[ 864 j 



XCII. Remarks concerning Fourier's Theorem as applied to 

 Physical Problems. By Lord Rayleigh, O.M., F.R.S.* 



FOURIER'S theorem is o£ great importance in mathe- 

 matical physics, but difficulties sometimes arise in 

 practical applications which seem to have their origin in the 

 aim at too great a precision. For example, in a series of 

 observations extending over time we may be interested in 

 what occurs during seconds or years, but we are not con- 

 cerned with and have no materials for a remote antiquity or a 

 distant future; and yet these remote times determine whether 

 or not a period precisely defined shall be present. On the 

 other hand, there may be no clearly marked limits of time 

 indicated by the circumstances of the case, such as would 

 suggest the other form of Fourier's theorem where every- 

 thing is ultimately periodic. Neither of the usual forms of 

 the theorem is exactly suitable. Some method of taking off 

 the edge, as it were, appears to be called for. 



The considerations which follow, arising out of a physical 

 problem, have cleared up my own ideas, and they may 

 perhaps be useful to other physicists. 



A train of waves of length A,, represented by 



yj r = e 2ni(ct+^ ...... (1) 



advances with velocity c in the negative direction. If the 

 medium is absolutely uniform, it is propagated without dis- 

 turbance ; but if the medium is subject to small variations, 

 a reflexion in general ensues as the waves pass any place x. 

 Such reflexion reacts upon the original waves ; but if we 

 suppose the variations of the medium to be extremely small, 

 w r e may neglect the reaction and calculate the aggregate 

 reflexion as if the primary waves were undisturbed. The 

 partial reflexion which takes place at x is represented by 



tfy = e 27ri ( ct -^(j)(a:)dx ,e^ ix ll\ . . . (2) 



in which the first factor expresses total reflexion supposed to 

 originate at x = 0, $(x)dx expresses the actual reflecting 

 power at x, and the last factor gives the alteration of phase 

 incurred in traversing the distance 2x. The aggregate re- 

 flexion follows on integration wdth respect to x ; with 

 omission of the first factor it may be taken to be 



C + iS, (3) 



* Communicated by the Author. 



