The above is equivalent to observing the waves at a fixed point as a 

 function of time, and aJl knowledge of the direction of travel of the waves is 

 lost since ^Tir 



[A(p., e)]^de = [A(fjL)]2 



The procedures for analyzing waves as a function of time at a fixed point 



have been described by Pierson and Marks [1952], and Ijima [1956] has 



carried out quite a number of such analyses in Japan with very interesting 



results. The same techniques are being used by Lewis [1955] to analyze 



the spectra of model waves and ship motions in a towing tank. The wave 



pole records will be analyzed using the methods described by Pierson and 



Marks [1952]. 



If t' is chosen to be zero, then an average over space can replace 



an average over time and space and the result is 



X Y 



/Try 



(8.9) Q(x',y') = lim ^^ J j t,(x, y) ti(x + x', y + y') dxdy 



X-*oo _ X „ Y 



Y-»oo ^ 2 

 oo GO 



i / / [A*(c, p)]2cos (ex' + (3y')dpda 



~ 2 



-oo oo 



In equation (8.9) the same right hand side results if -x' and -y' are substi- 

 tuted for x' and y', and therefore Q(x', y') = Q(-x',-y'). 



The above is equivalent to observing the waves at an instant of time over 

 an area. Some knowledge of the direction of travel of the waves is lost. Con- 

 sider, for example, a progressive simple sine wave observed at an instant 



62 



