221] GROUP-VELOCITY. 383 



of a finite depth h the average energy transmitted per unit 

 time is 



which is, by (4), the same as 



Hence the rate of transmission of energy is equal to the group- 

 velocity, d(kc)/dk ) found independently by the former line of 

 argument. 



These results have a bearing on such questions as the ' wave- 

 resistance' of ships. It appears from Art. 227, below, in the 

 two-dimensional form of the problem, that a local disturbance 

 of pressure advancing with velocity c [< (gh)^] over still water of 

 depth h is followed by a simple-harmonic train of waves of the 

 length (%7r/k) appropriate to the velocity c, and determined there- 

 fore by (3); whilst the water in front of the disturbance is sensibly 

 at rest. If we imagine a fixed vertical plane to be drawn in the rear 

 of the disturbance, the space in front of this plane gains, per unit 

 time, the additional wave-energy \gpa?c, where a is the amplitude 

 of the waves generated. The energy transmitted across the plane 

 is given by (8). The difference represents the work done by the 

 disturbing force. Hence if R denote the horizontal resistance 

 experienced by the disturbing body, we have 



R = 



As c increases from zero to (gh)*, kh diminishes from 05 to 0, and 

 therefore R diminishes from J#/oa 2 to ()(. 



When c > (gJift, the water is unaffected beyond a certain small 

 distance on either side, and the wave-resistance R is then zeroj. 



* Lord Kayleigh, " On Progressive Waves," Proc. Lond. Math. Soc., t. ix., p. 21 

 (1877); Theory of Sound, t. i., Appendix. 



t It must be remarked, however, that the amplitude a due to a disturbance of 

 given character will also vary with c. 



I Cf. Sir W. Thomson " On Ship Waves," Proc. Inst. Mech. Eng., Aug. 3, 1887; 

 Popular Lectures and Addresses, London, 1889-94, t. iii., p. 450. A formula equi- 

 valent to (10) was given in a paper by the same author, " On Stationary Waves in 

 Flowing Water," Phil. Mag., Nov. 1886. 



