382 PROPAGATION OF SOUND IN NARROW TUBES OF VARIABLE SECTION [402
As an example let us take the case of a tube of revolution for which y, being equal to y0 over the regular part, becomes y0 + Sy between x1 and xs. We have
Also*
1 _ 1 Jdx
^-
J 1/o 2/0
V o-R , . Pafrt/'s^2 7
and 5-+-o-=(*s-a'i)+ ]2— +jh; rrfa?> ......... (38)
2cr 2 J*, I V^/o/ 4 \diK ) } ^ '
the terms of the first order in 8y disappearing. Thus in the exponent of (35)
. . .(39)
of which the right-hand member, taken with the positive sign, expresses the retardation of the transmitted wave due to the departure from regularity.
* Theory of Sound, § 308.sufficient approximation. Thus