New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

Abstract

Steady two-dimensional surface capillary-gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained.This article is part of the theme issue 'Nonlinear water waves'.

Extreme water-wave profile recovery from pressure measurements at the seabed

“…This illustrates the robustness and applicability of the reconstruction method of Clamond (2013), which does not invoke intermediate conformal changes of variables, and which accordingly offers potential for further development towards capturing more physically complex scenarios, such as the recovery of nonlinear rotational water waves. For extreme and near-extreme waves, the advantage of working in the physical plane, instead of the conformal one, is outlined in Clamond (2018); the present paper provides further support to this claim.…”

Recovery of steady rotational wave profiles from pressure measurements at the bed

“…Similarly, as for irrotational waves in Clamond (2013, 2018), it is useful to introduce the holomorphic function where is an arbitrary constant.…”

General procedure for free-surface recovery from bottom pressure measurements: application to rotational overhanging waves

“…As for irrotational waves, it is useful (Clamond 2013, 2018; Clamond et al. 2023) to introduce the anti-derivative of , where is an arbitrary constant.…”