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



and as I' is the original pi, it means that in this case, only certain 

 sounds whose pitch depends upon the particular reflector used are 

 experimented on. (22) as an equation in V has an infinite number 

 of real roots. The first 6 roots (which are given on p. 266, Vol. II. 

 of Rayleigh On Sound) are 3-832, 7-015, 10174, 13324, 16471, 

 19'616. For a given value of I the small numbers correspond to 

 the low notes, the large numbers to the high notes, to which the 

 present investigation applies. 



11. From (15) putting i = 0we get 



U, = l-\ z +~-&z. = Jo{u) (23). 



By Art. 5 the velocity of an air particle normal to any paraboloid 

 u = a constant is 



dU 1 ( u 



du \u + vj 



multiplied by a time factor. [We must carefully notice that we 

 are using u and v here in the sense of the second part of Art. 6, 

 that is for u! and v'.] It is readily seen from (23) that when 

 u = 0, that is, when we approach and reach the axis Ox (fig. 3), 

 this velocity vanishes. Any concentration or magnification of the 

 sound as we approach the axis depends mainly on dVj/dv, to which 

 we next proceed. 



12. The value of V is got from (12). We will first consider 

 the significance of the non-logarithmic solution, which is 



^ = l-|+2i^-&c. = /„(*,) (24). 



The intensity of the sound at any point on the axis is measured by 

 the velocity along the axis, of the air particle at that point, and 

 this is seen from Art. 5 to be 2pdVJdv multiplied by a time factor. 

 We can readily shew from (24) that dVJdv has a maximum value 

 near the point determined by v 2 = 8/3, but whether this gives the 

 absolutely greatest value of dVJdv would require a separate in- 

 vestigation to discover. That there is a point on the axis of 

 absolutely greatest sound intensity, or " focus of reflection " as we 

 may call it, we shall see from what follows, but in the case of zero 

 or small values of A , to determine its exact position would probably 

 be a troublesome business, and perhaps not worth while to under- 

 take, as in these cases the magnification of the sound appears to 

 be small. 



For large values of v the series (24) is unsuitable for calculation, 



