276 TIDAL WAVES. [CHAP. VIII 



This integral measures the volume, per unit breadth of the canal, 

 of the portion of the wave which has up to the instant in question 

 passed the particle. Finally, when the wave has passed away, the 

 particle is left at rest in advance of its original position at a 

 distance equal to the total volume of the elevated water, divided 

 by the sectional area of the canal. 



169. We can now examine under what circumstances the 

 solution expressed by (9) will be consistent with the assumptions 

 made provisionally in Art. 166. 



The restriction to infinitely small motions, made in equation 

 (3), consisted in neglecting udujdx in comparison with du/dt. In 

 a progressive wave we have du/dt = cdujdx ; so that u must be 

 small compared with c, and therefore, by (15), 77 small compared 

 with h. 



Again, the exact equation of vertical motion, viz. 



Dv dp 

 ?Di = -te-W 



gives, on integration with respect to y, 



+i 2)v 



This may be replaced by the approximate equation (1), pro 

 vided /3 (h + 77) be small compared with grj, where {3 denotes 

 the maximum vertical acceleration. Now in a progressive wave, 

 if X denote the distance between two consecutive nodes (i.e. points 

 at which the wave-profile meets the undisturbed level), the time 

 which the corresponding portion of the wave takes to pass a 

 particle is X/c, and therefore the vertical velocity will be of the 

 order TJC/X*, and the vertical acceleration of the order ?;c 2 /X 2 , where 

 77 is the maximum elevation (or depression). Hence the neglect 

 of the vertical acceleration is justified, provided A 2 /X 2 is a small 

 quantity. 



Waves whose slope is gradual, and whose length X is large 

 compared with the depth h of the fluid, are called long waves. 



* Hence, comparing with (15), we see that the ratio of the maximum vertical to 

 the maximum horizontal velocity is of the order h/\. 



