Radiation and Vibrations in Conical Pipes. 75 
advance of a progressive wave. Its amplitude, however, 
suffers diminution by varying inversely as r. But, owing to 
this diminution or attenuation with advance, we have in the 
other equations the factors 1/r and also 1/r 2 , one applying to 
a sine and the other to a cosine function. Hence the speed 
it and the displacement f each exhibit, during advance^ an 
additional slight change of phase (as the sine and cosine 
terms are differently diminished), beyond that always pre- 
sent in a progressive wave. But, where r is great enough 
to make 1/kr negligibly small, (21) and (22) become 
app roximate ly 
and 
u= C -cosk(r—at), (23) 
f= -£ sin 4(r- at) (24) 
hr 
Reflexion at Pole. — Let us now regard the two spherical 
waves of (18) as a converging one and a diverging one 
to which the other gives rise by reflexion at the pole or 
centre of the system. And let it be required to determine 
the relation between / and f 2 so resulting. The total 
current across the surface of a sphere of radius r is &irr 2 u 
and, for r = 0, must vanish, since all is symmetrical about 
the pole or origin of coordinates. That is, u cannot be 
infinite and so make r 2 u finite for r=0. But, if ^.irr 2 u 
vanishes at r = for all values of /, so also will ^irr^dujdt 
vanish. And this condition is easier to fulfil analytically. 
Thus from (6) and (18) we have 
4:TTi- 2 du/dt ■- —4:7ra-r 2 ds/dr 
= kTra 2 \f l (r — at)+f 2 (r + at)}—±T7a-r{f\\r~at)+fJ(r + aty h 
where f denotes the first derived function of/. 
Hence, putting du/dt = for ? , =0, we find 
0=f 1 (-at)+f 2 (at) (25) 
as the relation between/ and / 2 . But we see from (18) that 
the right side of (25) is the value of (rs) for r = 0. Hence, 
we may write as the condition at the pole 
rs=0 for r=0, (26) 
or, rs must vanish with r. 
Thus, at the pole, a condensation is reflected as a rare- 
faction and vice versa, somewhat as in the case of reflexion at 
the open end of a parallel pipe. 
Conical Pipes. — To apply equation (17) to conical pipes 
