448 SURFACE WAVES. [CHAP. IX 



hour, with a wave-length of two-thirds of an inch. Combined with the 

 numerical results already obtained, this gives, 



for c= 27'8 32-5 37'1 41 -8 46 '4 



in centimetres and seconds. 



If we substitute from (7) in the general formula (Art. 221 (2)) 

 for the group-velocity, we find 



. dc 



Hence the group- velocity is greater or less than the wave- velocity, 

 according as X > X m . For sufficiently long waves the group- velocity 



is practically equal to |c, whilst for very short waves it tends to 

 the value fc*. 



A further consequence of (2) is to be noted. We have hitherto tacitly 

 supposed that the lower fluid is the denser (i.e. p>p'), as is indeed necessary 

 for stability when 7\ is neglected. The formula referred to shews, however, 

 that there is stability even when p<p', provided 



/ T \i 

 A<2, / ^ ^ 



i.e. provided X be less than the wave-length X m of minimum velocity when the 

 denser fluid is below. Hence in the case of water above and air below the 

 maximum wave-length consistent with stability is 1'73 cm. If the fluids 

 be included between two parallel vertical walls, this imposes a superior limit 

 to the admissible wave-length, and we learn that there is stability (in the two- 

 dimensional problem) provided the interval between the walls does not exceed 

 86 cm. "We have here an explanation, in principle, of a familiar experiment in 

 which water is retained by atmospheric pressure in an inverted tumbler, or 

 other vessel, whose mouth is covered by a gauze with sufficiently fine meshes f. 



247. We next consider the case of waves on a horizontal 

 surface forming the common boundary of two parallel currents 

 U, U'. 



* Cf. Lord Rayleigh, L c. ante p. 383. 



t The case where the fluids are contained in a cylindrical tube has been solved 

 by Maxwell, Encyc. Britann., Art. " Capillary Action," t. v., p. 69, Scientific Papers, 

 t. ii., p. 585, and compared with some experiments of Duprez. The agreement is 

 better than might have been expected when we consider that the special condition 

 to be satisfied at the line of contact of the surface with the wall of the tube has 

 been left out of account. 



