Importance of the continuous spectrum in the excitation of sheared surface gravity waves

TL;DR

This paper investigates how the continuous spectrum influences the excitation of long surface gravity waves in sheared flows, revealing its significant contribution to surface response especially at moderate Froude numbers.

Contribution

It demonstrates the importance of the continuous spectrum in surface wave excitation and provides an analytical solution linking it to dispersion relation methods.

Findings

Continuous spectrum can dominate surface response at Froude numbers around unity.

Analytical similarity solution for surface displacement evolution is derived.

Continuous spectrum contribution exceeds normal mode amplitude in certain conditions.

Abstract

The initial value problem is solved for the excitation of long surface gravity waves in a continuously sheared flow. This reveals the presence of a continuous spectrum along side the standard normal modes of gravity wave propagation. An analytical similarity solution for the evolution of the free surface displacement from the continuous spectrum is found for the impulse response to surface excitation. It is demonstrated that the continuous spectrum contribution can be a significant fraction of the surface response, with the amplitude of the continuous spectrum exceeding that of the upstream gravity wave mode for Froude numbers of order unity. The Landau damped mode description of the continuous spectrum is found to provide a link between methods using dispersion relations for phase speeds within the range of the velocity profile, and the variable-shear profiles that do not admit normal modes in this range.