Throat distortion

Air is not perfectly linear. At the sound pressures found in a horn throat the wave steepens as it travels and generates second harmonic. This sets a floor that no driver can improve on: air compresses adiabatically, so pressure and volume are related by a curve and not a straight line. The crest of a strong wave travels slightly faster than the trough and the waveform leans forward as it goes.

In an exponential horn

The pressure falls as the horn expands, so most of the distortion is generated in the first few centimetres. For a long exponential horn the result, with throat pressure $p_t$ (rms), atmospheric pressure $P_0$ and ratio of specific heats $\gamma = 1.4$, is

The throat pressure follows from the acoustic power $W_A$ and the throat area $S_t$ through the intensity $I_t$:

Inserting numbers for air gives the form found in the textbooks, with intensity in watts per square metre:

What to take from it

• Distortion rises in proportion to frequency and to the square root of power: +10 dB of level costs about 3.2 times the distortion.
• A larger throat lowers the intensity. Doubling the throat diameter halves the distortion at the same power.
• A higher cut-off means a faster flare, so the pressure drops sooner. This is one reason not to use a horn with a much lower cut-off than the crossover needs.
• Conical horns and waveguides expand faster at the throat and generate less; hypex horns with low $T$ generate more.

A. L. Thuras, R. T. Jenkins & H. T. O’Neil, “Extraneous Frequencies Generated in Air Carrying Intense Sound Waves”, J. Acoust. Soc. Am. 6 (1935).
L. L. Beranek, Acoustics (1954), §9.8.