# Understanding frequency in cuboid acoustic resonator

We have a box with a speaker in the corner:

There is a formula for a frequency: $$\nu=\sqrt{\nu _x^2+\nu _y^2+\nu _z^2}$$, where $$\nu _x$$,$$\nu _y$$ and $$\nu _z$$ can be any resonant frequencies in each direction $$\left(\nu _n=\frac{n c}{2 L}\right)$$. My question is what does this frequency represent? If the air is in a superposition of standing waves in $$x$$, $$y$$ and $$z$$ directions, shouldn't then each point move in some Lissajous path? In which case what is the meaning of thif frequency? Am I understanding something wrong? Also the same thing appears in the planck law derivation. Also the experiment shows, that $$\sqrt{\nu _x^2+\nu _y^2+\nu _z^2}$$ frequencies actually get emplified.

• The air is not in a superposition of $x$, $y$ and $z$ modes (that is a sum of those modes) – then they would not have a defined eigenfrequency (except in the degenerate case where $\nu_x = \nu_y$ or so). Rather they are the products of such modes (and those are again valid modes with said frequency $\nu$, as can be seen by solving the wave equation by a separation ansatz: $\phi(x,y,z,t) = \phi_x(x) \phi_y(y) \phi_z(z) \phi_t(t)$). Feb 5, 2023 at 22:58

The frequencies $$\nu_x$$, $$\nu_y$$ and $$\nu_z$$ represent waves traveling in parallel to the walls of the box, and bouncing off the opposite walls. They are called axial modes.

If the wave travels at an angle and bounces off two or all three pairs of the walls, the mode is called tangential or oblique.

The frequency of a mode can be calculated as

$$\nu = \frac{c}{2}\sqrt{ \left(\frac{n_x}{L_x}\right)^2 + \left(\frac{n_y}{L_y}\right)^2 + \left(\frac{n_z}{L_z}\right)^2 }\ ,$$ where $$L_x$$, $$L_y$$ and $$L_z$$ are the dimensions of the box, and a combination of natural numbers $$(n_x, n_y, n_z)$$ represent the mode. For axial modes, only one of them is nonzero; two for tangential, and three for oblique. This is where the formula you gave: $$\nu = \sqrt{\nu_x^2 + \nu_y^2 + \nu_z^2}$$ comes from.

That frequency represents the longest (half) wavelength in air that can be fitted into a box with those dimensions. It's the diagonal mode with velocity nodes at the two diagonal corners and a velocity maximum in the center.

If you sweep the speaker's frequency and measure the sound loudness inside the box, you'll see a peak corresponding to that frequency in the box's response curve. this is amplification by resonance.

Here is another experiment to perform:

Take a hammer and tap each side of the box in the center. You will discover that each different-sized side of the box has its own resonant frequency. If you drive the speaker at one of these frequencies you will note that the box does a poor job of blocking sound from escaping from the box at that frequency.