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[Philosophical Magazine, Vol. xxxi. pp. 89  96, 1916.]
UNDER this head there are two opposite extreme cases fairly amenable to analytical treatment, (i) when the changes of section are so slow that but little alteration occurs within a wave-length of the sound propagated and (ii) when any change that may occur is complete within a distance small in comparison with a wave-length.
In the first case we suppose the tube to be of revolution. A very similar analysis would apply to the corresponding problem in two dimensions, but this is of less interest. If the velocity-potential $ of the simple sound be proportional to eikat, the equation governing <f> is
x '
drz     r dr     das2
where x is measured along the axis of symmetry and r perpendicular to it. Since there are no sources of sound along the axis, the appropriate solution is*
(j)  J0 {r 4/(ds/da)* + &2)} F (so),  .....................(2)
in which F, a function of oc only, is the value of $ when r = 0.
At the wall of the tube r = y, a known function of x; and the boundary condition, that the motion shall there be tangential, is expressed by
__d4dy + d$^Q                               (g^
dx doc    dr       '..............................^ '
in which (r = y)
dd>    dF    i
+ ft8   -T- +
ci^r.         s,    \ rinr**               i   riw        92    4,2 \ /-7/y>2              /    /v/v*
tf/w          ^J      \Uw                /     \JUvU         ju       J3      \vOi.v                /       tttt/
yf*..t.*\ P, f f*+ktfp_
dr        2U*2^    J     "r22.4U2
Compare Proc. Lond. Math. Soc. Vol. vn. p. 70 (1876); Scientific Papers, Vol. i. p. 275.e diameter of the sphere. When e is nearly equal to unity,