The growth of wind-waves in finite depth

TL;DR

This paper extends Miles' theory to finite depth, deriving a wave growth rate that accounts for water depth, revealing how wave growth varies with wave age and depth, and aligning with empirical observations.

Contribution

The paper introduces a finite depth extension of Miles' theory, deriving a depth-dependent wave growth rate and characterizing wave growth limits in shallow and deep waters.

Findings

Wave growth rates are similar to deep water for small wave age.

Wave growth diminishes to zero at large wave age in finite depth.

Explicit calculations of limiting wavelength and phase speed in shallow and deep water.

Abstract

In order to study the growth of wind waves in finite depth we extend Miles' theory to the finite depth domain. A depth-dependent wave growth rate is derived from the dispersion relation of the wind/water interface. A suitable dimensionless finite depth wave age parameter allows us to plot a family of wave growth curves, each family member characterized by the water depth. Two major results are that for small wave age, the wave growth rates are comparable to those of deep water and for large wave age, a finite-depth wave-age-limited growth is reached, with wave growth rates going to zero. The corresponding limiting wave length and limiting phase speed are explicitely calculated in the shallow and in the deep water cases. A qualitative agreement with well-known empirical results is established and shows the robust consistency of the linear theoretical approach.