Diffraction of surface water waves by an undulating bed topography in the presence of vertical barrier

Abstract

In this paper, the problem of diffraction of surface water waves by an undulating bed and a vertical barrier is examined. Here, two different barrier configurations are studied, namely, (i) a partially immersed barrier and (ii) a bottom standing barrier. Perturbation analysis containing a dimensionless parameter which describe the smallness of bottom undulation is employed to find the solution of the mixed boundary value problem (BVP). This analysis reduces the BVP into two BVPs up to first order. The zeroth order BVP represents the problem of scattering of water waves by a vertical barrier. Using eigenfunction expansion method this zeroth order BVP leads to a pair of dual series relations which are solved by the application of least-squares method. The first order BVP containing the solution of the zeroth order BVP is solved by suitable application of the Green's function technique. From the solution of the first order BVP, the physical quantities, namely, the first order reflection and transmission coefficients are obtained in terms of integrals involving the shape function representing the bottom undulation and the solution of the zeroth order BVP. A patch of sinusoidal ripples which closely corresponds to some obstacles produced by nature due to ripple growth and alluviation of sand is considered for which the explicit expressions for the first order reflection and transmission coefficients are obtained. The variation of these coefficients on the barrier length, gap above or below the barrier, ripple amplitude and the number of ripples at the bottom is illustrated graphically. The other important factors of the study, namely, the hydrodynamic force and the moment are also studied and depicted graphically. Furthermore, the energy balance relation which ensures the correctness of the numerical results, is derived and verified.