Computation of the Motion of Long Water Waves 



assumed hydrostatic pressure distribution beneath the wave, the 

 assumption not being valid for finite amplitude waves. Camfield and 

 Street [ 1969] used a refined correction term, but the results were 

 not entirely satisfactory in either case. 



Peregrine [ 1967] and Madsen and Mei [ 1969a] derived approxi- 

 mate, nonlinear, governing equations for the propagation of long waves 

 over slowly varying bottom topography. In these equations the verti- 

 cal component of motion is integrated out of the computation so a 

 single-space-dimension problem is all that remains. Madsen and 

 Mei [ 1969a] showed that, while they and Peregrine used different 

 approaches and solution methods, the equations are the same when 

 presented in the same variables. Furthermore, Madsen and Mei 

 [ 1969a] explained that these nonlinear equations, obtained under 

 assumptions similar to those leading to cnoidal waves in the case of 

 horizontal bottoms , give a uniformly valid description of long wave 

 problems as long as breaking does not occur. In particular, their 

 equations were derived under the condition that the Ursell parameter 



2 



U* = ^=0(1) (1) 



where r\Q is a measure of wave amplitude, Lg is a characteristic 

 wave length and do is the water depth. Thus, the nonlinear, govern- 

 ing equations are of the Korteweg-deVries type (KdV) that have per- 

 manent solutions, e.g. , cnoidal waves, for the case of horizontal 

 bottoms. Madsen and Mei [ 1969a] demonstrated that, although the 

 equations pertinent to each of the three groups of long waves (Airy 

 where U^ » 1 , KdV where U^ = 0(1) and Linear where U^ « 1) 

 have different mathematical solutions, the features characterizing 

 each group are all contained in the equations derived under the 

 assumption of waves of the KdV type. 



The method of characteristics was applied by Madsen and Mei 

 [1969a, 1969b] to solve their equations for initial, boundary value 

 problems involving solitary waves and periodic waves on plane slopes 

 and a shelf; their equations, like those of Peregrine [ 1967] , are 

 applicable to general , uneven bottoms . Street, et al. [1969] pre- 

 sented a numerical model APPSIM and results based on the Peregrine 

 [ 1967] method, but employing initial and boundary conditions similar 

 to those of Madsen and Mei [ 1969b] . These methods reproduced the 

 nonlinear breakdown on a shelf of a solitary wave, the breakdown 

 having previously been observed only in experiraents [Street, et al . , 

 1968]. Furthermore, comparison shows quantitative agreement 

 amongst these methods and relevant experiments. Run-up cannot 

 be calculated with these models which employ the vertical beach (or 

 island) face that was used by Vastano and Reid [ 1967] . However, 

 Peregrine's [1967] derivation included the two horizontal space 

 dimensions, while Madsen and Mei [1969a] did not. Accordingly, 

 as an extension of APPSIM, a quasi-three-dimensional model 



149 



