6.7 Standing Waves and Musical Instruments

Standing waves form by superposition

A standing wave results when two identical waves traveling in opposite directions interfere, often from a wave reflecting back on itself. Points that never move are nodes; points of maximum displacement are antinodes. The pattern appears fixed in space, not traveling, because the interference is constant over time.

Strings fixed at both ends

A string fixed at both ends must have nodes at both ends. The fundamental (first harmonic) has length L = λ/2, so λ₁ = 2L. Higher harmonics follow λₙ = 2L/n, giving frequencies fₙ = nv/(2L), where n = 1, 2, 3… — all integer multiples of the fundamental are allowed.

Pipes: open vs. closed

An open-open pipe (antinodes at both ends) supports all harmonics: fₙ = nv/(2L), like a string. A closed-open pipe (node at closed end, antinode at open end) supports only odd harmonics: fₙ = nv/(4L), n = 1, 3, 5… This is why closed pipes sound different, lacking even harmonics.

Key terms

Standing wave

A wave pattern formed by superposition of two identical waves traveling in opposite directions, creating fixed nodes and antinodes.

Node

A point on a standing wave with zero displacement at all times.

Antinode

A point on a standing wave with maximum displacement amplitude.

String fixed at both ends: allowed harmonics

All integer harmonics (n = 1, 2, 3…); fₙ = nv/(2L).

Closed-open pipe: allowed harmonics

Only odd harmonics (n = 1, 3, 5…); fₙ = nv/(4L), due to node at closed end and antinode at open end.

Try an AP-style question

A 0.5 m string fixed at both ends has wave speed 200 m/s. What is its fundamental frequency?

1. 100 Hz
2. 200 Hz
3. 400 Hz
4. 50 Hz