Gravity–capillary wave-making resistance on deep water

Abstract

For supercritical cases (forcing speed > the minimum phase speed, 0.23 m/s), the problem of two-dimensional linear, inviscid gravity–capillary waves generated by a moving delta-function type pressure source is well known. Using harmonic functions or Fourier transform, Lamb [Hydrodynamics, 6th ed. (Cambridge University Press, 1993)] and Rayleigh [Proc. London Math. Soc. s1-15(1), 69–78 (1883)] detailed the steady-state full-space wave-profile solution using an artificial viscosity. Whitham [Linear and Nonlinear Waves (Wiley-Interscience, 1974)] presented the same solution for the region that is far-from-the-forcing using a slowly varying exponential function. For the same problem, but, considering not only supercritical but also subcritical cases, and, without using the artificial viscosity, the present work provides a detailed solution procedure to find full-space wave-profile solutions based on Fourier transform where complex integration is needed; different analytical expressions of the same wave profile will be provided depending on different paths. The associated wave-making resistance is shown to be equal to the integral of the product of a moving pressure source, and the resultant wave slope and is calculated in two ways. One is a direct calculation in the physical domain, which requires the wave-profile solution, and the other is an indirect calculation in the wavenumber domain, which does not require the wave-profile solution. For supercritical cases, short and long sinusoidal waves are calculated ahead of and behind the pressure source. The associated wave-making resistance decreases toward a certain constant as the forcing speed increases and the associated required power features a minimum at the forcing speed of 0.3027 m/s. For subcritical cases, a simple symmetric dimple is calculated and the wave-making resistance becomes zero due to its symmetry.