Resonance Blows the Amplitude Up
Anyone who has been on a swing knows resonance in their body. Kick your feet at random and the swing wobbles awkwardly, but kick exactly in time with the swing's return and you climb higher and higher. Add a small push in the same rhythm every time, and the swinging piles up and up until it's enormous. This is resonance. In lesson C2 we said every oscillating system has its own natural frequency ω₀. Apply a periodic force Fcos(ωt) from outside, and the size of the response depends on how close the force's frequency ω is to ω₀. The moment ω matches ω₀, the amplitude shoots up as if exploding. Tuning a radio, a bridge collapsing, a microwave heating water — all of it is this single phenomenon. In this lesson you'll turn the drive frequency yourself and watch, with your eyes, when and how the amplitude explodes.
First, set the stage. The oscillating system in C2 was a closed system: touch it once and it rings on its own and dies down. Now we add an external force. Someone from outside keeps pushing in a steady rhythm. In the equation, an Fcos(ωt) appears on the right: m x'' + c x' + k x = Fcos(ωt). Here ω is the pusher's rhythm, the drive frequency — separate from ω₀, the rhythm the system itself wants to swing at. Press play. The red force arrow pushes left and right, and the mass, dragged by that force, swings. Watch closely and the mass swings at exactly the same rhythm as the force, but follows a touch late. This lag is called the phase delay. Wait a moment and the initial raggedness settles down, and it locks into a steady state, swinging regularly with a fixed amplitude. What we want to know is exactly how big that steady-state amplitude is.
Now the key question. Keep the strength F of the push the same and change only the rhythm ω — how does the amplitude change? Even pushing with the same strength, the result is completely different depending on the rhythm. Plot drive frequency ω on the horizontal axis and steady-state amplitude on the vertical, and you get this resonance curve. Drag ω and raise it slowly. When ω is very low, the force is so sluggish that the mass simply follows it; the amplitude is small and flat. Raise ω toward ω₀ and the amplitude climbs steeply — each push's rhythm matches the rhythm the system wants to swing at, so energy piles up efficiently. Raise ω past ω₀ and now the force is too fast for the mass to keep up, and the amplitude shrinks again. In the end, the amplitude is a curve that forms a peak near ω₀. Rhythm is everything.
Look closely at the exact spot where the peak forms, ω = ω₀. It's the point where the drive frequency matches the system's natural frequency precisely. Compare the three cases with the buttons. When ω is below or above ω₀ the mass swings moderately, but when ω = ω₀, pushing with the same strength makes the swinging overwhelmingly larger. Why here, of all places? Think of the swing. Push at the very moment the swing is moving forward and the pushing force adds 100% to the motion. When the rhythm is off, some pushes help and some hinder, and they eat into each other. At ω = ω₀ every push adds in perfectly the same direction, so energy doesn't leak away and keeps piling up. That's how a small force builds an enormous amplitude. This is the heart of resonance.
So does the amplitude really grow without limit? Damping stops it. The damping ζ from C2, that loss which burns energy, holds down the explosion of resonance. Drag ζ and change it. When damping is very small, the peak is sky-high and sharp. The energy that piles up each cycle barely leaks, so it grows enormous only at exactly the ω₀ rhythm. Increase the damping and the peak drops and broadens. The loss is large, so the amplitude can't pile up indefinitely, and it responds moderately over a wide range around ω₀. The sharpness of this peak is called the Q factor. High Q (small damping) responds narrowly and keenly; low Q responds broadly and bluntly. A radio needs high Q so neighboring stations don't bleed in and it locks onto one frequency clearly; a bridge needs low Q so it won't collapse even if the wind happens to match. Design is, in the end, a question of where to place this Q.
This single phenomenon hides all over the world. The swing is on the friendly side: adding a small force in rhythm to enjoy a large amplitude — using resonance on purpose. The bridge is on the frightening side. In 1940 the Tacoma Narrows Bridge collapsed when a periodic force made by the wind matched the bridge's natural frequency and the amplitude grew uncontrollably. That's why bridges and buildings are designed so their natural frequency doesn't overlap common external forces, with enough damping to keep Q low. The radio is on the clever side. Countless broadcast signals arrive at the antenna at once, but tune the circuit's natural frequency to the station you want and only that signal is amplified large by resonance and singled out. Turning the dial means changing the circuit's ω₀. Resonance, used well, is a tool; suffered unknowingly, a catastrophe.