Interfacial waves from pressure forcing: revisiting classical theories from an IVP perspective

TL;DR

This paper revisits classical interfacial wave theories by analyzing pressure-driven waves using an initial-value problem approach, highlighting the role of time-dependent solutions in wave pattern formation.

Contribution

It introduces an IVP framework for interfacial waves, contrasting with classical steady-state methods, and extends analysis to two-fluid interfaces validated by nonlinear simulations.

Findings

Short capillary waves form ahead of the pressure source.

Long gravity waves develop behind the pressure source.

Time-dependent solutions decay algebraically and influence wave patterns.

Abstract

A localised overpressure translating at a uniform speed greater than a critical value acts at the interface between two deep fluid layers with different densities. We analyse the resulting wave patterns using an initial-value problem formulation within the linearised, inviscid, potential flow framework. The steady-state interface exhibits short capillary waves ahead of the forcing and long gravity waves behind it, arising from an asymmetric cancellation of Fourier components in the far field. The time-dependent part of the solution, decaying algebraically with time, plays a crucial role in this mechanism. This contrasts with classical steady approaches, which require additional conditions to select a unique solution. We extend this approach to a two-fluid interface and validate the predictions against nonlinear simulations.