The time rate of generation of heat is thus Rc2, or R (dQ ⁄ dt)2, when the units in which R and c are expressed are such as to make this quantity a rate of doing work in the true dynamical sense. This is the rate at which the sum of energy already found is being diminished, and so the equation
holds, or leaving out the common factor dQ ⁄ dt, the equation
This last equation was established by Thomson, and is precisely that which would be obtained for a pendulum bob of mass L, pulled back towards the position of equilibrium with a force Q ⁄ C, where Q is the displacement from the middle position, and having its motion damped out by resisting force of amount R per unit of the velocity.
It is more instructive perhaps to take the oscillatory motion of a spiral spring hung vertically with a weight on its lower end, as that which has a differential equation equivalent to the equation just found. When the stretch is of a certain amount, there is equilibrium—the action of the spring just balances the weight,—and if the spring be stretched further there will be a balance of pull developed tending to bring the system back towards the equilibrium position. If left to itself the system gets into motion, which, if the resistance is not too great, is added to until the equilibrium position is reached; and the motion, which is continued by the inertia of the mass, only begins to fall off as that position is passed, and the pull of the spring becomes insufficient to balance the weight. Thus the mass oscillates about the position of equilibrium, and the oscillations are successively smaller and smaller in extent, and die out as their energy is expended finally in doing work against friction.
If the resisting force for finite motion is very great, as for example when the vibrating mass of the pendulum or spring is immersed in a very viscous fluid, like treacle, oscillation will not take place at all. After displacement the mass will move at first fairly quickly, then more and more slowly back to the position of equilibrium, which it will, strictly speaking, only exactly reach after an infinite time. The resisting force is here indefinitely small for an indefinitely small speed, but it becomes so great when any motion ensues, that as the restoring force falls off with the displacement, no work is finally done by it, except to move the body through the resisting medium.
The differential equation is applicable to the spring if Q is again taken as displacement from the equilibrium position, L as the inertia of the vibrating body, 1 ⁄ C as the pull exerted by the spring per unit of its extension (that is, the stiffness of the spring), and R has the same meaning as before.
In this case of motion, as well as in that of the pendulum, energy is carried off by the production of waves in the medium in which the vibrator is immersed. These are propagated out from the vibrator as their source, but no account of them is taken in the differential equation, which in that respect is imperfect. There is no difficulty, only the addition of a little complication, in supplying the omission.
The formation of such waves by the spiral spring vibrator can be well shown by immersing the vibrating body in a trough of water, and the much greater rate of damping out of the motion in that case can then be compared with the rate of damping in air.