What distinguishes standing waves from travelling waves, and why does resonance matter?
When two identical waves travel in opposite directions through the same medium, they superpose to form a standing wave — a pattern of nodes (no displacement ever) and antinodes (maximum displacement). Unlike a travelling wave, a standing wave stores energy in a fixed spatial pattern rather than transporting it. This is how musical instruments work: a string or a column of air is constrained to vibrate only at frequencies that fit exactly within its boundaries.
Resonance occurs when a driving force matches the natural frequency of a system. At resonance, energy is transferred efficiently, and the amplitude of oscillation grows. Damping limits this growth: without any damping, amplitude at resonance would be infinite. Heavy damping suppresses the resonance peak entirely; critical damping returns the system to equilibrium as quickly as possible without oscillating — the principle behind car suspension.
A guitar string of length 0.65 m has a wave speed of 400 m s−1. Calculate the frequencies of the first three harmonics. Sketch the standing wave patterns for each harmonic, marking nodes and antinodes.
Key equations and rules
Standing wave: two identical waves travelling in opposite directions
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The Organic Chemistry Tutor
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WNY Tutor — worked problems
A string tied to a sinusoidal oscillator at P - standing wavesA stretched string fixed at each end has a mass of