The Physics
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Opus in profectus

# Resonance

## Discussion

Radio transmitters (including television transmitters, satellite uplinks, and cellular phones) broadcast at or near a particular frequency. We are swimming in a sea of different radio frequencies from both natural and artificial sources. How does your radio (television, satellite dish, or cellular phone) select the correct frequency from among all the others for decoding?

Even as a boy, I observed that one man alone by giving these impulses at the right instant was able to ring a bell so large that when four, or even six, men seized the rope and tried to stop it they were lifted from the ground, all of them together being unable to counterbalance the momentum which a single man, by properly timed pulls, had given it.

Galileo Galilei, 1638

This needs to be paraphrased. "An example of how local soil conditions can greatly influence local intensity is given by catastrophic damage in Mexico City from the 1985, MS 8.1 Mexico earthquake centered some 300 km away. Resonances of the soil-filled basin under parts of Mexico City amplified ground motions for periods of 2 seconds by a factor of 75 times. This shaking led to selective damage to buildings 15-25 stories high (same resonant period), resulting in looses to buildings of about \$4.0 billion and at least 8,000 fatalities."

Driven harmonic oscillator

The amplitude of a driven harmonic oscillator increases as the driving frequency approaches the natural frequency. In an undamped system, the amplitude will be infinite.

 A = 1 x = ω = ƒ √((1 − x2)2 + 4γ2x2) ω0 ƒ0

Fall of the Broughton suspension bridge, near Manchester (1831)

It appears that, on the day when this accident happened, the 60th regiment had had a field-day on Kersall Moor, and about 12 o'clock were on their way back to their quarters. The greater part of the regiment is stationed in the temporary barracks in Dyche-street, St. George's Road, and took the route through Strangeways; but one company, commanded, as it happened singularly enough, by Lieut. P.S. Fitzgerald, the son of the proprietor of the bridge, being stationed at the Salford barracks, took the road over the suspension bridge, intending to go through Pendleton to the barracks. Shortly after they got upon the bridge, the men, who were marching four abreast, found that the structure vibrated in unison with the measured step with which they marched; and as this vibration was by no means unpleasant, they were inclined to humour it by the manner in which they stepped. As they proceeded, and as a greater number of them got upon the bridge, the vibration went on increasing until the head of the column had nearly reached the Pendleton side of the river. They were then alarmed by a loud sound something resembling an irregular discharge of fire-arms; and immediately one of the iron pillars supporting the suspension chains, viz. that which was to the right of the soldiers, and on the Broughton side of the river, fell towards the bridge, carrying with it a large stone from the pier, to which it had been bolted. Of course that corner of the bridge, having lost the support of the pillar, immediately fell to the bottom of the river, a descent of about sixteen or eighteen feet; and from the great inclination thereby given to the road-way, nearly the whole of the soldiers who were upon it were precipitated into the river, where a scene of great confusion was exhibited. Such of them as were unhurt got out as well as they could, some by scrambling up the inclined plane which the bridge presented, and others by wading out on the Broughton side; but a number were too much hurt to extricate themselves without assistance, which was immediately rendered by their comrades.

Philosophical Transactions, 1831