where p = uh and q = vh are the volumetric flow rates per unit width. Numeri- 

 cal integration methods, testing, and some application examples of these 

 equations are reviewed below. 



Recently, Hauguel (1980) obtained somewhat different results by assuming 

 that the vertical velocity increases linearly through the water column and 

 averaging the Navier-Stokes equations over this depth. It should be noted 

 here that such an assumption for the vertical velocity, w is equivalent to 

 assuming a uniform horizontal velocity profile based on continuity principles. 

 If the bed stress term is neglected, the x-direction motion equation is 



^ + A (£^) + J_ (M) + JL r (.Stk + a. 2] 

 3t 3x ^h -* 9y ^h '^ 3x ^^ 2 ^ a^*" J 



= (g + b + f )h II (147) 



where p, q, and h are as previously defined, z is the bed elevation above an 

 arbitrary datum, and the new terms a and b characterize the vertical accel- 

 erations due to wave steepness and bed-slope variations. They are defined 

 as 



a = ^ (148) 



dt^ 



b = ^ (149) 



dt2 



where the total or substantive derivative is given by 



A = J_ + P.J_ + a_9_ (150) 



dt 9t h 3x h 9y 



The derivation is said to follow that given by Serre (1953)^^ for cnoidal 

 waves. A similar expression is given for the y-direction momentum. This 

 approach may permit steeper waves to propagate farther over irregular bathy- 

 metry. 



e. Limitations of Boussinesq Theory . The range of application of 

 equations (144), (145), and (146), based on a comparison with first-order 

 cnoidal-wave theory and Dean's stream-function wave theory (Dean, 1974) , 



^^SERRE, F., "Contribution a letude des ecoulements permanents et variables 

 dans les canaux," La Houille Blanahey 1953, pp. 374-388, 830-872 (not in 

 bibliography) . 



^^DEM, R.G., "Evaluation and Development of Water Wave Theories for 



Engineering Application, "SR-1. Vols. I and II, U.S. Army, Corps of Engineers, 

 Coastal Engineering Research Center, Ibrt Belvoir, "Wa. , 1974 (not 

 in bibliography). 



150 



