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4 Physical Mechanisms Forming the Wave Spectrum in Deep Water
where N; = N(k;) is the spectral density of the wave action; T(k, k 1 , k 2 , k3 )
is the core function of the non-linear interaction between wave components;
and J(k) and b(O") are the Dirac delta-function describing the resonance interaction conditions between four wave components:
The resonance condition is shown schematically in Fig. 4.1.
(4.2a)
(4.2b)
Hasselmann (1962) interpreted the integral ( 4.1) in the terms of a quadrupole interaction between three active wave components (defining the interaction intensity) and the fourth passive component, receiving energy without
directly affecting the interaction.
The most important property of the kinetic equation (4.1) is the preservation of the following three integral values:
total wave action:
A= J N(k) dk;
(4.3a)
total energy:
E = J O" N(k) dk
(4.3b)
y
X
Fig. 4.1. Four-wave interaction diagram (Hasselmann, 1963)