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Wave Reflection inside Enclosures, 3D

The 3D wave equation is

Assume harmonic sound wave imply

Take the time differential operation, imply

And get the Helmoltz equation,

Assume the pressure is of format:

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

let the dispersion equation :

substitute into the Helmoltz equation:

By separating the variable, imply:

Assume the corresponding solution of the equation of the form,

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

,,

and ,,

From the equation of momentum conservation in x direction

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

and

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

From boundary condition at x=0, imply:

From boundary condition at x=L_x, imply:

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

and

Similarly k_y, k_z,are:

and

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

and , where

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

where

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

where

Therefore the frequency is :

where

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.