Reflection and transmission of water waves in a two‐layer fluid flowing through a channel with undulating bed

Abstract

AbstractWithin the framework of two‐dimensional linear water wave theory, we consider the problem of normal surface wave propagation over small undulations in a channel flow consisting of a two‐layer fluid with the upper layer bounded by a fixed wall, considered as an approximation to the free surface, and the lower one by a bottom surface which has small undulations. The effects of surface tension at the surface of separation is neglected. Assuming irrotational motion, a perturbation analysis is employed to calculate the first order corrections to the velocity potentials in the two‐layer fluid by using Fourier transform appropriately and also to calculate the reflection and transmission coefficients in terms of integrals involving the shape function c (x) representing the bottom undulation. Two special forms of the shape function are considered for which explicit expressions for reflection and transmission coefficients are evaluated. For the specific case of a patch of sinusoidal ripples having the same wave number throughout, the reflection coefficient up to the first order is an oscillatory function in the quotient of twice the interface wave number and the ripple wave number. When this quotient is one, the theory predicts a resonant interaction between the bed and the interface, and the reflection coefficient becomes a multiple of the number of ripples. But the theory breaks down at resonance and it is applicable only to infinitesimal reflection when the reflection coefficient cannot assume large value, and away from resonance. Hence, the results demonstrated here is valid for up to the near resonant cases only. Again, when a patch of sinusoidal ripples having two different wave numbers is considered, the resonant interaction between the bed and the interface attains in the neighborhood of two (singular) points along x‐axis (when the ripple wave numbers of the bottom undulation become twice as large as interface wave number). The theoretical observations are presented in graphical form.