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  Acoustic Plane Wave
  Complex Exponential Representation
  Plane Wave Properties
  Medium Acoustic Impedance
  Plane Wave Superimposition

Acoustic Plane Wave

For a 1D acoustic plane wave in x direction,:

image

where

p is a function of x and t.
c is the speed of propagation.

The general solution of the equation is an arbitrary wave propagation along positive x direction and negative x direction.

image

For ct-x, the propagation of pressure fluctuation should be repeated after the period ct-x, therefore after time t, the wave should propagate a distance x in positive direction outwardly.

But for ct+x, the propagation of pressure fluctuation should be repeated after the period ct+x, therefore after time t, the wave should propagate a distance x in negative direction inwardly.

Since p is a function of x and t, it is characteristic by a double periodicity.

In time domain

image

where

ω is angular velocity
ƒ is frequency
ωT is period

And in physical domain:

image

where

k is wave number
c is speed of wave propagation λ is wave length

Consider a solution of the form:

image

where

ω is the angular frequency
k=ω/c
a1 is the magnitude constant, therefore:

image ,
image

and

image ,

image

Substituting into the 1D wave equation and the form of solution is confirmed.
And a solution of the same form in negative x direction is:

image

Therefore the general harmonic wave solution of wave equation is:

image

Similarly, consider a solution of the form:

image

where

ω is the angular frequency
k=ω/c,
a2 is the magnitude constant,
therefore:

image ,

image

and

image ,

image

Substituting into the 1D wave equation and the form of solution is confirmed also.

Since the 1D wave equation is linearized by making the differential coefficient is of first order only, the sum of two solution forms is also a solution of the wave equation. The differences between two solutions are the phase angle and the magnitude.

image

The general expression of a harmonic function is:

image

and it can be decomposed into sinusoidal and cosinusoidal components in quadrature of 90 degrees out of phase:

image

These two components can be individual harmonic function with arbitrary phase angle φ as in the first two solutions. Or they can be linked solution in a linear system in which the phase angle is not an arbitrary and is related as in the above solution:

Complex Exponential Representation

Since in complex system,

image

The solution of wave equation can be expressed in a complex exponential form,

image

where

à is Complex function of the form a+jb
exp[j(ωt-kx)] is ej(ωt-kx)
j is √(-1)

Since it is a complex function, the real acoustic pressure function can be represented by the real part of the complex expression.
Therefore

image

By comparing with the wave equation solution, imply:

image

Therefore, the general complex harmonic wave solution of 1D wave equation is :

image

Plane Wave Properties

Consider a plane wave propagates along positive x direction:

image or image

From the linearized conservation of momentum, the relationship between acoustic pressure and acoustic velocity is:

image

Therefore:

image

The differences in magnitude between acoustic pressure and acoustic velocity are the relative magnitude and the relative phase angle.
Wave propagation is a kind of forced vibration, from the linearized conservation of momentum,  Then

image or  image

Imply :

image

Therefore the relationship between acoustic pressure and acoustic velocity is:

image

In general:

image

Medium Acoustic Impedance

The acoustic impedance of a medium is defined as the ratio of acoustic pressure to the acoustic velocity, Then

image

For air, z=400

Plane Wave Superimposition

Consider two harmonic plane waves propagate of the same frequency along positive x direction and negative x direction respectively. The pressure fluctuation of two harmonic plane waves i.e. A along positive x direction and B along negative x direction are

image

If the two harmonic plane waves are in phase at x=0, then the phase angle at x is equal to zero also. Therefore the pressure fluctuation of two harmonic plane waves at x are

image

The total sound pressure at x is

image

According to the linearized equation of motion,

image

Imply,

image

From the linearized equation of motion, imply

image

Therefore, the vector sum of the particle velocity at x is

image

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References

  1. Michael P. Norton; Denis G. Karczub,, 2003, Fundamentals of Noise and Vibration Analysis for Engieer, Cambridge, United Kingdom
  2. G. Porges, 1977, Applied Acoustics, Edward Arnold Limited, Britain
  3. Douglas D. Reynolds, 1981, Engineering Principles of Acoustics:; Noise and Vibration Control, Allyn and Bacon, USA
  4. Conrad J. Hemond, 1983, Engineering Acoustics & Noise Control, Prentice-Hall, USA
  5. F. Fahy, 2001, Foundations of Engineering Acoustics, Academic Press, UK
  6. D.A. Biew; C.H. Hansen, 1996, Engineering Noise Control: Theory and Practice, E & FN Spon, New York
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ID: 100900021 Last Updated: 7/30/2012 Revision: 2 Ref:

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