Standing Wave on String
λ_n = 2L/n — nodes & antinodes.
Key Notes
Standing wave: pattern formed by superposition of two equal-amplitude waves traveling in opposite directions.
y(x,t) = 2A·sin(kx)·cos(ωt) — separates space and time dependence.
Has fixed NODES (zero amplitude) and ANTINODES (max amplitude). Particles oscillate but pattern doesn't propagate.
Distance between consecutive nodes: λ/2. Same for consecutive antinodes.
Distance from node to nearest antinode: λ/4.
Found on: vibrating strings (musical instruments), open/closed pipes, microwave ovens, laser cavities.
All points between two nodes oscillate IN PHASE; on opposite sides of a node, π out of phase.
Energy transport: ZERO in a perfect standing wave (no net flow).
Formulas
Standing wave (transverse)
Space and time separable.
Node positions
Where sin(kx) = 0.
Antinode positions
Where sin(kx) = ±1.
Distance node-to-antinode
Quarter wavelength.
Important Points
Standing wave doesn't TRANSPORT energy — energy bounces back and forth between two sides.
Nodes are SEPARATED by λ/2.
All points oscillate with the SAME f and ω, but at different AMPLITUDES.
Boundary conditions FIX the allowed wavelengths: only specific λ fit ⇒ DISCRETE modes (harmonics).
Pluck a guitar at center ⇒ excites odd harmonics; pluck off-center ⇒ excites both.
Lasers: standing waves between mirrors at specific wavelengths ⇒ cavity modes.
Standing Wave on String notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.