244 MR. C. J. T. SEWELL : EXTINCTION OF SOUND' IN A VISCOUS ATMOSPHERE 



3. The Incidence of Plane Waves of Sound upon an Obstructing Cylinder. We 

 are now in a position to consider the effect of a cylindrical obstacle upon a train of 

 waves propagated in a direction perpendicular to the axis of the obstacle and incident 

 upon it. 



We take the axis of the obstructing cylinder as axis of z, and suppose the incident 

 sound to be propagated in the negative direction along the axis of x. Then, as in (12) 

 of the last article, we may assume for the incident sound the expressions 



^ = e' hl , Vo = 0. 

 Expanding in series of Bessel functions, we obtain 



< = J (//r) + S 2i"J,, (hr) cos n$, i// = ..... (1). 



H = 1 



The scattered waves will be symmetrical about the axis of x, or = 0, and 

 consequently we may assume for them the forms 



, = A D (hr)+ 2 {A, D n (hr) cos nS}, 



= i 



^=1 {B n D n (A:r)sin^} ........... (2). 



At the surface of the obstacle the radial and tangential components of the velocity 

 must vanish ; hence we must have 



or 



"57T ) ^7. 



0$ 0$ 



when > = a, if a is the radius of the cylinder. 



In order that the boundary conditions (3) may be satisfied, we must have 



or 



A 7i D ' (ha) = haJ ' (ha) 



A D 1 (/ i )= -3, (ha) (4), 



and in general for n> 



AJu* D,/ (ha) + nB n D B (ka) = - 2i n haJ n ' (ha) 

 nA n D,, (ha) + ~B n ka D/ (ka) = - 2 in J B (ha) 



These equations (4) and (5) are sufficient to determine the various constants in the 

 expressions (2) for the scattered sound. In the process of approximating to the values 

 of these constants by means of equations (4) and (5), we shall confine ourselves to the 

 case when ha is a small quantity ; in other words, we shall assume that the dimensions 

 of the obstacle are small compared with the wave-length of the incident sound. The 

 other case, when the dimensions of the obstacle are large in comparison with the 



