Air & wavelength

Every dimension of a horn is ultimately a fraction of a wavelength, so the two numbers worth having at your fingertips are the speed of sound and the wavelength at the frequency in question.

Air

For dry air at sea-level pressure, treated as an ideal gas at temperature $\vartheta$ in degrees Celsius:

with $P_0 = 101\,325\ \text{Pa}$ and $R = 287.05\ \text{J/(kg·K)}$. Their product $\rho c$ is the characteristic impedance of air, the yardstick against which every throat and mouth impedance on this site is normalised. Sound travels about 0.6 m/s faster for each degree of warming, so a horn tuned in a cold workshop shifts by roughly 3 % in a hot venue.

Wavelength

• $\lambda/\pi$ is the mouth diameter at which $ka = 1$: the size where a mouth starts to radiate efficiently and, a little higher, to beam.
• $\lambda/4$ is the shortest horn or pipe that resonates at this frequency, and the path difference that puts two sources 90° apart.
• $\lambda/2$ is the spacing at which two drivers begin to interfere off axis.

Humidity raises the speed of sound slightly, by about 0.3 % at 20 °C between dry and saturated air; it is ignored here.