Nonlinear acoustics in non-uniform infinite and finite layers

Abstract

The propagation of weakly nonlinear acoustic waves in a non-uniform medium is treated. It is assumed that the waves are one-dimensional. Non-uniformities arising from variable cross-section and stratification are included. The effect of non-uniformities on unidirectional waves on an infinite interval and resonant waves on a finite interval is discussed for a near-uniform reference state (geometrical acoustics limit) and for stronger non-uniformities in the finite-interval case. Nonlinearities are taken into account up to quadratic and, wherever necessary, cubic order in the wave amplitude. Unidirectional waves in the geometrical acoustics limit can formally be reduced to the behaviour in a uniform system described by a kinematic wave equation with constant coefficients. For illustration acceleration waves in a weakly non-uniform medium are treated. The resonance case in the geometrical acoustics limit is closely related to resonance in a uniform system so that the methods developed for that situation require only slight modification. For larger influence of non-uniformity the geometrical acoustics limit does not apply and the resonance problem may lead to a Duffing oscillator type of behaviour.