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On the syllabus: A-Level Physics
In 1940, the Tacoma Narrows Bridge in Washington State collapsed — not from overloading, not from metal fatigue, but because the wind blew at exactly the wrong frequency. The bridge literally shook itself apart. Meanwhile, an opera singer can shatter a wine glass just by holding a sustained note. What do a bridge and a wine glass have in common?
In 1940, the Tacoma Narrows Bridge in Washington State collapsed — not from overloading, not from metal fatigue, but because the wind blew at exactly the wrong frequency. The bridge literally shook itself apart. Meanwhile, an opera singer can shatter a wine glass just by holding a sustained note. What do a bridge and a wine glass have in common?
Every physical structure has natural frequencies at which it prefers to vibrate. When energy arrives at exactly one of those frequencies, the vibrations grow without bound — a phenomenon called resonance. Standing waves are what resonance looks like from the inside, and understanding their patterns is what separates safe bridge design from catastrophic failure.
Standing waves form when two identical waves travel in opposite directions and superpose. For a string fixed at both ends, only wavelengths where a whole number of half-wavelengths fit exactly in the length L are resonant — these are the harmonics.
The fundamental frequency (n=1) has half a wavelength fitting the string length: λ₁ = 2L. The 2nd harmonic has one full wavelength (λ₂ = L); the 3rd has λ₃ = 2L/3. Wave speed in a string: v = √(T/μ) where T is tension and μ is mass per unit length. Higher tension → higher v → higher frequencies. Thicker strings → higher μ → lower frequencies. This is why guitar strings are wound with metal (to increase μ while staying thin) and why tightening a tuning peg raises pitch.