Mass 



9t 9x 9x 



Motion 



9h , 9u . 9h , 1 ,? 93u 



9t 9x dx 3 „ 2'^4. 



Korteweg and de Vries (1895)'*^ equations (KdV) . These are of less generality 

 and will be omitted in what follows. However, exact solution of the KdV 

 equations are termed first-order cnoidal waves. The solitary wave is a 

 limiting case of a cnoidal wave. Thus these classical wave theories which 

 are found to duplicate experimentally determined surface profiles and velo- 

 city distributions of real waves in shallow water have their roots in 

 Boussinesq theory, or vice versa. 



The basic Boussinesq equations in one dimension on a horizontal bottom 

 can be written (neglecting surface or bed stresses) in Eulerian form 



(133) 



(134) 



where h = n+d with d the Stillwater depth and n the instantaneous water sur- 

 face variation. The velocity u is a depth-averaged instantaneous value. 

 The equations are identical to the long wave equations except for the 

 mixed derivative Boussinesq term on the right-hand side of equation (134) . 

 It is instructive to briefly review various theories which omit certain 

 terms in equation ,(134) . 



a. Waves of Permanent and Nonpermanent Form . If the convective accel- 

 eration and Boussinesq terms are neglected, a wave of permanent form moving 

 with speed c = (gd)'2 results. Without only the Boussinesq term, each part 

 of the solution travels at a speed c = u±(gh)'2 so that the high parts tend 

 to overtake the low sections with time. The wave travels with nonpermanent 

 form, continually steepens, and eventually breaks. This tendency is termed 

 amplitude dispersion. If, however, only the convective acceleration term is 

 omitted, a linearized version of the Boussinesq equations results which in- 

 cludes vertical acceleration of the wave orbital motion. The celerity is of 

 the form 



c = (gd)^/(l - 4^2d2/3L2)^ (135) 



where L is the wavelength of any wave component in the solution. Thus each 

 wave component travels at its own speed depending on its length. This ten- 

 dency is termed frequency (or wavelength) dispersion. The full equations 

 with all terms maintain a balance between amplitude and frequency dispersion 

 only for the limiting case of the solitary wave. They also simulate a pro- 

 gressive wave of permanent form (first order, cnoidal) without excessive 

 dispersion as long as the wave does not propagate indefinitely (Peregrine, 

 1972). 



'^''korteweg, D.J., and de VRIES, G., "On the Change of Form of Long Waves 

 Advancing in a Rectangular Canal and on a New Type of Long Stationary 

 Waves," Phil. Mag., 5th Series, Vol. 39, 1895, pp. 422-443 (not in 

 bibliography) . 



146 



