Nonlinear forced and free vibrations in acoustic waveguides

Abstract

Nonlinear acoustic theory is used to show that the cutoff frequencies and the resonant frequencies of modes in acoustic waveguides of finite length depend upon the mode amplitude. The amplitude ε_n, of mode n > 0, produced by a periodic piston motion of amplitude δ_n, is also determined. It is found that at and near a resonant frequency, ε_n≏δ_n^1/3 while at and near the cutoff frequency ε_n≏δ_n^1/2. Away from these frequencies ε_n≏δ_n, as linear theory predicts. Thus the infinite amplitude, which linear theory yields at cutoff and resonance, is avoided. The results are obtained simply by using the result for the propagation constant as a function of mode amplitude given by J. B. Keller and M. H. Millman [J. Acoust. Soc. Am. 49, 329–333(1971)].