Third-order theory for bichromatic bi-directional water waves

Abstract

A new third-order solution for bichromatic bi-directional water waves in finite depth is presented. Earlier derivations of steady bichromatic wave theories have been restricted to second-order in finite depth and third-order in infinite depth, while third-order theories in finite depth have been limited to the case of monochromatic short-crested waves. This work generalizes these earlier works. The solution includes explicit expressions for the surface elevation, the amplitude dispersion and the vertical variation of the velocity potential, and it incorporates the effect of an ambient current with the option of specifying zero net volume flux. The nonlinear dispersion relation is generalized to account for many interacting wave components with different frequencies and amplitudes, and it is verified against classical expressions from the literature. Limitations and problems with these classical expressions are identified. Next, third-order resonance curves for finite-amplitude carrier waves and their three-dimensional perturbations are calculated. The influence of nonlinearity on these curves is demonstrated and a comparison is made with the location of dominant class I and class II wave instabilities determined by classical stability analyses. Finally, third-order resonance curves for the interaction of nonlinear waves and an undular sea bottom are calculated. On the basis of these curves, the previously observed downshift/upshift of reflected/transmitted class III Bragg scatter is, for the first time, explained.