3D Acoustical Wave Eqn: Closed Soln in Rectangle Coordinates

[Watts]

I am trying to find a closed solution to the three dimensional acoustical wave equation in rectangle coordinates \[<br /> \frac{{\partial ^2 p}}{{\partial x^2 }} + \frac{{\partial ^2 p}}{{\partial y^2 }} + \frac{{\partial ^2 p}}{{\partial z^2 }} = \frac{1}{{c^2 }} \cdot \frac{{\partial ^2 p}}{{\partial t^2 }}<br /> \]<br />. The wave is propagating along the x axis. I have a generic solution but I don’t have a closed solution subject to the required boundary conditions I need. I am assuming the wave begins propagation at the origin and travels to the end point of L and the amplitude is restricted to the width and height of A and B. I am assuming it is propagating in a rectangular tube. The variable p is the acoustic pressure and c is the velocity of the wave. If anybody could help I would appreciate it.

 

[shyboy]

you need to specify carefully what is the boundary conditions to find out what is a possible solution of the wave equation. The boundary condition for the logitudinal (sound) waves is that the pressure gradient at the wall is zero. Non zero pressure gradient means non zero flow of particles (displacement), but there cannot be any particle flow from or into the wall.
for the rectangular waveguide the solution of the wave equation may be assumed as a product of the solutions which depend only on x, y, or z only. So you may find the boundary conditions for y and z axis , and there is no boundary condition for x. But if you want a propagating wave then the wave along x should have a real wave vector.

 

[Watts]

you need to specify carefully what is the boundary conditions to find out what is a possible solution of the wave equation. The boundary condition for the logitudinal (sound) waves is that the pressure gradient at the wall is zero. Non zero pressure gradient means non zero flow of particles (displacement), but there cannot be any particle flow from or into the wall.

I am assuming the walls are rigid except for the end of the tube that is at L.

 

[Watts]

But if you want a propagating wave then the wave along x should have a real wave vector.

I am assuming the wave is propagating along x. By saying a real wave vector I think you mean a non complex vector.

 

[shyboy]

if you fix the waveguide length, then you have a resonator and boundary conditions for x, too. The boundary conditions for an open end is that there is no pressure change at it.

 

[Antiphon]

I am trying to find a closed solution to the three dimensional acoustical wave equation in rectangle coordinates \[<br /> \frac{{\partial ^2 p}}{{\partial x^2 }} + \frac{{\partial ^2 p}}{{\partial y^2 }} + \frac{{\partial ^2 p}}{{\partial z^2 }} = \frac{1}{{c^2 }} \cdot \frac{{\partial ^2 p}}{{\partial t^2 }}<br /> \]<br />. The wave is propagating along the x axis. I have a generic solution but I don’t have a closed solution subject to the required boundary conditions I need. I am assuming the wave begins propagation at the origin and travels to the end point of L and the amplitude is restricted to the width and height of A and B. I am assuming it is propagating in a rectangular tube. The variable p is the acoustic pressure and c is the velocity of the wave. If anybody could help I would appreciate it.

You have to write the general solution (which include all possible waves)
and then impose boundary conditions on that. When you do that, some
of the unknowns in the general solution will take on specific values, and
these specific values will give you the wavenumbers of the allowable waves
in the guide. It will also tell you everything about which wavelengths can
fit into the guide and which ones can't (dispersion relationship).

The general solution will be an infinite summation of Fourier modes in x, y and z
where each individual mode satifys the boundary conditions (and so the infinite sum does as well).

I leave the details to you!

Good luck.

Edit: Transform the wave equation into the frequency domain and solve
it there. It's much easier than doing it in the time domain. In the frequency
domain, the equation becomes

\[<br /> \frac{{\partial ^2 p}}{{\partial x^2 }} + \frac{{\partial ^2 p}}{{\partial y^2 }} + \frac{{\partial ^2 p}}{{\partial z^2 }} = k^2 \cdot p<br /> \]<br />.

With this solution in hand (with the bouandary conditions already imposed)
you transform it back into the time domain in the usual Fourier way.