On 7 November 1940, the Tacoma Narrows Bridge (L = 853 m, opened July 1940) collapsed after wind at ~67 km/h created periodic vortices at 0.2 Hz, exactly matching the bridge's natural frequency (period 5 s). Resonance drove the deck amplitude from ~0.6 m to 8.8 m in under an hour before the structure failed. The event redefined bridge engineering worldwide.
Three quick questions from earlier lessons. Pulling old material back to mind before you learn something new makes the new material stick better, so this is not busywork.
A guitar string is plucked. The ends are fixed. What determines which frequencies the string can vibrate at? Write your prediction.
Warm-up, in a standing wave, the points of zero displacement are called:
Core Content
On 7 November 1940, at 11:00 am, the Tacoma Narrows Bridge deck began twisting violently. Wind vortices at exactly 0.2 Hz were adding energy to the deck with every cycle. The deck's reflected oscillation met the next incoming impulse in phase, each cycle the amplitude grew a little more, from 0.6 m to 8.8 m in under an hour. What the engineers watching from shore were seeing was a standing wave pattern with nodes near the towers and an antinode at mid-span.
When two waves of identical frequency, amplitude and speed travel in opposite directions along the same medium, their superposition produces a standing wave. The nodes remain permanently stationary; the antinodes oscillate with maximum amplitude. The distance between adjacent nodes (or adjacent antinodes) is $\lambda/2$.
A standing wave forms when two identical waves travel in opposite directions and superpose; nodes (zero displacement, permanent) are separated by $\lambda/2$, and antinodes (maximum displacement) sit midway between nodes. Standing waves do not transport net energy.
Pause, copy the highlighted definition and spacings into your book before moving on.
Drive the wave superposition lab — its standing-wave mode lets you hunt for the driving frequencies that build a pattern on a 2.00 m string, and park a probe on a node to measure how much energy passes it. It is in Lesson 4 , in the step called The Superposition Principle .
The distance between two adjacent nodes in a standing wave equals:
We just saw that standing waves have fixed nodes and antinodes with no net energy transport. That raises a question: what happens if an external periodic force continuously feeds energy into a standing wave at exactly the right frequency? This card answers it → resonance: amplitude grows dramatically when the driving frequency matches the natural frequency.
Every mechanical system has one or more natural frequencies at which it oscillates after being disturbed. When an external periodic driving force is applied at the same frequency, energy is continually added in phase with the oscillation, the amplitude grows dramatically. This is resonance.
At resonance the energy input matches energy losses, so the system can sustain large, steady oscillations. If damping is low and the driving force is not removed, amplitude can grow until the system fails (as in the Tacoma Narrows bridge).
Resonance occurs when an external driving frequency equals a system's natural frequency, causing the amplitude to grow dramatically as energy is added in phase each cycle. Insufficient damping can lead to structural failure (Tacoma Narrows Bridge, 1940).
Add the highlighted resonance definition to your notes before the check below.
Standing waves carry energy in the same direction as the component waves.
Resonance occurs when the driving frequency equals the natural frequency of a system.
Activities
Sketch two full standing wave patterns for a string fixed at both ends: (a) the fundamental (1 loop), (b) the second harmonic (2 loops). Label all nodes and antinodes for each.
Describe one real-world example of resonance. Explain: (a) what the natural frequency belongs to, (b) what the driving force is, (c) what happens to amplitude at resonance.
A standing wave is set up on a string 1.2 m long. The pattern has 4 loops. (a) How many nodes are there? (b) How many antinodes? (c) What is the wavelength of the standing wave?
Which of these is the odd one out relating to standing waves?
A string 0.6 m long vibrates in its fundamental mode. The wavelength of the standing wave is:
Resonance occurs when:
UnderstandBand 3(3 marks) 1. Explain how a standing wave is formed and identify what is meant by nodes and antinodes.
ApplyBand 4(3 marks) 2. A string of length 0.80 m vibrates with 3 loops. Calculate the wavelength and state how many nodes are present.
AnalyseBand 5(4 marks) 3. Using the Tacoma Narrows Bridge collapse as an example, explain how resonance can cause mechanical failure. In your answer refer to natural frequency, driving frequency, and amplitude.
Q1 (3 marks): A standing wave forms when two waves of identical frequency, amplitude and speed travel in opposite directions and superpose. Nodes are fixed points of zero displacement, the waves always cancel there. Antinodes are fixed points of maximum displacement, the waves always reinforce there.
Q2 (3 marks): $\lambda = 2L/n = 2 \times 0.80/3 = 0.533$ m. Number of nodes = 3 + 1 = 4 nodes.
Q3 (4 marks): The Tacoma Narrows Bridge (1940) had a natural frequency of about 0.2 Hz. Steady wind created periodic vortices (a driving force) matching this frequency. Because the driving frequency equalled the bridge's natural frequency, energy was continuously added in phase, resonance. With insufficient damping, amplitude grew from centimetres to 8.5 m, exceeding the bridge's structural limits and causing collapse.
The Tacoma Narrows Bridge collapsed on 7 November 1940 because wind vortices at 0.2 Hz (period 5 s) matched the bridge's natural frequency exactly. Resonance drove the 853 m deck to 8.8 m amplitude, a standing wave mode with antinodes at mid-span and nodes near the towers. Low damping meant energy input exceeded losses, and the amplitude grew until structural failure.
Your Think First prediction about guitar string harmonics connects directly: the string's fixed ends are nodes, and only frequencies where an integer number of half-wavelengths fits between the ends ($f_n = nv/2L$) are sustained. The Tacoma Narrows bridge obeyed the same standing-wave resonance, just at catastrophic scale.
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