Mr Sharpe, On the Reflection of Sound at a Paraboloid. Ill 



Ten terms of this series give us, I believe, about "061. 

 The third part I cos a; 2 . \ogxdx is the most troublesome. 



By breaking it up into two parts from e to (57r/2) 5 , and from 

 (57r/2)^ to oo , we can express each part in converging, but rather 

 complicated, series. By the following plan, however, we can 

 make a rough approximation to the value of the integral. 



Integrating by parts so as to get descending powers of x we 

 get series which soon diverge, but which begin by converging. 

 The result is — '164 with an error that cannot exceed + *02. 



/• 00 



Thus I cos x 2 . log xdx = — "104 nearly. 



The other integral in (33) can be treated in a similar manner. 



16. By combining equations (26) and (32) we can express 



cos v . ftp and sin v . /v* 



as known linear functions of V 1 and V 2 . Thus restoring the 

 original notation in (5) P and Q can be chosen as the same 

 known multiple of 



cos pvj(pv)* and smpv/( pv)i respectively, 



and we have a single outgoing wave. It appears from (26) and 

 (32) that the amplitude of this wave will be larger the more nearly 

 the integrals in (33) agree in value. 



17. To get the strength of the source we must first find the rate 

 at which air crosses an element Pp (fig. 2) of the paraboloid PV. 

 F being the velocity potential, and using the old notation this will 

 be found to be 2irvdu x dF/dv. We must then put v = and 

 integrate with regard to u from up to the original I the semi- 

 latus rectum of the reflector. The result will be (remembering 

 that from Art. 7 vd V 2 /dv = 1 when v = 0, and for brevity omitting 

 the time factor) 



2tt ( Ujdu. 



Jo 



At any given point in LO the strength will be proportional to 

 phi- pHi? 



The series begins to converge at once for points where phi 2 < 12. 

 It will be readily seen that the strength is a maximum near the 

 point where pu = 2, and that this point of maximum strength is 



9—2 



