DEPTH-LIMITED SIGNIFICANT WAVE HEIGHT: A SPECTRAL APPROACH 



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 C. Linwood Vinaent 



I . INTRODUCTION 



Research into the shape of wind wave spectra in finite-depth water has 

 suggested an expression for the upper limit on the energy density as a func- 

 tion of depth and frequency (Kitaigorodskii, Krasitskii, and Zaslavskii, 1975). 

 In this report this expression is integrated over the part of the spectrum 

 expected to contain energy to estimate a limit on the energy, E, in the wind 

 wave spectrimi and to define a depth-limited significant wave height, Ho:^ 



H^ = 4.0(E)l/2 (1) 



More precisely, the quantity estimated is the variance of the sea surface to 

 which E is directly related. Following convention, E and E(f) denote 

 energy and energy density spectrum although the true units of computation are 

 length squared and length squared per hertz. The term zero-moment wave height, 

 I^o will be used to denote 4.0(E)^'2. H1/3 is the average height of the one- 

 third highest waves. H£ denotes values of Hj^^^ that are depth limited. In 

 deep water, H^q ^^ approximately ^i/-i> but this is not necessarily true in 

 shallow depths. Hj refers to the depth-limited monochromatic wave. The vari- 

 ation of H£ with depth, h, is investigated and compared with the mono- 

 chromatically derived depth-limited wave height, H^j. Because Hj^j^ and H1/3 

 are about equal in deep water, they are both frequently called significant 

 wave height. 



This report briefly reviews the theoretical development of the limiting 

 form for spectral densities as a function of water depth and presents field 

 evidence supporting this form. The simple derivation of the depth-limited 

 energy and significant wave. height is then given, followed by field and labora- 

 tory data evaluating the prediction equation. Unless otherwise noted, the 

 developments of this report are restricted to wave conditions described by a 

 wave spectrum of some width such as an active wind sea or a decaying sea. 



II, THEORETICAL BACKGROUND 



Phillips (1958) suggested that there should be a region of the spectrum 

 of wind-generated gravity waves in which the energy is limited by wave steep- 

 ness. Phillips derived an expression for the limiting density in deep water: 



Em(f) = ag2f-5(2TT)-'* (2) 



where a was considered to be a universal constant. Field studies reviewed 

 by Plant (1980) demonstrated that equation (2) adequately describes the part 

 of the wind sea spectrum above the peak frequency of the spectrum. However, 

 Hasselmann, et al. (1973) indicated that the equilibrium coefficient a is 

 not constant but varies systematically with wave growth leading the authors 

 to speculate that resonant interactions in the spectrum force the spectrum to 

 evolve to the form of equation (2). Toba (1973) suggested that the equilibrium 

 range form might be proportional to U^f"** in order to remove the variation 

 of a. 



