9t 9 X 



0) 



where F {o" ,<f>, x, t) is the local energy spectrum at position "x and time t, 

 Vg (o- ,<^) is the group velocity of the component with circular frequency cr and 

 relative direction (f> and the function G {cr ,<f> , ~k, t) represents all processes which 

 are adding to or subtracting energy from F. The complete G-function is not within 

 the realm of present knowledge, but it is possible to define a linear form of G that 

 will be adequate for our immediate purposes: 



G (ff,,;^, X, t)= a (cr,^, T, t)+ ^(o- ,^,X t) . F (cr,^, x", t) . (2) 



a and P correspond to wave growth mechanisms that are linear and exponential, 

 respectively, in time (space). Such mechanisms might, respectively, be explained 

 by the "resonance theory" of Phillips (1957) and the shear-flow theory of Miles (1957). 

 Reviev/s of these theories are given in the literature (e.g. Longuett-Higgins, et al., 

 1963) and will not be repeated here. Equation (1) then is the linear form of the energy 

 equation with G given by (2) and will be considered valid until non-linear and/or 

 dissipative effects take over. 



it is clear that an appropriate form of equation (1) can be used to obtain estimates 

 of a and P provided it is possible to fulfil one of three conditions during the initial 

 growth phase: 



(i) F (cr , <^, x", t) and W {yT, t) known for sufficient t and x^ 



(ii) F (o- , <^ , t) and W (Vgt, t) knov/n for sufficient t and ^ equal the 



group velocity of the cTq, ^^ component, 



(iii) F (cr , ^,~x) and W (x) known for sufficient "x" and stationary for specified t. 



The first condition essentially estimates a and P by a hindcasting technique and 

 generally involves the solution of a nonlinear integro-differentiai form of 

 equation (1). Details of such an analysis are presented by Bamett, 1966. 



The second circumstonce is identical to the case considered by Snyder (1965) and 

 summarized in Snyder & Cox (1966). In this experiment a series of wave recorders was 

 towed at constant speed downwind from the lee of Eleuthera Island in the Bahamas. As 

 observed by the moving recorders, a singularity in the spectral transformation relating 

 the true frequency and direction of the wave to its apparent frequency and direction 

 allowed an estimate of the intensity of a single spectral component. This component 

 had a group velocity aqua! to the towing velocity. Spectral growth curves were 



