532 



XI HYDRODYNAMICS. 



Let us now discuss the form of the wave -profile 135) when the 

 restriction that the height of the waves is small in comparison with 

 the wave-length is removed. The equation of the surface is 



This may be conveniently done by means of a graphical construction, 



Fig. 168. Let us 



construct two 

 curves, with the 

 coordi- 

 the 



runnng 

 nates X, Y, 

 first the logarith- 

 mic curve 



Fig. 168. 



X = e* Y 

 which must be 



and the second the straight line X = p . 7 



B smkx 



separately constructed for each value of x. At the intersection of the 

 line and curve , we have 



Y+Be* Y sin 7^ = 0, 



so that the value of Y thus obtained may be taken for the y coordinate 



of the wave -profile with the abscissa x. As x varies, the line swings 



back and forth about the X-axis, and we see that when sin&# is 



positive there is one intersection of the line and curve, while if sin ~kx 



is negative there are two, giving two values of y, both positive. 



Any positive y is greater in absolute value than the corresponding 



< / negative for the 



\ / / symmetrical posi- 



\ I \ / tion of the line. 



/ / Thus the unsym- 



\ / metrical nature of 



"'V_..X trough and crest 



is made evident. 

 Beginning with 



_^> -\^^ ^ ^\^^ x = 0, the two 



' f ^^ ^^"-~~^_ _- ^^ ^~ values of y are 



Fi g- 169. one zero, the other 



infinity, and as x increases, y has a single negative value. When x=-^- = > 

 y is again zero and infinity, and as x increases the two values of y, both 



Q 



positive approach each other until y = A, then recede until y = L 



The form of the curve as constructed in this manner is shown in 

 Fig. 169, the lower branch representing the wave -profile. If B is 

 greater than a certain quantity the values of y between certain limits 



