and that 



~ = T- 



I ( -1)" (2a)" T^ 



CD 



I 

 n:o 



CO 



n:0 



2n -t 



2n +5 



n ^ 2n 



y ( -1)" (2a) " T^ 



2n +3 J 



>z 



(34) 



The first few terms of this expansion are 



T=T;[ 



5 7 '^L 9 '^c ^ 



3 5 1- 7 1. 



(35) 



and under the condition that B. be small, a reasonable approximation 



1 

 is that 



T = T; -73/5 = 0.77 T 



(36) 



as given by Pierson, Neumann, and James (1955). 



2. Interpretation of the Average "Period" 



The average "period" as observed by stop watch, or as computed 

 from a wave record, can be an extremely misleading statistic. It 

 overemphasizes the short "periods" and neglects the long "period". 

 The maximum energy in the spectrum is always at a higher value than 

 is indicated by the average "period". 



The significant "period", that is, the average period of the one-third 

 highest waves, may equal the average "period" or it may be a trifle 

 higher because of the neglect of shorter ^'periods" in the average. 

 However, it is even more doubtful a statistic because its relation 

 to the wave energy spectrum is not known. 



For either the average "period" or the significant "period", the 

 computation of the average wave crest "speed" or the average "wave 

 length" cannot be carried out by the use of the classical formulas as 

 will be shown later. The classical formulas apply only to the true period 

 of a simple harmonic progressive wave. 



The average "period" can be used to determine the state of de- 

 velopment of the sea for a given wind velocity. It can be used to check 

 a given forecast of the wave spectrum if only a sea is present. However, 

 spectra of many different shapes can yield the same average "period"; 

 and the average "period" and the significant height do not completely 

 characterize a given state of the sea. 



35 



