How the Physics of Resonance Shapes Reality

Almost anytime physicists announce that they’ve discovered a new particle, whether it’s the Higgs boson or the recently bagged double-charm tetraquark, what they’ve actually spotted is a small bump rising from an otherwise smooth curve on a plot. Such a bump is the unmistakable signature of “resonance,” one of the most ubiquitous phenomena in nature.

Resonance underlies aspects of the world as diverse as music, nuclear fusion in dying stars, and even the very existence of subatomic particles. Here’s how the same effect manifests in such varied settings, from everyday life down to the smallest scales.

In its simplest form, resonance occurs when an object experiences an oscillating force that’s close to one of its “natural” frequencies, at which it easily oscillates. That objects have natural frequencies “is one of the bedrock properties of both math and the universe,” said Matt Strassler, a particle physicist affiliated with Harvard University who is writing a book about the Higgs boson. A playground swing is one familiar example: “Knock something like that around, and it will always pick out its resonant frequency automatically,” Strassler said. Or flick a wineglass and the rim will vibrate a few hundred times per second, producing a characteristic tone as the vibrations transfer to the surrounding air.

A system’s natural frequencies depend on its intrinsic properties: For a flute, for instance, they are the frequencies of sound waves that exactly fit inside its cylindrical geometry.

The Swiss mathematician Leonhard Euler solved the equation describing a system continuously driven near its resonant frequency in 1739. He found that the system exhibited “various and wonderful motions,” as he put it in a letter to fellow mathematician Johann Bernoulli, and that, when the system is driven precisely at the resonant frequency, the amplitude of the motion “increases continually and finally grows out to infinity.”

Driving a system too hard at the right frequency can have dramatic effects: A trained singer, for instance, can shatter a glass with a sustained note at its resonant frequency. A bridge resonating with the footsteps of marching soldiers can collapse. But more often, energy loss, which Euler’s analysis neglected, prevents the motion of a physical system from growing unchecked. If the singer sings the note quietly, vibrations in the glass will grow at first, but larger vibrations cause more energy to radiate outward as sound waves than before, so eventually a balance will be achieved that results in vibrations with constant amplitude.

Now suppose the singer starts with a low note and continuously glides up in pitch. As the singer sweeps past the frequency at which the wineglass resonates, the sound momentarily grows much louder. This enhancement arises because the sound waves arrive at the glass in sync with vibrations that are already present, just as pushing on a swing at the right time can amplify its initial motion. A plot of the sound amplitude as a function of frequency would trace out a curve with a pronounced bump around the resonant frequency, one that’s strikingly similar to the bumps heralding particle discoveries. In both cases, the bump’s width reflects how lossy the system is, indicating, for instance, how long a glass rings after it is struck once, or how long a particle exists before it decays.

But why do particles behave like humming wineglasses? At the turn of the 20th century, resonance was understood to be a property of vibrating and oscillating systems. Particles, which travel in straight lines and scatter like billiard balls, seemed far removed from this branch of physics.

The development of quantum mechanics showed otherwise. Experiments indicated that light, which had been thought of as an electromagnetic wave, sometimes behaves like a particle: a “photon,” which possesses an amount of energy proportional to the frequency of the associated wave. Meanwhile, matter particles like electrons sometimes exhibit wavelike behavior with the same relation between frequency and energy.