Scattering of oblique waves by a thin vertical wall with a submerged gap - Das, P.; Banerjea, S.; Mandal, B.N. (Archives of Mechanics, 1996)
This paper is concerned with scattering of an obliquely incident train of surface water waves by a thin vertical wall with a submerged gap. Utilizing Havelock's expansion of water wave potential, two integral equations, one involving the horizontal component of velocity across the gap and the other involving the difference of velocity potential across the wall, are obtained. The quantities of physical interest, namely the reflection and transmission coefficients, are related to the solutions of these integral equations. For the case of normal incidence of the wave train these integral equations have exact solutions. These exact solutions provide one-term Galerkin approximations to the solutions of the corresponding oblique incidence integral equations. Identifying the reflection and transmission coefficients as some inner products involving the solutions of these integral equations and exploiting the properties of self-adjointness and positive semi-definiteness of the integral operators defining the integral equations, the one-term approximations result in some lower and upper bounds for the reflection and transmission coefficients. Numerical evaluation of these bounds for any angle of incidence and any wave number reveals that they are very close to each other, and as such they produce good approximations to the exact values of the quantities of physical interest. For the special case of normal incidence this method produces numerical results which are in good agreement with the results available in the literature obtained by other methods.