Comparison between the Gol’dberg number and the Morfey-Howell Q/S for finite-amplitude plane waves

Abstract

In this paper we compare the two most popular indicators of nonlinearity, namely, the Gol’dberg number [Sov. Phys. Acoust. 2, 346-350 (1956)] and the Morfey-Howell Q/S [AIAA J. 19, 986-992 (1981)] for the case of an initially-sinusoidal, finite-amplitude plane wave in a thermoviscous medium. The Gol’dberg number Γ, defined as the ratio of the absorption length at the fundamental frequency to the shock formation distance, is a parameter that single-handedly dictates the entire history of the wave [for example, weak (Γ < 1) and strong (Γ >> 1) waves]. However, it fails to quantify the strength of nonlinearity that each individual harmonic “sees” at any given point. In other words, not every harmonic sees the apparent strength of nonlinearity according to Γ. The Q/S, on the other hand, is defined for each harmonic as a function of propagation distance, and hence keeps track of the ever-changing strength of nonlinearity during the course of propagation. One interesting ramification of this is that using Q/S it is possible to pinpoint the distance at which the value of nonlinearity relative to absorption actually becomes Γ for the fundamental component.