English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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 △2ø− 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.

A fresh look on old analytical solutions for water waves on a constant slope