309-310] DAMPING OF AIR- WAVES. 571 



and the physical equation is, if the transfer of heat be neglected, 



........................... (3), 



where c is the velocity of sound in the absence of viscosity. 

 Eliminating p and s, we have 



d z u _ 2 d?u 4 d s u ... 



dt*~ c fo* + * v da?d" 



To apply this to the case of forced waves, we may suppose that 

 at the plane x a given vibration 



u = ae iart .............................. (5) 



is kept up. Assuming as the solution of (4) 



u = a j*t+P* ........................... (6), 



we find yS 2 (c 2 + fiVo-) = -o- 2 ..................... (7), 



whence /3=l-| .................. (8). 



If we neglect the square of i>cr/c 2 , and take the lower sign, this 

 gives 



Substituting in (6), and taking the real part, we get, for the waves 

 propagated in the direction of ^-positive 



u = ae~ x!l cos a- 





(- 



V C, 



where Z = fc 3 /^ 2 (11). 



The amplitude of the waves diminishes exponentially as they 

 proceed, the diminution being more rapid the greater the value of 

 a. The wave-velocity is, to the first order of i/<r/c 2 , unaffected by 

 the friction. 



The linear magnitude I measures the distance in which the 

 amplitude falls to l/e of its original value. If X denote the wave- 

 length (27rc/cr), we have 



it is assumed in the above calculation that this is a small ratio. 



