Effect of surface tension on the propagation of water waves by a line source in the presence of a nearly vertical cliff

Present investigation is concerned with the propagation of obliquely incident water waves by a pulsating line source in presence of a fixed nearly vertical rigid cliff, submerged in deep water, assuming the surface tension effect at the free surface. For the perfectly vertical cliff, it's effect on the source is similar to another source located at the image point of the main source regarding to the vertical cliff. However, because of the bended figure of the cliff, there will be extra effects. Assuming the surface tension effect at the free surface, these effects have been obtained up to first order term to the wave amplitude at infinity ($A_{1}$) and the velocity potential ($\phi_{1}$) for deep water by employing a simplified perturbation theory followed by an adequate Havelock's expansion of water wave potential. Considering the two particular shapes of the nearly vertical cliff viz.(i) $c(y)=y\exp(-\lambda y)$ and (ii) $c(y)=\alpha\sin\beta y$, these corrections are also found interms of the integrals involving the shape function of the cliff in presence of surface tension at the free surface. Neglecting the effect of surface tension at the free surface, the approximate solution of the corresponding problem can be found. The solution of the corresponding two dimensional problem can also be derived by a known substitution.