Linear water waves in isosceles triangular domains

We study the frequencies and normal modes of linearized surface gravity water waves in triangular domains. We assume free surface potential flow and use the Schwartz–Christoffel conformal map to transform the normal mode equation to an eigenmode problem for functions on the interval (−1,1). The transformed operator involves the Hilbert transform and a conformal factor that encodes the geometry of the fluid domain. We use Chebyshev collocation to discretize the equation and present results for isosceles triangles with inclined walls of slope of 45° or higher. The length of the horizontal side representing the free surface is kept constant. A main observation is that all frequencies decrease with the slope. We also see that in the high-frequency regime, the squared frequencies undergo a constant shift that depends on the angle of the inclined walls. At the same time, the high-frequency asymptotic slope of the squared dispersion relation is seen to be independent of the angle of the inclined walls. Convergence by refining the discretization becomes slower as the angle of the inclined wall is decreased, but we argue that the observed patterns interpolate between exact results known for a triangular domain with 45° inclined walls and the limiting case of a domain with vertical walls. We also obtain qualitative information on the change of the shape of the normal modes with the inclined wall angle.