BY SMALL OBSTACLES OF CYLINDRICAL AND SPHERICAL FORM. 243 



provided < and /> are functions satisfying the equations 



+ f"V,V ......... (6). 



If we assume a time factor e" ( , these equations take the forms 



( Vl 2 + F) < = (), (V, 2 + F),/, = . . ..... (8), 



where h 2 and F are given by 



It* = o>/((?+$iv<r), k 2 =-i<r/v ..... (9), (10). 



We shall for convenience suppose h to be equal to that root of equation (9) which 

 reduces to a-jc when v is zero ; k will be taken to be equal to (cr/^) 1 2 .r^" r '. 



With these conventions the solution of the equations (8), which represents waves 

 of sound diverging from the origin, is given by 



<f> = A D (hr) + 2 A,,D n (hr) cos (n3 + a,,) 



n = l 



GO 



i/; = B,,D (r) + 2 B n D n (kr) sin 

 where for convenience the time factor has been omitted and where D,, () is given by 



D.() = |{Iog2- y -Jr)J.(0-Y.(0}, 



J B () and Y n (4) have their usual significance as Bessel functions, and A n , B,,, . and 

 /? are arbitrary constants. 



It need hardly be remarked that to obtain the actual expressions for < and \p it is 

 necessary to multiply the expressions contained in (11) by e l<rt , and to equate < and // 

 to the real parts only of these products. For the sake of brevity we shall usually 

 omit, when possible, the time factor e iat . 



Since, in the case of air, v is a small quantity, when expressed in cm. sec. units, it 

 is clear from (9) and (10) that for all audible sounds \k\ will be large compared with 

 \h\. In fact, at any ordinary distance r from the origin the i/< terms in (11) will 

 become insensible owing to the factor exp. { (^cr/v) l2 r}. 



For plane waves propagated in the negative direction of the axis of x, the solution 

 will be given by 



$ = Ce' A V ( , \fj = (12), 



where, as before, only the real part of <j> is to be taken into account. 



2 i 2 



