← Theme C · Wave Behaviour
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Theme C · Wave Behaviour

Chapter 2 · Resonance & Doppler

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Standing wave
Two identical waves travelling opposite ways superpose: fixed nodes (no motion) and antinodes (max motion).
String harmonics (both ends fixed)
λn=2Ln,fn=nv2L\lambda_n = \frac{2L}{n},\quad f_n = \frac{nv}{2L}
n=1,2,3,…n = 1, 2, 3, \dots
Pipe (one end closed)
fn=nv4Lf_n = \frac{nv}{4L}
Odd harmonics only (n=1,3,5n = 1, 3, 5).
Resonance
Driving a system at its natural frequency → large amplitude (e.g. bridges, instruments).
Doppler effect
f′=fvv∓vsf' = f\frac{v}{v \mp v_s}
Source approaching (−) → higher pitch; receding (+) → lower pitch.
Diffraction & interference
Waves bend around gaps; overlapping waves add (constructive) or cancel (destructive).
Path difference (interference)
Constructive: path difference = nλn\lambda. Destructive: (n+12)λ(n+\tfrac12)\lambda.
Nodes vs antinodes
Standing wave: adjacent nodes are λ/2\lambda/2 apart; antinode↔node is λ/4\lambda/4.
Open pipe (both ends open)
fn=nv2Lf_n = \frac{nv}{2L}
All harmonics, like a string.
Doppler — be careful with sign
Approaching source ⇒ v−vsv-v_s (smaller denominator ⇒ HIGHER f). Receding ⇒ v+vsv+v_s (lower f).
Key SI units
ff: Hz · vv: m s⁻¹ · LL: m · λ\lambda: m. Speed of sound ≈ 340 m s⁻¹ in air.
Common traps
Using 4L for a string (it's 2L); forgetting closed pipes only have ODD harmonics; wrong Doppler sign.