Wave motion in a viscous fluid of variable depth Part 2. Moving contact line

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Abstract
An earlier derivation (Miles 1990a) of the partial differential equation for the complex amplitude of a gravity–capillary wave in a shallow, viscous liquid of variable depth and fixed contact line is extended to accommodate a meniscus with a moving contact line at which the slope of the meniscus is assumed to be proportional to (but not necessarily in phase with) the velocity. The motion of the contact line implies capillary dissipation, which is absent for a fixed contact line. The results are applied to the normal reflection of a wave incident from a region of uniform depth on a beach of uniform slope. The reflection coefficient has the form R = R1RνRc, where R1 is the coefficient for an ideal fluid, and Rν and Rc comprise the respective effects of viscosity and capillarity.

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