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Resonance

No description of SHM would be complete without some discussion of the general phenomenon of resonance, which has many practical consequences that often seem very counterintuitive.^13.10 I will, however, overcome my zeal for demonstrating the versatility of Mathematics and stick to a simple qualitative description of resonance. Just this once.

The basic idea is like this: suppose some system exhibits all the requisite properties for SHM, namely a linear restoring "force"   $Q = - k \, q$  and an inertial factor  $\mu$. Then if once set in motion it will oscillate forever at its "resonant frequency"   $\omega = \sqrt{k \over \mu}$,  unless of course there is a "damping force"   $D = - \kappa \mu q$  to dissipate the energy stored in the oscillation. As long as the damping is weak  [ $\kappa \ll \sqrt{k \over m}$],  any oscillations will persist for many periods. Now suppose the system is initially at rest, in equilibrium, ho hum. What does it take to "get it going?"

The hard way is to give it a great whack to start it off with lots of kinetic energy, or a great tug to stretch the "spring" out until it has lots of potential energy, and then let nature take its course. The easy way is to give a tiny push to start up a small oscillation, then wait exactly one full period and give another tiny push to increase the amplitude a little, and so on. This works because the frequency  $\omega$  is independent of the amplitude  q₀.  So if we "drive" the system at its natural "resonant" frequency  $\omega$, no matter how small the individual "pushes" are, we will slowly build up an arbitrarily large oscillation.^13.11

Such resonances often have dramatic results. A vivid example is the famous movie of the collapse of the Tacoma Narrows bridge, which had a torsional [twisting] resonance^13.12 that was excited by a steady breeze blowing past the bridge. The engineer in charge anticipated all the other more familiar resonances [of which there are many] and incorporated devices specifically designed to safely damp their oscillations, but forgot this one. As a result, the bridge developed huge twisting oscillations [mistakes like this are usually painfully obvious when it is too late to correct them] and tore itself apart.

A less spectacular example is the trick of getting yourself going on a playground swing by leaning back and forth with arms and legs in synchrony with the natural frequency of oscillation of the swing [a sort of pendulum]. If your kinesthetic memory is good enough you may recall that it is important to have the "driving" push exactly ${\pi \over 2}$ radians [a quarter cycle] "out of phase" with your velocity - i.e. you pull when you reach the motionless position at the top of your swing, if you want to achieve the maximum result. This has an elegant mathematical explanation, but I promised . . . .