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DISCUSSION 



M. S. Longuet-Higgins 



The paper presented by Dr. Munk has rightly stressed the importance of meas- 

 uring the whole two-dimensional spectrum of the sea surface. I would like to describe 

 to you more fully the method he has mentioned that we are developing at the National 

 Institute of Oceanography, and to point out some of its advantages. 



The principle of the method is as follows. Suppose the surface elevation is 

 represented in the form 



t(x,y,t) = 2 C n cos {u n x + v n y + a n t + e„), (8) 



n 



where (x, v)are horizontal co-ordinates, and t is the time. The wave-numbers (u n , v n ) 

 are assumed to be distributed densely throughout the (u, v) -plane, and the frequency 

 a n is a function only of the wavelength; 



<x n = er(u»H- !>»»)* (9) 



as in free gravity waves. The phases e n are randomly distributed between and 2-n- 

 and the amplitudes C n are such that over any small interval of wave-number du dv 



2 y 2 Cn 2 = E(u,v)dudv, (10) 



dudv 



where E(u, v) represents the spectrum of the surface. If, by some means, we take the 

 intersection of the surface with a vertical plane x sin 9 — y cos 9 then the resulting 

 curve will have an autocorrelation function ip(R) given by 



*(R) = ?(x,yM(x + X,V + Y,t) = j / E{u,v) cos (uX + vY)dudv, (11) 



— 00 — 00 



where 



X = R cos 9, Y = R sin 9 (12) 



Assuming this function to be obtained for all directions 9 and hence all values of X 

 and Y, we may, by taking the cosine transform, obtain the function 



y 2 [E(u,v) +E(-u, -»)], (13) 



which is the even part of the energy spectrum. 



Suppose also that we are able to measure d£/dt at all points along the plane 

 section. Then from the relation 



— (x,y,t)£(x + X,y + Y,t) = - I I a{u,v)E(u,v) sin (uX + vY)dudv (14) 

 dt 



56 



