Consider the observation of the heights of all waves that pass a fixed 

 point. Such an observation coiild be made instrumentally by a wave- pole 

 recorder, or it could be made just as easily by an observer if he 

 knew that this was the correct procedure. Some of the theoretical 

 properties of such a series of observations are known, and therefore 

 their accuracy can be determined. The theory which is to be given is 

 therefore based on the observation of the heights of all waves which 

 pass a fixed point of observation. Once the distribution of the heights 

 of all Avaves is known, it then becomes possible to omit the observation 

 of some waves in a precisely defined way and still obtain reliable 

 results. 



The theory caji be extended to cover the properties of the heights 

 of all waves that pass (or are passed) by a moving point. Thus a point 

 fixed in azimuth and distajice relative to a nnoving ship can be used 

 just as well as a stationary point. 



The heights of the waves that pass a fixed point are lower than the 

 heights of the highest part of each short- crested wave since the side 

 of a short- crested wave can pass the point and the highest part can 

 pass at a distance from the point of observation. Since these heights 

 are the same as would be encountered by a ship under way, they are 

 also of practical importance. 



B. The Theory of A Wave Record 



An ocean wave record is a sample from a quasi- stationary Gaussian 

 process which is completely described by its energy spectrum. Much is 

 known about the theory of such Gaussian processes since they have been 

 studied extensively in electronics and in communication theory by Rice 

 (1944), Wiener (1949). and Tukey (1949). 



1. The Envelope 



There are a nunnber of ways to define the envelope of a wave 

 record. For one way that is used, it can be shown that the envelope will 

 touch each crest of the wave record only if the wave spectrum is 

 narrow, and that the envelope is always distributed according to equation 

 (1) as discussed below. For another way that is used, the envelope is 

 defined to touch each horizontal part of the record, but then the 

 probability distribution of the envelope reflects ripples and other minor 

 (for this application) irregularities and only reduces to equation (1) for 

 narrow spectra,* 



♦Personal communication, R. A. Wooding; see also "Wind Generated 

 Gravity Waves" by W. J. Pier son, Jr. (1954). 



