The Wrong Frequency
The problem
7 November 1940. The Tacoma Narrows Bridge twists itself apart in a 42 mph wind, filmed the whole way down. Every physics textbook since has told the same story: wind vortices shed off the deck at the same frequency as the bridge’s natural twisting motion, pumped it like a swing pushed at just the right moment, and resonance did the rest.
Check the story with the same tool an engineer of the day would have reached for.
The bridge deck was a solid plate girder just 8 ft deep — shallow and solid rather than an open truss. This wasn’t the ordinary choice for a bridge this size: the Washington State Toll Bridge Authority had originally proposed a 25-ft-deep stiffening truss, the kind used on most suspension bridges before and since, but engineer Leon Moisseiff argued it down to a fraction of that depth for cost and looks. An open truss avoids this problem two ways at once — its lattice lets wind pass through rather than piling up behind a solid face, and its greater depth gives it far more resistance to twisting. Take both of those away, as Tacoma’s design did, and you get a solid, bluff-shaped, torsionally soft deck: a shape that sheds vortices at a frequency given by the Strouhal relation, using these figures:
- Strouhal number for this girder’s cross-section: S = 0.11
- Wind speed at the time of collapse: 42 mph
- Girder depth, the characteristic dimension: D = 8 ft
- 1 mph = 5280/3600 ft/s
The bridge’s actual twisting motion that morning, filmed and timed by witness F.B. Farquharson, ran at 0.2 Hz — one full twist roughly every five seconds.
Find the vortex shedding frequency, in Hz. Is it close enough to 0.2 Hz for resonance to be the explanation?
Notes
- Work entirely in feet and seconds: convert the wind speed to ft/s first.
- The Strouhal relation is
f = S · V / D. S = 0.11 is specific to this girder’s cross-section, not a universal constant — a different bluff shape gets a different value. - This one formula can’t by itself explain what did bring the bridge down. That takes the fuller aeroelastic flutter analysis in Billah & Scanlan, American Journal of Physics 59(2), 1991 — the paper that ran exactly this comparison to overturn the textbook story. What the formula here can settle is narrower, and still worth knowing: whether “simple resonance” survives contact with arithmetic.
Hint
Click for a hint
Convert wind speed to ft/s, then apply the Strouhal relation:
V (ft/s) = 42 × 5280 / 3600
f = S · V / D
One chain of multiplications and divisions, no addition or subtraction.
Solution
Click to reveal the solution
Wind speed in ft/s:
V = 42 × 5280 / 3600 = 61.6 ft/s
Vortex shedding frequency:
f = S · V / D = 0.11 × 61.6 / 8 = 0.847 Hz
Answer: 0.847 Hz.
That is about 4.2 times — roughly two octaves — higher than the 0.2 Hz actually filmed that morning. Vortex shedding at 42 mph was cycling far too fast to be pushing the bridge’s twist in lockstep, and this is the same mismatch Billah and Scanlan (1991) used to retire the “simple resonance” version of the story. The real mechanism, established by their fuller analysis, was self-excited torsional flutter: an aerodynamic instability in which the bridge’s own motion — not a periodic outside push — sustained and amplified itself. That part goes past a single formula; what this arithmetic settles is only that resonance-at-a-matched-frequency does not.
A second, independent line points the same way. Larsen (2000) gives the critical wind speed for this kind of torsional flutter as roughly Uc ≈ 4·f·B (f the torsional frequency, B the deck width); the wind actually blowing on 7 November 1940 works out to U ≈ 8·f·B — about twice the threshold, not a coincidence sitting on some knife-edge frequency match. Two different ways of reading the numbers, same conclusion: this was a wind speed well past a real instability, not a lucky (or unlucky) match of frequencies.
Calculator solutions
HP-42S
One chain, no memory registers needed.
0.11 ENTER
42 *
5280 *
3600 /
8 /
Display: 0.847
Elektronika MK-61
Same chain. Enter every number starting with a digit, never a bare decimal point: 0.11 is typed
0 . 1 1.
0.11 ENTER
42 ×
5280 ×
3600 ÷
8 ÷
Display: 8.47 -01
This is a genuine hardware quirk, not a typo: every intermediate value along the way (4.62, 24393.6, 6.776) displays clean, but the final division on the real chip’s BCD arithmetic doesn’t land on an exact 0.847 the way typing 0.847 directly does — it normalizes to scientific notation instead. Read as 8.47 × 10⁻¹, it’s the same 0.847.
HP-15C
Same structure as the 42S:
0.11 ENTER
42 ×
5280 ×
3600 ÷
8 ÷
Result: 0.8470
HP-41C
0.11 ENTER
42 ×
5280 ×
3600 ÷
8 ÷
Display: 0.8470
HP-48GX
A plain stack entry — no PI, no squaring, so neither of the RPL traps from challenges 3 and 5
comes up here.
0.11 ENTER
42 ×
5280 ×
3600 ÷
8 ÷
Result: .847
HP-25
The machine powers up in FIX 2, which would round the answer to 0.85. f FIX 4 first shows the
full four digits to compare against the other machines.
f FIX 4
0.11 ENTER
42 ×
5280 ×
3600 ÷
8 ÷
Display: 0.8470
TI-84
.11*42*5280/3600/8
Result: 0.847
Casio fx-991
Same expression, natural display mode:
0.11×42×5280÷3600÷8
Result: 0.847
Slide Rule
Every step is multiplication or division, so the rule handles all of it.
- Convert the wind speed. 42 × 5280 ÷ 3600. Set the index of C at 42 on D, read 5280 on C’s position… more directly: multiply 4.2 × 5.28 on C and D to get ≈ 2.22 (22.2, tracking the exponent by hand), then divide by 3.6: set 2.22 on D, bring 3.6 on C over it, read ≈ 6.16 at the index — that’s 61.6 ft/s.
- Apply the Strouhal number. Multiply 0.11 × 61.6: set the left index of C at 1.1 on D, move the cursor to 6.16 on C, read ≈ 6.78 (which is 6.776 to a hair more precision than the rule gives).
- Divide by the girder depth. 6.78 ÷ 8: set 6.78 on D, bring 8 on C over it, read at the index: ≈ 0.847.
Answer: ≈ 0.847 Hz, hand-checked against the verified 0.847 result above — a rare case where a slide rule’s three honest digits happen to match the exact value.
Leon Moisseiff, the engineer who designed Tacoma Narrows, was also the consulting engineer for the Golden Gate Bridge’s deck and suspension structure — but Golden Gate was built with an open stiffening truss, not Tacoma’s shallow, solid plate girder, so it never carried the same specific vulnerability. It still had its own scare: designed for winds up to 68 mph, it met gusts of 69 mph on 1 December 1951 and swayed and rolled badly enough to close the bridge for the evening. A later engineering study put the original structure’s critical flutter speed at 31 m/s — about 69.34 mph, essentially the same wind that actually hit it. The 1953–54 retrofit that followed added bracing specifically to stiffen the deck in torsion.
Solvable on: HP-42S, HP-15C, HP-41C, HP-48GX, HP-25, Elektronika MK-61, TI-84, Casio fx-991, Slide Rule