DYNAMICAL THEOEY OF SOUND 



We have seen that a true simple-harmonic vibration may 

 be regarded as the orthogonal projection of uniform motion in 

 a circle. An analogous representation of the modified type (10) 

 is obtained if we replace the circle by an equiangular spiral 

 described with constant angular velocity ri about the pole 0, in 

 the direction in which the radius vector r decreases*. The 

 formula (10) is in fact equivalent to # = rcos#, provided 



r = ae~ il \ 6 = n't + (11) 



Eliminating t we have 



r=ae~ f "> (12) 



where = (n'r)~ l , a. ae^ . This is the polar equation of the 

 spiral in question. The curve in Fig. 12 corresponds in scale 

 with Fig. 11. 



In most acoustical applications the fraction k/2n, or 1/nr, is 

 a very small quantity. 

 In this case, the dif- 

 ference between n and 

 ri is a small quantity 

 of the second order, 

 and may usually be ig- 

 nored ; in other words, 

 the effect of friction on 

 the period is insensible. 

 It may be noted that 

 the quantityl/nr, whose 

 square is neglected, is 

 the ratio of the period 

 27T/71 to the time 2-7TT 

 in which the amplitude is diminished in the ratio e~ * or -^. 



If k be greater than 2n the form of the solution of (3) is 

 altered, viz. we have 



*, (13) 



Fig. 12. 



y 



whence 

 if 



Ae 



-at 



(14) 



* This theorem was given in 1867 by P. G. Tait (18311901), Professor of 

 Natural Philosophy at Edinburgh (18601901). 



