TRANSFOI^IATION OF MONOCHROMATIC WAVES FROM DEEP 

 TO SHALLOW WATER 



hy 

 Bernard Le Mehaute and John D. Wang 



I. INTRODUCTION 



An understanding of many nearshore phenomena relies on the ability to 

 predict the local wave climatology, given a deepwater wave description. 

 For example, a quantitative description of longshore sediment transport is 

 based on a knowledge of the wave characteristics in the surf zone. This 

 report presents methods for determining the changes in the characteristics 

 of a wave traveling over a variable bottom from deep water to shallow 

 water. 



The acute sensitivity of the rate of littoral transport to wave 

 breaking characteristics implies an accurate determination of these 

 characteristics. The problem has numerous facets: 



(a) Given a deepwater unidirectional monochromatic wave, what 

 are the breaking wave angle, depth of breaking, breaking wave height, 

 and related quantities? 



(b) -Given a multidirectional deepwater incident wave spectrum, 

 what is the distribution of breaking wave characteristics and the 

 "equivalent" monochromatic wave used to determine the littoral drift? 



(c) How should a synoptic wave climatology be treated in order 

 to determine the rate of littoral drift and related quantities? 



Only the first problem is addressed in this report. The relevant 

 literature is reviewed, and a new hybrid wave theory is proposed to 

 determine wave breaking characteristics on a sloped plane beach. 



II. NONLINEAR WAVE TRANSFORMATION 



1. Nonlinear Wave Shoaling- 



It is generally assumed that the wave motion over a gentle slope 

 is the same as that on a horizontal bottom, and that there is no re- 

 flection nor wave profile deformation. The wave motion is then 

 determined so that the rate of transmission of energy or energy flux 

 is constant over varying depth. 



The average energy flux through a vertical plane of unit width 

 perpendicular to the wave propagation is 



/■t+T ,n 

 Pav = T J J ^^^^1^1 V^^ "dzdt (1) 



