Water-wave scattering by two submerged plane vertical barriers-Abel integral-equation approach

The classical problem of surface water-wave scattering by two identical thin vertical barriers submerged in deep water and extending infinitely downwards from the same depth below the mean free surface, is reinvestigated here by an approach leading to the problem of solving a system of Abel integral equations. The reflection and transmission coefficients are obtained in terms of computable integrals. Known results for a single barrier are recovered as a limiting case as the separation distance between the two barriers tends to zero. The coefficients are depicted graphically in a number of figures which are identical with the corresponding figures given by Jarvis (J Inst Math Appl 7:207-215, 1971) who employed a completely different approach involving a Schwarz-Christoffel transformation of complex-variable theory to solve the problem.

Proportional navigation with delayed line-of-sight rate

This brief discusses the convergence analysis of proportional navigation (PN) guidance law in the presence of delayed line-of-sight (LOS) rate information. The delay in the LOS rate is introduced by the missile guidance system that uses a low cost sensor to obtain LOS rate information by image processing techniques. A Lyapunov-like function is used to analyze the convergence of the delay differential equation (DDE) governing the evolution of the LOS rate. The time-to-go until which decreasing behaviour of the Lyapunov-like function can be guaranteed is obtained. Conditions on the delay for finite time convergence of the LOS rate are presented for the linearized engagement equation. It is observed that in the presence of line-of-sight rate delay, increasing the effective navigation constant of the PN guidance law deteriorates its performance. Numerical simulations are presented to validate the results.