The Tacoma Bridge Did Not Collapse by Resonance. The Arithmetic Says So

Wind of about 19 m/s blowing past a deck 2.4 m deep sheds vortices at roughly 1.5 Hz. The Tacoma Narrows deck twisted at 0.2 Hz when it failed on 7 November 1940. The two numbers differ by a factor of about 8. If the textbook story were true, they would be equal.

That story says the wind shed vortices at the bridge's natural frequency, the deck resonated, and the amplitude grew until the span broke. I think it fails a one-line check, and the rest of this post shows the check, the data behind it, and the assumption that moves the answer most. I computed everything below by hand, without the Lab. Every input is listed so you can redo it.

The question

Does vortex shedding at the Strouhal frequency match the torsional frequency seen on the day of the collapse? If it does, the resonance story survives this test. If it misses by a large factor, the story needs a different driver.

Billah and Scanlan made the same argument in the American Journal of Physics in 1991, under the title "Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks" [1]. Their complaint, in their words as quoted by a reader of the paper, is that texts are vague about what the exciting force was and how it acquired the necessary periodicity [2]. My aim here is narrower than theirs. I only redo the frequency comparison, because it needs no wind tunnel.

Data and where it came from

Three inputs matter.

1. Torsional frequency. Farquharson, who was on the bridge, measured 12 cycles per minute, a period of 5 s [3]. That is 0.20 Hz. The same Physics Today piece notes that classroom videos digitised in 1982 show 18 cycles per minute, a telecine conversion error [3]. So any frequency read off a classroom video is suspect. Use the 12 per minute value.
2. Deck depth. The stiffening girders were 8-foot-high plate girders [4]. That is 2.44 m. A solid girder is a bluff body, so I take the girder depth as the length scale $D$ that the wind sees.
3. Wind speed. Secondary summaries put the wind at 42 mph during the final hours, which is 18.8 m/s. I did not trace this figure to Farquharson's own report, so treat it as recalled, not re-read. The sensitivity section shows the conclusion does not depend on it.

The Strouhal number is my assumption, not a measurement. I use $St = 0.2$, the usual order of magnitude for a bluff body at high Reynolds number. I have no cited value for this particular H-shaped section.

Where this fails: the Strouhal number of a real deck depends on its cross-section, its edge details and the turbulence in the oncoming wind. A single number of 0.2 is a placeholder, and I vary it below.

Method

Vortex shedding from a bluff body of depth $D$ in a wind of speed $U$ has frequency

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

The resonance claim needs $f_s = f_{\text{tors}}$ with $f_{\text{tors}} = 0.20\ \text{Hz}$. The unit conversions:

$$D = 8\ \text{ft} \times 0.3048\ \text{m/ft} = 2.44\ \text{m}$$ $$U = 42\ \text{mph} \times 0.44704\ \text{m/s per mph} = 18.8\ \text{m/s}$$

Result

Insert the numbers:

$$f_s = \frac{0.2 \times 18.8\ \text{m/s}}{2.44\ \text{m}} = 1.54\ \text{Hz}$$

The ratio to the observed twist is

$$\frac{f_s}{f_{\text{tors}}} = \frac{1.54\ \text{Hz}}{0.20\ \text{Hz}} = 7.7$$

Two inverse questions give the same message. First, what wind speed makes the shedding frequency equal 0.2 Hz?

$$U_{\text{match}} = \frac{f_{\text{tors}}\, D}{St} = \frac{0.20 \times 2.44}{0.2}\ \text{m/s} = 2.4\ \text{m/s} \approx 5.5\ \text{mph}$$

That is a breeze, not the gale in which the deck failed. Second, what Strouhal number would make resonance hold at 18.8 m/s?

$$St_{\text{needed}} = \frac{f_{\text{tors}}\, D}{U} = \frac{0.20 \times 2.44}{18.8} = 0.026$$

That is about 8 times smaller than the bluff-body value I assumed. I know of no bluff section with a Strouhal number that low.

I give the result as one significant figure on the ratio: about 8, and certainly between 3 and 10 under the ranges below. I do not quote an interval tighter than that. The inputs are known to perhaps 10 to 50 percent, and the method is a scaling law.

Where this fails: this test shows only that classic lock-in at the observed torsional frequency is not supported by the shedding frequency. It does not prove the mechanism by itself. It also does not say vortex shedding played no role at lower wind speeds or in other modes. A reader who wants to defend resonance must name a different mode with a frequency near 1.5 Hz, and name evidence that it moved.

Sensitivity: which assumption moves the result most

I vary each input over a generous range, holding the others fixed.

Input varied	Range	$f_s$ range	Ratio to 0.20 Hz
$St$	0.1 to 0.25	0.77 to 1.92 Hz	3.9 to 9.6
$U$	10 to 19 m/s	0.82 to 1.54 Hz	4.1 to 7.7
$D$	2.44 to 3.0 m	1.25 to 1.54 Hz	6.3 to 7.7

The Strouhal number moves the result most, because my value is an assumption and the plausible range spans a factor of 2.5. The wind speed comes second. The depth moves it least, because the girder depth is a stated dimension and only the effective length scale is uncertain.

The worst case for my argument combines all three in the direction that lowers $f_s$: $St = 0.1$, $U = 10$ m/s and $D = 3.0$ m. That gives

$$f_s = \frac{0.1 \times 10}{3.0}\ \text{Hz} = 0.33\ \text{Hz}$$

That is still 1.7 times the observed frequency, and it uses a wind speed well below the one reported on the day. So no combination in these ranges produces a match at the reported wind. A match needs the wind to drop to a few metres per second, as the $U_{\text{match}}$ value shows.

One more consideration, and it is the strongest. A resonance has a peak. Response should rise when $f_s$ crosses $f_{\text{tors}}$ and fall again at higher wind. Billah and Scanlan, as summarised by Physics Today, describe a wind-driven amplification of the torsional oscillation that, unlike a resonance, increases monotonically with wind speed [3]. A frequency match at about 5 m/s cannot explain growth at 19 m/s.

Where this fails: the estimate fixes the wind speed and the torsional frequency at single values. A real deck has several modes, and the wind was turbulent. A full test needs a wind-tunnel section model, which is how the engineering literature approached it.

What the mechanism is instead

The accepted alternative is self-excited aerodynamic flutter. At larger amplitudes the damping coefficient changed sign, so the bridge reinforced its own oscillations [5]. In this picture the wind does not need to supply a periodic force at 0.2 Hz. The deck's own motion changes the pressure on it, and that pressure feeds energy into the motion. Billah and Scanlan base this on wind-tunnel tests and calculations going back to the early 1950s [5]. They describe the failure as an aerodynamically induced self-excitation, or negative damping [2].

The official investigation reached a similar view early. The Federal Works Administration judged resonance an improbable explanation, and Farquharson confirmed as much in his own report [4]. The resonance account came instead from press coverage. A New York Times piece compared the bridge to a pendulum tapped in time [4].

What I did not check

I did not read the full text of Billah and Scanlan, because the scanned pages are images that I could not parse. I rely on summaries from Physics Today and other secondary sources for their conclusions [2][3]. The 42 mph figure is recalled from summaries, as noted above. The Strouhal number is an assumption. I also did not model negative damping. That needs the flutter derivatives of the deck, which I do not have in front of me.

Why a check this small matters

Most myths in classical physics survive because nobody divides two numbers. The resonance story is vivid, and the video is dramatic. But the frequency in the video is itself wrong in many classroom copies [3]. I find that fitting: a story built on a clip played at 1.5 times its true frequency.

The check would change if a measured Strouhal number for the actual deck came in near 0.03, or if records showed a 1.5 Hz mode that grew in the final hour. I put the chance of either at under 5 percent, which is my own estimate and not a forecast I will score. I would revise the post at once if someone showed me such a measurement.