Variational approximations for gravity waves in water of variable depth

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Abstract
Eckart's (1952) second-order, self-adjoint partial differential equation for the free-surface displacement of monochromatic gravity waves in water of variable depth h is derived from a variational formulation by approximating the vertical variation of the velocity potential in the average Lagrangian by that for deep-water waves. It is compared with the ‘mild-slope equation’, which also is second order and self-adjoint and may be obtained by approximating the vertical variation in the average Lagrangian by that for uniform, finite depth. The errors in these approximations vanish for either κh ↓ 0 or κh ↑ ∞ (κ ≡ ω2/g). Both approximations are applied to slowly modulated wavetrains, following Whitham's (1974) formulation for uniform depth. Both conserve wave action; the mild-slope approximation conserves wave energy, but Eckart's approximation does not (except for uniform depth). The two approximations are compared through the calculation of reflection from a gently sloping beach and of edge-wave eigenvalues for a uniform slope (not necessarily small). Eckart's approximation is inferior to the mild-slope approximation for the amplitude in the reflection problem, but it is superior in the edge-wave problem, for which it provides an analytical approximation that is exact for the dominant mode and in error by less than 1.6% for all higher modes within the range of admissible slopes. In contrast, the mild-slope approximation requires numerical integration (Smith & Sprinks 1975) and differs significantly from the exact result for the dominant mode for large slopes.

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