Hasselmann 



in solids and now developed to a standard scattering formalism in various fields 

 of physics. The theory has recently also found a number of geophysical applica- 

 tions (8). 



The energy transfer arises from interactions between combinations of 

 Fourier components whose wave numbers kj, , k^^ and frequencies oj^, . . . , w^ 

 satisfy the transfer conditions 



and 



L 



(6) 



where s^ = ± l . Equation (5) follows from the homogeneity of the physical sys- 

 tem and applies to all interactions. An energy transfer between the interacting 

 components occurs only if the additional resonance condition of Eq. (6) is also 

 satisfied. 



The net energy transfer is found by summing the contributions from all 

 combinations of resonant interactions. The final expression consists of a num- 

 ber of integrals containing various spectral products, which can conveniently be 

 divided into two classes. Integrals in which the transfer conditions of Eqs. (5) 

 and (6) occur as S factors are associated with scattering processes, the remain- 

 ing integrals are associated with parametric processes. 



To distinguish between the various transfer terms, it is further convenient 

 to introduce a notation based on transfer diagrams. The transfer diagram for a 

 scattering process consists of a number of wavenumber vectors k j , . . . , k^_ j 

 entering a vertex and a single wave component k^^ leaving the vertex. The com- 

 ponents satisfy the transfer conditions of Eqs. (5) and (6). Components associ- 

 ated with a negative sign s. = -l are indicated by a cross stroke. 



The transfer diagram for a parametric process consists only of ingoing 

 components. There are no side conditions on the wavenumber s or frequencies. 

 Parametric processes occur only in interacting systems in which the total 

 energy and momentum of the wave fields are not conserved. They have no 

 counterpart in the theory of wave-wave interactions, but there is a close analogy 

 with the interactions occurring in nonlinear parametric amplifiers. 



The structure of the various transfer expressions can be deduced from the 

 transfer diagrams with the aid of a single transfer rule: the rate of change of 

 the energy spectrum of any wave component in a transfer diagram is proportional 

 to the product of the spectral densities of the ingoing components. Thus for any 

 interacting system, the set of all transfer expressions for a particular wave field 

 w is obtained by applying the transfer rule to all wave components w in all possi- 

 ble transfer diagrams (Fig. 1). 



588 



