ments is also necessary to enable the naval architect to design ships with the best per- 

 formance in waves. 



What is the present position with regard to the measurement of waves at sea? 

 A shipborne wave recorder is available (Tucker, 1955) which enables the frequency 

 composition and the mean direction of wave travel to be measured (Cartwright, 1956), 

 but as far as the authors are aware no one has yet succeeded in measuring the complete 

 wave characteristics in frequency and direction with anything approaching the necessary 

 detail, and this remains one of the most important problems in wave studies. We shall 

 now discuss the problem in more detail, and then some possible methods of measure- 

 ment. 



THE 2-DIMENSIONAL WAVE SPECTRUM 



We have seen that the wave spectrum is continuous in direction. However, for 

 most problems, both the conception and the mathematics become considerably simpler 

 if we start by considering the sea as being made up of a large number of simple wave- 

 trains, so that the elevation £ (x,y,t) above a point in the mean sea surface with coordi- 

 nates (x,y) is 



?(x,y,t) = 2 C„ cos (w»i — k n x cos 0„ — k n y sin d„ + e„) (1) 



n 



where k„ — 27r/wavelength, ©„ = -iri n and e„ is a phase angle which is assumed to 

 be random and usually disappears when appropriate averages are taken. At some stage 

 in the problem the expressions can be written in terms of ViC n 2 , which is proportional 

 to the energy per unit area of the sea surface associated with the n th wavetrain. If 

 the energy of all the wavetrains contained in an interval 88, 8f is summed, then 



2 %C n *~E(f,d)8d8f, (2) 



i8Sn 



where E(f,8) is the energy per unit frequency range and per unit angle. The number 

 of elementary wavetrains can now be considered to be infinite, so 88 and 8f to tend to 

 zero, so that 



2 )iCj -> JE(f,d)d6df. (3) 



71 



This process was used, for example, by Rice (1944) to solve one-dimensional prob- 

 lems. E(f,0) is called the spectral density function, and is the characteristic of the 

 waves which we desire to measure. 



For a given /, E(f,6) is periodic in 8 with a period of 2tt, and may therefore 

 be represented as a Fourier series of harmonic components: 



E(f,6) = OoCf) + 2 M/) cos nd + &„(/) sin nd] (4) 



n 



where n has integral values from 1 to oo. «„(/) and b,,(f) define the n th harmonic 

 and are given by 



1 /•»' 



a n (f) = - / E(f,6) cos rum (5) 



7T ./ 

 1 [** 



&„(/) = - / E(f,d) sin nOdd (6) 



7T J 



Most methods for measuring E(f,8) do not give a direct estimate of its value 

 (smudged by the instrumental resolving power) but rather values of the individual 



50 



