Three-dimensional linear solution for wave propagation with sloping bottom

This paper presents a three-dimensional analytic linear wave solution for surface gravity wave propagation over a sloping bottom that is valid for small, but realistic, slopes. The sloping-bottom linear model is compared to published laboratory data and to predictions of two-dimensional, constant-bottom nonlinear theories. The model is shown to describe the measured wave-height growth in the wave transformation region up to a limiting local Ursell number U_r of 0.35-1.0, depending on the wave type, although, as a linear model, it does not predict the harmonics observed in that range. For U_r<0.35, the harmonics can generally be neglected and the sloping-bottom linear theory agrees closely with both the published wave-height data and third-order Stokes nonlinear theory. As a three-dimensional linear model, superposition can be invoked to synthesize and relate wave structure in the transformation region to complex incident ocean spectra with both wind wave and swell components that arrive with a range of incidence angles. As such, the sloping-bottom linear model presented here should be a convenient useful tool for ocean modeling through a significant portion of the wave transformation region