246] MINIMUM WAVE-VELOCITY. 447 



shewing that for any prescribed value of c, greater than c m , there 

 are two admissible values (reciprocals) of X/X m . For example, 

 corresponding to 



c/c m = T2 1-4 1-6 1-8 2-0 



we have 



, f 2-476 3-646 4-917 6*322 7 873 

 1 m ~{ -404 -274 -203 -158 -127, 



to which we add, for future reference, 



sin- 1 c ni /c = 5626 / 45 35 38 41 33 45 30. 



For sufficiently large values of X the first term in the formula 

 (3) for c 2 is large compared with the second ; the force governing 

 the motion of the waves being mainly that of gravity. On the 

 other hand, when X is very small, the second term preponderates, 

 and the motion is mainly governed by cohesion, as in Art. 245. As 

 an indication of the actual magnitudes here in question, we may 

 note that if X/\ m &amp;gt;10, the influence of cohesion on the wave- 

 velocity amounts only to about 5 per cent., whilst gravity becomes 

 relatively ineffective to a like degree if X/X m &amp;lt; -fa. 



It has been proposed by Lord Kelvin to distinguish by the 

 name of ripples waves whose length is less than X m . 



The relative importance of gravity and cohesion, as depending on the 

 value of X, may be traced to the form of the expression for the potential 

 energy of a deformation of the type 



The part of this energy due to the extension of the bounding surface is, per 

 unit area, 



Tr^o /X 8 .................................... (ii), 



whilst the part due to gravity is 



p ) 2 .................................... (iii). 



As X diminishes, the former becomes more and more important compared with 

 the latter. 



For a water-surface, using the same data as before, with # = 981, we find 

 from (5) and (6), 



X m = l-73, c m = 23 2, 



the units being the centimetre and the second. That is to say, roughly, the 

 minimum wave-velocity is about nine inches per second, or -45 sea-miles per 



