Spectrograms of ship wakes: identifying linear and nonlinear wave signals

TL;DR

This paper investigates the features of spectrograms of ship wakes, combining simulations and analysis to distinguish linear and nonlinear wave signals, and explaining discrepancies observed in high-speed ferry data.

Contribution

It introduces a combined linear and nonlinear analysis approach to interpret complex spectrogram features of ship wakes, including effects of ship acceleration.

Findings

Linear dispersion relation explains primary spectrogram components.

Nonlinear effects account for additional observed features.

Ship acceleration influences spectrogram characteristics.

Abstract

A spectrogram is a useful way of using short-time discrete Fourier transforms to visualise surface height measurements taken of ship wakes in real world conditions. For a steadily moving ship that leaves behind small-amplitude waves, the spectrogram is known to have two clear linear components, a sliding-frequency mode caused by the divergent waves and a constant-frequency mode for the transverse waves. However, recent observations of high speed ferry data have identified additional components of the spectrograms that are not yet explained. We use computer simulations of linear and nonlinear ship wave patterns and apply time-frequency analysis to generate spectrograms for an idealised ship. We clarify the role of the linear dispersion relation and ship speed on the two linear components. We use a simple weakly nonlinear theory to identify higher order effects in a spectrogram and, while the high speed ferry data is very noisy, we propose that certain additional features in the experimental data are caused by nonlinearity. Finally, we provide a possible explanation for a further discrepancy between the high speed ferry spectrograms and linear theory by accounting for ship acceleration.