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

Weakly nonlinear, bi-periodic patterns of waves that propagate in the (Formula presented.) -direction with amplitude variation in the (Formula presented.) -direction are generated in a laboratory. The amplitude variation in the (Formula presented.) -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 (Formula presented.), the comparison with the elliptic function solution of the sNLSE has significantly less error when the ratio of (Formula presented.) -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 (Formula presented.) -direction, obtained from a Stokes-type expansion of interacting, symmetric wavetrains. There is no evidence of instability growth in the (Formula presented.) -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 (Formula presented.), which is examined via a qualitative stability calculation that allows symmetry breaking in that direction.