Shanshan Xu, Frédéric Dias
The first studies of water waves climbing a beach of constant slope were restricted to standing wave solutions. They all used potential flow theory. Various methods can be found in the literature. Hanson (The theory of ship waves. Proc. Roy. Soc. Lond. A Mat., 1926, 111, 491–529) and Stoker (Water Waves. Interscience, New York, 1957) gave the standing wave solution for slope angles of π /2n with n an integer. Stoker (Water Waves. Interscience, New York, 1957), Peters (Water waves over sloping beaches and the solution of a mixed boundary value problem for Δ2ø – k2ø = 0 in a sector. Commun. Pur. Appl. Math., 1952, 5, 87–108), and Isaacson (Water waves over a sloping bottom. Commun. Pur. Appl. Math., 1950, 3, 11–31) provided solutions for beaches of arbitrary slope angles. Due to the lack of numerical tools at the time, results were completely based on the theory of functions of complex variables, which is sometimes tedious and not easy to compare with modern numerical evaluation of analytical solutions. Here, we present four old solutions of standing waves and then evaluate the analytical solutions numerically to visualize results and perform comparisons. The run-up of waves in arbitrary water depth is also discussed.
Keywordspotential flow theory, standing wave, run-up, sloping beach.
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