Analysis of the linear version of a highly dispersive potential water wave model using a spectral approach in the vertical

For variable bottom bathymetries, the shoaling properties of the model are studied numerically, exhibiting good agreement with results from Stokes linear theory in the case of mild bottom slopes, using a sufficiently high value of NT with respect to the offshore relative water depth. For an offshore water depth of kh=10 (i.e. more than 3 times the deep water limit), accurate wave heights in shallow water (kh=0.25) are obtained with NT=6 (or higher). Finally, the linear version of the model is validated with comparisons to analytical solutions of the reflection and transmission coefficients of regular waves over Roseau-type bathymetric profiles. Two bottom profiles are considered, including one with a steep slope, whose maximum value reaches about 1:0.7 (i.e. an angle of about 54.9 deg.). Using NT=7, small differences (<0.4%) with the analytical solution are observed for the four considered cases, confirming the ability of the linear model to represent accurately the effects of steep bottom gradients on wave propagation dynamics.