Do Beach Profiles Under Nonbreaking Waves Minimize Energy Dissipation?

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. Plain Language Summary Beach profiles describe the sea's depth as a function of the distance from the shore. Waves sculpt the beach profile as they propagate, dissipating energy in the process; nonbreaking waves do so almost exclusively via friction with the seabed. From decades of data collection, it is well known that in equilibrium profiles the depth varies with the distance offshore according to a power law, but the reason for this is much less understood. In this paper, we revisit and advance previous research in order to prove theoretically that the reason may lie behind a principle of energy economy (found ubiquitously throughout nature). In particular, nonbreaking waves appear to sculpt beach profiles into equilibrium shapes that tend to minimize the energy being thereafter dissipated by the waves themselves. [ABSTRACT FROM AUTHOR]