309-310] DAMPING OF AIR-WAVES. 571 



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



p=p^^p,s (3), 



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

 Eliminating p and s, we have 



d*u A*u . d*u 



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

 at the plane x = a given vibration 



u = ae i&amp;lt;Tt (5) 



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



u = aeP rt+ f&amp;gt; x (6), 



we find /3 s (c 2 + ^Vcr) = - cr 2 (7), 



whence 8=4- . ^ ... , 



c V 6 c 2 



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

 gives 



c 3 c 3 



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 



H 



where l = $&/vo* ........................ (11). 



The amplitude of the waves diminishes exponentially as they 

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

 &amp;lt;7. The wave-velocity is, to the first order of i/&amp;lt;7/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/&amp;lt;r), we have 



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



