A topological study of gravity waves generated by moving bodies using the method of steepest descents

TL;DR

This paper introduces a topological approach using steepest descents to analyze gravity waves generated by moving bodies without simplifying their geometry, linking flow features to Riemann surface topology.

Contribution

It presents a novel methodology that relates physical flow geometry to Riemann surface topology, enabling analysis of wave effects from geometrical modifications without linearization.

Findings

The approach successfully models wave patterns around complex geometries.

Changes in body shape correspond to topological modifications in the Riemann surface.

Demonstrated on flow past a ship and over an angled step.

Abstract

The standard analytical approach for studying gravity free-surface waves generated by a moving body often relies upon a linearization of the physical geometry, where the body is considered asymptotically small in one or several of its dimensions. In this paper, a methodology that avoids any such geometrical simplification is presented for the case of flows at low speeds. The approach is made possible through a reduction of the water-wave equations to a complex-valued integral equation that can be studied using the method of steepest descents. The main result is a theory that establishes a correspondence between a given physical flow geometry, with the topology of the Riemann surface formed by the steepest descent paths. Then, when a geometrical feature of the body is modified, a corresponding change to the Riemann surface is observed, and the resultant effects to the water waves can be derived. This visual procedure is demonstrated for the case of two-dimensional free-surface flow past a surface-piercing ship and over an angled step in a channel.