  We study normal modes for the linear water wave problem in infinite straight
channels of bounded constant cross-section. Our goal is to compare semianalytic
normal mode solutions known in the literature for special triangular
cross-sections, namely isosceles triangles of equal angle of 45 and 60 degrees,
see Lamb [17], Macdonald [19] , Greenhill [11], Packham [23], and Groves [12],
to numerical solutions obtained using approximations of the non-local
Dirichlet-Neumann operator for linear waves, specifically an ad-hoc
approximation proposed in [25], and a first order truncation of the systematic
depth expansion by Craig, Guyenne, Nicholls, and Sulem [6]. We consider cases
of transverse (i.e. 2-D) modes and longitudinal modes, i.e. 3-D modes with
sinusoidal dependence in the longitudinal direction. The triangular geometries
considered have slopping beach boundaries that should in principle limit the
applicability of the approximate Dirichlet-Neumann operators. We nevertheless
see that the approximate operators give remarkably close results for transverse
even modes, while for odd transverse modes we have some discrepancies near the
boundary. In the case of longitudinal modes, where the theory only yields even
modes, the different approximate operators show more discrepancies for the
first two longitudinal modes and better agreement for higher modes. The ad-hoc
approximation is generally closer to exact modes away from the boundary.
