Abstract
We reformulate the variational problem describing equilibrium beach profiles in the thermodynamic approach of Jenkins and Inman. A first integral of the resulting Euler-Lagrange equation coincides formally with the Friedmann equation ruling closed universes in relativistic cosmology, leading to a useful analogy. Using the machinery of Friedmann-Lema\^{\i}tre-Robertson-Walker cosmology, qualitative properties and analytic solutions of beach profiles, which are the subject of a controversy, are elucidated.
Highlights
Since the early work of Bruun [1], the profile of a beach, measured from the shore seaward and perpendicular to the shoreline, has been one of the most studied features of coastal morphology
Contrary to Ref. [4], we first reformulate the variational principle in terms of the beach profile h(x), instead of its inverse x(h),5 which uncovers two analogies
The first is an analogy with the mechanics of a point particle in one-dimensional motion, which provides a graphic way of deducing basic qualitative properties of the solutions
Summary
Since the early work of Bruun [1], the profile of a beach, measured from the shore seaward and perpendicular to the shoreline, has been one of the most studied features of coastal morphology. [4] consists of maximizing the rate of energy dissipation of both breaking and nonbreaking waves This extremization leads to an elegant variational principle formulation of the problem and to an associated Euler-Lagrange equation for the curves describing the equilibrium beach profiles. While it is understandable that the cosmological analogy was missed in the literature because of the enormous gap between the communities of cosmologists and ocean scientists, it is surprising that another, rather obvious, analogy between any beach profile ordinary differential equation (ODE) and the one-dimensional motion of a point particle was missed While this second analogy is much less useful than the first one, it provides some insight on the qualitative nature of the solutions of the beach profile equation, and we discuss it briefly. We follow the notation of Ref. [10]; the signature of the space-time metric is − + ++, and we use units in which Newton’s constant G and the speed of light c are unity
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