# Resonant cavity for sound waves

I would appreciate some assistance in understanding resonant cavities. I would like to be able to calculate the resonant frequencies of sound waves in a resonant cavity. I would think to use the following equation for a box

$$\omega _{nml}=c\pi \sqrt{\left(\frac{n}{a} \right)^2+\left(\frac{m}{b} \right)^2+\left(\frac{l}{d} \right)^2}$$

where a,b,d are the modes and n,m,l are the dimensions of the box and c is the speed of sound. Is this correct? Also I am not sure what form a 3d sound wave might take in a box. This would be helpful to know. Additional what boundary conditions are present? In the case of EM waves I know we need to consider the E field going to zero at the walls, are there any similar boundary conditions in the case of a sound wave?

• Ooh yes, I forgot to mention. But I think in your equation $a,b,d$ are the dimensions and $n,l,m$ are the modes ;) Commented Jun 18, 2022 at 13:05

The equation for the resonant frequencies are correct. To get to that equation you indeed need to impose the boundary conditions such that at the boundaries the wave vanishes, which is at $$x=y=z=0$$, $$x = a$$, $$y=b$$ and $$z=d$$.
The differential equation to solve for an acoustic or sound wave is $$\partial^2_x \psi +\partial^2_y \psi +\partial^2_z \psi-\frac{1}{c^2}\partial^2_t \psi = 0.$$ This you can do by separation of variables: $$\psi = X(x)Y(y)Z(z)T(t)$$. You should be able to find the solutions yourself, after which you can impose the boundary conditions. These conditions will pick out certain resonant frequencies, which will satisfy the equation you wrote down in the question.