In defense of the Morfey-Howell single-point nonlinearity indicator: An impedance-based interpretation

Since the Morfey-Howell Q/S was proposed as a nonlinearity indicator for propagation of intense broadband noise [AIAA J. 19, 986-992 (1981)], there has been considerable debate as to its meaning and utility. Perhaps the most contentious argument against Q/S is about its validity as a single-point nonlinearity indicator: the importance of nonlinearity is often judged by observing cumulative effects over some propagation distance, whereas Q/S is based on a pressure waveform at a single location. Studies to address these criticisms have emerged over the years, most recently by Reichman et al. [J. Acoust. Soc. Am. 139, 2505-2513 (2016)] in support of Q/S. In this paper, we show that the Burgers equation (from which Q/S was originally derived) can be recast in terms of specific impedance, linear absorption and dispersion coefficients, and normalized quadspectral (Q/S) and cospectral (C/S) densities. The resulting interpretation is that Q/S and C/S represent the additional absorption and dispersion, introduced by the passage of a finite-amplitude wave to the existing linear absorption and dispersion. In other words, a nonlinear wave process alters the apparent material properties of the medium, the extent of which can be used as a single-point indicator of the relative strength of nonlinearity.