186 DYNAMICAL THEOEY OF SOUND 



The solution of (14) may also be effected concisely by means 

 of imaginary quantities. Thus in investigating forced simple- 

 harmonic vibrations of prescribed frequency we assume that 



fc _ ftgi nt+ mx (19) 



whence, on substitution, 



n 2 



(20) 



c 2 -I- 1 ivn 



The ratio vn/c* is usually very small; thus for n = 1500 its value 

 is, with previous data, about 1*8 x 10~ 7 . Hence 



Taking the lower sign, which corresponds to waves travelling in 

 the direction of ^-positive, and rejecting the imaginary part of 

 (19), we find 



............... (22) 



provided l=3(?/2vn* ...................... (23) 



This represents a system of waves generated to the right of 

 the origin by a prescribed motion f = a cos nt at this point (as 

 by a piston in a tube if we neglect the friction at the sides). 

 The waves advance, with (sensibly) the usual velocity c, but 

 diminish exponentially in amplitude as they proceed*. The 

 linear magnitude I measures the distance over which the waves 

 travel before the amplitude is diminished in the ratio \\e. In 

 terms of the wave -length we have 



I = (Sc/STr 2 !/) . X 2 , .................. (24) 



or, with previous data, I = 9'56X 2 x 10 3 . The effect of viscosity 

 in stifling the vibrations is therefore very slight except in the 

 case of sounds of very high frequency and consequently short 

 wave-length. Even for \ = 10 cm. the value of / is nearly 10 

 kilometres. When we come to the discussion of three- 

 dimensional waves it will be clear that the effect of viscosity 

 may for most purposes be ignored in comparison with the 

 diminution of intensity due to spherical divergence. It is, 

 however, of some interest to observe that there is a physical 



* This calculation was first made by Stokes (1845). 



