Observations and Measurements of Ocean Waves 63 



as well as height and length of the swells which were observed. This table 

 contains only observations for winds from to 5 Beaufort, in order to be 

 reasonably sure that only swell and no waves produced by strong winds 

 were observed. The proof that generally this was the right procedure is shown 

 in the bottom lines of the table, where the average period of waves is given 

 for each wind speed, and there is no correlation between the two quantities. 

 The characteristics of sea motion (state of the sea) with different wind 

 speeds, which are based on the statistical wave values, can be derived from 

 the most important spectral features of wind generated ocean waves. Such 

 characteristics can, of course, only be given for fully arisen seas, because 

 the ocean wave pattern at the time of generation depends not only upon 

 the wind speed r but also upon the wind duration t and the fetch F. To a fully 

 arisen sea belong minimum values F m and t m , given in the right of Table \0a. 

 This table contains the most important wave values for fully arisen seas; 

 it is, besides other tables, based upon the energy spectrum for all states of 

 generation for given values of F and / (Pierson, Neumann and James 

 1953) of utmost practical importance for the prediction of sea-motion 

 characteristics. 



9. The Mathematical Formulation of the Actual Ocean Wave Pattern 



The state of the sea surface at a fixed point and at a certain time can, 

 without doubt, be described by the interference of a very large number of 

 harmonic waves of relatively small amplitudes, which progress in different 

 directions. They have different frequencies and become superposed with 

 random phases. It has been attempted to give a general representation of 

 the sea motion by means of a step-by-step approximation. The investigations 

 by Pierson (1952) are of importance in this connection, which he suggested 

 from procedures in theoretical statistics and in part carried out himself. 

 Pierson presents the motion of the sea surface at a fixed point (zero point) 

 by the following integral as a pure function of time 



oo 



r)(t) = jcos[at^d(a)]} {[h(a)fda} . (111.21) 



o 



This unusual integral cannot be solved by common methods; it represents 

 a mathematical abstraction which can be approximated with desirable ac- 

 curacy by a partial sum. h 2 (a) is the energy spectrum according to (III.4), 

 6(a) is of the character of a random phase; all values between and 2n are 

 of equal probability and are independent of one another. To each a there 

 belongs a certain da, where 



W[a < b{a) < 2na\ = a , (01.22) 



that is, a is the probability of d(a) falling within the range 0-2jra. In order 



