Ogilvie 



Such a representation yields the same energy spectrum for all values of x and 

 all functions e(co). It is supposed that any particular stationary long-crested 

 sea can be represented by such a formula and conversely that any realization of 

 this formula (through an arbitrary choice of e(w)) can occur. It should be noted 

 that the relationship between wave number and frequency is just that which ob- 

 tains for small -amplitude deep-water waves. 



The "integral" in Eq. (1) has needlessly caused much controversy and con- 

 fusion. St. Denis and Pierson carefully defined it as the limit of a sequence of 

 partial sums, in a manner common in noise theory, in the theory of gust-loading 

 on airplanes, etc. The conventional integral sign is always symbolic, denoting a 

 limiting operation on a sequence of partial sums. In the present situation, the 

 operation is not the usual Riemann integration, but it is quite properly defined 

 provided that one is certain of the existence of the limit (or of the convergence) 

 of the defining sequence. Proof of this point is a problem in the calculus of 

 probability and will not be discussed here. 



In the theory of random noise, there is another standard representation of 

 the time history of a random variable with a given spectrum. Instead of using 

 a cosine function with random phase (as in (1)), one uses a sum of sine and 

 cosine functions of (Kx - cot), with random amplitudes which are uncorrelated 

 with each other. This stochastic model was applied to sea waves by Cote (1954). 

 It is entirely equivalent to the random phase model, the choice between the two 

 depending primarily on the relative convenience of deriving various probability 

 properties of the sea. 



From these stochastic models, one can derive all kinds of interesting con- 

 clusions about the sea, some of which will be true. But our interest is primarily 

 with the ship. St. Denis and Pierson suggested that, looking at (1) as a sum of 

 many sinusoidal waves, we should determine the response of the ship to each 

 component, and then the response of the ship to the actual sea would be just the 

 sum of the responses to the component waves. The process of finding the re- 

 sponse to a regular sinusoidal wave is a completely deterministic process, of 

 course, but in summing (or integrating) these responses we carry the stochastic 

 nature of the seaway over to the ship motions. In particular, if we use the ran- 

 dom phase model for the sea, the ship response should be expressible by an 

 integral like that in (1). 



The remainder of this section will be devoted to an investigation of the evi- 

 dence for accepting this supposition of St. Denis and Pierson, i.e., that the ship 

 response to a random sea is just the sum of its responses to the various fre- 

 quency components. The accumulation of such evidence during the last few 

 years is striking, and, prejudging the case somewhat, I believe that the chapter 

 which was opened by St. Denis and Pierson in 1953 is now almost concluded. 



The most straightforward approach to verifying the superposition principle 

 for ship responses is to conduct model tests in different sea conditions. In each 

 test the wave height and motions spectra are measured, and, from these, the 

 amplitude of the frequency response (f.r.) functions of the ship can be calculated. 

 If different conditions yield the same f.r. functions, then the ship can be de- 

 scribed as responding "separately" to each frequency component, the total 



10 



