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 Acoustic 3D Plane Wave
  Continuity Equation, 3D
  Euler's Equation of Inviscid Motion, 3D

Acoustic 3D Plane Wave

The 1D plane wave can be extended into a 3D plane wave by considering a rectangular volume with coordinates x, y, z.

Continuity Equation, 3D

For a control volume, from the principle of conservation of mass, the instantaneous rate of change of mass in a control volume equals to the net mass flux flow into or out of the control volume, therefore:

image

As in 1D plane wave, the linearized form is :

  image

And the time derivative of the equation is:

image

Euler's Equation of Inviscid Motion, 3D

For a control volume, from principle of momentum conservation, the instantaneous rate of change of net momentum of a control volume equals to the net applied force and the net momentum change due to the momentum flux flow into or out of the control volume. The applied force in this case is pressure only and no other forces, no gravity, no viscous force etc., then:

In x direction:

image

As in 1D plane wave, the linearized form is :

  image

And the position derivative of the equation is

image

Similarly in y and z direction is

image and image

And get::

image

Equate two equations::

image

Wave Equation, 3D

Applying the equation of state and substitute the speed of propagation, imply.

image

Plane Wave in 3D

Assume harmonic sound wave, imply

image

substitute the harmonic function into the 3D wave equation and take the time differential operation, imply:

image

and get the Helmholtz equation:

image

Assume the pressure is of format:

image

substitute the pressure function into the Helmholtz equation, and take the Laplacian differential operation:

image

let the dispersion equation :

image

substitute into the Helmholtz equation:

image

By separating the variable, imply:

image

The corresponding solutions are:

image

substute all into the pressure function and get the solution of wave equation is:

image

Therefore a propagating wave can be decomposed into three components along three axes. A positive sign implies a forward wave from the source to the free space. While a negative sign implies a reflected wave. With the dispersion of the value, k into three components along the three axes


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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: 101000014 Last Updated: 10/16/2010 Revision: 0 Ref:

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