Loading & acoustics

Mouth size

Minimum mouth area for a given low-frequency limit in free space, on a floor, against a wall or in a corner.

Sheet: 08 / 25
Units: Metric
Air: 20 °C

Fig. 1Mouth to scale, round and rectangular

Mouth area $S_m$: 1.30m²
Round $d_m$: 129cm
Rect. width: 140cm
Rect. height: 93.2cm
Wavelength $\lambda_c$: 5.72m
Rad. resistance at $f_c$: 0.230ρc/Sm

Fig. 2Radiation impedance seen by this mouth

ResistanceReactancedrag on plot to read

Baffled-piston model of equal area, with boundary images included. The resistance should be well on its way to 1 by the lowest frequency you care about.

How it works

A horn works by matching the high impedance at the throat to the low impedance of open air. If the mouth is too small for the wavelength, the air in front of it behaves like a mass rather than a resistance and a good part of the wave is reflected back into the horn. The result is the familiar ripple in response and impedance just above cut-off.

The rule of thumb

The traditional design rule asks for a mouth circumference of one wavelength at the lowest frequency when the horn radiates into free space:

C_m = \lambda_c \quad\Rightarrow\quad S_m = \frac{\lambda_c^{2}}{4\pi}, \qquad \lambda_c = \frac{c}{f_c}

Each reflecting boundary next to the mouth acts as a mirror. The horn and its image radiate together, so the mouth can be halved for every boundary:

S_m = \frac{\lambda_c^{2}}{4\pi}\cdot\frac{\Omega}{4\pi}, \qquad \Omega = 4\pi,\; 2\pi,\; \pi,\; \tfrac{\pi}{2}\ \text{sr}

A bass horn on the floor against a wall needs a quarter of the free-space mouth; pushed into a corner, an eighth. This is why corner horns can reach so low from a cabinet of domestic size.

What Fig. 2 shows

The radiation impedance of a rigid piston of radius $a$ in a large baffle is

Z_r = \frac{\rho c}{S_m}\left[1-\frac{J_1(2ka)}{ka} + j\,\frac{H_1(2ka)}{ka}\right]

with $J_1$ the Bessel and $H_1$ the Struve function. With the rule-of-thumb mouth, the resistance at $f_c$ is only about a quarter of $\rho c/S_m$ and reaches roughly 0.7 an octave higher. The rule is a minimum, not an optimum: a larger mouth gives a smoother bottom octave, and a smaller one can be acceptable when some ripple is tolerable or the flare ends in a generous round-over.

For rectangular mouths keep the aspect ratio moderate; beyond about 3:1 the narrow dimension loses pattern control early.

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How it works

The rule of thumb

The traditional design rule asks for a mouth circumference of one wavelength at the lowest frequency when the horn radiates into free space:

C_m = \lambda_c \quad\Rightarrow\quad S_m = \frac{\lambda_c^{2}}{4\pi}, \qquad \lambda_c = \frac{c}{f_c}

Each reflecting boundary next to the mouth acts as a mirror. The horn and its image radiate together, so the mouth can be halved for every boundary:

S_m = \frac{\lambda_c^{2}}{4\pi}\cdot\frac{\Omega}{4\pi}, \qquad \Omega = 4\pi,\; 2\pi,\; \pi,\; \tfrac{\pi}{2}\ \text{sr}

A bass horn on the floor against a wall needs a quarter of the free-space mouth; pushed into a corner, an eighth. This is why corner horns can reach so low from a cabinet of domestic size.

What Fig. 2 shows

The radiation impedance of a rigid piston of radius $a$ in a large baffle is

Z_r = \frac{\rho c}{S_m}\left[1-\frac{J_1(2ka)}{ka} + j\,\frac{H_1(2ka)}{ka}\right]

For rectangular mouths keep the aspect ratio moderate; beyond about 3:1 the narrow dimension loses pattern control early.

Mouth size

Inputs

How it works

The rule of thumb

What Fig. 2 shows

Inputs

How it works

The rule of thumb

What Fig. 2 shows

§ How it works

The rule of thumb

What Fig. 2 shows

§ How it works

The rule of thumb

What Fig. 2 shows

How it works

How it works