Physics · oscillations
A weight on a spring, a swing, a string, a bridge, a skyscraper, the crystal in a watch — they all oscillate by the same laws. Release springs and pendulums, watch friction damp out the motion, pump up a swing with nothing but pushes, sweep the frequency through resonance, pour energy back and forth between pendulums and excite standing waves.
A weight on a spring and a pendulum side by side — with an oscilloscope recording their motion.
Pull the weight down or swing the bob out and let go. The spring's period doesn't depend on how far you stretch it — check it.
For small swings, a pendulum's period hardly depends on the amplitude. For large ones, it grows.
Curve: exact formula (elliptic integral). Dot: the period measured on the simulated pendulum above.
A point goes around at constant speed. Its shadow on the vertical axis moves exactly like a weight on a spring — that's harmonic motion.
The angle grows steadily: φ = ωt, and the height of the shadow is A sin ωt. More about the sine: Trigonometry.
Friction and drag eat up the energy of oscillation. How much damping is just right?
Increase the damping and release the weight each time. At what c does it come back fastest without overshooting?
The bigger Q, the weaker the damping and the longer something “rings.” The amplitude drops by a factor of e (to about 37%) after roughly Q/π oscillations. Rough values, orders of magnitude.
Logarithmic scale: equal steps mean multiplying — from 1 to 10¹⁰ is ten steps of ×10. An atomic clock “rings” about 10 billion times longer than a car's suspension.
Push a system in rhythm with its own oscillations, and the amplitude grows many times over.
Move the frequency slider slowly and wait for the amplitude to settle. Where does the weight go wildest?
Earlier series of measurements (for a different damping) stay as faint traces — compare the heights and widths of the peaks.
Push when the swing is at the back and just starting to head forward. Goal: a 1 m swing.
The LC circuit in a receiver “rings” at only one frequency. Turning the knob changes the capacitance C — and picks the station.
Station names and frequencies are made up. Model: a 0.10 µH coil, circuit quality factor about 60.
Not everything called resonance really is. For each example — what was actually going on.
The bridge in the US twisted in a wind of about 65 km/h and collapsed on November 7, 1940. The wind wasn't “pushing in rhythm” — it was fairly steady. The twisting deck itself changed the airflow so that the wind pumped in more energy with every wobble. That's aeroelastic flutter, not simple forced resonance. You can't save such a bridge by choosing a different wind frequency — you have to change the shape and stiffness of the deck.
Right after opening, the footbridge began to sway from side to side. When we walk, we shift our weight slightly from foot to foot — about once a second sideways. Once the bridge started to wobble, pedestrians matched their steps to its swaying, which rocked it even more. It was closed after two days, fitted with dampers and reopened in 2002.
The suspension bridge at Broughton near Manchester collapsed while soldiers marched across it in step. The rhythmic marching is thought to have set the structure swaying, although sources also point to a flaw in the bridge. Ever since, troops cross bridges out of step (“break step”).
A wine glass has its own tone (tap it). A very loud, pure sound at exactly that frequency can shake it until it cracks. It rarely works: you need thin glass and enormous volume, usually from a loudspeaker. Singers have managed it only in a handful of demonstrations.
Microwaves at 2.45 GHz jerk water molecules around as they try to turn with the alternating field; the “friction” turns energy into heat. That's dielectric heating. Water absorbs microwaves over a broad range of frequencies, and 2.45 GHz is not its resonance — the band was chosen partly because the waves penetrate a few centimeters into food.
In a strong magnetic field, hydrogen nuclei spin like tops and precess at about 42.6 MHz per tesla (64 MHz in a 1.5 T scanner). Radio waves at exactly that frequency are absorbed by them and, once switched off, given back. From that signal a computer builds an image of the inside of the body.
An antenna picks up hundreds of stations at once. A circuit with a coil and a capacitor has a natural frequency f = 1/(2π√(LC)) and responds strongly only to that — the rest is muted. Try the knob above.
Connect two pendulums with a little spring, and they'll pass energy back and forth.
Swing the left pendulum and watch: after a while it stops, and the right one takes over all the energy.
Two metronomes on a board resting on cans. Each ticks a little differently — and yet after a while they tick together.
Two tones with close frequencies take turns reinforcing and canceling each other — the loudness “wobbles” f₁ − f₂ times per second.
Many connected oscillating bodies already make a wave. And a wave trapped between two ends has its favorite shapes — modes.
Jerk the end — a hump will run along the chain and bounce off the wall upside down. Then turn on the drive and look for the frequencies at which a standing wave forms.
A vibrating plate sprinkled with sand: the grains bounce where the plate vibrates and gather on the nodal lines, which stay still.
Every building has its own natural frequency. A tall one — low; a short one — high. The ground shakes at the frequency you choose.
Oscillations keep time, buzz, beat in our chests — and sometimes break machines.
Laundry bunched into one lump is a spinning weight. When the drum's speed hits the natural frequency of the suspension, the whole machine shakes.
From a skyscraper swaying to an atom vibrating — twelve orders of magnitude. Typical, approximate values.
An unbalanced car wheel at 100 km/h turns about 14 times a second. When that rhythm hits the natural frequency of the suspension or steering, the steering wheel starts to shake — usually only within a certain speed range. The cure: balancing the wheel.
We walk at about 2 steps per second, and fans jump 2–3 times a second. Footbridge designers try to keep natural frequencies out of that range, or add dampers.
As it speeds up, the drum passes through the suspension's resonance — it shakes harder for a moment, then calms down at high speed. Newer machines detect imbalance and redistribute the laundry first.
Air vortices peel off behind a tall building, alternately from the left and right side. When the vortex rhythm approaches the building's natural frequency, it sways across the wind. That's why skyscrapers have twisted or cut-off corners, and dampers.
The heart has a rhythmic “pacemaker” — its own electrical oscillator that stimulates the muscle about 60–80 times a minute (Heart). A hummingbird beats its wings dozens of times a second, and a mosquito several hundred times — its wings make that thin, irritating whine.
Oscillation is motion back and forth around an equilibrium position. It needs a force that always pulls back, and inertia, which makes the body “overshoot” equilibrium. The simplest case is a weight on a spring. The spring pulls harder the more you stretch it — that's Hooke's law.
This kind of motion is simple harmonic motion. Its period doesn't depend on the amplitude: a weight pulled twice as far has twice as far to go, but also twice the force, so it comes back in the same time. Gravity only shifts the equilibrium position of a vertical spring (by mg/k) and doesn't change the period.
In a pendulum, the restoring force is the component of the weight mg sin θ. For small angles sin θ ≈ θ, and the pendulum also oscillates harmonically. The mass cancels out, so the period depends only on the length and on g. On the Moon the same pendulum swings about 2.5 times slower. At large angles sin θ is noticeably smaller than θ, the restoring force weakens and the period grows.
Air resistance, friction and the oil in a shock absorber take energy away. The simplest model is a damping force proportional to velocity:
When something pushes a system periodically at frequency f, after a while the system oscillates at exactly f, not at its own natural frequency f₀. The amplitude depends on how close f is to f₀:
At resonance, the driving force is in step with the body's velocity, so it adds energy every cycle — until the damping losses equal what the drive supplies. Hence the rule of the swing: push in rhythm with its own swings, at the moment it starts moving away from you.
Two identical pendulums connected by a spring have two modes of oscillation: in-phase (the spring doesn't deform, ω₁ = √(g/l)) and anti-phase (the spring works, ω₂ = √(g/l + 2κ)). Any other motion is a sum of these two. When you swing one pendulum, the two modes run slightly differently and take turns reinforcing and canceling each other, and the energy flows from pendulum to pendulum. You hear the same thing as beats between two tones.
A chain of many beads is already a medium that waves can travel through. A wave trapped between the ends reflects and overlaps with itself. At the mode frequencies a standing wave forms, with nodes that stay still. A string, a Chladni plate and a building have many such modes.
Simplifications: springs are massless and perfectly linear; the pendulum is a point mass on a light, rigid rod; damping is proportional to velocity. The swing is a pendulum with slight drag, and a push is a constant force for 0.35 s. The metronomes have an escapement modeled as a van der Pol term, and the board rolls on the cans with slight resistance. Chladni figures are computed with the simplified formula cos(mπx)cos(nπy) ± cos(nπx)cos(mπy); a real plate has somewhat different shapes and frequencies. A building is a single oscillating “weight” with the rule of thumb T ≈ 0.1 s × number of floors (real buildings can differ quite a bit), and the damper in the model is exaggeratedly heavy so the effect is clear. Motion is computed by a custom 4th-order Runge-Kutta integrator (the chain — semi-implicit Euler) with a small fixed step.