When waves traveling toward each other overlap, a standing wave is produced. Let us understand the “frequencies that readily produce sound” (resonance conditions) of strings and air columns.
Basic Knowledge
A standing wave is formed when two waves with equal amplitude, frequency, and wavelength travel in opposite directions and overlap.
・Node: A point whose displacement is always 0. Spacing = λ/2
・Antinode: A point where the displacement is maximum. Spacing = λ/2 (nodes and antinodes are separated by λ/4)
Natural vibrations of a string（fixed at both ends, string length L）:
Since both ends are nodes, L = n(λ/2)（n = 1,2,3,…）
→ λ = 2L/n，fn = nv/(2L)（v: wave speed on the string）
n=1 is the fundamental vibration（fundamental tone）, and n=2,3,… are harmonics.
Natural vibrations of an air column:
・Open pipe at both ends: Both ends are antinodes → L = n(λ/2) → fn = nv/(2L)（v: speed of sound）
・Closed pipe at one end: The closed end is a node and the open end is an antinode → L = (2n−1)(λ/4) → fn = (2n−1)v/(4L)（only odd harmonics）
Standing wave（Superposition of identical waves traveling in opposite directions. Nodes and antinodes are fixed）
Node / Antinode（A point whose displacement is always 0 / A point with maximum displacement）
Fundamental（The lowest-order resonance state. n = 1）
Harmonic（A resonant sound whose frequency is an integer multiple of the fundamental tone）
Open pipe・Closed pipe（An air column open at both ends / closed at one end. Their resonance conditions differ）
Further Study (Background・Significance)
The wave speed on a string is v = √(S/ρ)（S: tension，ρ: linear density）. The principle is that tightening a guitar string more strongly (larger S) produces a higher sound, while a thicker string (larger ρ) produces a lower sound.
The reason a pipe closed at one end produces only odd harmonics is that the boundary conditions of a node at the closed end and an antinode at the open end are satisfied only by λ = 4L/(2n−1). A flute is an open pipe at both ends (even harmonics are also produced), while a clarinet behaves like a pipe closed at one end (odd harmonics are strong); this provides the physical basis for their different timbres.
The fact that the pitch rises by 1 octave (frequency doubles) at the 12th fret of a guitar, where the string length is halved, can also be directly explained by the resonance conditions of a string.
- Standing wave: the spacing between nodes and between antinodes is λ/4
- String (fixed at both ends): L = nλ/2 → fn = nv/(2L)
- Open pipe at both ends: same condition as a string
- Pipe closed at one end: L = (2n-1)λ/4 → only odd harmonics
- Wave speed on a string v = √(S/ρ)（determined by tension and linear density）
Points to Note (Easily Confused)
① The resonance conditions for an open pipe at both ends and a string have the same form, but their wave speeds differ (string: wave speed on the string; open pipe: speed of sound). ② A pipe closed at one end produces only odd harmonics (even harmonics are not produced). ③ The spacing between nodes is λ/2 (λ/4 is the spacing between a node and an antinode). ④ Harmonic series: when the fundamental tone is f1, the harmonics are 2f1, 3f1, …, not f1/2.
Practice
- Find the fundamental frequency and the frequency of the second harmonic of a 0.60 m string with a wave speed of 120 m/s.
- Find the lowest resonant frequency of a pipe closed at one end (length 0.25 m) (speed of sound 340 m/s).
- State the meanings of “node” and “antinode” in a standing wave, and express the spacing between adjacent nodes using the wavelength λ.