Interference Phenomena. 521 



(fig. 2) at a time t iy and lasting during an interval of time t, 

 the displacement produced by the velocity being c = vt. 



The value of f(at—<ys) will vanish in this case unless 

 at—ys = at 1 , which condition determines some point along s 

 at which the disturbance is sensible, provided that t lies 

 between t 1 —yl/a and t^ + ylja. The integral (1) therefore 

 becomes 



hac cos /3 r u , N . -, hac cos & ra vr ,. , x . , -, 

 — - cos [ q a(t-t l)M = __ cos [M SH JW. 



The equation applies while t lies between the stated limits ; 



for smaller or greater values of t the disturbance vanishes. 



Taking account of the omitted factor, the disturbance is seen 



lie cos /3 

 to begin with an impulsive displacement, ^, followed 



by a simple vibration continuing for as many periods as there 

 are lines on the grating, ending with a permanent displace- 

 ment c. While the disturbance lasts it is of the same nature 

 as a homogeneous vibration having a wave-length y^/N, and 

 as N/Z is the distance between the lines of a grating we may 

 express our result as follows : — 



The disturbance produced by a single impulse leaving the 

 focus of the collimator when resolved by a grating and tele- 

 scope will, at the focus of the telescope, consist of a succession 

 of oscillations equal in number to the lines of the grating and 

 having the same period as that homogeneous vibration which, 

 leaving the collimator, would have its principal maximum at 

 the focus of the telescope. 



It would be wrong to speak of such vibrations as homo- 

 geneous, although for a certain time they may be analytically 

 represented by a sine function. This has repeatedly been 

 pointed out by Lord Rayleigh, and a good deal of the difficulty 

 which has been felt on the subject is due to our natural 

 inclination to consider as homogeneous a train of waves which 

 for a finite space coincides with the curve y — cos kx. Such a 

 train is the more homogeneous the greater the space for which 

 the equation holds, and our grating will give at the focus of 

 the telescope light which is the more homogeneous the greater 

 the number of lines. 



12. It will be instructive to take as a third example the 

 case of such a finite succession of waves as we have just 

 spoken of, and see what the grating will make of it. 



We put f(at) = for all values of t smaller than ^ and 

 greater than t 2 , while between these values of t the velocity 

 f(ai) shall be equal to p cos pt. If at the time t 1 there is no 

 discontinuity of displacement, we must put cos^>^ = l ; and 

 if the train consists of m complete waves, pfe — ^i) = 27rw. 



