Non-dissipative effects of breaking on the wave field - Breaking and Dissipation of Ocean Surface Waves - page 346

A variant of this approach, the ratio of counted breaking to non-breaking waves as a
function of the steepness of the windward (rear) face of individual dominant waves, is
shown in Figure 8.6 c. This figure shows the relative probabilities of breakers, conditioned
on their rear face steepness. As one would expect, the larger the wave steepness, the more
frequently they break. The ratio reaches 54% for waves in the steepness range 0.25-0.3
(percentage of breaking waves, if defined as the ratio of the number of breakers to the total
number of waves, is 35%).
Next, a phase-average technique was used to analyse links between surface waves and
pressure oscillations induced by them in the case of breaking and no-breaking. This phase-
average technique was described and widely utilised in the wind-input study by Donelan
et al. ( 2006 ). It employed the average pressure conditionally sampled on the phase of the
surface elevation, in order to obtain a statistically robust relation between phases of the
surface and pressure waves.
Phase-averaging techniques have been used for a variety of applications (see e.g.
Hristov et al. , 1998 ). Here, this conditional-averaging method employs time series of the
phase of a reference signal (surface wave) at a particular frequency to obtain an average
profile of various flow variables sampled on the phase of the reference (e.g. pressure). For
example, if the mean profile of the wave at a particular frequency (e.g. 1 Hz) is sought
out of a non-filtered wave record, the record should be bandpass-filtered in a narrow band
around the chosen frequency and then used to obtain time series of the phases of the 1 Hz
surface elevations, by means of a Hilbert transform or wavelet analysis (see Section 3.7 for
a description of those). In other words, at each instant we now know what was the phase
of the 1 Hz surface oscillations.
The phase record can then be used to conditionally sample the original record to choose
values of surface elevations, pressure, velocity, etc., in selected phase bins. That is, for
each phase bin, for example, from 175 ◦ to 185 ◦ , whenever a value of the wave-phase
belongs to this bin in the phase time series, we notice a respective value of the pressure
at the same moment, from the pressure time series, and then add up and average these
pressure readings. The mean and standard deviation within the phase bins then yield the
conditionally-averaged pressure-flow variable on the phase of the wave component chosen.
This method provides an interesting and instructive insight into the behaviour of wave-
induced pressure fluctuations relative to the surface waves here. It is a powerful data-
analysis tool, operative at frequencies and signal-to-noise ratios well beyond the limit
where co-spectral analysis fails to find any correlation between two related signals.
Figure 8.7 shows the breaking enhancement of the wind input to the waves from a
phase-averaged perspective. In all the subplots, the upper lines are phase-average for 1132
breakers, the lower lines - for 5215 non-breaking waves, and the middle lines - for the
non-segregated 6347 waves.
A Hilbert transform was used to determine the phase of the individual dominant waves.
This required bandpass filtering the wave height signal around the spectral peak f p in the
spectral band f p ±

1 f p . We note that bandpass filtering changes the wave height and
steepness significantly, and this is important here, particularly for the breakers, which are

0 .