420 



SURFACE WAVES. 



[CHAP, ix 



necessarily one of elevation only, and denoting by a the maximum height above 

 the undisturbed level, we have 



c?=ff(h + a) .................................... (ix), 



which is exactly the empirical formula for the wave-velocity adopted by 

 Russell. 



The extreme form of the wave will, as in Art. 231, have a sharp crest of 

 120 ; and since the fluid is there at rest we shall have c 2 =2ga. If the 

 formula (ix) were applicable to such an extreme case, it would follow that 

 a=h. 



If we put, for shortness, 



y-h = ^ h*(/i + a)/3a = b* ........................ (x), 



we find, from (viii) 



the integral of which is 



rj = a sech 2 #/2& ......... ........................ (xii), 



if the origin of x be taken beneath the summit. 



There is no definite ' length ' of the wave, but we may note, as a rough in- 

 dication of its extent, that the elevation has one-tenth of its maximum value 

 when #/6=3'636. 



y 



0-3- 

 0-2 - 

 0-1 



O 05 



The annexed drawing of the curve 



1-0 



1-5 



represents the wave-profile in the case a \h. For lower waves the scale of y 

 must be contracted, and that of x enlarged, as indicated by the annexed 

 table giving the ratio 6/A, which determines the horizontal scale, for various 

 values of a/h. 



It will be found, on reviewing the above investigation, that 

 the approximations consist in neglecting the fourth power of 

 the ratio (A-f a)/26. 



If we impress on the fluid a velocity c parallel to x we 

 get the case of a progressive wave on still water. It is not 

 difficult to shew that, when the ratio a/h is small, the path 

 of each particle is an arc of a parabola having its axis vertical 

 and apex upwards*. 



It might appear, at first sight, that the above theory is 

 inconsistent with the results of Art. 183, where it was shewn 

 that a wave whose length is great compared with the depth 

 * Boussinesq, I. c. 



