Capillary–gravity-wave resistance of a translating, pulsating line source

A pressure disturbance in steady rectilinear translation applied at a capillary–gravity interface radiates waves only above a minimum speed c Subscript italic min c min cₘᵢₙ , where it experiences a wave resistance; below this threshold it generates no waves, and the response is merely a localised deformation of the interface around the source, free of wave resistance. Modulating the source amplitude at a frequency upper Omega Ω shifts the forcing to the sideband pair k upper V plus or minus upper Omega k V ± Ω kV± , where k k k is the wavenumber and upper V V V is the translation speed, opening resonance conditions absent from the steady problem; radiation then sets in below the steady threshold c Subscript italic min c min cₘᵢₙ . Here, we derive the mean wave resistance left angle bracket upper R right angle bracket ⟨ R ⟩ R of a line source in closed form, and show it to be strictly positive at every non-zero speed upper V V V