Capillary-gravity wave resistance of a translating, pulsating source

Abstract

A disturbance in steady rectilinear translation along a capillary-gravity interface radiates waves only above a minimum speed $c_{min}$, 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 $\Omega$ shifts the forcing to the sideband pair $kV\pm \Omega$, where $k$ is the wavenumber and $V$ is the translation speed, opening resonance conditions absent from the steady problem; radiation then sets in below the steady threshold $c_{min}$. Here we derive the mean wave resistance $⟨R⟩$ of a line source in closed form, and show it to be strictly positive at every nonzero speed $V$: the drag-free below-threshold regime disappears for any $\Omega \ne 0$. Crossing the radiation transitions---where the group velocity matches the source speed---produces integrable resonant peaks in $⟨R⟩$, whose exponents we derive, including at the cusp where two transition branches merge. Below threshold, where a steadily moving source feels no drag, $⟨R⟩$ rises from zero as ${\Omega }^5$ (at fixed $V$). Throughout, surface tension is essential to $⟨R⟩$: because it controls the short, dominant capillary-scale waves, discarding it misrepresents $⟨R⟩$ over almost the entire speed-frequency plane.

10 pages