An alternative dispersion equation for water waves over an inclined bed.

*(English)*Zbl 1137.76323A classical approach to extending the validity of Airy’s dispersion relation for surface gravity waves by K. O. Friedrichs [Commun. Appl. Math., New York 1, 109–134 (1948; Zbl 0030.37901)] to gentle slopes (of special inclinations) is here re-examined with extended small-slope asymptotics using the full linear harmonic function theory combined with the method of steepest descent. A new dispersion relation emerges that appears to give significantly increased accuracy over sloping beds when tested on the plane beach problem with various forms of the mild-slope equation (MSE) and global error reductions of the order 50% are noted in some ‘from deep to shore’ computations. Unlike the classical formula, the new formula predicts a discontinuous wavenumber at a place where the bottom slope is discontinuous. Preliminary tests examining the reflection coefficient with the basic (early version) MSE over ramp-type profiles indicate that this is not a major problem and numerical results using wavenumber calculated by the new dispersion relation are qualitatively similar to those of the modified MSE (MMSE) developed in P. G. Chamberlain and D. Porter [J. Fluid Mech. 291, 393–407 (1995; Zbl 0843.76006)]. When the new formula is applied (with mass conservation) to the MMSE on the ramp, results are almost identical to those of a full linear model for inclines having a gradient up to 8:1.