2. Modern Approaches . 



Since 1969, two fundamentally different yet related theoretical ap- 

 proaches to predict coastal hydrodynamics have emerged. Both rest on solid 

 physical laws of mass (continuity) and momentum (motion) conservation that 

 form the basis of Newtonian fluid mechanics. Most importantly, both assume 

 the velocity profile to be uniform over the water depth for all points in the 

 flow. No flow direction reversals with depth can be resolved (Fig. 9,b) so 

 that circulations about horizontal axes are impossible. 



a. Radiation Stress Theory . The equation of continuity and two horizon- 

 tal momentum equations are depth-integrated and time-averaged to account for 

 the excess lateral momentum thrust present in wind waves on the coast. Local 

 net velocity components and the average MWL are the dependent variables of 

 interest. The theory is now more than 12 years old and has undergone consid- 

 erable development and refinement. It requires a priori specification of wave 

 height fields throughout the area of interest. This is usually accomplished 

 by normal wave refraction procedures (Snell's Law), combined with diffraction, 

 reflection, and wave-current interaction estimates, as required. Most of this 

 chapter (Sees. II to VI) is devoted to a thorough review of the extensive 

 literature that has been published since 1969 on this subject. A key aspect 

 is the time-averaging process inherent in the theory so that this could be 

 labeled a time-average theory. For simple geometries, analytical (closed form) 

 solutions are possible. 



b. Boussinesq Theory . Based on ideas that go back much further 

 (Boussinesq, 1872) •^^, the vertical acceleration and streamline curvature effects 

 in wind waves give rise to lateral momentum fluxes. They appear as additional, 

 mixed and higher derivative terms in the horizontal, long wave momentum equations 

 that are depth-averaged but not time averaged. Local instantaneous velocity 

 components and the instantaneous water surface fluctuations are the dependent 

 variables of interest. The first engineering applications to coastal swell 



wave propagation in two dimensions appeared in 1978, and only outside the 

 breaker zone. Considerable research and development work remains to use the 

 method for calculation of coastal currents, circulations, and rip currents. No 

 additional wave field specifications are required since the waves propagate in 

 space and time as a fundamental part of the solution. The limited amount of 

 published information on this method is reviewed in Section VII. Numerical 

 integration methods are required to obtain solutions. 



For further insight into the physics and mathematics surrounding waves in 

 the coastal zone, Lundgren (1976) is recommended. He refers to the Boussinesq 

 approach as T-Methods (time-step methods) . 



19 BOUSSINESQ, M.J. , "Theory of Waves and Surges Which Propagate the Length 

 of a Horizontal Rectangular Canal, Imparting to the Fluid Contained Within 

 the Canal Velocities That Are Sensibly the Same from the Top to the Bottom," 

 Jduvnat of Pure and Applied Mathematics ^ Vol. 17 (2nd Series), Feb. 1872. 

 For English translation, see VASTAITO, A.C.J. , and lOTSGALL, J.C.H., Refer- 

 ence 76-2-T, Dept. of Oceanography, Texas A&M University, College Station, 

 Tex., Mar. 1976 (not in bibliography). 



65 



