Abstract
Equilibrium beach profiles are explained in the context of a thermodynamic theory in which energy dissipation rate due to shoaling waves is extremized through a variational principle. The issue of whether this extremum corresponds to a maximum or a minimum of the action integral under the assumptions made is answered by computing the second variation of the integral around the equilibrium profile: it turns out to be a minimum.
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Notes
Strictly speaking, Larson et al. (1999), Jenkins and Inman (2006), and Maldonado and Uchasara (2019) consider instead a functional for the inverse beach profile x(h), but it is more convenient to consider \(J\left[ h(x) \right] \) as in Eq. (1) (see Faraoni (2019) for a discussion). In the end, since the function h(x) is always monotonic and invertible, these two approaches are equivalent.
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Acknowledgements
The author would like to thank two referees for valuable comments. This work is supported, in part, by Bishop’s University and by the Natural Sciences & Engineering Research Council of Canada (Grant No. 2016-03803).
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Faraoni, V. On the extremization of wave energy dissipation rates in equilibrium beach profiles. J Oceanogr 76, 459–463 (2020). https://doi.org/10.1007/s10872-020-00556-4
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DOI: https://doi.org/10.1007/s10872-020-00556-4