  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.
