Spherical sound-waves of finite amplitude

Owing to the use of acoustical outputs up to 500 watts in modern public-address loud-speaking apparatus, it is necessary to consider the distortion due (mainly) to the adiabatic pressure-volume relationship for air being non-linear. The physical principles underlying the transmission of sound-waves of finite amplitude in expanding waves are epitomized, and the differential equation for a horn of any cross-section is stated. The equation is solved for spherical-wave propagation, which includes the case of a conical horn having any solid angle up to 4π. Formulae are given which enable the shapes of the particle amplitude and pressure waves to be calculated to a second approximation. As the sound travels down the horn, the crests of the waves tend to overtake the troughs, and harmonics are created, the power associated therewith being transferred from the fundamental. The ratio of the power in the second harmonic to that in the fundamental is obtained, and comparison made with that for a uniform tube and an exponential horn. The analysis is illustrated by numerical examples, and design formulae are deduced from which the area of the horn-throat to keep the distortion below a prescribed limit can be calculated. Lamb's analysis for a uniform tube has been extended from the second to the third approximation.