Shallow Water Wave Transformation through External Factors 117 



velocity component. Scott Russell's solitary wave may be regarded as an ex- 

 treme case of Stokes's oscillatory wave of permanent type, the wave length 

 being great compared with the depth of the canal. The theory showed that 

 the wave profile is not a trochoid as assumed by Scott Russell, but given 



by the formula 



y 

 n = osech 2 ^- , (V.4. 7) 



lb 



in which b is a quantity depending on the wave height A and the depth h. 

 Figure 54 shows the wave profile of the solitary wave and the theoretical wave 



o 

 Fig. 54. Profile of the solitary wave. 



line according to equation (V.4). With increasing wave amplitudes the 

 agreement is not so good now. Mc Cowan (1891) found sharper pointed 

 crests. The waves are no longer permanent, and the limiting value of the 

 waves a/h was found to be 0-68 h, in which case the velocity is given by 

 c 2 = \-56 gh (see also Michell, 1893; and Gywgther, 1900). 



The theory of the solitary wave starts as usual with the equations of motion 



du 1 dp dw 1 dp 



— = , — = g (V.5) 



dt q dx dr o 8z 



and the equation of continuity 



du dw 



- + - = 0. (V.6) 



dx dz 



The origin of the co-ordinates is at the bottom of the ocean and i] is the elevation of the water 

 above the undisturbed level h. If U is the average horizontal velocity component in the whole water 

 column of a height h+rj, then we can give the equation of continuity the form 



dn 8U 



- + (h+r l ) 7 - = 0, (V.7) 



dt dx 



in which 



h + rj 



(h+t])U= j udz. 







We can now integrate the second equation of motion from any level where there is a pressure p 

 to the surface z = h+i], where the pressure p = 0. From the equation (V.6) we can obtain by 

 integration from z = 0, where w = 0, to z 



