Resonance — why some frequencies are special — Physics

Before this

This leads to

Damped and forced oscillations · coming soon
Resonance in circuits · coming soon
The electromagnetic spectrum

A bridge that thousands of people shook by walking

On 10 June 2000, a new footbridge opened across the River Thames in London: the Millennium Bridge. Thousands of people walked across it that day. As the crowd grew, the bridge began to sway from side to side, and people had to hold the rails to keep their balance. Two days later it was closed. It stayed closed for almost two years.

Nobody was jumping. Nobody was pushing hard. Each person was simply walking, and a walker pushes on the ground with a force far smaller than the weight of a steel bridge. So how did ordinary footsteps make a bridge move?

The answer is not in how hard the bridge was pushed. It is in when. The last lecture ended with a question: what happens when you drive a pipe, a string or a bridge at one of its allowed frequencies from outside? The answer is resonance, and this lecture explains it.

Every object has its own rhythm

Pull a child on a swing back and let go. The swing moves to and fro at its own steady rate. Pluck a guitar string, tap a wine glass, or hit a tuning fork, and each one vibrates at its own frequencies. These are natural frequencies: the frequencies at which an object oscillates when it is disturbed once and then left alone. We write the lowest one as $f_{0}$ .

You have already met natural frequencies. In the harmonics lecture, a string fixed at both ends could vibrate only at $f_{1}, 2 f_{1}, 3 f_{1}$ and so on: its natural frequencies. A pipe has its own list. A swing has essentially one. A bridge has several: it can bounce, sway sideways or twist, each at its own frequency.

An object left alone in this way is in free oscillation. Now push it again and again, at a rate you choose. That is a forced oscillation, and the rate at which you push is the driving frequency. After a short start-up, the object moves at the driving frequency, not its own. The question is how large the movement becomes.

Pushing a swing

Think about what one push does. A push adds energy only if it acts in the same direction as the swing is moving. If you push forwards while the swing is coming forwards, you add energy. If you push forwards while the swing is coming back towards you, you take energy away.

Now push at the swing's natural frequency, once each cycle, always just as it starts to move away from you. Every push arrives at the right moment. Every push adds energy. Cycle after cycle, the energy adds up, so the swing goes higher and higher.

Push at a different frequency, and the timing slowly drifts. For a few cycles your pushes help. Then they arrive at the wrong moment and take energy away. Over many cycles, the gains and losses nearly cancel. The swing never goes very high, however long you push.

That is it. That is the whole idea. When the driving frequency matches a natural frequency, every push adds energy at the right moment, so the energy, and the amplitude, build up. This is resonance.

So the Millennium Bridge did not need strong pushes. It needed pushes at the right rhythm. A walker's body sways a little to the side with each step, left then right, so it pushes the ground sideways once for every two steps. That is about once a second. The bridge's sideways natural frequency was also about once a second. And as the bridge began to move, people unconsciously fell into step with it, which made their pushes even better timed (Arup; Steven Strogatz and colleagues, reported by NBC News).

Stop and think

You push a swing three times as hard as before, but at half its natural frequency. Will it go higher than gentle pushes at exactly its natural frequency?

Show the answer

Almost certainly not. For a lightly damped swing, the simulation below gives about 4 times the amplitude of a slow push for the hard pushes. The gentle pushes at the natural frequency give about 17 times. Timing beats strength. Pushes at the wrong rhythm spend half their time taking energy away. (Pushing on every second swing is different: each push still arrives at the right moment, so the swing still builds, just more slowly.)

Why the amplitude stops growing

If every push adds energy, why does a swing at resonance not go higher forever? Because every real system also loses energy. Air resistance, friction at the hinge and flexing of the materials turn some of its energy into heat in every cycle. This loss is called damping.

A bigger swing loses more energy per cycle, because it moves faster through the air and rubs harder. So the amplitude grows until the energy lost in each cycle equals the energy added by each push. Then it stays the same. With light damping, that balance comes only at a very large amplitude. With heavy damping, it comes early, at a small one.

Try it below. The model is a mass on a spring with a natural frequency of exactly 1 Hz, driven by a force that pushes it backwards and forwards. The amplitude is measured against a very slow push, which simply moves the mass to and fro by 1 unit.

Simulation — driving an oscillator

Choose a driving frequency and how much damping there is, then press Play to watch the motion build up over 40 seconds. The lower graph marks your choice on the resonance curve.

Driving frequency: 0.80 Hz (natural frequency 1.00 Hz)

Damping:

A mass on a spring with a natural frequency of 1.00 Hz, driven backwards and forwards. The table gives its steady amplitude, compared with the amplitude of a very slow push.

Driving frequency	Light damping	Medium damping	Heavy damping
0.50 Hz	1.3	1.3	1.2
0.80 Hz	2.8	2.5	1.4
1.00 Hz	16.7	5.0	1.25
1.20 Hz	2.2	2.0	0.95
2.00 Hz	0.33	0.33	0.29

With light damping, driving at 1.00 Hz takes about 12 seconds to reach 90% of its steady amplitude. With heavy damping it takes under a second, but the amplitude is far smaller.

What should you have seen? With light damping, driving at 1.00 Hz builds a motion about 17 times larger than a slow push, but it takes over 10 seconds to grow. Move the frequency 20% away and the amplitude falls below 3. With heavy damping, the peak almost disappears, and it sits a little below 1 Hz.

Resonance

When a system is driven at a frequency equal to one of its natural frequencies, it absorbs energy most effectively and its amplitude becomes very large. This is resonance.

The amplitude grows until the energy lost to damping in each cycle equals the energy added by each push.

Light damping: a tall, narrow peak at the natural frequency. Heavy damping: a lower, broader peak, slightly below the natural frequency.

The graph of steady amplitude against driving frequency is called a resonance curve. The figure shows three of them for the same system.

Resonance curves for one system with three amounts of damping. Amplitude is measured in units of the amplitude of a very slow push. Light damping gives a tall, narrow peak (about 17) exactly at the natural frequency

f_{0}

. Medium damping gives a peak of about 5. Heavy damping gives a low, broad hump of about 1.4, a little below

f_{0}

A narrow peak means a system is selective: it responds strongly only to frequencies very close to its own. A broad peak means it responds a little to a wide range. Both are useful. A radio must be selective, so that it picks up one station and ignores the next one. A car's suspension must not be, so engineers give it heavy damping with shock absorbers: a bump should make the car settle at once, not bounce again and again.

Resonance you have already met

In the last lecture, a tuning fork sounded above a tube of air, and the tube was loud only at certain lengths. That was resonance. The fork was the driver, at a fixed frequency. Changing the length of the air column changed its natural frequencies. When one of them matched the fork, the air in the tube was driven in step, and the sound grew loud.

The sound box of a guitar is driven too, but its natural frequencies are many and broad, so the strings drive it over a wide range. Its large surface moves a lot of air, so the sound is louder than a bare string. The box adds no energy of its own. It only passes the string's energy into the air more effectively.

A famous collapse that was not resonance

Many books describe a second bridge as the classic example of resonance. On 7 November 1940, the Tacoma Narrows Bridge in Washington State, USA, twisted violently in the wind and its main span fell into the water below. It had been open for only four months.

But the wind that day was steady, at about 42 miles per hour (about 67 km/h). A steady wind does not push at any particular frequency, so it could not drive the bridge in time with its natural frequency. What happened instead is called flutter. Once the deck began to twist, the twist changed the angle at which the wind met it. That changed the wind's force in a way that pushed the twist further. The bridge's own motion set the timing of the push. It twisted at about 0.2 Hz, and with each cycle the wind fed in more energy than the bridge could lose (Billah and Scanlan, American Journal of Physics, 1991).

The difference matters. In resonance, an outside driver has its own rhythm, and the system responds when the rhythms match. In flutter, nothing outside has a rhythm. The system creates its own, and the energy input grows with the motion. That is why engineers today test models of large bridges in wind tunnels, and why the replacement bridge, opened in 1950, was built much stiffer against twisting.

Sort the cases below to test the difference.

Sorter — is it resonance?

Tap a case, then tap a box: is it resonance, or something else? Read the reason after each one.

Eight cases sorted by whether they are resonance: an outside driver at the system's natural frequency.

Resonance. The Millennium Bridge swaying as walkers push sideways about once a second, matching its sideways natural frequency. A wine glass shattered by a singer holding its ringing note. An MRI scanner's radio waves at the natural frequency of hydrogen nuclei in its magnetic field. A resonance tube sounding loudly when its air column matches the tuning fork. A swing pushed once every 2.5 s when its natural period is 2.5 s.
Not resonance. The Tacoma Narrows Bridge twisting in a steady wind: flutter, where the motion sets its own timing. A guitar string ringing after one pluck: free oscillation, with no driver. A car bouncing once after a bump and then settling: free oscillation, heavily damped.

What should you have learned? Large movement alone does not prove resonance. Ask where the rhythm comes from. If an outside driver supplies it, and it matches a natural frequency, that is resonance.

Working with resonance

Worked Example 1 — Timing the pushes

A child on a swing takes 2.5 s for each complete swing, there and back. At what frequency should you push? What happens if you push only on every second swing?

The natural frequency is $f_{0} = \frac{1}{T} = \frac{1}{2.5 s} = 0.4 Hz$ . So push at 0.4 Hz: once every 2.5 s, each time the swing starts to move away from you.

On every second swing, you push once every 5 s, at 0.2 Hz. Each push still arrives at the right point in the cycle, so it still adds energy. The swing builds up, but more slowly, because it gets half as many pushes. Pushing every 3 s would not work: the timing would drift, and pushes would start to take energy away.

Check: $0.4 Hz \times 2.5 s = 1$ push per cycle. ✓

Worked Example 2 — The wine glass

A thin glass rings when it is tapped. In 2005, the singer Jaime Vendera broke a crystal glass with his unamplified voice on a television science programme. The glass rang at about 556 Hz, and his note reached about 105 decibels (Scientific American; Mental Floss). Ordinary speech is about 50 decibels. Why does a conversation never break a glass, even if a voice passes through 556 Hz?

Two reasons. First, timing must last. A passing sound at 556 Hz adds energy for only a few cycles. A held note keeps adding energy in step, cycle after cycle, while damping removes only a little. Second, the push must be large enough. The amplitude grows until damping removes as much energy as each cycle adds. A quiet voice reaches that balance long before the glass bends far enough to crack.

So it took the right note, held, and very loud: about 55 decibels louder than speech, which is more than 100 000 times the sound intensity. Even then, several glasses survived first.

Worked Example 3 — Resonance inside the body

An MRI scanner places the body in a strong magnetic field. Hydrogen nuclei in a field of $B$ teslas have a natural frequency of about $42.58 \times B$ MHz. A common scanner has $B = 1.5 T$ . Find the frequency of the radio waves it must send, and their wavelength. Take the speed of radio waves as $3.0 \times 1 0^{8} m/s$ .

Natural frequency: $42.58 \times 1.5 \approx 63.9 MHz$ , which is $63.9 \times 1 0^{6} Hz$ (Radiopaedia gives 42.58 MHz per tesla).

Wavelength: $λ = \frac{v}{f} = \frac{3.0 \times 1 0 ^{8}}{63.9 \times 1 0 ^{6}} \approx 4.7 m$ .

The scanner sends radio waves at exactly this frequency, and the nuclei absorb energy strongly, because they are being driven at their natural frequency. When the waves stop, the nuclei give the energy back as a faint radio signal, and the scanner builds an image from it. Check: $63.9 \times 1 0^{6} \times 4.7 \approx 3.0 \times 1 0^{8} m/s$ . ✓

Worked Example 4 — Fixing the Millennium Bridge

Engineers had two ways to stop the sway. They could change the bridge's natural frequency, so that walkers no longer matched it. Or they could add damping, so that the energy from each step was removed before it built up. Which did they choose?

Making the bridge stiff enough to move its sideways frequency well away from walking would have changed its slim shape completely. So they added damping. They fitted 37 viscous dampers, like large shock absorbers, to control the sideways motion, and 52 tuned mass dampers to control vertical motion. The work cost about £5 million, and the bridge reopened in February 2002 (Arup, via Wikipedia).

On the resonance curve, the dampers turned a tall, narrow peak into a low, broad one.

Three classic mistakes

Trap 1 — "Resonance means a large amplitude"

Resonance is about matching frequencies, not size. A heavily damped system at resonance moves very little. And a large amplitude can come from other causes, such as one big push or flutter. Define resonance by the cause: a driving frequency equal to a natural frequency.

Trap 2 — "Tacoma Narrows collapsed because the wind matched its frequency"

The wind was steady and had no frequency to match. The bridge failed by flutter, in which its own twisting made the wind push it harder. Many textbooks still repeat the resonance story. Use the Millennium Bridge, a wine glass or a swing as your example of resonance instead.

Trap 3 — "Each object has one natural frequency"

A swing has essentially one, but most objects have several. A string has $f_{1}, 2 f_{1}, 3 f_{1}$ and more. A bridge can bounce, sway and twist, each at its own frequency. Resonance can happen at any of them.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

A pendulum takes 1.6 s for each complete swing. (a) Find its natural frequency. (b) A student pushes it every 0.8 s. Will the amplitude build up steadily? Explain.

Hint 1
$f_{0} = \frac{1}{T}$ .

Hint 2
Pushing every 0.8 s means pushing twice per cycle. Where is the pendulum, and which way is it moving, at the second push?

Full answer
(a) $f_{0} = \frac{1}{1.6} \approx 0.63 Hz$ (exactly 0.625 Hz). (b) No. The pushes come at 1.25 Hz, twice the natural frequency. If one push helps, the next comes half a cycle later, when the pendulum is moving the other way, so it takes energy away. The gains and losses cancel, and the amplitude does not build. Pushing every 1.6 s, or every 3.2 s, would work. Check: $1.6 \div 0.8 = 2$ pushes per cycle. ✓
Two invented systems, P and Q, are driven at different frequencies. Their steady amplitudes (in mm) are given below. (a) Estimate each one's natural frequency. (b) Which is more heavily damped? Give two reasons.

Driving frequency (Hz) P (mm) Q (mm)

10 2 3

15 4 5

20 40 7

25 5 6

30 2 4

Hint 1
The natural frequency is where the amplitude peaks.

Hint 2
Compare the height of each peak, and how quickly the amplitude falls on either side of it.

Full answer
(a) Both peak at about 20 Hz, so both have a natural frequency near 20 Hz. (b) Q is more heavily damped. Its peak is much lower (7 mm against 40 mm), and it is broader: Q's amplitude at 15 Hz and 25 Hz is close to its peak, while P's falls to about a tenth of its peak. A low, broad peak means heavy damping.
Newer MRI scanners use a field of 3.0 T. (a) Find the frequency of the radio waves needed for hydrogen nuclei. (b) Find their wavelength. (c) Explain why a 1.5 T scanner's radio waves would not work in a 3.0 T scanner.

Hint 1
The natural frequency is about $42.58 \times B$ MHz.

Hint 2
Use $λ = \frac{v}{f}$ with $v = 3.0 \times 1 0^{8} m/s$ .

Full answer
(a) $42.58 \times 3.0 \approx 127.7 MHz$ . (b) $λ = \frac{3.0 \times 1 0 ^{8}}{127.7 \times 1 0 ^{6}} \approx 2.3 m$ . (c) In 3.0 T the nuclei's natural frequency is about 127.7 MHz. Radio waves at 63.9 MHz would be far from it, so the nuclei would hardly absorb any energy. Resonance needs the frequencies to match. Check: doubling the field doubles the frequency and halves the wavelength. ✓
An invented washing machine spins its drum faster and faster, up to 1200 revolutions per minute. A slightly uneven load pushes the machine once per revolution. The machine on its feet has a natural frequency of 4 Hz. (a) At what spin speed, in revolutions per minute, does the machine shake most? (b) Explain why it shakes hard for a moment, then runs smoothly at full speed. (c) Suggest two ways a designer could reduce the shaking.

Hint 1
The driving frequency in Hz is the number of revolutions per second. 4 Hz is 4 revolutions per second.

Hint 2
At full speed, compare the driving frequency with 4 Hz. Look at the resonance curve far to the right of the peak.

Full answer
(a) $4 \times 60 = 240$ revolutions per minute. (b) As the drum speeds up, the driving frequency rises through 4 Hz. Near 240 revolutions per minute the pushes match the natural frequency, so the shaking grows. The drum passes quickly through this speed, so the amplitude never has time to reach its full steady value. At 1200 revolutions per minute the driving frequency is 20 Hz, five times the natural frequency, far to the right of the peak, where the amplitude is small. (c) Add damping, for example dampers between the drum and the frame, so the peak is lower and broader. Or change the natural frequency, for example with stiffer springs or a heavier base, and pass through it as quickly as possible. Check: $1200 \div 60 = 20 Hz$ . ✓

Driving frequency (Hz)	P (mm)	Q (mm)
10	2	3
15	4	5
20	40	7
25	5	6
30	2	4

The bigger idea

Resonance is about timing, and timing lets a small push do a great deal. A walker moved a steel bridge, a voice broke a glass, and weak radio waves are absorbed by nuclei deep inside the body. In every case, the driver's energy is small in each cycle. It succeeds because it arrives in step, cycle after cycle.

This topic began with standing waves, and found that a string or a pipe has a list of natural frequencies. Resonance is the reason that list matters: it is the list of frequencies at which the object will respond strongly. Engineers spend a great deal of effort keeping bridges, buildings and machines away from it, or damping it. Others use it on purpose, in musical instruments, radio receivers and medical scanners. At Pre-University you will write the equations of a driven, damped oscillator and derive the resonance curve. The next topic, the electromagnetic spectrum, includes the radio waves that every receiver must pick out by resonance.

Key concepts in this lesson

A natural frequency is a frequency at which a system oscillates when it is disturbed once and left alone, and most systems have several.
Resonance happens when a system is driven at one of its natural frequencies, because every push then adds energy at the right moment.
The amplitude at resonance grows until the energy removed by damping in each cycle equals the energy added by each push.
Light damping gives a tall, narrow resonance peak, and heavy damping gives a lower, broader peak a little below the natural frequency.
The Millennium Bridge, a wine glass and an MRI scanner are examples of resonance, while the Tacoma Narrows collapse was flutter, a self-excited motion in a steady wind.
Engineers control resonance by moving a natural frequency away from the driving frequency or by adding damping.