Dual integral equations -- revisited
The authors consider the following dual integral equations: \[ \left\{\begin{aligned} \int_0^\infty A(k)L(k,y) dk &=-R\exp(-Ky),\quad y\in G,\\ \int_0^\infty kA(k)L(k,y) dk &=iK(1-R)\exp(-Ky),\quad y\in B, \end{aligned}\right.\tag{1} \] where \(L(k,y) = k\cos ky-K\sin ky,\quad G=(0,\infty)-B\), \(A(k)\) is an unknown function, and \(R\) is an unknown constant. This integral equation arises in the well-known problem of scattering water waves in deep water by a vertical barrier under the assumption of linearized theory. The solution of (1) is obtained here by noting the behaviour of the second equation of (1) at the end points of the interval \(G\), which can be determined from physical consideration. Equation (1) is then reduced to a singular integral equation whose kernel involves Cauchy and logarithmic type singularities, the solution of which is known. Knowing the solution of (1), the solution of the corresponding scattering problems is obtained in a closed form.
scattering of water waves by a vertical barrier
singular integral equation with logarithmic and Cauchy kernel
scientific article; zbMATH DE number 1930013