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



It is known that the equations Jo (u) — and -v- Jo (u) = have 



an infinite number of real roots, so that I has an infinite number 

 of values appropriate to the present solution. The same can be 

 said of the original pi (Art. 39), and (as we suppose p constant) of 

 the latus rectum of the Reflector. It will be interesting to find 

 the various points on the axis where the air- velocity is a maximum, 

 and especially the position and intensity of the greatest maximum. 

 We shall suppose the vibration stationary, then it can be shewn 

 (Art. 5) that the air- velocity in the direction OV (fig, 8) at any 



point V on the axis is 2 -r- x time factor, which =2-^ Jo(v) x 



time-factor. We shall, however, for brevity always omit the 

 time factor. The position on the axis to the right of where 

 the air-velocity is a maximum is determined by the equation 



-Y-„ Jo (v) = 0, which is the same thing as -^ J\ («) = 0. 



Now the zeros and turning points of Ji {x) are given by the 

 following Table, where, for simplicity, we have retained only two 

 places of decimals. 



When a; = 1-84, 5-33, 8-54, 11'71, 14-86, 18-00, &c. 



Then -^ Jo {x) = — J^(x) = 



-•58, +-34, -'27, +-23, -'20, -h 19, &c., 

 and the roots of Jj (x) — are given by 



x = 0, 3-83, 7-01, 10-17, 13-32, &c. 



Looking at fig. 8, if P' be the optical image of P in OR', we 

 see that the « and v of P will be equal to the v and u of P' 

 respectively, and when A = 0, it will be found (see Art. 5) that 

 the sound motion to the left of OR' is the image of the sound motion 

 to the right of OR'. It follows that if in fig. 8 we draw the parabola 

 L'R' the image of LR', there will be no air-velocity normal to 

 L'R'; and L', as tuell as L, will be a point of rest. 



We can now see something as to the nature of the air motion 

 and sound intensity, at any rate at points along the axis of the 

 reflector LR'. To the right of OR' (see fig. 9) along the axis Ox 

 we have an infinite series of points Vi, Vo, Vg, ^4, etc., whose abscissae 

 are Ovi=l-84, 0^2=5-33, etc. The ordinates of the wavy curve 

 represent the magnitudes and directions of the corresponding 

 maximum air-velocities. 



For instance, at Vj the magnitude is -58 to the left, 



Vo^ „ „ -34 „ right, 



Vs „ „ -27 „ left, and so on ; 



''1, f'a, *'3j '"45 etc., are intermediate points of rest. 



