Wave resistance for capillary gravity waves: Finite-size effects

We study theoretically the capillary-gravity waves created at the water-air interface by an external surface pressure distribution symmetrical about a point and moving at constant velocity along a linear trajectory. Within the framework of linear wave theory and assuming the fluid to be inviscid, we calculate the wave resistance experienced by the perturbation as a function of its size (compared to the capillary length). In particular, we analyze how the amplitude of the jump occurring at the minimum phase speed c_min=(4 gγ/ρ)^1/4 depends on the size of the pressure distribution (ρ is the liquid density, γ is the water-air surface tension, and g is the acceleration due to gravity). We also show how for pressure distributions broader than a few capillary lengths, the result obtained by Havelock for the wave resistance in the particular case of pure gravity waves (i.e., γ=0) is progressively recovered.