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



article is correct only when A is so large that A~% may be re- 

 garded as a small quantity. 



31. We have proved (Art. 30) that the point on the axis of 

 greatest sound intensity or focus of reflection is a little on the left 

 of the point A (fig. 4) given by v = A. As A is supposed large, 

 we will consider it practically to coincide with it. 



But (Art. 6) we must express this in terms of the original 

 notation and see what it means, which is that pv — A/p = 0, A being 

 a root of equation (40) which also we may consider as expressed in 

 the original notation, so that the focus of reflection is really 

 given by 



v. = - 2 = 20A, see fig. 4 and Art. 5 (80). 



This shews that if A be given, and we experiment upon a number 

 of different notes, the focus of reflection will be nearer the geo- 

 metrical focus of the reflector for high notes than for low ones. 

 This perhaps accounts for the great audibility of high notes in 

 experiments of the kind described in Art. 1, for we see that even 

 if A be large (which it apparently must be, to make or to account 

 for foci of reflection) yet the largeness of p 2 may draw the point A 

 near to 0. For instance p 2 may be as large as 3600 for very high 

 notes (one foot and one second being taken as units). 



32. We will next make some observations on the magnifi- 

 cation of the sound produced by a sound-receiving parabolic 

 reflector. By Art. 5, using v in the sense of the latter part of 

 Art. 6, it can be shewn that the velocity of any air-particle in the 

 axis is equal to 2pdVJdv multiplied by a time factor. In what 

 follows we will for brevity omit the time factor. Then by (41) the 

 air velocity at any point in LO (figs. 2 or 4) is 2pA, and when we 

 speak of sound magnification at any point we mean that we 

 compare the air velocity at that point with this 2pA. 



First then at P 2 (fig. 4) by (46) we shall find that 



The sound magnification = , a nearly (81)- 



This is an interesting and remarkable result, for by Art. 6 we 

 see that it is independent of p or the same for all notes experi- 

 mented on. It ought however in fairness to be pointed out that 

 (46) and so (79) were only obtained on the supposition that v/A 

 is small, which means (Art. 6) that p 2 v/A is small, so that the 

 above statement about independence is generally more true for 

 low notes than for high ones, unless A be excessively large. 



