II. PRINCIPLES OF RADIATION STRESS 



In a series of papers beginning in 1960, Longuet-Higgins and Stewart 

 (1962, 1964) laid down the principles and gave the physical meaning behind 

 the concepts of radiation stress for water waves. About the same time, 

 Lundgren (1962, 1963)^° described how water waves can exert an extra wave 

 thrust in the horizontal direction. The term radiation stress is borrowed 

 from EM wave theory where a radiation pressure impinges on a surface. It is 

 actually a misnomer in that depth integration gives a wave- induced excess 

 pressure force per unit length (in excess of the hydrostatic pressure force) 

 and not a true stress (force per unit area). However, the term stress implies 

 a directional quantity which is true for this wave-induced thrust and radiation 

 stress is now an accepted term in the literature. Transformations applicable 

 to true stresses can be used. As will be shown later in this chapter, these 

 radiation stresses resulting from time-averaging gravity wave orbital motions 

 are also completely analogous to Reynolds stresses resulting from time-averaging 

 turbulent flow motions . 



The radiation stress principle has been used to develop theoretical, ana- 

 lytic expressions for the following coastal phenomena: 



(a) Wave setdown and setup (Sec. Ill) 



(b) Uniform longshore current profiles (Sec. IV) 



(c) Nonuniform longshore current profiles (Sec. IV) 



(d) Nearshore circulation systems and rip currents (Sec. V) 



In addition, it is the basis for numerical integration schemes over variable 

 bathymetry and boundaries (jetties, breakwaters) where analytic solutions are 

 not available (Sec. V). Finally, principles of radiation stress theory have 

 been extended to nonlinear and irregular waves as discussed in Section VI. 



1. Progressive Waves in Uniform Water Depths . 



a. Radiation Stress Components . The summary that follows is essentially 

 that from the work of Longuet-Higgins and Stewart (1964). A monochromatic wave 

 is shown propagating in the X-direction in Figure 19. A relatively standard 

 set of nomenclature is adopted and defined at the beginning of this report. 



(1) Normal Stress . The component S^ of the radiation stress along 

 a wave ray is defined as the time-average value of the total flux of horizontal 

 momentxjm across a vertical plane minus the Stillwater hydrostatic pressure 

 force. Thus 



^XX 



n 



(p+pu^)dz 







p dz (1) 



-d -d 



^''See some corrections in Danish Technical University, ISVA, Index to Reports, 

 Rept. No. 20, Dec. 1969, Lyngby, Denmark (not in bibliography). 



66 



