1915] THE CONE AS A COLLECTOR OF SOUND 363
Apart from what may happen afterwards, there is a preliminary question at the mouth. In the passage from plane to spherical waves there is a phase-disturbance (between the centre and the edge) to be reckoned with, represented by
where R is the length of the cone, and 6 the semi-vertical angle. That this may be a small fraction of \ itself a small fraction of the diameter of the mouth (2RO), it is evident that 6 must be very small.
We may now consider the incidence along the axis (x) of plane waves of simple type. Within the cone, supposed to be complete up to the vertex, the vibrations are stationary, and since no energy passes into the cone, the same must be true of the plane waves just outside — at any rate over the greater part of the mouth. The velocity potential just outside may therefore be denoted by
A|T = cos kat . cos (kx + e),
making at the mouth (as = 0)
ty = cos kat . cos e, d^rjdx = — k cos kat . sin e. On the other hand, in the cone
. sn T , , = A ~i — cos kat, kr
making at the mouth (r = R)
sin kR , d^lf , , (cos kR sin kR
~~ cos kat> = IcA
Equating the two values at the mouth of ^ and dty/dx; or dty/dr, we get
. sinkR . , {coskR sinkR
cos e = A — pg- , — sin e = A •< 7 p -kR ( kR
A2 L &w2kR sirfkR
I — _ J i JL — 7 „ -r-«n T J-
fCJiii fO jft
When kR is considerable, the second and third terms may be neglected, whatever may be the particular value of kR, so that for a long enough cone
A — kR simply,
in which k = 2?r/X. Here A is the maximum value of ty at the vertex of the cone, and the maximum value of ty in the stationary waves outside the mouth is unity, the particular place where this maximum occurs being variable with the precise value of kR.
The increase of i/r, or of the condensation, at the vertex of the cone as compared with that obtained by simple reflection at a wall is represented by the factor kR, which, under our suppositions, is a large number.e relation