On the variation of bi‐periodic waves in the transverse direction

Abstract

AbstractWeakly nonlinear, bi‐periodic patterns of waves that propagate in the ‐direction with amplitude variation in the ‐direction are generated in a laboratory. The amplitude variation in the ‐direction is studied within the framework of the vector (vNLSE) and scalar (sNLSE) nonlinear Schrödinger equations using the uniform‐amplitude, Stokes‐like solution of the vNLSE and the Jacobi elliptic sine function solution of the sNLSE. The wavetrains are generated using the Stokes‐like solution of vNLSE; however, a comparison of both predictions shows that while they both do a reasonably good job of predicting the observed amplitude variation in , the comparison with the elliptic function solution of the sNLSE has significantly less error when the ratio of ‐wavenumber to the two‐dimensional wavenumber is less than about 0.25. For ratios between about 0.25 and 0.30 (the limit of the experiments), the two models have comparable errors. When the ratio is less than about 0.17, agreement with the vNLSE solution requires a third‐harmonic term in the ‐direction, obtained from a Stokes‐type expansion of interacting, symmetric wavetrains. There is no evidence of instability growth in the ‐direction, consistent with the work of Segur and colleagues, who showed that dissipation stabilizes the modulational instability. Finally, there is some extra amplitude variation in , which is examined via a qualitative stability calculation that allows symmetry breaking in that direction.