E'(f,5) = — E(f,^) 

 Pg 



(4) 



The expression on the right is more formally known as the variance per unit 

 crestlength of the water surface displacement. Thus, Equation 4 is sometimes 

 called the sea surface variance. It is still proportional to the energy, as 

 shown. 



44. The energy represented by Equation 4 does not expressly depend on 

 frequency, direction, or phase, just amplitude. Frequency and direction are 

 retained (as arguments) on the right side of the equation for two reasons. 

 First, when more than one wave train is present, energy is said to be dis- 

 tributed in the frequency-direction domain. More general expressions of 

 energy will have frequency and direction appear on the left side of the 

 equation. Second, for a numerical or physical model to simulate a sea, the 

 amplitude can be found (if the energy is known) from Equation 4, but frequency 

 and direction must be known for Equation 1 to be complete. 



45. Phase 4> does not appear in the energy definition because energy 

 is an average over one wave cycle. Hence, phase information is not usually 

 represented in energy spectral representations. In applied modeling of linear 

 wave fields, phases are typically assigned random values from the range to 

 360 deg (or to 2n radians) . 



46. A real sea, in terms of linear theory, is the sum of many wave 

 components, all having a form like the right side of Equation 1 but with 

 different amplitudes, frequencies, and directions (and phases). It is 

 convenient to consider frequency and direction in discrete terms. For either 

 frequency or direction, one can divide the full range of values into a number 

 of increments (or bands) of finite size and assign any property within a given 

 band to the frequency or direction at the center of the band. This dis- 

 cretization is convenient for bookkeeping (i.e., computer storage or data 

 plotting) and is a consequence of conventional harmonic analysis which forms 

 consistent patterns in the way data are communicated. Since the increments 

 can be made arbitrarily small, this leads to no theoretical loss of 



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