Water wave scattering by two sharp discontinuities in the surface boundary conditions

The problem of water wave scattering by two sharp discontinuities in the surface boundary conditions involving infinitely deep water is examined here by reducing it to two coupled Carleman-type singular integral equations. The discontinuities arise due to the presence of two types of non-interacting materials floating on the surface, one type being in the form of an infinite strip of finite width sandwiched between another type. The non-interacting materials form an inertial surface which is a mass-loading model of floating ice and is regarded as a material of uniform surface density having no elastic property. The two integral equations are solved approximately by assuming the two discontinuities to be widely separated, and approximate analytical expressions for the reflection and transmission coefficients are also obtained. This problem has applications in wave propagation through strips of frazil or pancake ice modelled as floating inertial surfaces. Numerical results for the reflection coefficient are depicted graphically against the wave number for different values of the surface densities of the two types of floating materials. The main feature of the graphs is the oscillatory nature of the reflection coefficient and occurrence of zero reflection for an increasing sequence of discrete values of the wave number. A direct analytical treatment to solve the integral equations numerically, when the separation length between the two discontinuities is arbitrary, is also indicated. For the case of more than two discontinuities the solution methodology of the corresponding scattering problem is described briefly