118 Shallow Water Wave Transformation through External Factors 



and from the integration of the second equation of motion we have 



Q 2(h + ri) dt* 



(V.8) 



If we substitute this value for p in the first equation of motion (V. 5) and if we integrate 



this relation over the whole water column from until h + i], and neglecting small quantities, we 



obtain, 



cU 8U dti h d s ri 



— +U—+g-^ - = 0. (V.9) 



8t 8x 8x 3 8x8t 2 



We can substitute for the time t the velocity of propagation c of the wave disturbance and if 

 we give the disturbance the general form of rj =f(x—ct) and U = F(x—ct), eliminating the first 

 and second differential, we obtain 



8U 8U 8 2 rj 8h) 



— = — c — and — = c a — , 

 8t 8x 8t 2 8x 2 



and equation (V.9) gives us 



U s c 2 h 8 2 n , 



cU grj =0. 



8x\ 2 3 8x 2 



(V.10) 



Since there were 7] = 0, also U = the value between parentheses is always equal to 0, so that 

 in 'general we have 



U 2 c 2 h 8 2 n 



cU - -gr\- — = 0. 



2 3 8x 2 



(V. 11) 



We can eliminate U through the equation of continuity (V. 7) and equation (V. 10) and then 

 have U = [cl(h+7])]t] and from equation (V. 11) we obtain in first approximation 



c 2 = gh 



3 n h 2 8 2 rj 



1+- 7 + 



2 // 3r] 8x 2 



(V.12) 



This is the formula for the velocity of propagation of a wave disturbance ii according to Bous- 

 sinesq; this velocity is dependent on the wave form /; and also on the curvature of the water surface. 

 It is possible to get a wave profile i] of the permanent type, and in which the influences of ti and 

 d 2 ri/8x 2 cancel each other; the terms can then be replaced by a constant a. The velocity of propaga- 

 tion of the disturbance would then be c — \?[gh(\ -\-a)]. 

 This condition would give a differential equation 



8h) 9>f 3a// 



2/; 3 



Ir 



= . 



Multiplying this by 8t]\8x and integrating, and assuming that the bottom of the wave is a horizontal 

 straight line (for r\ = 0, 8rij8x = 0) we obtain 



(V.13) 



r) = a for the crest of the wave and here also 8t]j8x must be equal to 0. This condition is satisfied 

 if according to equation (V.13) r\ = ah. For the velocity of propagation of the wave we obtain 



c = V[g(h+a)]. (V.14) 



This is the equation Scott Russell derived from his experiments. Except for r\ = and for the value 



