RECENT PROGRESS IN HYDRODYNAMICS. 87 



and showed that if V is the velocity of propagation of a wave-form of 

 length ^TTJli, the velocity of the group is d (kY)ldk. In the 'Proceedings 

 of the Mathematical Society,'^ Rayleigh extends Reynolds's theory to other 

 types of waves, and attempts to show how his expressions for the velocity 

 may result from the theory of the latter. If the velocity of the wave 

 vary as the ?Ath power of its wave-length, the velocity of group-propa- 

 gation is (1 — n) times the velocity of the wave (V) ; thus, for deep-water 

 waves it is ^V ; for aerial waves, V (beats travel with the same velocity 

 as the notes) ; and for waves due to capillary action on the surface 

 it is f V. 



The theory of these latter waves has been given by Thomson. ^ The 

 velocity of propagation Avith air of density a, above water whose super- 

 ficial tension is T is v/(<7,\/2t + 27rT,//\) where pi = g(l - ff)/! + o-) 

 and T, = T/(l + a). So long as the wave-length is less than 27r V(TJgi) 

 the velocity of propagation increases as the wave-length diminishes, and 

 the capillary tension has most effect in producing the motion, while if the 

 wave-length is greater, the velocity of propagation increases with the 

 wave-length, and gravity has most effect. Thomson proposes to confine 

 the name ripples to fluid waves whose wave-length is less than this 

 critical value. The velocity of propagation is then a minimum and 

 ^/(2^/ C^TiTi) ). For water, this gives a velocity of 23 centimeters per 

 second for a wave-length of 17 centimeters. This result agrees well with 

 some experiments made by him on sea water. When wind acts on the 

 surface, the results are modified, the wave-velocity with a wind-velocity 

 equal to V is equal to ffV/(l + rr) ± {w^ — (7V2(1+ a) "^jJ where w is 

 the velocitj^, without wind. The discussion of this formula leads to 

 interesting results ; for instance, that the surface of still water is unstable 

 if the velocity of the wind exceeds 



^^'^/^-^)) 



This accounts for the fact that a small breath of Avind sweeping over the 

 surface of still water dims the surface only while it lasts, for the capillary 

 waves die away at once through the viscosity of the water. The values of 

 the velocity of propagation of capillary waves on water without air, have 

 been found by Kohicek,^ apparently without knowledge of Thomson's 

 work. 



Some very elegant illustrations of Huyghen's principle as applied to 

 liquid waves have been given by Hirst.* He has developed the differen- 

 tial equations for the line of ripple due to any centres, or lines of 

 disturbance, which give small waves, as, for instance, the ripples just 

 mentioned, on the surface of fluid moving in any manner ; and has 

 applied them particularly to the case where the centre of disturbance 

 describes a circle uniformly on still water. The results are compared 

 with experiment, with full agreement between theory and practice. 



The theories of straight waves in a vertical square cylinder, and of 



• ' On progressive waves,' Proc. Land. Math. Soc, ix. p. *21. He states that the 

 theory is originally due to Stokes. 



= ' Hydrokinetic Solutions,' parts 3, 4, 5, PMl. Mag. (4) slii. p. 368. 

 ' ' Ueber den Einfluss des capillaren Oberflachen'druckes auf die Fortpflanzungs- 

 geschwindigkeit von Wasserwellen,' Wied. Ann. v. p. 42.5. 



* ' On Eipples and their relation to the velocities of currents,' PMl. Mag. (4), xxi. 

 p, 1 and 188. 



