7.7 Modes of a rectangular cavity

A rectangular box of dimensions $L_x \times L_y \times L_z$ with rigid walls supports a discrete set of standing-wave modes. Each mode is a 3-D standing wave labeled by three integers $(n_x, n_y, n_z)$ specifying how many half-wavelengths fit along each axis.

The modes

Separation of variables in the wave equation, with pressure antinodes at each rigid wall, gives the mode shapes

p_{(n_x, n_y, n_z)}(x, y, z) \;=\; \cos\!\left(\frac{n_x \pi x}{L_x}\right) \cos\!\left(\frac{n_y \pi y}{L_y}\right) \cos\!\left(\frac{n_z \pi z}{L_z}\right),

with $n_x, n_y, n_z = 0, 1, 2, \ldots$ Not all zero — the all-zero mode is just a uniform pressure shift, not an acoustic oscillation. The allowed frequencies are

f_{(n_x, n_y, n_z)} \;=\; \frac{c}{2}\, \sqrt{\frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2}}.

This is the eigenfrequency equation for the rectangular cavity. Each mode has its own frequency; each frequency is determined by the geometry.

Mode types

Three types of modes, distinguished by how many of the indices are zero:

Axial modes: exactly one $n_i > 0$ . E.g., $(2, 0, 0)$ — second harmonic along the $x$ -axis. There are $3 \cdot (N - 1)$ axial modes with $n \le N$ .
Tangential modes: exactly two $n_i > 0$ . E.g., $(1, 1, 0)$ . The mode pattern lies in a horizontal plane.
Oblique modes: all three $n_i > 0$ . The mode shape is genuinely 3-D.

In rooms, axial modes are the strongest and easiest to identify; tangential and oblique modes are weaker but more numerous at higher frequencies.

Example: a small box

A rectangular room of $5 \times 4 \times 3$ m. Some low modes:

Mode	Frequency (Hz)	Type
$(1, 0, 0)$	$343/10 = 34.3$	axial $L_x$
$(0, 1, 0)$	$343/8 = 42.9$	axial $L_y$
$(1, 1, 0)$	$54.9$	tangential
$(0, 0, 1)$	$343/6 = 57.2$	axial $L_z$
$(2, 0, 0)$	$68.6$	axial
$(1, 0, 1)$	$66.7$	tangential
$(1, 1, 1)$	$73.0$	oblique
$(2, 1, 0)$	$80.9$	tangential
…	…	…

The modes get dense quickly. By 200 Hz there are hundreds of modes; by 1 kHz tens of thousands.

Why these matter

In a real room, sound from a source excites all these modes. The room colours the sound: at frequencies near a mode peak, sound builds up; at frequencies between modes, sound is weaker. The boomy bass of small rooms — a corner where a 35 Hz hum from a refrigerator is uncomfortably loud — is a low-axial-mode antinode.

At low frequencies, modes are widely separated and individually identifiable. At high frequencies, they overlap into a quasi-continuous reverberant field and the discrete-mode picture stops being useful. The transition between these regimes happens at the Schroeder frequency, where the modal density (modes per Hz of bandwidth) becomes large enough that adjacent modes overlap within their damping bandwidth.

Mode density

The number of modes below frequency $f$ in a 3-D rectangular cavity of volume $V = L_x L_y L_z$ , surface area $S$ , and total edge length $L_e$ is approximately

N(f) \;\approx\; \frac{4\pi V f^3}{3 c^3} \;+\; \frac{\pi S f^2}{4 c^2} \;+\; \frac{L_e f}{8 c}.

The volume term dominates at high frequencies: number of modes $\propto f^3$ . The mode density $dN/df = 4\pi V f^2 / c^3 + \ldots$ grows as $f^2$ .

For our $5 \times 4 \times 3$ m room ( $V = 60$ m³):

At $f = 100$ Hz: $N \approx 7$ modes.
At $f = 500$ Hz: $N \approx 900$ modes.
At $f = 5$ kHz: $N \approx 900{,}000$ modes.

At low frequencies, room modes dominate the perception. At high frequencies, the room is approximately a continuous diffuse field — the regime treated by statistical room acoustics. Next lesson.