256 MR. C. J. T. SEWELL : EXTINCTION OF SOUND IN A VISCOUS ATMOSPHERE 

 Hence, if p. be the refractive index as modified by the wires 



...... (6). 



Hence we have 



...... (7) 



where p denotes the ratio, assumed small, of the volume occupied by the wires to 

 the total volume. 



Let us now consider the case when \a is great. In this case we have, from (17) 

 and (18) 3 



-i {2 ^2 (Xa)-' + 2 (Xa)- a +f*AV}]. 



Substituting this expression for A^iAo in (3), we obtain for the resultant at of 

 all the secondary vibrations coming from the stratum dx 



fa = IIHTT dx . ha 2 [1 + 2 ^/2 (Xa)" 1 f h 2 a 2 (log ^ 



of which the real part is 



-^mr.dx.ha? {2^/2 (Xffl)-' + | (Xa)- 2 +frrAV} cos (lix + trt) 

 + nir.dx.ha 2 {l + 2 v / 2 (Aa)- 1 -f# l a 3 (log&a+y+-&)} sin (hx + <rt) . (8). 

 To this is to be added the expression for the primary waves 



fa = cos (hx + vt). 



The coefficient of cos (hx + at) is thus altered by the obstacles in the layer dx from 

 unity to 



dx. 



Hence, if E be the energy in the incident waves, we have 



dE/E = -mrha? {2^/2 (Xa^' + f (Xa)- 2 + f7rAV} dx. 



Integrating this, we obtain 



E = Eoe , 

 where 



a = ,nrha* {2 ^2 (\a)~ l +l (\a)~ 2 + f ir/tV} . 



When o-/c and (a-fv) 112 are substituted for h and X respectively, this takes the form 



Vc 4 ..... (9). 



