where g is the acceleration of gravity, and h is the mean water depth. The dispersion 

 relation defines the phase speed of each component, allowing each separate component 

 to move independently, without interaction with one another. Any Eulerian current 

 is not included in this relation, and is generally ignored in spectral methods. 



The discrete variance spectrum, jB, can be computed from a water surface or sub- 

 surface pressure record through the use of variations of the Fast Fourier Transform 

 (Dean and Dalrymple 1991; Newland 1993). A point measurement provides no in- 

 formation about the direction of propagation of the waves, 9m- It is usually assumed 

 that all the waves are propagating in the same direction. Once the amplitude, fre- 

 quency, wave number and phase of each individual wave are defined, the kinematics 

 and dynamics of the wave field can be computed by superimposing the kinematics of 

 each individual wave, as predicted by linear wave theory. 



Directional Spectra When measurements are taken by an array of instruments, 

 the directional nature of the sea can be described by a directional spectrum, ^(w, 9). 

 In this case, the water surface is represented by a large number of linear waves of 

 different frequencies and directions: 



M N 

 rj{x,y,t) = ^ ^am,n cos {km{x cos 6n + ysm9n) -UJmt + 0!m,n) (1-4) 



7n=l n=l 



where: 



am.n = ^2S{Um,0n)AuAe (1.5) 



and Alj and A9 are the sample spacing of the discrete spectrum in frequency and 

 direction space. tUm and km are once again related by the linear dispersion relation 

 (Eq. 1.3). The directional spectrum is usually broken down into two parts, the one 

 dimensional variance spectrum, E{lu), and the directional spreading function (DSF), 



D{u:,ey. 



S{u,9) = E{u)D{lu,9) (1.6) 



where E{uj) is the variance spectrum defined above, and 



D{u,9)d9duj = l (1.7) 



J — oo J —n 



