Scattering of long water waves in a canal with rapidly varying cross-section in the presence of a current

TL;DR

This paper analytically examines how long water waves scatter in a rapidly changing canal with a current, revealing phenomena like negative energy waves, overreflection, and spontaneous wave generation in various flow regimes.

Contribution

It provides a comprehensive analytical framework for wave scattering in variable cross-section canals with currents, including conditions for negative energy waves and spontaneous wave generation.

Findings

Negative energy waves can appear during scattering.

Overreflection and over-transmission occur under certain conditions.

Spontaneous wave generation can arise in trans-critical accelerating flows.

Abstract

The analytical study of long wave scattering in a canal with a rapidly varying cross-section is presented. It is assumed that waves propagate on a stationary current with a given flow rate. Due to the fixed flow rate, the current speed is different in the different sections of the canal, upstream and downstream. The scattering coefficients (the transmission and reflection coefficients) are calculated for all possible orientations of incident wave with respect to the background current (downstream and upstream propagation) and for all possible regimes of current (subcritical, transcritical, and supercritical). It is shown that in some cases negative energy waves can appear in the process of waves scattering. The conditions are found when the overreflection and over-transmission phenomena occur. In particular, it is shown that a spontaneous wave generation can arise in a trans-critical accelerating flow, when the background current enhances due to the canal narrowing. This resembles a spontaneous wave generation on the horizon of an evaporating black hole due to the Hawking effect.