This allows for linearized equations to be used (below 30dB). Gravitational forces can be neglected so that the equilibrium (undisturbed state) ressure and density take on uniform values, 0 and, ρ 0 throughout the fluid. 3

4 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 Small-amlitudes assumtion: article velocity is small. Particle velocity is small, and there are only very small erturbations (fluctuations) to the equilibrium ressure and density The equation of state relates the internal forces to the corresonding deformations. Since the heat conduction can be neglected the adiabatic form of this (constitutive) relation can be assumed. The equation of continuity relates the motion of the fluid to its comression or dilatation. The equilibrium equation relates internal and inertial forces of the fluid according to the Newton s second law. To derive the one-dimensional wave equation, let's look at the motion of a small volume of fluid. We can relate its motion to the sring-mass system from the revious section. f we aly a ressure gradient to the fluid volume, V, (such as an acoustic ressure from an acoustic wave) it will move and comress the volume of fluid. The ressure on the left face of the fluid block is (x ), while that exerted on the right face is (x ). f there is a differential ressure, Δ, then the fluid block might move to the right, and, as the block accelerates, it will change to volume, V F. We will make some assumtions regarding the movement of the block: i. The rocess is adiabatic no heat is lost or gained by the resence of the acoustic wave. This is a reasonable assumtion because for acoustic wave frequencies in the ocean, the wavelength is too long and thermal conductivity of seawater too small for significant heat flow to take lace. ii. Changes in article dislacement of the fluid from equilibrium are small. iii. The fluid column is not deformed (shear deformation) by differential ressure. To fully describe the motion of sound in the fluid from first rinciles, we will examine well known Physics laws Newton s Second Law, an equation of state, and conservation of mass. These laws, couled with the assumtions above rovide a robust and owerful model for underwater sound. Newton s Second Law Newton's Second Law is customarily used by examining the forces in a articular direction and then summing them as vectors. n the case of our fluid volume, V, the forces in the x direction are: F ( x ) A ( x ) A A x 4

5 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 This net force across the volume is equal to the mass times the acceleration of the volume. The mass is found by multilying the initial density by the initial volume (Δx = x -x ) m A x The acceleration in the x direction is the second time derivative of average dislacement s ax t and s s s Substituting into Newton s Second Law, becomes, and rearranging gives or more aroriately F x ma A Ax x x s t s t a x s t n the final result, acoustic ressure was used since the derivative of the static ressure is zero. Additionally, the instantaneous density and dislacement for an infinitesimally small volume are substituted. Equation of State and Conservation of Mass Even though we think of liquids mostly as incomressible fluids, in reality, they are not. The Bulk Modulus of Elasticity describes how much the volume of the liquid changes for a given change in ressure. n equation form this is: B ( x) ( x) a V V / V V / V F The significance of the negative sign in above equation is that when a is ositive, then V F <V and ΔV is negative. Using solid geometry we can develo an exression to relate the acoustic ressure to the dislacement of the small volume in the above figure. mlied in this argument is the law of conservation of mass. We are not allowing any of the medium to escae the volume, nor are we allowing any additional mass to see in. V V A( x s) F Ax ( ) VF V V A x s Ax s V V Ax x 5

6 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 Thus substituting in the last two equations and rearranging the definition of the Bulk Modulus of Elasticity: V B V or more correctly a a B x a s s B x Substituting this last result into our revious relationshi between ressure and dislacement: a s s s B B x t x x t. The One Dimensional Wave Equation Substituting the conclusion from conservation of mass and equation of state into Newton s Second Law results in the one-dimensional wave equation that we can use to describe the dislacement, s, from their rest osition of articles in a medium, with resect to time and osition. This equation is a artial differential equation with a solution that varies with time and osition. As with the mass-sring system equations, if we can find an equation that satisfies this second order differential equation, the equation could be used to describe the motion of the articles in the medium. s s s s x B t x c t This equation cal also written in termes of ressure form. Therefore, x c t One solution that we will use was described above as a lane wave and has the form: s( x, t) s sin( kx wt) 0 To check the validity of this solution we must take the aroriate second derivatives: s0sin( kx wt) s0k sin( kx wt) x s0sin( kx wt) s0w sin( kx wt) t Substitution into the wave equation s0k sin( kx wt) s0w sin( kx wt) B or k w B 6

7 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 Rearranging and recalling that the seed of the wave, c = ω/k, w k c B This is a fairly rofound result. t tells us that the lane wave solution for article dislacement is a good solution rovided the seed of the wave is not arbitrary, but exactly equal to the square root of the bulk modulus divided by the density. When the bulk modulus and density of water are used, a nominal value for the seed of sound in water is 500 m/s. This agrees with measured results. Had we used an equation of state for a gas instead of a liquid, we would have arrived at a similar result following a similar rocedure. The lane wave solution would still solve the wave equation, but the wave seed would become: nrt nrt c c m m When tyical room temerature numbers are used, this results in a nominal seed of sound in air of 340 m/s. The rules of differential equations make no statement about the uniqueness of a solution to the wave equation. Many other solutions exist as well. Had the solution been exressed as a cosine vice a sine, the wave equation would still have been satisfied. Additionally, comlex exonentials could have been used as a solution due to Euler s identity. s( x, t) s e i( kxwt ) 0 This exression is really shorthand for the real (or imaginary) art of the comlex exonential. A Gaussian ulse of the following form also satisfies the wave equation. s( x, t) s e kxwt wt 0 Additionally, if a certain frequency wave satisfies the differential equation, all multiles or harmonics of that frequency must also work. s( x, t) s sin( nkx nwt) 0 Rules for differential equations also secify that linear combinations of solutions are also solutions. This is called the rincile of suerosition. A method using the theory develoed by a French mathematician named Fourier will allow disturbances of almost any shae to be constructed using series of harmonic. 3. Plane Waves A lane wave is the simlest wave tye. Plane wave is a wave whose wave fronts (surfaces of constant hase) are infinite arallel lanes of constant amlitude normal to the hase velocity vector (direction of roagation). 7

8 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 The characteristic roerty of a lane wave is that each acoustic variable has constant amlitude and hase on any lane erendicular to the direction of roogation. Since the surfaces of constant hase for any diverging wave become nearly lanar far from their source, w emay exect that the roerties of diverging waves will, at large distances, become very simimlar to those of lane waves. f the coordinate system is chosen that the lane wave roogates along the x axis, than the wave equation reduces to x = c x This equation is a second order Partial Differential Equation (PDE) in both sace and time. t can be solved using the technique of searation of variables. P can be reresented by means of Fourier analysis as a sum of simle harmonic functions. So we assume: Substituting this function into wave equation, we have the equation for sound ressure distribution in sace: x ( ) k ( x) 0 x This is called the Helmholtz equation which often arises in the study of hysical roblems involving artial differential equations (PDEs) in both sace and time. The general solution of Helmholtz equation is ikx ( x) Ae Be From equation (x,t), we have ( x, t) ( x) e it i( tkx) i( tkx) ( x, t) Ae Be For a wave roagating in the x direction in an infinite sace without reflection: ( x, t) Ae i( t kx) ikx n this form, in atial distribution at a fixed time t: Acos( kx) Acos( x) And, temoral distribution at a fixed osition x: Acos( t) Acos( kct) Acos( t) Acos( ft) T Fixed Time Fixed Location 8

9 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 Exercise : The ressure in a rogressive sound waves is given by the equation =.4sin (π(x 330t)) Where x is in meters, t in seconds and in N/m. Find a) the ressure amlitude, b) frequency, c) wavelengh and d) seed of the wave. 4. Sherical waves Sherical waves can be thought of emitted by a oint source. They roagate sherically in all directions. The two dimensional analogue are water waves that occur as the results of a local distortion, for examle a stone falling into the water. Due to symmetry reasons the sound ressure and the amlitude of the sound article velocity have to be constant on a sherical surface with arbitrary radius and a center that coincides with the source oint. The vector of the sound article velocity oints in radial direction outwards. The wave equation in sherical coordinates: Let Ф=r, then equation becomes r r r c t r c t This equation () is similar to the lane wave equation and its solution is Ae Be j( tkr ) j( tkr) From Ф=r, we have the general solution of sherical wave equation: A B e e r r j( tkr ) j( tkr ) B = 0 when there is no reflection. The wave imedance for a sherical wave can be derived as below: u A e j( tkx) t r r r 9

10 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 Solving for u, we get And, u jkr c jkr jkr z c u jkr 5. Alternate Views for Describing an Acoustic Wave The Pressure Field So far, we have viewed sound moving in a fluid as a harmonic traveling wave, considering only article dislacements. This is not a unique view. Just as an electromagnetic wave can be seen as an oscillating electric field or and oscillating magnetic field, so too can a sound wave be seen as an oscillating ressure field, an oscillating velocity field or an oscillating density field. Of course, the fundamental difference remains that the electromagnetic wave is always a vector field, while the sound wave in a fluid is generally a scalar field. Using the solution for the wave equation, s(x, t) = s 0 sin(kx ωt), we can find the equations for two of these fields. First the we will find the acoustic ressure. Previously we found the relationshi of the acoustic ressure a, and the dislacement of the small volume from the equation of state. Using this we get: s ( s0 sin( kx wt)) a B B Bs0k cos( kx wt) x x The first imortant observation about the ressure field relative to the dislacement field is that they are 90 degrees out of hase with each other. This means that when the article dislacement of the medium is at a maximum, the acoustic ressure is at a minimum. Additionally, when the dislacement is zero, the maximum acoustic ressure is: Bs k c s k a, mak 0 0 By convention, acousticians refer not to use an engineering modulus, B, instead substituting B=ρc. 6. Alternate Views The Velocity Field and Secific Acoustic medance The article velocity is not the wave velocity. The seed that the wave travels is a function of the medium and is a constant. The seed of sound, c, is given by the equations: B w c f T k The article velocity of the medium, on the other hand tells us how fast the molecules in the fluid are moving. t is found by simly taking the time derivative of the equation describing the osition of the medium, the lane wave solution. s u( x, t) s0sin( kx wt) s0wcos( kx wt) t t where u s w s ck mak 0 0 0

11 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 As we know that, the driving force in an acoustic wave is the ressure and the rate at which articles in the medium ass a articular oint is the velocity. t is no accident that we define the secific acoustic imedance as the ratio of the ressure to the article velocity. z ( x, t) u( x, t) For the case of a lane wave we have found exressions for both the ressure and velocity fields. c s0k cos( kx wt) z c u s ck cos( kx wt) 0 This equation is also derived from ressure equations as following; Then, it gives u i( t kx) Ae u t x x c z c u The secific acoustic imedance relates the characteristics of a sound wave to the roerties of the medium in which it is roagating. Nominal values for the density, ρ, and the wave seed, c, for water are ρ = 000 kg/m3 and c = 500 m/s. Do not be confused into thinking that secific acoustic imedance is always the roduct of density and the seed of sound. This is only true for a lane wave. For other geometries, for instance a sherically sreading wave, the secific acoustic imedance is a different exression even in the same fluid. 7. More on Continuity of Mass The Density Field When motivating the wave equation, it was mentioned that the mass in our test fluid volume was not changing. Secifically, the initial mass in osition is the same as that in osition F. V V F F Recalling our exression for the equation of state and substituting, V VF V F F F a B B B B B V V F F F F We find that the fractional change in density, is directly roortional to the ressure. This F fractional change in density is called a condensation variable. t is often written, (, ) 0 x t 0 a s0k cos( kx wt) B We have develoed four different descritions for a traveling acoustic lane wave; article dislacement, article velocity, acoustic ressure and fractional change in density. Particle dislacement is 90 degrees out of hase with the other three, but all four discritions travel with the same wave seed and have the same eriod and wavelength. All four can be used to roerly model acoustic effects.

12 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, Energy and ntensity in a Sound Wave Missing in the discussion of wave equations and their solution is any mention of energy. We started the semester with a review of simle harmonic (sinusoidal) motion. The reason we did this should be aarent to you by now. As a lane wave traverses any medium, all secific article locations undergo simle harmonic motion as the wave asses by. Because of this, we can use the basic SP equations for kinetic and otential energy of the medium. The only modification is to relace mass with density so as to calculate energy density or energy er unit volume. This is a logical modification since the medium carrying the wave is continuous. t would make no sense to identify a articular iece of mass, nor the total mass. The equations for kinetic and otential energy density in a simle harmonic oscillator are resectively as follows k u khookes mw s ws v v Since we have equations for article dislacement and article velocity, we can simly substitute these into the above. k u s0wcos( kx wt) w s0 cos ( kx wt) w s w s0sin( kx wt) w s0 sin ( kx wt) t should be clear that the total energy is the sum of the otential and kinetic energy and that when the kinetic energy is maximum the otential energy is zero and vice versa. The question of how the energy is artitioned deends on when you ask the question. The average energy in a simle harmonic oscillator is calculated using the following definition for a eriodic function: f ( t) f ( t) dt T T 0 For kinetic and otential energy we find that since the time average of sin ( ( t)) cos ( ( t)) k w s cos ( kx wt) u 4 w s sin ( kx wt) u 4 0 max 0 max lane wave are the same, each being exactly one half the total energy of the harmonic oscillator. The total average energy density of the wave is then, k u u u 4 4 max max max Using the acoustic imedance,

13 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 u z c allows us to write the total energy in terms of maximum ressure. a,max c Acoustic ntensity Acoustic intensity,, is defined as the amount of energy assing through a unit area er unit time as the wave roagates through the medium. Energy moved er unit time is ower which has units of Watts. ntensity then must have units of Watts/m. Power Work Forcexdislacement x ressurexvelocity Area time Area Areaxtime This unit analysis suggests acoustic intensity can be calculated from the roduct of acoustic ressure and article velocity. au where sin( ) a a,max kx wt and u u sin( ) max kx wt but u c a ( x, t) ( x, t) c One imortant thing to note is that since the acoustic ressure is a time-varying quantity, so is the intensity. We will use a more meaningful quantity, the time average acoustic intensity. The average intensity of an acoustic wave is the time average of the ressure over a single eriod of the wave and is given by the equation: Since the time averaged acoustic ressure is a c a a,max a,max c, the average acoustic intensity can be written: This result looks remarkably similar to the average energy density of a traveling lane wave. n fact this is not accidental. f you consider a lane wave as a cylinder of length cdt and cross section A, the total energy in this cylinder, de, would be the roduct of the energy density and the volume. Rearranging we see an alternative exression for average acoustic intensity, de c A dt 3

14 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 Since a,max c, the average acoustic intensity is again a,max c This result is leasing in that it agrees with an analogy suggested earlier between voltage and ressure. n your electrical engineering class, you learned that electric ower was voltage squared divided by imedance. Average ower was found using P Z Now we have found that average acoustic ower er unit area is simly acoustic ressure squared divided by secific acoustic imedance. V m ax P A a,max z This sheds light on why the modifier secific recedes acoustic imedance. By analogy, secific acoustic imedance, z, must be acoustic imedance Z, divided by area. To further make use of electrical engineering backround, time averaged ressure may also be determined by: therefore max a rms a rms c a,max rms c Lastly, from this oint further, unless otherwise noted, when we refer to the intensity of the wave, we actually mean the time-averaged intensity. 9. Suerosition of Pressure Waves Any number of harmonic waves of the same frequency travelling in different directions can combine to roduce one wave travelling in one direction. Consider and j t Ae j t Ae Then the summation of two waves is A e j t where A A A A A cos A sin tan A Acos 4

15 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 The suerosition of two sine waves of equal frequency and amlitude but oosite directions results in a standing wave. For a mathematical investigation the two waves are introduced in comlex writing: Ae j( t kx) and Be j( t kx ) Combining the two waves, we have: j j( t kx) j( t ) ( A Be ) e Be cos kx Since the amlitudes are same A=B, then the equation reduces to The suerosition of the two waves is no longer a roagating wave but a harmonic oscillation that is modulated in sace with cos(kx). As a consequence at certain locations maxima and at other locations minima arise. Rewrite in trigonemetrik form leads to ( ) j t Be cos kx sw in ( x, t) ( x, t) re P sin( f t kx) P sin( f t kx) in re By using trigonometric relations: P sin( f t) cos( kx) P cos( f t)sin( kx) sw in in P sin( f t) cos( kx) P cos( f t)sin( kx) n a closed acoustic tube, the waves have equal amlitudes: re re sw in ( x, t) ( x, t) P cos( kx) sin( f t) re in where Amlitude of Standing Wave = Pin cos( kx) Time-variation of Standing Wave = sin( f t) Standing waves-location of Nodes The maximum amlitude of an element of the medium has a minimum value of zero when x satisfies the condition cos kx = 0, that is, when kx =, 3, 5 (n )..., 5

16 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 These oints of zero amlitude are called nodes. Thus, the ositions of the nodes are given by x = 4, 3 (n ),. = 4 4 n =,,3,. Node intervals are n The element with the greatest ossible dislacement from equilibrium has an amlitude of A, and we define this as the amlitude of the standing wave. The ositions inthe medium at which this maximum dislacement occurs are called antinodes. The antinodes are located at ositions for which the coordinate x satisfies the condition cos kx =, that is, when Because k =/, these values for kx give x [( n) ] (n) xn ; 4 4 kx =,, 3... x = 3,,,. = n n = 0,,,3. Exercise : (a) Show that the suerosition of the waves = A sin(kx ωt) and = +A sin(kx + ωt) is a standing wave. (b) Where are its nodes and antinodes? 6

17 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, 08 Problems. (a) Calculate the sound ressure level (db re μpa) of an acoustic disturbance in water, at 50kPa static ressure, at which the instantaneous total ressure becomes negative. (b) What is the acoustic article velocity amlitude if the disturbance is a lane wave. (c) What is the acoustic article velocity amlitude if the disturbance is a oint source, m from the measurement location and the frequency is 000Hz.. A sound wave roagates a oint about 50 meters below the surface of a calm sea. The instantaneous ressure at the oint is given by: where t is inseconds and in Pascals. = 6x sin(400πt), a) What is the value of static ressure at the oint? b) What is the value of maximum (or eak) acoustic ressure at the oint? c) What is the root-mean-square acoustic ressure? d) What is the acoustic ressure when t=0,.5,.5, 3.75, 5.00 milliseconds? e) What is the average acoustic intensity of the sound wave? (The density of the water is 000 kg/m 3 and the sound seed is 500 m/sec.) f) What is the intensity level, L, in db re μpa? 3. A lane acoustic wave is roagating in a medium of density ρ=000 kg/m 3. The equation for a article dislacement in the medium due to the wave is given by: s = (x0 6 )cos(8πx 000πt), where distances are in meters and time is in seconds. a) What is the rms article dislacement? b) What is the wavelength of the sound wave? c) What is the frequency? d) What is the seed of sound in the medium? e) What is the value of maximum (or eak) article velocity? f) What is the value of maximum acoustic ressure? g) What is the secific acoustic imedance of the medium? h) What is the bulk modulus of the medium? i) What is the acoustic intensity of the sound wave? j) What is the acoustic ower radiated over a 3 m area? 4. A lane acoustic wave travels to the left with amlitude 00 Pa, wavelength.0 m and frequency 500 Hz; = 00 Pa cos πx m + 300πt sec while another lane wave travels tothe right with amlitude 00 Pa, wavelength ½ m and frequency 750 Hz: = 00 Pa cos πx m + 500πt sec a) Find the rms average total ressure. Your answer will not deend on distance x. b) f = 0 cos(kx ωt) and f = 0 cos(kx ωt +φ, find the rms average total ressure. 7

18 Lecture notes on OPAC0- ntroduction to Acoustics Dr. Eser OLĞAR, Given the following equation for an acoustic wave, originating from a source in the ocean x, t = Pa cos πx 3 m 30πt sec Determine the following: a) The wavelength b) The rms ressure of the wave c) What is the frequency of the wave? d) The time averaged intensity of the acoustic wave. 6. f the article dislacement can be found to be: s x, t = m cos πx 3 m 30πt sec a) What is the value of the eak article velocity? b) What would be the maximum acoustic ressure if the Bulk Modulus of Elasticity of the medium were.0x0 9 N/m? 7. Given that the acoustic ressure at a distance r from a small source radius r 0, and surface velocity amlitude U=U 0 e iwt is of the form = A r a) Find an exression for the article velocity at any arbitrary distance from the surface. b) Show the constant A is given by; A=jw r 0 ρ U 0 c) Find the acoustic ower radiated by this source at 00 Hz if U 0 =m/s and r 0 =5 cm. ejω (t r/c) 8. A lane wave with an intensity of 50 mw/cm and a frequency of 3 MHz is roagating in connective tissue (blood). What is the ressure, article dislacement, and velocity for this continuous wave? 9. A beam of lane waves in water contains 50W of acoustic ower distributed uniformly over a circular cross-section of 50 cm diameter. The frequency of the waves is 5 kc/s. Determine (a) the intensity of the beam, (b) the sound ressure amlitude, (c) the acoustic article velocity amlitude, (d) the acoustic article dislacement amlitude and (e) the condensation amlitude. Assume that the velocity of sound in water is 450 m/s. 0. For the faintest sound that can be heard at 000 Hz the ressure amlitude is about 05 N/m. Find the corresonding dislacement amlitude. Assume that the velocity of sound is 33 m/s and the air density is. kg/m 3.. Two sound waves of equal ressure amlitudes and frequencies traverse two liquids for which the velocities of roagation are in the ratio 3: and the densities of the liquids are in the ratio 3:4. Comare the (a) dislacement amlitudes, (b) intensities and (c) energy densities.. One sound wave travels in air and the other in water, their intensities and frequencies being equal. Calculate the ratio of their (a) wavelength, (b) ressure amlitudes and (c) amlitudes of vibration of articles in air and water. Assume that the density of air is.93 kg/m 3, and sound velocity in air and water is 33 and 450 m/s, resectively. 8