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Did resonance really bring down the Tacoma Narrows Bridge?

A film from 1940 shows a suspension bridge twisting itself apart in the wind, and for decades it has been shown to students as the picture of resonance. The film is real. The explanation usually attached to it is not, and the difference is a lesson in oscillations that most textbooks skip.

Short answer

No, not in the textbook sense. Resonance needs a push that repeats in step with the bridge's own vibration, and on 7 November 1940 the wind was steady. The deck's twisting drew energy from that steady wind and grew by itself, a self-excited motion engineers call flutter. In the terms of a JEE problem, the damping turned negative.

By 9 min read
Contents
  1. What happened
  2. What resonance needs
  3. Why it was not resonance
  4. What flutter is
  5. Fair to the textbook
  6. What changed
  7. Practice
Where and whenTacoma Narrows, a strait of Puget Sound in Washington State, USA. The bridge opened on 1 July 1940 and fell on 7 November, four months later.

When it opened on 1 July 1940, the Tacoma Narrows Bridge had the third-longest main span of any suspension bridge in the world: 853 m (2,800 ft). It was also unusually light. The original design stiffened the deck with trusses 7.6 m (25 ft) deep. The consulting engineer, Leon Moisseiff, replaced them with solid steel plate girders only 2.4 m (8 ft) deep, which saved a great deal of steel. The deck was about 1/350 as deep as the main span was long.

It moved from the start. In moderate winds the deck rose and fell in long, slow waves, and the bridge was soon nicknamed Galloping Gertie. For four months it carried traffic like that.

What happened on 7 November 1940?

Natural frequencyThe frequency at which a structure vibrates when it is disturbed and left alone. A bridge has many, one for each shape it can move in. Tacoma's twisting shape had one of about 0.2 Hz.

On the morning of 7 November a steady wind blew across the strait. Engineers measured about 68 km/h (42 mph) near the east end of the bridge. The deck was moving up and down, as it often did. At about 10 a.m., after a cable band at the middle of the span slipped, the motion changed and the deck began to twist. The two halves of the main span rotated in opposite directions about a still line at the centre, one edge of the roadway rising as the other fell, about once every five seconds: 0.2 Hz.

The bridge was closed to traffic. The last person on it, the newspaper editor Leonard Coatsworth, abandoned his car on the tilting deck and crawled to safety; his daughter's dog, Tubby, was left in the car. At about 11 a.m. the main span broke up and fell into the water. No person died.

The 1940 Tacoma Narrows Bridge and the twist that destroyed it The 1940 Tacoma Narrows Bridge seen from the side. Two towers hold a main cable that sags between them; vertical hangers carry a thin deck. The main span between the towers is 853 metres; the deck girder is only 2.4 metres deep, about one three-hundred-and-fiftieth of the span. Drawn in red over the main span: the twisting that destroyed it. The two halves of the span turned in opposite directions about a still line at mid-span, about once every five seconds, in a steady wind of about 68 kilometres an hour. main span 853 m Deck girder only 2.4 m deep:about 1/350 of the main span The twist: the two halves turnedopposite ways, once every 5 s steady wind across the deck, about 68 km/h The 1940 Tacoma Narrows Bridge and the twist that destroyed it The 1940 Tacoma Narrows Bridge seen from the side. Two towers hold a main cable that sags between them; vertical hangers carry a thin deck. The main span between the towers is 853 metres; the deck girder is only 2.4 metres deep, about one three-hundred-and-fiftieth of the span. Drawn in red over the main span: the twisting that destroyed it. The two halves of the span turned in opposite directions about a still line at mid-span, about once every five seconds, in a steady wind of about 68 kilometres an hour. main span 853 m Deck girder only 2.4 m deep:about 1/350 of the main span Red: the twist. The two halvesturned opposite ways, once every 5 s Steady wind across the deck,about 68 km/h
The bridge and its last motion. A deck 2.4 m deep across an 853 m span. In red, the twist: the two halves turned opposite ways about a still line at mid-span.

Notice what is missing from that account: anything in the wind that repeats every five seconds.

What does resonance actually need?

SyllabusJEE Advanced 2026: “Forced and damped oscillation (in one dimension), resonance.” JEE Main 2026 keeps resonance in the LCR circuit, which obeys the same equation. ESE Civil Paper II: “Free and forced vibrations of single degree and multi degree freedom system.”

Resonance is what happens when a periodic force drives an oscillator at close to its natural frequency. For a mass on a spring with damping, pushed by a force that repeats with angular frequency \( \omega \), the equation of motion is

$$ m\ddot{x} + b\dot{x} + kx = F_0 \cos \omega t $$

and once the start-up motion has died away, the mass oscillates at the driving frequency with amplitude

$$ A = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (b\omega/m)^2}}, \qquad \omega_0 = \sqrt{k/m} $$

Three things follow, and each is a test the Tacoma story has to pass. The motion happens at the driving frequency \( \omega \), whatever the structure would prefer. It is large only when \( \omega \) is close to \( \omega_0 \). And even at \( \omega = \omega_0 \) it stays finite, \( A = F_0/(b\,\omega_0) \), because damping removes energy as fast as the push supplies it.

So the textbook story needs the wind to push the deck in step with its twist, about once every five seconds.

Why was it not resonance?

SourceK. Y. Billah and R. H. Scanlan, “Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks”, American Journal of Physics 59 (1991).

Because the wind did no such thing. A wind of 68 km/h is close to steady. Its gusts are random, spread over many frequencies with none of them dominant. Random gusts shake a structure a little, which engineers call buffeting, but they do not build one frequency up to destruction.

The one thing wind does in rhythm is shed vortices. As air flows past a blunt shape, swirls peel off alternately from its top and bottom edges, and each one gives the body a small push. Their frequency rises with the wind:

$$ f_s = \mathrm{St}\,\frac{U}{D} $$

where \( U \) is the wind speed, \( D \) the depth of the obstacle and St, the Strouhal number, a constant for the shape. For Tacoma's 2.4 m deck in a 68 km/h wind, estimates put the shedding near 1 Hz: five times faster than the twist. A push at 1 Hz cannot pump up a motion at 0.2 Hz.

Check the numbersEngineers compare the wind speed U with the twisting frequency f times the deck width B. On 7 November, U ≈ 8fB: 8 × 0.2 Hz × 11.9 m ≈ 19 m/s, the 68 km/h measured. A 2022 study finds that ratio was past the point at which this deck shape becomes unstable in twisting.

And this twisting does not behave like resonance as the wind changes. Resonance peaks at one driving frequency and falls away on either side. The twisting that destroyed Tacoma, as Billah and Scanlan point out, sets in above a critical wind speed and grows stronger as the wind rises. That is the signature of a different mechanism.

Resonance against flutter: what goes in, and what comes out Two panels, each with an input above and the motion it produces below. Resonance: the input is a push that repeats, in step with the motion; the motion grows, then levels off, because damping takes out what the push puts in. Flutter: the input is a steady wind with no rhythm at all; the motion grows at the structure's own frequency and keeps growing, because it draws more energy from the wind every cycle. Resonance input: a push that repeats motion grows, then levels off: damping takes outwhat the push puts in Flutter input: a steady wind motion keeps growing: each cycle draws moreenergy from the wind Resonance against flutter: what goes in, and what comes out Two panels, each with an input above and the motion it produces below. Resonance: the input is a push that repeats, in step with the motion; the motion grows, then levels off, because damping takes out what the push puts in. Flutter: the input is a steady wind with no rhythm at all; the motion grows at the structure's own frequency and keeps growing, because it draws more energy from the wind every cycle. Resonance input: a push that repeats motion grows, then levels off: damping takes outwhat the push puts in Flutter input: a steady wind motion keeps growing: each cycle draws moreenergy from the wind
What goes in, and what comes out. Resonance needs an input with a rhythm, and damping caps the result. Flutter needs only a steady flow, and nothing caps it once it starts.

So what is flutter?

Self-excited oscillationAn oscillation kept going by a steady source of energy, with its timing set by the oscillating body itself. A flag in a steady breeze and a violin string under a bow are everyday examples.

Flutter is a self-excited oscillation: the motion creates the force that drives it. As Tacoma's deck twisted, it changed the angle at which the wind met it, and the air pressure on the deck shifted in a way that pushed the twist further, a fraction of a cycle behind. Each cycle the deck took a little more energy from the wind. A flag flapping in a steady breeze does the same, and so does a violin string under a steadily drawn bow.

You can write it in the language of your syllabus. Let \( \theta \) be the angle of twist, \( I \) the deck's moment of inertia about its axis, \( \kappa \) its stiffness against twisting and \( b \) its own damping. Suppose the wind's moment on the deck is proportional to the rate of twist, \( c\,\dot{\theta} \), with \( c \) growing as the wind gets stronger. Then

$$ I\ddot{\theta} + b\dot{\theta} + \kappa\theta = c\,\dot{\theta} \quad\Longrightarrow\quad I\ddot{\theta} + (b - c)\,\dot{\theta} + \kappa\theta = 0 $$

This is the damped oscillator from your textbook, with \( b - c \) in place of the damping constant. Its solution is

$$ \theta(t) = \theta_0\, e^{-(b - c)t/2I} \cos(\omega' t + \phi) $$

In a light wind, \( c \) is smaller than \( b \) and the twist dies away. Past a critical wind speed, \( c \) is larger than \( b \), the exponent turns positive, and the twist grows at the deck's own frequency, \( \omega' \), close to \( \sqrt{\kappa/I} \). There is no driving frequency anywhere in the equation. Nothing in it limits the growth either: in a real structure, something breaks.

This one-line model is a simplification. Engineers describe the wind's moment with coefficients measured in wind tunnels, and on streamlined decks the twisting couples with up-and-down motion. But for a bluff, open deck like Tacoma's, Billah and Scanlan describe the failure in essentially these terms: the twisting motion lost its damping to the wind.

Resonance is a peak; flutter is a threshold Two graphs. Left, resonance: amplitude against driving frequency rises to a single peak at the natural frequency and falls away on either side; damping sets the height of the peak. Right, flutter: the net damping of the twisting falls as wind speed rises and crosses zero at a critical wind speed. Below that speed, twists die out; above it, net damping is negative and twists grow. Resonance: a peak natural frequency driving frequency amplitude damping sets the height of the peak Flutter: a threshold critical wind speed twists die out twists grow wind speed net damping + − past it, the wind feeds every twist Resonance is a peak; flutter is a threshold Two graphs. Left, resonance: amplitude against driving frequency rises to a single peak at the natural frequency and falls away on either side; damping sets the height of the peak. Right, flutter: the net damping of the twisting falls as wind speed rises and crosses zero at a critical wind speed. Below that speed, twists die out; above it, net damping is negative and twists grow. Resonance: a peak natural frequency driving frequency amplitude damping sets the height of the peak Flutter: a threshold critical wind speed twists die out twists grow wind speed net damping + − past it, the wind feeds every twist
A peak against a threshold. Resonance is large only near one frequency, and damping limits it. Flutter switches on above one wind speed, and nothing limits it.

Was the textbook story entirely wrong?

In the examJEE questions on resonance test the forced-oscillator equation, not this bridge. If a question offers resonance as the cause and nothing better, the setter means the textbook account: choose it, and keep the fuller answer for a question that lets you explain.

Not entirely, and it is worth being fair to it. The bridge did twist at one of its own natural frequencies, and in loose speech any large vibration at a natural frequency gets called resonance. The months of up-and-down motion before the collapse were a different matter: there, the rhythmic shedding of vortices may well have acted as a periodic push, and researchers still study how the two motions were linked.

What the story gets wrong is the cause of the collapse. There was no push in step with the twist, and the remedy the story implies, avoiding a matching frequency, would not have saved the bridge. A deck that resisted twisting, and let the wind through, would have.

What changed in how bridges are built?

Depth to span1940 deck girder: 2.4 m on an 853 m span, about 1 to 350. 1950 stiffening truss: 10 m, about 1 to 85.

The replacement bridge opened on 14 October 1950. Its deck hangs from a stiffening truss 10 m (33 ft) deep, with slots of open steel grating in the roadway, a design tested for years in a wind tunnel by Frederick Farquharson at the University of Washington. An open truss lets the wind pass through, and a deep frame resists twisting. Later designers went further: from the 1960s, many long bridges have used a closed steel box shaped like a wing, which the wind flows smoothly around.

The reason a box works is in the torsion chapter of strength of materials. An open section built of thin plates, like Tacoma's H-shape, resists twisting weakly. Its torsion constant is about

$$ J \approx \sum \tfrac{1}{3}\, w\, t^{3} $$

summed over the plates, where \( w \) is a plate's width and \( t \) its thickness. Close the same plates into a box and the torsion constant becomes

$$ J = \frac{4A^{2}}{\oint ds/t} $$

where \( A \) is the area enclosed by the walls. With thin plates the second is thousands of times the first. A stiffer deck twists less under the same moment and has a higher twisting frequency, which raises the wind speed at which flutter can begin.

Indian codeIS 875 (Part 3): 2015 treats a structure whose fundamental natural frequency is 1 Hz or less as flexible, and requires its dynamic response to wind to be checked.

Long-span bridges are now routinely tested as models in wind tunnels before they are built. India's wind code treats any structure whose fundamental frequency is 1 Hz or less as flexible, and asks for its dynamic response to wind to be checked. Tacoma's deck twisted at 0.2 Hz.

Three ways to build a bridge deck, seen end-on Three bridge deck cross-sections, with the wind blowing from the left. 1940: a flat deck between two solid plate girders 2.4 metres deep, an H shape the wind cannot pass through; it sheds swirls behind. 1950: the replacement's open truss, 10 metres deep, with slots in the roadway; the wind passes through, and the deep frame resists twisting. Since the 1960s: a closed steel box shaped like a wing; the wind flows smoothly round it, and a closed box is very stiff in twisting. Not to scale. 1940 solid plate girders, 2.4 m deep:the wind cannot pass throughand sheds swirls behind 1950 open truss, 10 m deep, slotsin the roadway: the wind passesthrough; the frame resists twist Since the 1960s closed steel box shaped likea wing: smooth flow, and verystiff in twisting wind from the left cross-sections, not to scale Three ways to build a bridge deck, seen end-on Three bridge deck cross-sections, with the wind blowing from the left. 1940: a flat deck between two solid plate girders 2.4 metres deep, an H shape the wind cannot pass through; it sheds swirls behind. 1950: the replacement's open truss, 10 metres deep, with slots in the roadway; the wind passes through, and the deep frame resists twisting. Since the 1960s: a closed steel box shaped like a wing; the wind flows smoothly round it, and a closed box is very stiff in twisting. Not to scale. 1940 solid plate girders, 2.4 m deep:the wind cannot pass throughand sheds swirls behind 1950 open truss, 10 m deep, slotsin the roadway: the wind passesthrough; the frame resists twist Since the 1960s closed steel box shaped likea wing: smooth flow, and verystiff in twisting wind from the left cross-sections, not to scale
Three decks, seen end-on. The 1940 H-shape blocked the wind and was weak in twisting. The 1950 truss let the wind through. A modern box deck lets it flow round and barely twists.

What to remember:

  • Resonance: a periodic push near the natural frequency. The motion follows the push's frequency, peaks near the natural frequency, and damping limits it.
  • Flutter: a steady flow and a motion that feeds itself. The motion is at the structure's own frequency, starts above a critical speed, and grows until something gives.
  • Tacoma: flutter in twisting, made possible by a deck that was shallow, solid and weak in torsion.

Try it: JEE Advanced, ESE and GATE

  1. JEE Advanced style. A model of a twisting bridge deck obeys \( I\ddot{\theta} + (b - c)\,\dot{\theta} + \kappa\theta = 0 \), with \( I = 2.5 \times 10^{6}\ \mathrm{kg\,m^{2}} \), \( \kappa = 4.0 \times 10^{6}\ \mathrm{N\,m\,rad^{-1}} \) and \( b = 4.0 \times 10^{5}\ \mathrm{N\,m\,s\,rad^{-1}} \). The wind's coefficient is \( c = \alpha U \), where \( U \) is the wind speed and \( \alpha = 2.0 \times 10^{4}\ \mathrm{N\,s^{2}\,rad^{-1}} \). (a) Find the twisting frequency. (b) Find the wind speed above which the twisting grows. (c) At \( U = 25\ \mathrm{m\,s^{-1}} \), how long does the amplitude take to double? These are model numbers, not the real bridge's.
  2. Concept check. A structure vibrates with growing amplitude. Which observation shows that the cause is self-excitation rather than resonance? (A) The amplitude grows with time. (B) The structure vibrates at one of its natural frequencies. (C) The force acting on it has no component at the frequency of vibration. (D) The vibration stops when the structure fails.
  3. ESE and GATE (short answer). Using the torsion constants of thin-walled sections, explain why a closed box deck resists torsional flutter better than an open section built from the same plates.
Answers

1. (a) \( \omega \approx \sqrt{\kappa/I} = \sqrt{1.6} \approx 1.26\ \mathrm{rad\,s^{-1}} \), so \( f = \omega/2\pi \approx 0.20\ \mathrm{Hz} \); the damping term changes this by well under 1%. (b) The twisting grows when \( c > b \), that is when \( U > \dfrac{b}{\alpha} = \dfrac{4.0 \times 10^{5}}{2.0 \times 10^{4}} = 20\ \mathrm{m\,s^{-1}} \), or 72 km/h. (c) At 25 m/s, \( c = 5.0 \times 10^{5} \), and the amplitude grows as \( e^{\gamma t} \) with \( \gamma = \dfrac{c - b}{2I} = \dfrac{1.0 \times 10^{5}}{5.0 \times 10^{6}} = 0.020\ \mathrm{s^{-1}} \). It doubles when \( \gamma t = \ln 2 \), after about \( 0.693/0.020 \approx 35\ \mathrm{s} \).

2. (C). Options (A) and (B) are true of both resonance and flutter, and (D) says nothing about the cause. Only (C) rules out a periodic push.

3. An open section's torsion constant, about \( \sum \tfrac{1}{3}\, w\, t^{3} \), is tiny for thin plates; a closed section's, \( 4A^{2}/\oint ds/t \), is thousands of times larger. The box's higher torsional stiffness raises its twisting frequency, which raises the critical wind speed for flutter, and its streamlined shape takes less energy from the wind.

Read: Oscillations and SHM notes · Theory of Structures notes

Practise: JEE chapter-wise tests · ESE Civil tests · GATE Civil tests

Sources

  • O. H. Ammann, T. von Kármán and G. B. Woodruff, The Failure of the Tacoma Narrows Bridge, report to the Federal Works Agency (1941).
  • K. Y. Billah and R. H. Scanlan, “Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks”, American Journal of Physics 59(2), 118–124 (1991).
  • “Vertical and torsional vibrations before the collapse of the Tacoma Narrows Bridge in 1940”, Journal of Fluid Mechanics (2022): Cambridge Core.
  • Washington State Department of Transportation, Tacoma Narrows Bridge history: the 1940 bridge, the collapse and the 1950 bridge.
  • “Tacoma Narrows Bridge failure, 1940”, Structure magazine.
  • Bureau of Indian Standards, IS 875 (Part 3): 2015, Design loads (other than earthquake) for buildings and structures, Part 3: Wind loads.
  • Syllabus lines: JEE Main 2026 and JEE Advanced 2026 (Physics), ESE 2026 Civil Engineering Paper II.

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