Spectral methods for forward-propagating water waves in conformally-mapped channels

The prediction of wave fields in domains with complicated geometries may be aided by the use of conformal-mapping, which simplifies the shape of the domain. In this conformal domain, parabolic models have been used previously to treat wave problems. In Cartesian coordinates, the angular spectrum model, based on a Fourier transform in the direction perpendicular to the principal propagation direction, has been shown to handle, in principle, a wider range of wave directions than the parabolic model. Here, the extension of the angular spectrum model to conformally-mapped domains with impermeable lateral boundaries is shown. Next, the Fourier-Galerkin method is developed for conformal domains; this is identical to the angular spectrum model in Cartesian coordinates, but differs in the conformal domain. Finally, a Chebyshev-tau model for conformal domains is developed, based on using Chebyshev polynomials rather than trigonometric functions as a basis. For all models, forward-propagation equations are derived, by splitting the governing elliptic equations into first-order equations. Examples of all methods are shown for a simple conformal mapping that permits the study of waves in a diverging channel and in a circular channel. The forward-propagation models are shown to be optimal for methods that use eigenfunctions for the lateral transform and less accurate for others.