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.

Wave Reflection inside Enclosures, 1D

A sound source radiates outwardly in any direction in a free source will always encounter reflecting surfaces unless the sound source is place in an anechoic chamber where all sound energy is almost absorbed with any reflection. And in such case then only wave equation should be satisified without any boundary conditions.

 Multiple reflection in an enclosure due to the wall creates standing waves with acoustic modes charactered by frequencies and mode shapes. These modes should satisfied the wave equation and boundary conditions simultaneously.

The boundary conditions are usually defined in terms of continuity between the wall motion and the particle velocity.

The wall characteristics are usually expressed in term of impedance. A rigid wall is used to describe a wall with higher impedance while a soft wall is used to describe a wall with lower impedance.

Duct, 1D mode,  with two closed ends

The 1D wave equation is

Assume harmonic sound wave imply

Take the time differential operation, imply

And get the Helmoltz equation,

Assume the solution of the equation of the form,

Assume the two closed ends are rigid wall, the boundary conditions at two ends are with particle velocity equals to zero, imply

and

From the equation of momentum conservation

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

From boundary condition at x=0, imply:

From boundary condition at x=L, imply:

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

and

Therefore the natural frequency and mode shape is:

,  and

Duct, 1D mode,  with two open ends

The 1D wave equation is

Assume harmonic sound wave imply

Take the time differential operation, imply

And get the Helmoltz equation,

Assume the solution of the equation of the form,

With two open ends, the boundary conditions at two ends are with pressure equals to zero, imply

and

From boundary condition at x=0, imply:

From boundary condition at x=L, imply:

Substitute k, A and B into the pressure function, and p<>0, imply:

and

Therefore the natural frequency and mode shape is:

,  and

Duct, 1D mode,  with one end open and one end  closed

The 1D wave equaton is

Assume harmonic sound wave imply

Take the time differential operation, imply

And get the Helmoltz equation,

Assume the solution of the equation of the form,

Assume the closed end is rigid wall, the boundary conditions at closed end is with particle velocity equals to zero, imply

From the equation of momentum conservation

At x = 0, u=0, therefore the time derivative of u is zero also, imply: o, imply:

From boundary condition at x=0, imply:

From boundary condition at x=L, imply:

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

and

Therefore the natural frequency and mode shape is:

,  and