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Sound in Enclosure 3D


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 Wave Reflection inside Enclosures, 3D
   Number of Modes

Wave Reflection inside Enclosures, 3D

The 3D wave equation is

image

Assume harmonic sound wave imply

image

Take the time differential operation, imply

image

And get the Helmoltz equation,

image

Assume the pressure is of format:

image

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

image

let the dispersion equation :

image

substitute into the Helmoltz equation:

image

By separating the variable, imply:

image

Assume the corresponding solution of the equation of the form,

image
image
image

Assume the rectangular enclosure are with rigid walls, the boundary conditions at 6 walls are with particle velocity equals to zero, imply

image ,image,image

and image,image,image

From the equation of momentum conservation in x direction

image

At x = 0 or x=Lx, u=0 at all time, therefore the time derivative of u is zero also, imply:

image and image

Therefore at x=0 and x=Lx, imply :

image

From boundary condition at x=0, imply:

image

From boundary condition at x=Lx, imply:

image

Substitute kx, A and B into the pressure function, imply:

imageand image

Similarly ky, kz,are:

image and image

And the corresponding pressure function in y and z direction are:

image and image , where image

Substitute all individual pressure functions into the total pressure function is:

imagewhere image

Substitute all individual wave number into the combined wave number is:

imagewhere image

Therefore the frequency is :

imagewhere image

Number of Modes

The number of modes increases dramatically with the increase of frequency and with the volume of the cavity and can be estimated by.

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: 101000022 Last Updated: 10/20/2010 Revision: 0 Ref:

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