A straight-line beach profile yields a larger value of Jenkins and Inman's energy dissipation integral than their elliptic cycloid solution, so the cycloid is not the maximizer.
(2020) Do beach profiles under nonbreaking waves minimize energy dissipation? Journal of Geophysical Research: Oceans
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On the thermodynamics-based equilibrium beach profile derived by Jenkins and Inman
A straight-line beach profile yields a larger value of Jenkins and Inman's energy dissipation integral than their elliptic cycloid solution, so the cycloid is not the maximizer.