Standing Waves — Formation
Overview
A standing wave forms when two identical waves travel in opposite directions in the same medium and superimpose. The wave does NOT travel — it oscillates in place. Formed when a wave reflects off a fixed end and interferes with the incoming wave. The pattern has permanent nodes and antinodes. Energy is stored in the standing wave — it does not propagate.
Formation of standing waves
Standing waves form when two identical waves travel in opposite directions and superpose. The result is a pattern of nodes (points of zero amplitude, always destructive) and antinodes (points of maximum amplitude, always constructive). The wave pattern does not travel — it 'stands'. Standing waves store energy but do not transfer it (unlike travelling waves).
Harmonics on a string (fixed at both ends)
For a string of length L fixed at both ends, the boundary condition requires nodes at both ends. The allowed wavelengths: λ_n = 2L/n (n = 1, 2, 3, ...). The frequencies: f_n = nv/(2L) = n × f₁. Fundamental (n=1): f₁ = v/(2L), λ₁ = 2L. Second harmonic (n=2): f₂ = 2f₁, λ₂ = L. Third harmonic (n=3): f₃ = 3f₁, λ₃ = 2L/3.
- ⚠Using λ = L instead of λ = 2L for the fundamental of a fixed-fixed string
- ⚠Forgetting the difference between open pipes (antinode at each end) and closed pipes (node at closed end)