136 Lord Kayleigh on 



given amplitude. But the phase* of the simple train approxi- 

 mately representing the given waves would vary from place 

 to place, slowly indeed but to any extent. 



Thus if we take, as analytically expressing the dependence 

 of the displacement upon time, 



H cospt + K. smpt, (1) 



where H and K are slowly varying functions of t, the fre- 

 quency may be regarded as constant, while the amplitude 

 </(H 2 + K 2 ) and the phase tan-^K/H) vary slowly but with- 

 out limit. It scarcely needs to be pointed out that a slow 

 uniform progression of phase is equivalent to a small change 

 of frequency. 



In one important class of cases the phase remains constant 

 and then, since a constant addition to t need not be regarded, 

 (1) is sufficiently represented by 



Bcospt (2) 



simply. If the changes of amplitude are periodic, we may 

 write 



H = H + Hi cos qt + H/ sin qt 



+ H 2 cos 2qt + H 2 ' sin 2qt + . . ., . . . (3) 



in which q is supposed to be small. The vibration (2) is 

 then always equivalent to a combination of simple vibrations 

 of frequencies represented by 



p, p+q, p—q, p + ty, p—%q, &c. 



Under this head may be mentioned the case of ordinary 

 beats, so familiar in Acoustics. Here 



H = II lC os^, (4) 



and H cosp^iHj cos (p + q)t + ^H 1 cos (p-q)t. . (5) 



It may be observed that although the phase is regarded as 

 constant, the change of sign in the amplitude has the same 

 effect as an alteration of phase of 180°. 



Another important example is that of intermittent vibra- 

 tions. If we put 



H = 2(l+cos^) 5 ...... (6) 



the amplitude is always of one sign, and 



Hcosp£=2 cosjof + cos (p + q)t + cos (p — q)t. . (7) 



Three simple vibrations are here required to represent the 

 effect. 



* AYhat is here called for brevity the phase is more properly the 

 deviation of phase from that of an absolutely simple train of waves. 



