Linear – Nonlinear Acoustic Wave Propagation Boundary in Air

In certain discussions of acoustic wave propagation, assumptions are introduced that lead to a linear approximation of the governing equations. Implicit in such assumptions is the notion of smallness of a parameter, the size of which is rarely given. We here give a value for the parameter, using a numerical procedure. This study models acoustic wave propagation in air, using a nonlinear acoustic wave equation in one space dimension, by a Riemann invariant based method of characteristics. We simulate propagation of a particle velocity wavelet along a straight pipe, closed at one end and open at the other. The initial amplitude of the wavelet lies between 0.1 and 200 m/s, assuming free space propagation. Data from the computation is analysed by exploiting the known ‘steepening’ of a part of the wavelet in finite-amplitude wave propagation. We have calculated the slope of that part of the wavelet, at various times, for wavelets of different initial amplitudes. The results show clearly a demarcation between linear- and nonlinear-acoustic wave propagation (that is, a demarcation between finite-amplitude and small-amplitude wave propagation.) This demarcation is controlled by the opposing influences of energy loss and nonlinear effects. For an acoustic wavelet, the initial amplitude 50 m/s in terms of particle velocity (or 0.02 m in terms of particle displacement, or 0.15 in terms of acoustic Mach number) marks the upper limit beyond which nonlinear effects cannot be ignored. That is how small the size of the initial disturbance should be, for the assumptions made while deriving a linear wave equation to be valid.