On the utility of C/S as an indicator of nonlinearity in finite-amplitude dispersive waves

Abstract

A frequency-domain nonlinearity indicator C/S, recently introduced by Ohm et al. [Proc. Mtgs. Acoust. 29, 045003 (2016)], is a companion indicator to the well-known Morfey-Howell Q/S [AIAA J. 19, 986–992 (1981)]. The two indicators can be regarded as “two sides of the same coin,” because C/S and Q/S are, respectively, defined by the real and imaginary parts of the cross-spectrum of the pressure and squared-pressure waveforms. Despite the common origin, however, C/S and Q/S describe the pertinent nonlinear wave process in a different light. Furthermore, for nonlinear waves in media with strong dispersion only C/S may be applicable, because the use of Q/S could lead to erroneous interpretations. In this paper, we demonstrate the utility of C/S in the context of the Korteweg-de Vries (KdV) equation, which is one of the most celebrated model equations for nonlinear waves in dispersive media. The success of C/S is juxtaposed with the limitation of Q/S in describing nonlinear dispersive wave processes, especially in the case of solitons.