652 Dr. G-. Green on a Fluid 



suppose that all these different qualities will be concerned 

 in the propagation of waves. We are accordingly led to 

 look for an analogue to the aather in a material medium, 

 in which the propagation of waves is governed by two 

 apparently dissimilar motive influences. This is of course 

 the case with water in which the waves travel under the 

 influence of gravity and of surface tension combined : and 

 we shall commence by showing that in water, under certain 

 circumstances, all waves of a particular type are propagated 

 at a very nearly uniform velocitj r . 



The well known expression for the velocity of a wave in 

 water of any depth, h, is 



U = \f \~ + Tm)tanh mh, 



(1) 



where 27r/m represents the wave-length under consideration, 

 and T the value of the surface tension. This expression 

 indicates that there is a minimum velocity of wave-pro- 

 pagation corresponding to a certain w 7 ave-length for which 

 the value of m is determined by the root of the equation — 



sinh 2mh 



2mh W 



The wave-length which corresponds to the minimum velocity 

 of w r ave-propagation we shall refer to as the critical wave- 

 length. For very deep water, the minimum velocity is 

 associated with a wave-length for which m 2 =g/'T. As the 

 depth is diminished the minimum velocity is associated 

 with a greater and greater critical wave-length. Ultimately 

 the critical 'wave-length itself is greater than the depth 

 when the water is very shallow. It is to be noticed, 

 however, that when the depth is diminished below a certain 

 value, not far different from half a centimetre, equation (2) 

 has no positive real root and there is then no minimum 

 wave-velocity. But in those cases where the depth is 

 moderate, and a minimum velocity of wave-propagation 

 obtains, the value of the minimum velocit}^ is given by 



sinh mh 



n , = ,.,„, ,_ v .... (3) 



mh 



where m is again determined by equation (2). It is easy 

 to verify from (3) that, when h is small, the speed of 

 all waves whose length exceeds the critical wave-length 

 is practically constant and equal to the limiting speed, \fgli, 

 of a long wave in water of depth h. We have therefore 



