1916] VIBRATIONS AND DEFLEXIONS OF MEMBRANES, BARS, AND PLATES 423
If we replace the reactions against acceleration by external forces, we may obtain the solution of a statical problem. When a membrane of any shape is submitted to transverse forces, all in one direction, the displacement is everywhere in the direction of the forces.
Similar conclusions may be formulated for the conduction of heat in two dimensions, which depends upon the same fundamental differential equation. Here the- boundary is maintained at a constant temperature taken as zero, and "persistences" replace the periods of vibration. Any closing in of the boundary reduces the principal persistence. In this mode there can be no internal place of zero temperature. In the steady state under positive sources of heat, however distributed, the temperature is above zero everywhere. In the application to the theory of heat, extension may evidently be made to three dimensions.
Arguments of a like nature may be used when we consider a bar vibrating transversely in virtue of rigidity, instead of a stretched membrane. In Theory of Sound, §184, it is shown that whatever may be the constitution of the bar in respect of stiffness and mass, a curtailment at either end is associated with a rise of pitch, and this whether the end in question be free, clamped, or merely " supported."
In the statical problem of the deflexion of a bar by a transverse force locally applied, the question may be raised whether the linear deflexion must everywhere be in the same direction as the force. It can be shown that the answer is in the affirmative. The equation governing the deflexion (w) is
?
dx*
>d*w
where Zdx is the transverse force applied at doc, and B is a coefficient of stiffness. In the case of a uniform bar B is constant and w may be found by simple integration. It suffices to suppose that Z is localized at one point, say at x = b; and the solution shows that whether the ends be clamped or supported, or if one end be clamped and the other free or supported, w is everywhere of the same sign as Z. The conclusion may evidently be extended to a force variable in any manner along the length of the bar, provided that it be of the same sign throughout.
But there is no need to lay stress upon the case of a uniform bar, since the proposition is of more general application. • The first integration of (1)
gives
d frtd2w\ I® „, n /9x
T-f-fi-Tl)- Zdx + C,.........................(2)
dx\ da?J Jo
and fZdm = 0 from x: = 0 at one end to x = b, and takes another constant value (Zj) from at = b to the other end at x = l A second integration now shows as is shown by the swellings between the disks. Indeed the propagation of any wave at all is inconsistent with uniformity of pressure within the jet.l as of the particles of fluid with each other, is more directly shewn by an experiment on the continuance of a column of mercury, in the tube of a barometer, at a height considerably greater than that at which it usually stands, on account oi the pressure of the atmosphere. If the mercury has been well boiled in the tube, it may be made to remain in contact with the closed end, at the height of 70 inches or more " (Young's Lectures, p. 626,1807). If the mercury be wet, boiling may be dispensed with and negative pressures of two atmospheres are easily demonstrated.