Linear and nonlinear propagation of water wave groups

Abstract

Specific waveforms with known analytical group shapes were generated, and their evolution observed, in a wave tank in the form of both transient wave groups and the cnoidal (cn) and dnoidal (dn) wave trains as derived from the nonlinear Schrödinger equation. Low‐amplitude transients behaved as predicted by linear theory. The cn and dn wave trains of moderate steepness behaved almost as predicted by the nonlinear Schrödinger equation. There is no adequate theory for the higher nonlinear transient wave groups. The functions of time at successive wave staffs were analyzed in terms of calculations of Fourier integral spectra to interpret the nonlinear behavior of these groups. Dispersed wave groups that were less nonlinear at the wave maker and that became highly nonlinear as they traveled along and coalesced provided an unusual data set. The effects of sum and difference frequencies increased. The apparent phase and group velocities increased for the higher frequencies. The Fourier integral spectra changed shape from one wave staff to the next over the entire range of frequencies that could be analyzed as the waves coalesced. In general, the spectra broadened, shifting energy to both lower and higher frequencies. This experimental exploration of the properties and evolution of transients is motivated by the possibility that the chance occurrence of steep transient wave groups on the ocean may be an important aspect of the evolution of a wind‐driven sea.