OPAC102. The Acoustic Wave Equation

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Ethelbert Mervyn Richards
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1 OPAC102 The Acoustic Wave Equation

2 Acoustic waves in fluid Acoustic waves constitute one kind of pressure fluctuation that can exist in a compressible fluid. The restoring forces responsible for propagating a wave are the pressure changes that occur when the fluid is compressed or expanded. Individual elements of the fluid move back and forth in the direction of the forces, producing adjacent regions of compression and rarefaction.

3 Sound Speed c Bulk modulus B density Air Water Steel Bulk Modulus 1.4(1.01 x 10 5) Pa 2.2 x 10 9 Pa ~2.5 x Pa Density 1.21 kg/m kg/m 3 ~10 4 kg/m 3 Speed 343 m/s 1500 m/s 5000 m/s Please Memorize!!!

4 Necessary Differential Equations to Obtain a Wave Equation Mass Continuity Equation of State Force Equation Newtons Second Law Assumptions: homogeneous, isotropic, ideal fluid

5 Assumptions for the Acoustic wave equation Equations governing acoustic phenomena are derived from general hydrodynamic equations and are generally quite complex, but can be simplified due to the following. We are dealing with a continuous medium. The medium (fluid) is homogeneous, isotropic, and perfectly elastic. Dissipative effects, that is viscosity and heat conduction, are neglected. This allows for linearized equations to be used (below 130dB).

6 Gravitational forces can be neglected so that the equilibrium (undisturbed state) pressure and density take on uniform values, p0 and %0, throughout the fluid. Small-amplitudes assumption: particle velocity is small. Particle velocity is small, and there are only very small perturbations (fluctuations) to the equilibrium pressure and density

7 The assumptions allow for the linearization of the following equations (which combined lead to the acoustic wave equation): The equation of state relates the internal forces to the corresponding 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 compression or dilatation. The equilibrium equation relates internal and inertial forces of the fluid according to the Newton s second law.

8 The equation of state For fluid media, the physical quantities describing the thermodynamic behavior of the fluid are related by the equation of state. The equation of state relates the internal forces to the corresponding deformations. Since the heat conduction can be neglected the adiabatic form of this (constitutive) relation can be assumed. The equation of state for a perfect gas is P rt (1) P: the total pressure in pascals (Pa); ρ: the density in kilograms per cubic meter (kg/m 3 ) T: the absolute temperature in degrees Kelvin (K); r: the specific gas constant

10 The equation of state Ideal Gasses: P = rtk P P o o P P o o P 1 P 2 o 2 o Real Fluids: P = P o o o V p B B Bs V B o P o

11 The equation of continuity To connect the motion of the fluid with its compression or expansion, we need a functional relationship between the particle velocity u and the instantaneous density ρ. Also called the equation of conservation of mass. u 0 (2) t is called the divergence operator (incorporates 3D vectors)

12 x u x x dx dz The equation of continuity dy u x dx dm dm dm dm x y dm dt dt dt dt y z u x dydz u x x dydz xdx dt dt dt u 0 t z dm u dmy dm dm x dm u x dydz u x dx dydz x x dt x dt dt x u u y u dm dt x y z x z dxdydz dxdydz dxdydz u u y u d x dt x y z x y x z z y z z

13 The wave equation The wave equation is then derived from the combination of the equation of state, the equation of continuity and the momentum equation. 2 2 is called the Laplace operator (incorporates 3D vectors) p 2 1 p c 2 2 t (4)

14 Plane waves Plane wave is a wave whose wave fronts (surfaces of constant phase) are infinite parallel planes of constant amplitude normal to the phase velocity vector (direction of propagation).

15 Plane waves In the case of plane wave propagation, only one spatial dimension (x) is required to describe the acoustic field. 2 2 p 1 x c t p (5)

16 Plane waves Equation (5) is a second order Partial Differential Equation (PDE) in both space and time. It can be solved using the technique of separation of variables. P can be represented by means of Fourier analysis as a sum of simple harmonic functions. So we assume: p( x, t) p( x) e jt (6)

17 Plane waves Substituting equation (6) into (5), we have the equation for sound pressure distribution in space: 2 px ( ) x 2 2 k p( x) 0 ( k / c is called wavenumber) (7) This is called the Helmholtz equation which often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time.

18 Plane waves The general solution of equation (7) is jkx p( x) Ae Be jkx (8) From equation (6) and (8), we have j( tkx) j( tkx) p( x, t) Ae Be (9) Forward propagating wave Backward propagating wave For a wave propagating in the x direction in an infinite space without reflection: p( x, t) j( t kx) Ae

19 Plane waves Spatial distribution at a fixed time t: 2 p Acos( kx) Acos( x) Temporal distribution at a fixed position x: 2 p Acos( t) Acos( kct) Acos( t) Acos(2 ft) T k 2 / / c 2 f 2 / T k: wavenumber λ: wavelength ω: angular frequency

20 Plane waves Fixed time Fixed location

21 Plane waves The wave impedance for a plane wave can be derived as below: u t p p x Ae j( t kx) u p c z p c u

22 Spherical wave When sound waves are generated by a small source in free space with no boundaries nearby, they spread out in all directions in a nearly spherical fashion.

23 Spherical wave The wave equation in spherical coordinates: 2 2 p 2 p 1 p r r r c t (10) Let Ф=pr, then equation (10) becomes (11) r c t

24 Spherical wave Equation (11) is similar to the plane wave equation and its solution is Ae j( tkr ) j( tkr) Be From Ф=pr, we have the general solution of spherical wave equation: (12) A B p e e r r j( tkr ) j( tkr ) (13) B = 0 when there is no reflection.

25 Spherical wave The wave impedance for a spherical wave can be derived as below: u t p z A r p r j( tkx) e 1 p jkr c u 1 jkr u c p jkr jkr

26 Wave summation Any number of harmonic waves of the same frequency travelling in different directions can combine to produce one wave travelling in one direction.

27 Wave summation p j t 1 Ae 1 j t p2 A2e j t p12 A12 e 12 A A A 2A A cos A sin 1 2 tan A1 A2cos

28 Standing waves Assume there are two waves of the same frequency and of amplitudes A and B, respectively, travelling in the two opposite directions: p 1 j( t kx) Ae 2 j( t kx ) p Be

29 Standing waves Combining the two waves, we have: j j( t kx) j( t ) p ( A Be ) e 2Be cos kx Standing wave: varies in amplitude with time, but remains stationary in space.

30 Standing wave: ), ( ), ( t x p t x p p re in sw ) sin( ) sin( kx t f P kx t f P re in 2 2 ) )sin( cos(2 ) cos( ) 2 sin( kx t f P kx t f P p in in sw By using trigonometric relations: ) )sin( cos(2 ) cos( ) 2 sin( kx t f P kx t f P re re Standing waves

31 Standing waves In a closed acoustic tube, the waves have equal amplitudes: Pin P re p sw p in ( x, t) p ( x, t) 2P cos( kx) sin( 2 f t) re in Amplitude of Standing Wave = 2P cos(kx) in Time-variation of Standing Wave = sin( 2 f t)

32 Standing waves node location Amplitude: 2P in cos(kx) When: cos(kx) 0 p sw 0 called NODES We have kx 2, 3 2, 5 2,, (2n 1) 2 ;?

33 ; ) (,,,, k n k k k x n The wave number is: 2 k ; ) ( n x n,...),, ( 3 n 1 2? Standing waves node points

34 Standing waves Anti-nodes Node intervals: x n1 x n [2( n 1) 4 1] (2n 1) ; 4 2 The anti-node point is between the node intervals: [ p sw ] peak 2 P in

36 Standing waves Piston vibration One position of the rod Two waves incident & reflected waves will superpose: with the same frequency but different time.

37 Directivity Index Generally, in a large open area an acoustic field is said to have far field characteristics. In such a case a single point source should typically radiate sound equally in all directions. In most cases however, the radiation of sound from a source will have directional properties. More on far fields and near fields as well as multiple sound sources and sound radiation in general later.

38 Directivity Index In order to account for noise source directivity in acoustic fields, a parameter known as the directivity factor, Q, is introduced. Q can be defined in terms of the mean sound pressure or mean sound intensity. Q is a ratio of the mean squared sound pressure in a particular direction to the mean squared sound pressure averaged over all directions. Q is also the ratio of the sound intensity in a particular direction to the sound intensity that would be produced at the same location by a perfectly spherical source radiating with the same acoustic energy.

39 Directivity Index In terms of sound intensity and acoustic energy (sound power) then: Q 4 r W 2 I I = sound intensity (W / area) W = total sound power (= I x area) Q is basically a ratio of the total area of potential sound transmission to the area of actual sound transmission

40 Directivity Index Recall that for a plane wave: Note: S = area And for a spherical wave:

41 Directivity Index The directivity index, DI, is related to the directivity factor, Q, by this equation. DI 10 log Q 10 For a spherical sound source in an open area, the directivity factor, Q = 1, (both areas are the same) and the directivity index then equals zero.

42 Directivity Index If a spherical sound source is placed near a hard flat floor or wall, all the sound energy can be assumed to radiate away from the wall or floor and through the hemisphere of area = 2πr 2 In this case, I W area W r 2W QW r 4r I And therefore, Q 2

43 Directivity Index If the directivity factor is Q = 2, the directivity index is, DI 10 log

44 Directivity Index Similarly, if a spherical sound source is placed on the floor near a wall, the sound energy radiated away from the source must all pass through an area = πr 2 Q 4 DI log 10

45 Directivity Index If a spherical sound source is placed on the floor near a corner between two walls, Q = 8, and: DI log 10

46 Directivity Index The directivity factor can be determined analytically or from experimental measurements of the acoustic pressure. The directional pressure distribution function, H(,φ) is defined by: H, is the azimuth angle and φ is the polar angle p(0) is the acoustic pressure on the axis, = 0. p p, 0

47 Directivity Index

48 Directivity Index The directivity factor can be evaluated from the directional pressure distribution function: Q H 2 4, sin d d However, if the pressure distribution is symmetrical, H(,φ) = H(), the integration wrt φ can be carried out directly (and equals ). 2

49 Directivity Index The directivity factor for a symmetrical source is then given by: Q 0 H 2 2 sin d

Longitudinal Waves. waves in which the particle or oscillator motion is in the same direction as the wave propagation

Longitudinal Waves. waves in which the particle or oscillator motion is in the same direction as the wave propagation Longitudinal Waves waves in which the particle or oscillator motion is in the same direction as the wave propagation Longitudinal waves propagate as sound waves in all phases of matter, plasmas, gases,