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 p—ikx 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/