Standing / Stationary Waves

Interactive Physics Presentation

What is a Standing Wave?

A standing wave is formed by the superposition of two identical progressive waves traveling in opposite directions. The result is a wave pattern with fixed nodes (zero displacement) and antinodes (maximum displacement).

Real-life examples: Vibrations on a guitar string, violin string, microwave oven, organ pipes.

Live Interactive Standing Wave

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● Blue dots = Nodes ● Red dots = Antinodes

Superposition Principle & Derivation

Two opposite progressive waves:

y₁ = a sin(ωt − kx)
y₂ = a sin(ωt + kx)

Superposition →

y = y₁ + y₂ = 2a sin(ωt) cos(kx) ← Standing wave equation

Or the most common form for a string fixed at both ends:

y(x,t) = 2A sin(kx) cos(ωt) where A = a (original amplitude)

Nodes and Antinodes

Node points (amplitude = 0) Antinode points (amplitude = maximum)
sin(kx) = 0 → kx = nπ → x = nL/(2n) = nL/n (n = 0,1,2...) cos(kx) = ±1 → kx = π/2, 3π/2, … → x = (2n+1)L/(2n)
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