254 ME. C. J. T. SEWELL : EXTINCTION OF SOUND IN A VISCOUS ATMOSPHEEE 



5. Extension to the Problem of a Number of Cylindrical Obstacles. Let us 

 suppose that there are n parallel wires per unit area of a plane perpendicular to their 

 common direction. At a distance r from the axis of any one of these wires, great 

 compared with the wave-length of the incident sound, the secondary waves due to 

 that wire will be given very approximately in all cases by 



fa - A DO (hr) + A! D! (hr) cos &. 

 Since hr is great, we may write for all values of n 



and (f) l takes the form 



jrtvrj 



= (A + A,i cos S) 



\iriir / 



Along the course of the primary waves ( = IT] this reduces to 



fa = (A -A ll ) 



(I)- 



Consider now the wires which occupy a thin stratum dx perpendicular to the course 



of the primary waves. Let AP be the section of this 

 stratum by a plane at right angles to the wires and let O be 

 the point, at which the vibration is to be estimated at a 

 P great distance from the stratum. 



If AP = 2, the element of area is dx . dz, and consequently 

 the number of wires cutting it is n . dx . dz. 



Also, if OP == r, AO = - .r, then r 2 = x 2 + z 2 and 

 z dz = rdr. 



The resultant at O of all the secondary vibrations, which 

 issue from the stratum, is by (1) 



Writing / = x + rj, we obtain 



In evaluating this last expression we may assume rj/x to be very small ; it then 



takes the form 



/ 2 y/3 r 00 



\ irh/ J o 



Now 



= 2 e~ lhv *dv. 

 Jo 



