Do beach profiles under nonbreaking waves minimize energy dissipation?

Abstract

The hypothesis that equilibrium beach profiles under nonbreaking waves minimize wave energy dissipation was considered by Larson et al. (1999 Coast. Eng. 36 59-85). Larson et al. approached the hypothesis as a variational problem, assuming a priori that the solution (the extremal profile) followed a power law with a freely tunable exponent, which was varied so as to extremize the relevant functional. Here, we revisit this hypothesis and solve the associated variational problem approximately via analytical means, without a priori assumptions on the mathematical structure of the solution. We remark that for the solution to be realistic, the problem formulation must consider additional constraints; for example, the bed slope angle must not exceed the sediment's angle of repose. Incidentally, the solution we derive recovers the power law prescribed by Larson et al., which is in turn backed by a large body of empirical evidence. However, the exponent of the power in our solution is not an arbitrarily free parameter; it depends on the parametrization of the bed shear stress (the main mechanism by which nonbreaking waves dissipate energy), and predicted values of the exponent are supported by previous research. The power law curve derived here agrees well with empirical data from field and laboratory, suggesting that a principle of energy economy may indeed underpin the particular shape adopted by beach profiles under nonbreaking waves. This theoretical study aims at promoting and aiding further tests of this hypothesis.

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