BY SMALL OBSTACLES OF CYLINDRICAL AND SPHERICAL FORM. 259 



Returning now to the consideration of the equations of vibration, we find from (8) 

 and (10) that the general solution suitable to divergent waves is given by 



(J (h 1 / -i \ / / T \ O ff^n 7 9 9i 4- T /* /7 \ O / CtJ 



, Wi* I ^ J /Ai _J_ 1 1 / I /*/V 1 . 11 /*/>^^ lJ / / t"l\ 



W = -x- 1 - -T Z "S l?T i)Jn-l \^')~^~ "** ^ yn+1 V"'' / K~^ "^J 



OX ?i=0 L u*& O3C \i 



with corresponding expressions for v and iv ; < here represents the general solution of 

 (7) suitable to divergent waves. 

 Hence 



= S f n (hr}4 n (19), 



n=0 



where <j> n is a solid harmonic of positive degree n. 



If the motion is in planes through the axis of x, and is symmetrical about that 

 axis, the solution takes the form 



U ^- 



with corresponding expressions for v and w. Further we have 



0= S A,/. (Ar) *?(/*), eu, = B H r"P, M (21), (22), 



11=0 



where A B and B B are arbitrary constants. 



We write as before 



k = Xe- 1 ' 1 *", where X = (o-/i>) 12 .... (23), (24). 



From (15) it is seen th&tf n (kr) contains the exponential factor e~ (t! * r '" ) " r or e~ Vs ' /2 *'' ; 

 consequently since Xr becomes very great within a short distance of the origin, it is 

 clear that, at a moderate distance from the origin, those parts of the expression (20), 

 which depend on the functions f n (kr), become inappreciable and may be neglected. 

 Hence, at a sufficient distance from the origin, we may write 



00 



(u, v, w) = grad <f>, where $ = 2 A B / B (hr) r"P B (/x). 



n=0 



7. The Incidence of Plane Waves of Sound upon an Obstructing Sphere. We 

 may now consider the effect of a spherical obstacle upon a train of plane waves of 

 sound. Suppose the centre of the obstructing sphere to be at the origin, and the 

 sound to be propagated in the negative direction along the axis of x ; then, as before, 

 we may assume for the incident waves 



(MO, t'o, w ) = grad ^ , where < = e' hz (1), (2). 



2 L 2 



