Analytic calculation of two-dimensional linear viscous gravity–capillary waves on deep water generated by a moving forcing at non-critical conditions: wave patterns and spatial decay rate

Abstract

Two-dimensional (2-D) steady viscous gravity–capillary wave patterns (minimum phase speed: cmin) by a moving Gaussian forcing (speed: V) above the surface of deep water are calculated by analytically solving a theoretically derived linear wave model equation, based on the spatial Fourier transform and Cauchy residue theorem. By considering viscous effects on the poles in the relevant complex region, the analytics results in a single expression of wave profiles for both supercritical (V > cmin) and subcritical (V < cmin) cases, i.e., at non-critical conditions. For supercritical cases, a pair of decaying short (capillary) and long (gravity) sinusoidal waves are calculated ahead of and behind the moving forcing, respectively. For subcritical cases, a small depression with a pair of elevated shoulders is calculated below the moving forcing. From the analytic expression of the spatial decay rate of the resultant wave, it is found that the viscous effect on the decaying wave pattern is prominent only in the capillary waves in the supercritical case, where the decay rate increases as the forcing speed increases.