resonance in differential equations
Anyone who has ridden a swing knows resonance in their body. Pump your legs at random and the swing wobbles halfheartedly, but pump right in time with its return and it climbs higher and higher. Add a small force in the same rhythm every time and the swinging piles up until it is huge. That is resonance. In the last lesson we saw that every vibrating system has its own natural frequency ω₀. Apply a periodic outside force Fcos(ωt), 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 it exploded. Tuning a radio, a bridge collapsing, and the body of an instrument making its sound louder all connect to this one phenomenon. In this lesson you change the driving frequency yourself and watch when and how the amplitude explodes.
First, set the stage. The vibrating system of the last lesson was closed: nudge it once and it swings and dies out on its own. Now add an external force: someone outside keeps pushing at a steady beat. In the equation, Fcos(ωt) appears on the right-hand side: m x'' + c x' + k x = Fcos(ωt). Here ω is the pusher's beat, the driving frequency, and it is separate from ω₀, the beat at which the system itself wants to swing. Press "Play" and the red force arrow pushes left and right while the gold mass on the green spring is dragged along by it. Look closely and the mass swings at the same beat as the force but lags slightly behind; that time difference is called the phase lag. In a real system, the initial jumble settles down into a steady state that swings regularly with a constant amplitude, and what the figure draws is exactly that steady state. What we want to know is how large that steady-state amplitude is.
Now the central question. Keep the pushing force the same strength and change only the beat ω: how does the amplitude change? Push with the same strength, and the result is completely different depending on the beat. Plot driving frequency ω on the horizontal axis and steady-state amplitude on the vertical axis, and you get this resonance curve. Drag ω slowly upward on the track below and the amplitude ratio at that frequency appears as a number. When ω is very low, the force is so sluggish that the mass simply follows it, and the amplitude is small and flat. Raise ω toward ω₀ and the amplitude climbs steeply, because the pushing beat lines up with the beat the system wants to swing at, so energy accumulates efficiently. The gold tick on the track marks that ω₀. Raise ω past ω₀ and now the force is too fast for the mass to follow, so the amplitude shrinks again. In the end, the amplitude traces a curve that peaks near ω₀. The beat is everything.
Look closely at the spot where the peak forms, ω = ω₀: the point where the driving frequency exactly matches the system's natural frequency. Compare the three cases with the three buttons: when ω is below or above ω₀ the mass swings moderately, but when ω = ω₀ the swinging becomes overwhelmingly large even though the push has the same strength. The amplitude ratio shown below makes the difference plain. Why here of all places? Think of the swing. Push at exactly the moment it moves forward and 100% of the push adds to the motion. When the beat is off, some pushes help and others hinder, and they eat into each other. At ω = ω₀ every push adds in the same direction, so energy does not leak away but keeps building. That is how a small force produces a huge amplitude. This is the heart of resonance.
So does the amplitude really grow without bound? Damping stops it. The damping ζ from the last lesson, the loss that eats away energy, holds down the explosion of resonance. Drag ζ along the track below: when the damping is very small, the peak is sky-high and needle-sharp. Almost no accumulated energy leaks away, so the response becomes enormous only at exactly the beat ω₀. Increase the damping and the peak becomes lower and broader. The losses are large, so the amplitude cannot pile up without limit, and the system responds moderately over a wide range near ω₀. The sharpness of this peak is called the Q factor, and its value appears below together with ζ. A large Q means small damping and a narrow, sharp response; a small Q means a broad, dull one. A radio needs a large Q to pick up one frequency clearly without neighboring stations mixing in, while a bridge needs a small Q so it does not collapse when the wind happens to hit the right beat. Design comes down to where to put this Q.
This one phenomenon hides all over the world. The swing is the friendly side: adding a small force in time to enjoy a large amplitude, using resonance on purpose. The bridge is the frightening side. In 1940 the Tacoma Narrows Bridge collapsed when the wind locked onto the bridge's twisting motion and kept pumping energy into it until the amplitude grew out of control. Strictly speaking it was flutter, in which the wind amplifies the vibration by itself, but the skeleton is the same: energy piling up because it arrives in step with the vibration. That is why bridges and buildings are designed so their natural frequencies stay clear of common external forces, with enough damping to keep Q low. The radio is the clever side. Countless broadcast signals arrive at the antenna at once, and tuning the circuit's natural frequency to the station you want makes that signal alone grow large through resonance, picking it out. Turning the dial means changing the circuit's ω₀. Used well, resonance is a tool; ignored, it is a disaster.