SOME VIEWS ON LORD KELVIN'S WORK 433 



where k determines the wave-length, X, of the particular group 

 of Fourier wave-trains which predominate at point % at time t. 

 The ambiguous sign in the denominator is to be chosen so as 

 to make the expression positive. By the principle of stationary 

 phase, the relation between k and % is 



x -t{f(k) + kf'(k)}-tU 

 where U is the group-velocity corresponding to wave-length A,. 

 Consider now the wave energy contained in the medium, at 

 time t, from the place where wave-length X predominates to the 

 place where wave-length X + SX predominates, that is, the energy 

 corresponding to wave-length X. The extent of the medium 



concerned, at time t, is 



S x = t { 2 f'(k) + kf"(k)} 5k 



and, as the energy per unit length of the medium is proportional 



to the square of the amplitude, the total wave energy in the 



medium associated with wave-length X is as follows : 



E8X = (Amplitude) 3 x AS^ where A is a constant 

 = constant X 8k = constant x SX/X 2 



Thus we arrive at the result that the energy corresponding to 

 wave-length X, and carried along through the medium, is 

 independent of the time elapsed from the beginning of motion, 

 and of the place in the medium where the Fourier trains of 

 wave-length A, predominate, and of the dispersive quality of the 

 medium itself. This means that the energy associated with each 

 wave-length remains unchanged during its distribution through- 

 out the medium, and is therefore the same at all times as the 

 energy, belonging to the wave-length considered, in the initial 

 pulse, before its resolution and transformation by the medium into 

 energy of wave-motion. The same is of course true for any form 

 of initial pulse, and the theorem is an illustration of the fact that 

 the group-velocity U is the velocity at which a certain quantity 

 of energy, that belonging to Fourier trains of wave-length X, 

 as given by Fourier's theorem, moves through the medium — a 

 theorem proved originally by Lord Rayleigh for the case of a 

 regular group of waves (Sound, vol. i. Appendix). One case of 

 the theorem is that with any form of initial pulse the maximum 

 energy per wave-length is always associated with the same wave- 

 length, and depends only on the form of the initial pulse itself. 

 The importance of the result in connection with radiation lies in 

 the fact that radiant energy, emitted at a fixed temperature, has 

 always the same distribution of the energy among the various 



