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



we should have to use the ascending convergent series for V^ 

 and V 2 . 



In Art. 34, beginning with an expression for F of the form 

 U-iYz (omitting the time factor), we got the expression (86) for 

 strength of source. But now, as from (92) and (93), we begin with 

 UjY 2 multiplied by 9^4/2 *J2 r(|), we shall get for the strength 

 (86) multiplied by this same factor. We must, however, re- 

 member that A in (86) has its original meaning, while in (92) and 

 (93) A means A/p, so that the final result is : Strength of source 



4>57tA% ~ m m 2 m 3 . ~| ._.. 



"16V27lTf) X L i ~l + 3(2l7~4730 2+ J { h 



where m = 5tt/8 and A has its original meaning (Art. 6). 



[Once more it may be well to remind the reader that in (91) A 

 means A/p and v means pv, and that in (92) and (93) A means 

 A/p, and U 1} V 1} V 2 are certain functions of u and v given by (15) 

 and (16) where u and v stand for pa and pv.] 



Comparing together (91) and (94) we observe that if A in- 

 creases, the strength of source varies as A 2 , whilst the magnification 

 varies nearly as e% nAlp , that is, in a much higher ratio, unless p be 

 comparable with A, which we do not suppose. Again, supposing 

 in (91) we keep A constant and suppose p to vary, we see that the 

 magnification varies as e^ nAlp /p^, which increases as p diminishes, 

 shewing that for a given large value of A, low notes are more 

 magnified by a reflector than high ones. 



Putting in (91) A /p for A and pv for v, we shall readily find 

 that the distance between the crests of two successive waves is 

 given nearly by 



p(l-A/2p*v) V >' 



Here if v is increased the distance is diminished, consequently 

 it would seem that as the wave moves outwards, the note gets slightly 

 sharper — only ultimately attaining the pitch defined by the letter 

 p. This peculiarity, if it exist, should be more observable with 

 low notes than with high ones. We may also notice that since by 

 (40), for a given value of n, A varies as I -1 , other things being the 

 same, the more tube-like be the paraboloid the larger will be A, 

 and so from (91) the larger will be the magnification of the sound 

 caused by the reflector. This agrees with a remark of LordRayleigh's 

 in Art. 280, Vol. n., of his Treatise on Sound, where he treats of 

 conical pipes with a source at one end. The same observation 

 might have been made at the end of Art. 32 on the Sound- 

 receiving- Reflector. 



