Module 7 · oscillation, sound and the wave idea
A mass on a spring, a child on a swing, a guitar string, the air in front of a speaker, a bridge that will not stop shaking, and every note you have ever heard. All of them are one equation repeated — and once you can see that equation, you own about a third of physics, because light and quantum mechanics are built out of the same idea.
Pull a mass on a spring down by 5 cm and release it. It oscillates. Pull it down by 20 cm and release it. It swings four times as far. Does one full cycle now take longer, shorter, or exactly as long as before? Commit to an answer at the gate below before reading on — the reason lies in one particular force law:
You double the amplitude of a mass on a spring — you pull it twice as far before letting go. The time for one complete oscillation:
You do not need calculus to see where this comes from, only one new symbol beside Module 6. Here ω is the rate at which a circle is swept out, in radians per second; one whole turn is 2π radians, so the period of the turn is T = 2π/ω, and a point on a circle of radius A at rate ω moves at speed v = ωA. Module 6 gave that point's centripetal acceleration as v²/A, which is ω²A, pointing at the centre.
Now shine a torch from the side of the circle so the moving point casts a shadow on a wall. The shadow slides up and down. Its position is A sin(ωt) — which is what sine means. The shadow's acceleration is the shadow of the point's arrow, so when the shadow sits at displacement x from the middle, its acceleration is −ω²x. Nothing in the real experiment goes round in a circle: the mass moves in a straight line. The circle is only a device for working out the answer, and the match holds only for as long as F = −kx does — stretch a spring past its linear range, or swing a pendulum through a large angle, and the shadow picture begins to fail.
Compare that with what Newton says about our spring: a = F/m = −(k/m)x. The two are the same statement if ω² = k/m, so ω = √(k/m). The mass on the spring moves exactly like the shadow of something going round a circle, and with T = 2π/ω its period is:
Two consequences worth carrying around. A pendulum one metre long takes almost exactly two seconds for a round trip, which is why grandfather clocks are the height they are. And the pendulum formula contains g, so a pendulum clock taken up a mountain runs slow — a fact used for two centuries to map the Earth's gravity before anyone had a better instrument.
Take any object sitting in a stable position — a marble at the bottom of a bowl, an atom in a crystal, a molecule's two atoms holding each other at a fixed distance. Draw the potential energy against displacement: because it is a stable minimum, the curve has a bottom, and every smooth curve looks like a parabola if you zoom in far enough on its minimum. A parabola shaped like ½kx² is exactly the energy of a spring. And the force on the object is how steeply its energy climbs when you displace it, always pointing back down the slope; for a parabola that slope grows in strict proportion to displacement, giving F = −kx, which is simple harmonic motion. That is why this one equation describes springs, pendulums, atoms in a solid, the vibration of a CO₂ molecule, an electron in a radio antenna, and the tuning of a bridge. It is not a coincidence and it is not that nature loves springs. It is that everything looks like a spring if you do not push it too hard.
Hang your keys from a shoelace. Time twenty full swings with a phone stopwatch and divide by twenty — timing twenty and dividing is how you beat your own reaction time. Now do it again with the lace half as long. The period should drop by a factor of √2 ≈ 1.41, not by 2. Then swap the keys for something much heavier on the same length and confirm that nothing changes at all.
Every child works out how to make a swing go high, and no child is taught it. You do not push harder; you push in time. Small pushes, delivered at exactly the swing's own rhythm, add up cycle after cycle until the amplitude is enormous. Push at the wrong rhythm and you spend the same energy fighting yourself.
Every oscillator has a natural frequency — the rate it wobbles at when left alone, ω₀ = √(k/m) for our spring. Drive it at that frequency and the response is dramatic. Drive it at any other and the response is feeble. That peak is called resonance, and its sharpness is set entirely by how much friction the system has.
Slide the drive frequency slowly up through 1.00 and watch the swing grow and then collapse again. Then drop the damping to its lowest and repeat: the peak becomes a spike, and even a tiny push at exactly the right frequency produces a huge amplitude. That spike is why engineers both exploit resonance and design carefully to keep away from it.
| Where resonance shows up | What is oscillating, and what is driving it |
|---|---|
| A swing | a pendulum, driven by a person who has learned its frequency by trial and error |
| Tuning a radio | an electrical circuit whose natural frequency you adjust with a dial until it matches one station and ignores all the others |
| A wine glass shattering | the rim's own vibration, driven by a sound wave at exactly its pitch until the glass flexes further than it can survive |
| The Millennium Bridge, London, 2000 | the deck swayed a little; walkers unconsciously adjusted their steps to match; that made it sway more. Strictly this is not resonance — no outside driver had the bridge's frequency, the walkers were driven by the bridge — but the same feedback that appears in the correction below. It closed after two days, and reopened only after 37 large shock absorbers and 52 tuned masses had been bolted underneath to soak up the sway. |
| An MRI scanner | placed in a strong magnetic field, the hydrogen nuclei in your body wobble like slightly tilted spinning tops, at one particular rate fixed by the field's strength. The scanner drives them at exactly that rate, then stops and listens as the wobble dies away. |
| Soldiers on a bridge | armies genuinely do break step crossing bridges, and have since a column of troops brought down the Broughton suspension bridge in 1831 |
You have probably seen the film of the Tacoma Narrows bridge twisting itself apart in 1940, captioned "resonance". Most physics textbooks said so for decades, and it is not quite right. The wind was steady, not oscillating at the bridge's frequency, so there was no external driver to resonate with. What happened was something else, called aeroelastic flutter — "aeroelastic" simply means air and bendiness acting on each other. As the deck twisted, it changed the way the air flowed over it. That changed flow pushed the deck further in the direction it was already twisting. So the twist created the push, and the push created more twist — a feedback loop in which the structure drives itself, with the steady wind doing nothing but supplying the energy. It is more dangerous than a resonance, because there is no wrong frequency to avoid. Worth knowing both because the physics is better and because it is a good lesson in how a tidy story outlives its own correction.
Now line up a row of oscillators and connect each to its neighbour. Wobble the first one. It drags the second, which drags the third, and a disturbance travels down the line. No matter travels — each oscillator stays where it always was, moving only up and down about its own resting place. But one thing does travel besides the shape: energy, passed from each oscillator to its neighbour by the little work it does dragging it along. That is what lets a shout made here rattle a window over there. So a wave is a pattern that carries energy through a medium, while the medium itself stays put.
A duck sits on the sea while waves roll past toward the shore. What does the duck do?
The crucial and surprising part: for a given medium, the speed is fixed and the wave does not get to choose it. Sound travels through air at about 343 m/s whether it is a whisper or a shout, a bass note or a whistle — which is lucky, or an orchestra would arrive at the back of the hall in the wrong order. So when the frequency goes up, the wavelength must come down to compensate. The source sets f; the medium sets v; λ is whatever is left over.
Tick the longitudinal box. Every picture of sound you have seen is a wiggly line, but air does not wiggle sideways — it cannot, because a gas has nothing to hold it in place sideways. Air moves back and forth along the direction the sound is going, crowding together (compression) and spreading apart (rarefaction). The wiggly line is a graph of pressure against position, not a picture of the air.
Your eardrum is not being waved from side to side. It is being pushed in and pulled out, thousands of times a second, by pressure changes of a few parts in a million. In ordinary conversation a metre away, that motion is about the width of a large molecule. For the very quietest sound you can hear, it is smaller than a single atom.
| What you perceive | What the wave is doing |
|---|---|
| Pitch — high or low note | frequency. Middle C is 262 Hz. Human hearing runs about 20 Hz to 20,000 Hz, and the top of that range falls with age. |
| Loudness | amplitude — how big the pressure swings are. Measured in decibels, a logarithmic scale: +10 dB is ten times the energy but sounds about twice as loud. |
| Timbre — why a violin and a flute playing the same note sound different | no real instrument makes a single pure frequency. It makes the note you name plus a set of quieter tones at two, three and four times that frequency, and the recipe of quieter tones differs from one instrument to another. Same lowest frequency, different shape of wave. Bench 4 shows where those extra tones come from. |
This one leans on Module 4. Sound travels by molecules bumping into their neighbours, so its speed must be tied to how fast those molecules are already moving. Module 4 gave that typical speed as vrms = √(3kT/m), where T is the absolute temperature, m is the mass of one molecule, and k is Boltzmann's constant — a fixed number of nature, nothing to do with the spring stiffness k used earlier in this module. Sure enough, the speed of sound in air is about 331 + 0.6T m/s with T in °C: 331 at freezing, 343 at 20 °C, 355 on a 40 °C Chennai afternoon. And note what is absent: pressure. Squeeze the air to twice the pressure at the same temperature and you double the density (more molecules to push) and double the springiness (they push back harder), and the two cancel exactly. Sound at the top of a mountain travels at the same speed as at sea level, as long as the temperature is the same. Helium is a different story — same temperature, but molecules seven times lighter and rattling in only three directions rather than five, so sound travels through it at about 1005 m/s, which is why a lungful of helium raises the pitch of the resonances in your throat and not the pitch of your vocal cords.
Here is the rule that makes waves interesting, and it is almost too simple to be worth stating: when two waves arrive at the same point, the displacements simply add. Crest on crest makes a double-height crest. Crest on trough makes nothing at all. Nothing is destroyed; the two waves pass through each other and carry on unchanged, which is why you can hear two people talking at once.
Three things fall out of that one rule, and the bench does all three.
First, what a phase difference is. Two waves of the same frequency need not arrive in step: one can be running a fraction of a cycle behind the other. Measure that fraction as an angle, counting one full cycle as 360°, and you have the phase difference. At 0° the two are crest on crest. At 180° each is half a cycle behind the other, so one has a trough exactly where the other has a crest — it is the same wave turned upside-down.
Now set the mode to phase and slide it to 180°. The sum vanishes: two real waves, each carrying real energy, adding to silence. This is not a trick: it is how noise-cancelling headphones work — a microphone hears the engine drone, the electronics produce the same wave upside-down, and your eardrum receives their sum, which is nothing. (Where does the energy go? It never arrives at your ear in the first place. Adding an upside-down wave changes the whole sound field, and the energy that would have reached your eardrum ends up elsewhere — reflected, absorbed inside the headphone, or simply travelling in another direction. Energy is conserved everywhere and always; what the headphones change is where it goes, not how much of it there is.)
Switch to beats and set the second frequency slightly above 1. The sum swells and fades, swells and fades. Here is why, and how often. Take one wave at 440 cycles a second and another at 443, and start them crest on crest: loud. The faster wave gains three whole cycles on the slower one every second, so after one sixth of a second it has gained half a cycle — crest against trough, which is silence. A sixth of a second later it has gained a full cycle, the two are back in step, and it is loud again. The loud-quiet-loud cycle happens three times a second, because three is the number of whole cycles the faster wave gains in a second. The number of swells per second is exactly the difference of the two frequencies: two notes 3 Hz apart give three throbs a second. Every guitarist in the world tunes by this: play the string against a reference, listen for the throbbing, and tighten until the throbbing slows to nothing. It is a way of measuring a tiny frequency difference with no instrument except your ear, and it is far more sensitive than trying to judge the two pitches separately.
Switch to standing. Now the two waves travel in opposite directions — which is what happens when a wave on a guitar string reflects off the fixed end and comes back. The sum no longer travels at all. Certain points, the nodes, never move; halfway between them the antinodes swing wildly. The pattern stands still and breathes.
A string clamped at both ends can only hold patterns with a node at each end, so only certain wavelengths fit: 2L, L, 2L/3, and so on. Since v = fλ and v is fixed by the string, only certain frequencies are allowed:
Read that box again with one substitution. Confine anything wave-like to a finite region, and only certain frequencies fit. Confine an electron to an atom, and only certain energies fit — which is why atoms have sharp spectral lines, why the periodic table has the shape it does, and why matter is stable at all. The word "quantum" just means that only certain values are allowed, and that is the same phenomenon as a guitar string's allowed notes. One warning about how far this picture goes, though: a guitar string is a real material object shaking in one dimension, and you can watch it do it. An electron's wave is three-dimensional, is not made of any material, and describes a probability rather than a displacement. The idea of allowed patterns carries across exactly; the picture of something wiggling does not. When you get to Module 22, come back and read this page first.
An ambulance goes past with its siren on and the pitch drops as it passes — not gradually the whole time, but sharply, at the moment it passes you. The siren itself never changes.
Why is the pitch higher while the ambulance is approaching?
Notice that the rings themselves are perfect circles, each one expanding at the same speed from the point where it was born. Nothing about the sound is different. What changed is only where each crest started, and that is enough.
Now push the source speed past 1.0. The source is outrunning its own sound. The circles pile up along a cone. When that cone sweeps across a listener, a great many crests arrive together as a single violent pressure jump: a sonic boom. Note that it is not one bang at the instant the aircraft passes the speed of sound. The cone is dragged behind the aircraft for as long as it stays supersonic, sweeping over the ground like an invisible wake, and every listener along the route gets their own boom as the cone reaches them.
A bat screams at 50 kHz and listens for the echo; the returning frequency tells it whether the moth is coming or going, and a few species of moth have evolved to jam the signal. A traffic radar gun bounces a radio wave off your car and measures the Doppler shift of the return. And the biggest one of all: light from almost every distant galaxy arrives with its spectral lines shifted toward the red — nearly the same effect, applied to light, except that for the most distant galaxies the stretching happens to the light in flight, because space itself is expanding, rather than because the galaxy is moving through space — and the further away the galaxy, the bigger the shift. That single observation, made by Hubble in 1929, is how we know the universe is expanding.
1. A pendulum clock that keeps perfect time in Chennai is taken to the Moon, where g is one sixth of Earth's. Compared with a real second, its pendulum now takes:
2. A guitar string plays a note of 220 Hz. You press it exactly halfway along, halving its length, and pluck again. The new frequency is:
3. You stand still and an ambulance drives past at constant speed with a constant siren. As it passes, the pitch you hear: