482 Lord Rayleigh on the Open Organ-Pipe 
of the extension to infinity this resistance is finite. For if 
x be the radius of a large sphere whose centre is at the mouth, 
the resistance between rv and 7+dr is dr/27r?; and the part 
correspording to the passage from a sufficiently great value 
of + outwards to infinity may be neglected. 
A parallel treatment of the problem in two dimensions, 
where inside the mouth the boundary consists of two parallel 
planes, appears to fail. The resistance to infinity, involving 
a) 
@ 
now \7~1dr, instead of (200, has no finite limit; and we 
must conclude that when the wave-length ()) is very great, 
the correction to the length becomes an infinite multiple 
of the width of the pipe. But it remains an open question 
how the correction to the length compares with »,—whether, 
for example, when X is given it would vanish when the width 
of the pipe is indefinitely diminished. 
The following consideration suggests an affirmative answer 
to the last question. If we start with a pipe of circular 
form and suppose the section, while retaining its area, to 
become more and more elliptical, it would appear that the 
correction to the length must continually diminish. But the 
question has sufficient interest to justify a more detailed 
treatment. | 
In Theory of Sound, § 302, the problem is considered of 
the reaction of the air upon the vibratory motion of a circular 
plate forming part of an infinite and otherwise fixed plane. 
For our present purpose the circular plate is to be replaced 
by an infinite rectangular strip extending from y=—o to 
y= + ,and in the other direction of width z. If dd/dn be 
the given normal velocity of the element dS of the plane 
and k=29/h, 
x iBee 2 dpe 
o= LE “ ds . . . . (1) 
gives the velocity-potential at any point P distant r from 
dS. In the present case d¢/dn is constant, where it differs 
from zero, and accordingly may be removed from under 
the integral sign. If a denote the velocity of sound, @ varies 
as eat, and if o be the density, we get for the whole variation 
of pressure acting upon the plate 
ff 8p dS= —a\\¢ dS = —ikao\\ ds. 
Thus by (1) 
ikr 
eedke (oo sso eee eee) 
Tw dn r 
