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



It is interesting to know what value of pu gives the numeri- 

 cally greatest value of J (pu). This is at once got from Table I. 

 in Gray and Mathews' Treatise on BesseVs Functions. It will be 

 observed that for brevity I have put down only two places of 

 decimals. The turning points of J (pu) are given by 



pu = 

 and then J (pu) = 1 



3-83 



702 



1017 



1332 



-•40 



+ •30 



-•24 



+ •21 



&c. 



We see at once from this that u = gives the point of numerically 

 greatest strength, which is the geometrical focus of the reflector. 



We will next suppose u = I to be a sound-receiving reflector, 

 and so the vibration to be stationary. We will endeavour to find 

 the "focus of reflection" (Art. 12) in the axis LOx (Fig. 2). 



By Art. 11 the velocity at a point U in LO (Fig. 2) is (putting 

 v = and omitting the time-factor) 



2d£7 1 dJ (pu) T 



Now the turning points of J x (pu) are given by 



pu= 1*84 



and then J 1 (pu) = '58 



5-33 

 -34 



8-54 



•27 



11-71 

 -•23 



&c. 



&c. 



.(96), 



so that pu — 1*84 gives the point in LO (Fig. 2) of numerically 

 greatest velocity. But since for all points in LO we must have 

 pu < pi, we must also have 



1-84 <pl, 



if there is to be a " focus of reflection " in LO. But p is given by 

 the condition (22), 



dU y 



= = J x (p>u), when u = I. 



d (pu) 

 But the zeros of the function J x (pi) are given by 



pl = 0, 3-83, 701, 1017, 13-32, &c (97). 



Passing over 0, the deepest sound is given by pi = 3'83, so 

 that even if the sound be as low as to be given by pi = 3 '83 there 

 will be a focus of reflection in LO. 



Again by Art. 11 the velocity at a point V in Ox (Fig. 2) is 

 (putting 16 = and omitting the time-factor) 



so that pv = l - 84 gives another " focus of reflection " in Ox. 



