These equations form the basis for all subsequent applications of radiation 

 stress principles that has been termed the first-order theory. The key as- 

 sumptions are linear wave theory and a time-average over one wave period. If 

 a steady state prevails, the averaging time can be over many wave periods. 



2. Other Waveforms . 



The principle stress equations (7) and (9) can be employed with other 

 waveforms and expressions for t], u and w. For example, for standing waves in 

 water of uniform depth and again using linear wave theory, Longuet-Higgins 

 and Stewart (1964) obtained 



8^^= E[2n(l + cos 2kx) - 1] (27) 



for the principle stress components. S is again identically zero. Although 

 linear theory has been used extensively and forms the basic theory, various 

 nonlinear wave theories and irregular waves with known spectral characteristics 

 have also been studied. These special theories are discussed in Section VI of 

 this chapter. The linear theory is reviewed in Sections III, IV, and V. 



III. MEAN WATER LEVEL CHANGES 



The radiation stress components are directly expressed in terms of the 

 wave parameters: wave height H, wavelength L, crest angle a, and the still- 

 water depth d. Waves approaching a sloping coastline or near structures will 

 undergo modifications in these parameters resulting from shoaling, refraction, 

 diffraction, reflection, and breaking processes. Spacial changes (gradients) 

 in the radiation stress components must result. Under steady-state influence 

 of the incident wave field and the modified wave field, a new time-averaged 

 equilibrium will be established for the time-averaged water level and time- 

 averaged currents present. For this steady-state situation, the equilibrium 

 equations are the momentum balance equations perpendicular to and parallel to 

 the shoreline and the continuity (mass conservation) equation. All forces and 

 stresses in these equations are mean values over the wave period T. Forces 

 are depth-integrated values per unit horizontal width. 



1. Normal Wave Incidence . 



When waves approach the coast at right angles with crests parallel to the 

 coastline, the principles of radiation stress theory are readily demonstrated 

 in terms of MWL changes. In this case for a =0, the x-direction stress S 

 reduces back to principle stress S „ and no shear-stress component exists. 



a. Momentum Balance . Consider a plain sloping beach with bottom contours 

 parallel to the wave crests as shown in Figure 21. For steady-state time-mean 

 conditions, the rate of change of the radiation stress must create a rate of 



72 



