yigh") 



23 CM/SEC- 



VUh) 



Figure ?. Wave velocity c versus wavelength X for waves of small amplitude on water. (Note that 

 the relative position of the points 1.7 cm. and 2h on the wavelength scale is variable; so is that 

 of the points 23 cm. /sec. and V(g/i) on the wave-velocity scale.) 



is then closer to c, and indeed one observes that the fading-out of wavecrests at the 

 front of a group of waves on a pond becomes less pronounced when the group moves 

 into the shallower water at the edge. 



4. Capillary waves 



For very small wavelengths surface tension becomes comparable with gravity 

 as a force tending to restore the horizontal condition of the surface, and then g must 

 be replaced (if J = surface tension, p = density) by 



T 4tt2 



g + 



p A* 



(5) 



throughout. Then (Fig. 1) c has a minimum of 23 cm/sec. at A = 1.7 cm. and tends 

 to infinity as A ~* 0. The group velocity U exceeds c for A < .1.7 cm. I should 

 remark, however, that for water of very small depth h (about half a centimetre) the 

 rise in c due to surface tension and the fall due to finite X/h occur for the same value 

 of A and largely cancel out, and there is then very little frequency dispersion.* This 



* Expanding c = 



*■=- 



T 2v 



+ 



X 



tanh 



IttIi 



X 



in powers of h/\ gives 



c = V (gh) + O (/z7\ 4 ) provided h = V 3T/ P g = 0.5 cm. for water. To illustrate the con- 

 flicting tendencies better, the curve is shown for a slightly larger value of the depth (0.55 cm.) 

 in Fig. 1. 



19 



