CAPILLARY WAVES. 



445 



245-246] 



By compounding two systems of standing waves, as in Art. 218, 

 we obtain a progressive wave-system 



?; = a cos (kso + crt) (7), 



travelling with the velocity 



cr 



c= k = 



or, in terms of the wave-length, 



.(9). 



The contrast with Art. 218 is noteworthy; as the wave-length 

 is diminished, the period diminishes in a more rapid ratio, so that 

 the wave-velocity increases. 



Since c varies as X~*, the group-velocity, Art. 221 (2), is in the 

 present case 



c-xj = |o (10). 



The fact that the group-velocity for capillary waves exceeds the 

 wave- velocity helps to explain some interesting phenomena to be 

 referred to later (Art. 249). 



For numerical illustration we may take the case of a free 

 water-surface; thus, putting /&amp;gt; = !, /o = 0, 2^ = 74, we have the 

 following results, the units being the centimetre and second *. 



246. When gravity is to be taken into account, the common 

 surface, in equilibrium, will of course be horizontal. Taking the 



* Cf. Sir W. Thomson, Math, and Phys. Papers, t. iii., p. 520. 



The above theory gives the explanation of the crispations observed on the 

 surface of water contained in a finger-bowl set into vibration by stroking the rim 

 with a wetted finger. It is to be observed, however, that the frequency of the 

 capillary waves in this experiment is double that of the vibrations of the bowl ; see 

 Lord Kayleigh, &quot; On Maintained Vibrations,&quot; Phil. Mag., April, 1883. 



