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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 $c_{\rm min}=(4 g γ/ρ)^{1/4}$ 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.