426 ON VIBRATIONS AND DEFLEXIONS OP • [411
We will apply this equation to the plate bounded by the lines y = 0,y = '7r, and extending to infinity in both directions along ac, and we suppose that external transverse forces act only along the line as = 0. Under the operation of these forces the plate deflects symmetrically, so that w is the same on both sides of x = 0 and along this line dw/doi = 0. Having formulated this condition, we' may now confine our attention to the positive side, regarding the plate as bounded at a? = 0.
The conditions for a supported edge parallel to x are
M/ = 0, dhv / df = 0 1 ........................... (4)
and they are satisfied at y = 0 and y = IT if we assume that w as a function of y is proportional to sin ny, n being an integer. The same assumption introduced into (3) with Z=Q gives
(d*/daP-na)aw = Q, .............................. (o)
of which the general solution is
i(j={(A+Bx}e-^x + (C+Dx}enx}smny, ............... (6)
where A, B, G, D, are constants. Since w — 0 when ac = +• oo , C and D must here vanish ; and by the condition to be satisfied when x = 0, B = nA. The solution applicable for the present purpose is thus
w = A smny . (1 + nx) e~~nx ......................... (7)
The force acting at the edge as = 0 necessary to maintain this displacement is proportional to
dV2w d? dw . d3w
in virtue of the condition there imposed. Introducing the value of w from (7), we find that
dsw I dx* = 2n*A sinny, ........ , .................. (9)
which represents the force in question. When n - 1,
w = A siny. (1 +x)e~x; ........................ (10)
and it is evident that w retains the same sign over the whole plate from oo = 0 to cc = oo . On the negative side (10) is not applicable as it stands, but we know that w has identical values at + SB.
The solution expressed in (10) suggests strongly that Resal's expectation is not fulfilled, but two objections may perhaps be taken. In the first place the force expressed in (9) with n=l, though preponderant at the centre y — \TT, is not entirely concentrated there. And secondly, it may be noticed that we have introduced no special boundary condition at oc = oo . It might be argued that although w tends to vanish when x is very great, the manner of its evanescence may not exclude a reversal of sign.when one side of the rectangle is relatively long. It seems therefore desirable to inquire more closely into this question.