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.
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.
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.
4.2 Magnitude of Wave-Wave Interactions
Definitions of wave steepness of an irregular wave train:
1
Nominal Wave Steepness: H1/ 3 k p .
2
Steepness of individual free waves ai ki 1 (i 1, 2,..N ).
General definition of wave steepness ai k j where
i may or may not equal to j. ai k j may 1, but may
close to or greater then unity.
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,
N
a ,
n 1
n
where an , n 1- N , can be different from
each other or the same.
The magnitude of an interaction is
of order N in wave steepness.
Based on this definition,
1) an interaction of second order involves two free waves;
2) 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.
Barring the situation where interactions involve
ai k j O(1) or >1, the magnitude of effects of
an interaction on wave characteristics is usually
much weaker when N is greater.
The most significant interaction is of second order in wave steepness.
It involves two interacting free waves.
k k 2 k1, (4.2.1a)
2 1. (4.2.1b)
+ Sum - frequency Interaction.
Difference - frequency Interaction.
In general, and k do not satisfy the dispersion relation
in deep or intermediate-depth water. Hence, the second-order
interactions only generated bound waves.
The interaction of third order in wave steepness
involves three interacting free waves.
k r ,i k 3 k 2 k 1 , i 1 4 (4.2.2a)
r ,i 3 2 1 . i 1 4, (4.2.2b)
In deep or intermediate-depth water, the combination of
the plus and minus signs indicates there possibly are four waves.
Among the four possible resulted waves, three (with the sequence
of arithmetic operations of , , and ) are bound waves.
However, the frequency and wavenumber of the remaining one
( ) given below may possibly satisfy the dispersion relation.
k 4 k 3 k 2 k1, (4.2.3a)
4 3 2 1. (4.2.3b)
1) If 4 and k4 do not satisfy the dispersion relation,
the remaining resulted wave is a bound wave.
2) If 4 and k4 satisfy the dispersion relation, it is a free wave.
This free wave and the original three interacting free waves
resonantly interact.
This resonant interaction may result in transfer energy among
them and is known as quartet resonant interaction. Thus, it
belongs to weak or resonant interaction. Equations (4.2.3a) &
(4.2.3b) are known as the resonance condition for the quartet
wave interaction.
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 :
1) 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
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.
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.
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).