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Full text of "Scientific Papers - Vi"

1916]   PROPAGATION OF SOUND IN NARROW TUBES OF VARIABLE SECTION   377
Using these in (3), we obtain an equation which may be put into the form
As a first approximation we may neglect all the terms on the right of (6), so that the solution is
A pikx   i   J) aikx
_, ........................... (7)
(7
where A and B are constants.    To the same approximation,
 +**'-     *dydF
^7   T  T n -l7   ---- , --- 7 
ofl9                   y dx dx
For a second approximation we retain on the right of (6) all terms of the order dty/da;9, or (dy]dxf. By means of (8) we find sufficiently for our purpose
dx        y dx dx- '
.      ^-4^1--
\da?                ~     dx? V (dx       y do?  '
Our equation thus becomes
dx*       2 do? dx2 in which on the right the first approximation (7) suffices.    Thus
yF (x) = ~ \eikx (Y (B + Ae~^x) dx - e^kx \Y(A + Be*kx} dosl ,     (10)
*j/i/6   (         *.'                                                                   J                                             J
T7    1 + pV d*y whore                                    F = - ^-r-
In (10) the lower limit of the integrals is undetermined ; if we introduce arbitrary constants, we may take the integration from - oo to x.
In order to attack a more definite problem, let us suppose that d*y/dx2, and therefore Y, vanishes everywhere except over the finite range from x = 0 to x  b, 1} being positive. When x -is negative the integrals disappear, only the arbitrary constants remaining ; and when x is positive the integrals may           /       tttt/