MECH 482 – Noise Control Week 2, Lecture 2 The Wave Equation

1. MECH 482 – Noise ControlWeek 2, Lecture 2The Wave Equation

2. Sound propagating in air • Let: = sound speed; • = the period of the source vibration; • = wavelength: the sound wave traveling distance in one vibration period. • Distance traveled Velocity = • Corresponding vibration period Constant • In air, the speed of sound is proportional to the temperature in degrees Celsius: Constant (m/s)

3. Sound propagating in air Compression Rarefaction Compression Rarefaction = 100 Hz At 1000 Hz At 10000 Hz

4. Sound propagating in air Sound frequency changing with sound wavelength Low frequency sound High frequency sound

5. Sound propagating in air – moving source Air particles are COMPRESSED by the sound source to such a degree that an AIR-BARRIER is formed.

6. Sound propagating in air – moving source (Mach number) rising pitch falling pitch Higher frequency Lower frequency To the observer (A) at the left, the frequency or pitch of the car is HIGHERthan it is to the observer (B) at the right.

7. 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.

8. Assumptions on the fluid medium 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. Only very small disturbances of the fluid particles take place. No (or very very small) dissipative effects, such as those from viscosity or heat conduction, take place. This allows for “linearized” equations to be used (below 130dB).

9. The equation of state (1) P: the total pressure in pascals (Pa) ρ: the density in kilograms per cubic meter (kg/m3) T: the absolute temperature in degrees Kelvin (K) r: the specific gas constant For fluid media, the physical quantities describing the thermodynamic behavior of the fluid are related by the equation of state. The equation of state for a perfect gas is:

10. The equation of continuity (2) Δ 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. is called the divergence operator (incorporates 3D vectors)

11. The momentum equation (3) From the conservation of the momentum carried by the fluid through which the sound wave propagates, we have

12. The wave equation (4) Δ The wave equation is then derived from the combination of the equation of state, the equation of continuity and the momentum equation. 2 is called the Laplace operator (incorporates 3D vectors)

13. 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).

14. Plane waves (5) In the case of plane wave propagation, only one spatial dimension (x) is required to describe the acoustic field.

15. Plane waves (6) 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:

16. Plane waves (7) Substituting equation (6) into (5), we have the equation for sound pressure distribution in space: 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.

17. Plane waves (8) (9) Forward propagating wave Backward propagating wave The general solution of equation (7) is From equation (6) and (8), we have For a wave propagating in the x direction in an infinite space without reflection:

18. Plane waves k: wavenumber λ: wavelength ω: angular frequency Spatial distribution at a fixed time t: Temporal distribution at a fixed position x:

19. Plane waves Fixed time Fixed location

20. Plane waves The wave impedance for a plane wave can be derived as below:

21. 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.

22. Spherical wave (10) (11) The wave equation in spherical coordinates: Let Ф=pr, then equation (10) becomes

23. Spherical wave (12) (13) Equation (11) is similar to the plane wave equation and its solution is From Ф=pr, we have the general solution of spherical wave equation: B = 0 when there is no reflection.

24. Spherical wave The wave impedance for a spherical wave can be derived as below:

25. 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.

26. Wave summation

27. Standing waves Assume there are two waves of the same frequency and of amplitudes A and B, respectively, travelling in the two opposite directions:

28. Standing waves Standing wave: varies in amplitude with time, but remains stationary in space. Combining the two waves, we have:

29. Standing waves Standing wave: By using trigonometric relations:

30. Standing waves In a closed acoustic tube, the waves have equal amplitudes: Amplitude of Standing Wave = Time-variation of Standing Wave =

31. Standing waves – node location Amplitude: When: called NODES We have ?

32. Standing waves – node points The wave number is: ?

33. Standing waves – Anti-nodes Node intervals: The anti-node point is between the node intervals:

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

36. 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.

37. 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.

38. Directivity Index • In terms of sound intensity and acoustic energy (sound power) then: • 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

39. Directivity Index • Recall that for a plane wave: • Note: S = area • And for a spherical wave:

40. Directivity Index • The directivity index, DI, is related to the directivity factor, Q, by this equation. • 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.

41. 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 = 2πr2 • In this case, • And therefore,

42. Directivity Index • If the directivity factor is Q = 2, the directivity index is,

43. 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 = πr2

44. Directivity Index • If a spherical sound source is placed on the floor near a corner between two walls, Q = 8, and:

45. 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: •  is the azimuth angle and φ is the polar angle • p(0) is the acoustic pressure on the axis,  = 0.

46. Directivity Index

47. Directivity Index • The directivity factor can be evaluated from the directional pressure distribution function: • However, if the pressure distribution is symmetrical, H(,φ) = H(), the integration wrt φ can be carried out directly (and equals ).

48. Directivity Index • The directivity factor for a symmetrical source is then given by:

49. Next Time Noise Criteria and Human Hearing