Shallow Water Wave Transformation through External Factors 119 



derived from equation (V. 13), the wave profile has no horizontal tangent, the wave disturbance 

 consists of a single crest. The wave profile relative to the middle of the crest has the form 



*("V?) 



,-asechM— -1/ — I. (V.15) 



For this wave profile (see Fig. 54) we have no definite "wave length", but we can establish, 

 an approximate wave extent by assuming the limit there where the elevation r\ is one-tenth of 

 its maximum value. This will occur for xx\'3a\h 3 = 3 - 636; if a = 2 m and h = 12 m, the length 

 of the wave disturbance will be 130 m. Boussinesq has shown that if the ratio ajh is small, the path 

 of each particle is an arc of a parabola having its axis vertical and its apex upwards, when the soli- 

 tary wave passes by. The wave disturbance is not periodic but there is a permanent displacement 

 of the water particles, which is associated with mass transport. Lamb (1932), describes Lord 

 Rayleigh's derivation of the solitary wave. The result is slightly different, as we have to substitute 

 in equation (V.15) the root \/[h 2 (h+a)] for/? 3 . The Rayleigh equation gives, therefore, for larger 

 values of the ratio a\h flatter slopes, but the difference is not very considerable. 



According to Lamb, the cnoidal waves of Korteweg and de Vries can be derived from the above- 

 mentioned formulas. The equation (V. 13) has the form 



and the solitary wave requires that the constant C is equal to zero. However, in a general way C is 

 different from 0, and the right side of this equation (V. 16) must vanish, both for the wave crest 

 and for the wave trough. This will give us a cubic equation of the form rf—afaf— ih 3 C = which 

 must have three real roots. As a > 0, this requires, as the discriminant shows, that c < — 4a 3 /9. 

 As C is negative, the two roots rjx = a and rj 2 = b must be positive and the third one r\ 3 = — abl(a + b) 

 negative. We then obtain easily 



a 2 +ab+b 2 3a 2 b 2 



ah = and C = 



a+b h 3 (a+b) 



and the equation (V. 16) takes the form 



a/A 2 3 / ab \ 



_) = _ (a _,„,_ 4) |, + _j. (V.17) 



If we put >/ = rtCos' 2 x+£ s ui 2 X> we obtain from the integration the equation 



IKx 



r] = b+(a-b)cn 2 , (V.18) 



A 



l -b 2 

 mod k = 



\ a 2 



+ 2ab 



At the same time we have 



V-: 



, h 3 (a+b) , 



/ 2 ^, ., and c = V[gh{l + a)]. 



If we know the values of three out of the four quantities, h, a, b, and A then we have determined 

 mathematically the value of the fourth one. Equation (V.18) represents the cnoidal waves (equa- 

 tion V.l). 



