446 SURFACE WAVES. [CHAP. IX 



positive direction of y upwards, the pressure at the disturbed 

 surface will be given by 



, e), 



P , , 



-, = -^ - gy = I ?=r + g J a cos kx . sin (at + e) 



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



at^^gk+W .................... (2). 



P+&amp;gt; /&amp;gt; + ? 



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

 waves, 



where we have written 



P ) = T ................ (4). 



In the particular cases of T l = 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 

 (2ir/&) diminishes from oo to 0, the speed (a-) continually increases, 

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

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



(5), 



and corresponds to a wave-length 



X m = 2^ m = 27r(r/ S r)i .................. (6)*. 



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



&amp;gt; 



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

 of Arts. 245, 246, was given by Sir W. Thomson, &quot; Hydrokinetic Solutions and 

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



