Chapter 4 Wave-Wave Interactions in Irregular Waves

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Chapter 4 Wave-Wave Interactions in Irregular
Waves
Irregular waves are approximately simulated by
many periodic wave trains (free or linear waves)
of different frequencies, amplitudes, and
advancing in different directions. Due to
nonlinear nature of surface water waves which is
described by free-surface B.Cs., free waves
interact among themselves. Free waves their
wavenumber and frequency obey the dispersion
relation. Bound Waves their wavenumber and
frequency do not satisfy the dispersion relation.
Bound waves result from wave-wave interactions
among free waves.
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4.1 Weak and Strong Wave-Wave Interaction
Physical phenomena resulting from strong
interactions are observable soon after free waves
start to interact. Those of weak interactions
become substantial only after hundreds of wave
periods (Su and Green 1981 Phillips 1979).
Weak interactions, also known as resonance wave
interactions, may occur when the frequencies and
wavelengths of interacting free waves satisfy the
corresponding resonance conditions. Strong
Interactions occur once interacting free waves
exist in the same area.
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Weak Interactions result in energy transfer among
free waves of different frequencies or
wavenumbers (Phillips 1960, Hasselmann 1962).
This mechanism is crucial to wave energy transfer
among free waves of different wave frequencies or
wavenumbers in the context of air-sea
interactions (Komen et al. 1994). Strong
interactions are observable immediately after the
interactions (among free waves) start. They
disappear except phase shifts after the
interacting free waves longer overlap (Yuen
Lake 1982). Strong interactions do not result in
long-lasting effects as the weak interactions
have on energy transfer among free waves.
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4.2 Magnitude of Wave-Wave Interactions
Definitions of wave steepness of an irregular
wave train
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Definition of magnitude in terms of wave
steepness A wave-wave interaction is in
general described or dictated by one or several
nonlinear forcing terms (in the free-surface
boundary conditions) involving the multiplication
of the amplitudes of free waves,
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• Based on this definition,
• an interaction of second order involves two free
  waves
• an interaction of third order involves three free
  waves in the forcing term
• 3) In general, an interaction of order involves
  N free waves.
• These N free waves are not necessary to be N
  different free waves. For example, they can be
  only one free wave in the multiplication by
  multiplying itself N-1 times.

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Interactions of fourth or higher orders They
are weaker and become significant only in very
steep Waves. Also involve both strong and weak
interactions.

• Types of Resonant Interactions
• Type (I) Interaction (or instability), can occur
  at 3rd order,
• Quartet wave interaction, predominantly
  2-D.
• 2) Type (II) Interaction, can occur at 4th
  order,
• Quintet wave interaction, predominantly 3-D

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4.3 Impact of Strong Interactions on Irregular
Waves
In a linear spectral method, nonlinear wave
interactions are ignored in the decomposition of
a measured wave field as well as in the
calculation of resultant wave properties. In
short, bound waves are treated as free waves of
the same frequency. When ocean waves are not
steep, the free waves are dominant in almost the
entire frequency range and a linear spectral
method may be a simple and fairly good
approximation. When ocean waves are steep, the
free waves near the spectral peak frequency still
remain dominant but the bound waves may become
dominant or comparable to the free waves in the
frequency ranges either much lower or higher than
the peak frequency.
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4.4 Various Perturbation Methods
Conventional Perturbation Methods Mode Coupling
Method (MCM), Stokes Expansion Zakharov
Equation Method (ZEM) Common Features Linear
Phases for free bound waves.
Potentials are
constructed based on a separation
variable method. Phase Modulation Methods
(PMM) Nonlinear Phases and Potential are not
constructed based on a SVM.
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• Differences Between MCM ZEM
• Expansion of the free-surface boundary
  conditions
• New variables in ZEM
• Continuous and discrete wavenumbers.
• The results obtained respectively using MCM and
  ZEM are virtually the same (Zhang Chen 1999).