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1916]                       NARROW TUBES  OF VARIABLE  SECTION                          379
or, since the ends of the disturbed region at 0 and b are cylindrical,
0 from which the term in /<?2#2 may be dropped.
That the reflected wave should be very small when the changes are sufficiently gradual is what might have been expected.    We may take (13)
in the form
7?      ,'   rb                                y   rb -[ f}-^,
~ = ~r     7e-^xdx = ~     ^2e~M*dx ............. (19)
A     2k JQ                      2/cJ0 y dx2                                ^
As an example let us suppose that from as = 0 to x  1}
y == y0 + 7} (1  cos mas),   ........................ (20)
where y0 is the constant value of y outside the region of disturbance, and w = 27r/6. If we suppose further that y is small, we may remove 1/y from under the sign of integration, so that
Independently of the last factor (which may vanish in certain cases) B is very small in virtue of the factors inz/kz and 77/2/0-
In the second problem proposed we consider the passage of waves proceeding in the positive direction through a tube (not necessarily of revolution) of uniform section o-j and impinging on a region of irregularity, whose length is small compared with the wave-length (X). Beyond this region the tube again becomes regular of section cr2 (fig. 1). It is convenient to imagine the
f b cos mx e-M*dx = ^L {1 - cos 2kb + i sin 2&Z>}.    . . .(21) J0                      4o
Fig. 1.
axes of the initial and final portions to be coincident, but our principal results will remain valid even when the irregularity includes a bend. We seek to determine the transmitted and reflected waves as proportional to the given incident wave.
The velocity-potentials of the incident and reflected waves on the left of the irregularity and of the transmitted wave on the right are represented
respectively by
(22)smitted wave.   It appears, rather unexpectedly, that there is a linear acceleration amounting to