446 SURFACE WAVES. [CHAP. IX 



positive direction of y upwards, the pressure at the disturbed 

 surface will be given by 



-= j* -gy= 



p at , . 



approximately. Substituting in Art. 245 (5), we find 



' = di ~ gy = "" \k + a cos ' sn 



-gk+ .................... (2). 



p+p 



Putting o- = kc, we find, for the velocity of a train of progressive 

 waves, 



Tl 



/ 7 I t "' 



/ 7 I 

 P + P P + P 



where we have written 



p'/p=s, TJ(p-p-)-T' ................ (4). 



In the particular cases of T = and g = 0, respectively, we fall 

 back on the results of Arts. 223, 245. 



There are several points to be noticed with respect to the 

 formula (3). In the first place, although, as the wave-length 

 (2-7T/A;) diminishes from oo to 0, the speed (cr) continually increases, 

 the wave-velocity, after falling to a certain minimum, begins to 

 increase again. This minimum value (c m , say) is given by 



and corresponds to a wave-length 



\ m = 2^ m = 2^(2"/^ .................. (6)*. 



In terms of \^ and c m the formula (3) may be written 



* The theory of the minimum wave- velocity, together with most of the substance 

 of Arts. 245, 246, was given by Sir W. Thomson, " Hydrokinetic Solutions and 

 Observations," Phil Mag., Nov. 1871; see also Nature, t. v., p. 1 (1871). 



