Reflection and radiation of capillary-gravity water waves

The necessary edge condition, which is used here, includes both dynamic variation of the contact angle and contact angle hysteresis. It is given by making the slope of the free surface at contact proportional to its velocity, however, viscosity has been ignored throughout. Six problems are studied. The first one is the damping of capillary-gravity waves inside a vertical and axisymmetric cylinder, where the frequency of these waves are calculated. The second problem is concerned with the study of the waves produced by a vertically oscillating cylinder, and determining the surface elevation, at both on and large distances away from the cylinder. The third problem is the horizontal oscillation of a cylinder partially immersed in the fluid, such that the cylinder and the fluid are both of infinite or finite depth, where again the surface elevation from the free surface at large distances and on the edge of the cylinder are evaluated. The fourth problem studies the scattering of a capillary-gravity wave by a surface-piercing circular cylinder, and the depending condition applied at the contact line between the fluid and the obstacle. Using a model for this condition that incorporates the effect of dynamic contact-angle variation, the wave field close to the obstacle and at large distances away are determined. The fifth problem studied, concerns the vertical and horizontal oscillation of a vertical cylinder, as well as the scattering of waves made by it, in shallow water. When the depth of the fluid is small compared with the wavelength, the simplifications of shallow-water theory can be applied and the results arrived at more readily than by the methods used for arbitrary depths. In each case, the surface elevation of the radiated waves at large distances away from the cylinder is obtained, as well as some special cases (when some of the parameters have extreme values). The sixth problem studied is the waves produced by a vertical plate, when it is forced to oscillate horizontally. The length of the plate is considered to be finite and the fluid is either of infinite depth or of the same depth as the length of the plate. Both the steady-state and the transient motion are studied. However, when the depth of the fluid is small, the simplifications of shallow-water theory can be applied. The surface elevation at large distances away and the amplitude of these waves are calculated for the cases of vertically moving boundary and for the reflection of an incident wave