Standing Waves & Resonance
Reflections can interfere with incoming waves to form standing patterns. Boundary conditions allow only particular modes, and periodic driving is strongest when it matches one of those natural frequencies.
Why do bounded systems prefer particular frequencies?
The reflected wave must fit the boundary conditions and remain self-consistent after repeated trips. Only certain wavelengths create stable node patterns, so only corresponding frequencies become normal modes.
The integer labels the harmonic mode.
Choose a mode, then drive the system through resonance.
For a string fixed at both ends, displacement must vanish at the endpoints. Allowed modes insert an integer number of half-wavelengths into the length.
A periodic driver adds energy most efficiently when its frequency is near a natural mode and its forcing overlaps that mode shape.
Real systems lose energy each cycle. Without damping, an ideal driven resonance could grow without bound in the linear model.
A perfect standing wave is the superposition of equal waves traveling in opposite directions; its nodes stay fixed in space.
For this ideal string, mode frequencies are integer multiples of the fundamental frequency f₁.
Strings, air columns, buildings, circuits, atoms, and optical cavities all exhibit mode structure and resonant response.