Wave Resistance for Capillary Gravity Waves: Finite Size Effects

TL;DR

This paper theoretically investigates how the size of a moving pressure disturbance affects wave resistance in capillary-gravity waves, revealing finite size effects and the transition to pure gravity wave behavior.

Contribution

It provides a detailed analysis of finite size effects on wave resistance in capillary-gravity waves within linear wave theory, extending previous models to include size-dependent phenomena.

Findings

Wave resistance depends on the size of the pressure distribution.

A jump in wave resistance occurs at the minimum phase speed, influenced by size.

For larger disturbances, the results approach those of pure gravity waves.

Abstract

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 $c_{{\rm min}}=(4 g \gamma /\rho)^{1/4}$ depends on the size of the pressure distribution ($\rho$ is the liquid density, $\gamma$ 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., $\gamma = 0$) is progressively recovered.