A mild-slope model for membrane-coupled gravity waves

A depth averaged equation is derived via variational principle to study the effect of varying bottom on membrane-coupled gravity waves caused by a floating membrane with spatially varying material properties. Variation of gradients for bottom topography, membrane tension and membrane mass density is assumed to be small. The surface energy generated by the membrane deflection contributes to the total energy for membrane-coupled gravity waves. An alternative derivation is also provided by making use of integration by parts. The model equation for the uniformly tensional membrane with uniform mass density has been solved numerically, for the membrane-coupled gravity wave scattering in two dimensions, using mass conserving jump conditions applied at the locations of possible bottom slope discontinuities. This model also describes the reflection characteristics for capillary-gravity waves when the membrane mass density alone is neglected. Reflected amplitudes of membrane-coupled as well as capillary-gravity waves by different bottom topographies are shown graphically for the physical parameters of surface tension, membrane tension and uniform mass of membrane. Moreover, the phenomenon of Bragg resonance is discussed numerically for sinusoidal bottom variations. Further, the model has been utilized to study the effect of depth variations beneath a floating membrane of finite length on the free-surface gravity waves.