Standing Waves on a String: Harmonics & Resonance | AP Physics 2

In open-open pipes, both ends are antinodes, with fundamental wavelength λ = 2L and harmonics at 1, 2, 3, 4... In closed-open pipes, the closed end is a node and open end is an antinode, with fundamental wavelength λ = 4L and only odd harmonics (1, 3, 5, 7...). The formulas differ from strings due to different boundary conditions.

For a pipe open at both ends, the fundamental resonance occurs when the air column length equals half the wavelength (L = λ/2), giving f₁ = v/(2L). Both ends are antinodes since particles can move freely. The second resonance occurs at one full wavelength (L = λ), giving f₂ = v/L = 2f₁. The third resonance occurs at three-halves of a wavelength (L = 3λ/2), giving f₃ = 3v/(2L) = 3f₁. Closed pipes produce only odd harmonics (f₁, 3f₁, 5f₁, 7f₁...), while open pipes produce all harmonics (f₁, 2f₁, 3f₁, 4f₁...). This difference arises from boundary conditions: closed pipes have a node at one end and antinode at the other, while open pipes have antinodes at both ends. Consequently, open pipes produce smoother, sweeter sounds with more variety, while closed pipes have a more distinctive, nasal quality.

Closed pipes (one end closed) have only odd harmonics: wavelengths are λ = 4L/n where n = 1, 3, 5, 7..., and frequencies are F1, 3F1, 5F1, 7F1... Open pipes (both ends open) have all integer harmonics: wavelengths are λ = 2L/n where n = 1, 2, 3, 4..., and frequencies are F1, 2F1, 3F1, 4F1... Strings fixed at both ends have the same harmonic series as open pipes: λ = 2L/n and F1, 2F1, 3F1, 4F1... The key difference is that closed pipes lack even harmonics due to the node at the closed end.

Resonance in air columns occurs when standing waves form. In open pipes (open at both ends), the first resonance occurs at L = λ/2 with antinodes at both ends. In closed pipes (closed at one end), the first resonance occurs at L = λ/4 with a node at the closed end and antinode at the open end. For closed pipes, resonances occur at L = (2n-1)λ/4, giving wavelengths λ = 4L/(2n-1). For open pipes, resonances occur at L = nλ/2, giving wavelengths λ = 2L/n. The fundamental frequency of an open pipe is twice that of a closed pipe of the same length.

Air columns (pipes) are classified as open (open at both ends) or closed (closed at one end, open at the other). Standing wave patterns differ: open pipes have antinodes at both ends with nodes in between, while closed pipes have a node at the closed end and antinode at the open end. For open pipes, the fundamental frequency is f = v/(2L), and all harmonics (n×f) are present. For closed pipes, the fundamental frequency is f = v/(4L), and only odd harmonics (n×f where n=1,3,5...) are present. The number of antinodes equals the number of half-wavelengths in the pipe length.

Stringed instruments produce sound through vibrating strings. The tension of a string directly controls its pitch: tighter strings vibrate faster and produce higher pitches, while looser strings vibrate slower and produce lower pitches. The position along the string also affects pitch—near the tuning pegs, strings are under maximum tension and produce very high pitches, while near the bridge, there is more space for vibration and lower pitches. Frets on instruments like ukuleles and guitars change the effective vibrating length of the string, allowing musicians to play different notes. This relationship between tension, vibration, and pitch is fundamental to all stringed instruments.

This comprehensive lesson explains how string instruments produce musical notes through wave physics. Sound is a mechanical wave requiring matter to travel, unlike electromagnetic waves like light. The wave speed on a string depends on tension (T) and linear density (μ) through the formula v = √(T/μ). Standing waves form when ripples overlap, with nodes at ends and antinodes in the middle. The fundamental frequency f₁ = v/(2L) determines the lowest pitch. Thicker strings have more mass per length, slowing waves and lowering pitch, while tighter strings increase tension, speeding waves and raising pitch. A worked example demonstrates calculating linear density (0.023 kg/m), wave speed (59 m/s), and fundamental frequency (45 Hz) for a guitar string. This physics explains why thinner strings produce higher notes and how musicians control pitch through string selection and tension.

String musical instruments produce sound through vibrating strings. The pitch of a string instrument depends on three factors: (1) Length of the vibrating string—longer strings produce lower pitch, shorter strings produce higher pitch; (2) Tension of the string—increase tension increases pitch, decrease tension decreases pitch; (3) Mass per unit length of the string—increase mass per unit length decreases pitch, decrease mass per unit length increases pitch.

String instruments (guitar, violin, cello, harp) produce sound through vibrating strings. The frequency of a vibrating string depends on three factors: tension (restoring force - higher tension gives higher frequency), mass per unit length (inertia - thicker/heavier strings give lower frequency), and total length (longer strings give lower frequencies). Guitar frets and violin fingerboards allow players to shorten vibrating length to play different notes on the same string.

Fourier synthesis demonstrates that complex waveforms are constructed by adding multiple sine waves called harmonics. A square wave contains the fundamental plus all odd harmonics (3rd, 5th, 7th, 9th, etc.) in phase, progressively creating steeper sides and flatter tops. A sawtooth wave contains all harmonics (both odd and even) in phase, creating its characteristic asymmetry. A triangle wave contains all odd harmonics but with alternating phase relationships (3rd, 7th, 11th in phase; 5th, 9th, 13th inverted). Using online Fourier synthesizers, you can visualize how adding harmonics to a fundamental creates different wave shapes. This demonstrates that any repeating waveform can be decomposed into or synthesized from sine wave components of different frequencies.

Complex sounds consist of harmonics—whole-number multiples of the fundamental frequency. Square waves contain only odd harmonics with amplitudes proportional to 1/n; triangle waves follow the same pattern but with 1/n² decay. Sawtooth waves include all harmonics (odd and even) with 1/n decay. Sine waves represent pure tones with no harmonics. Joseph Fourier proved any periodic waveform can be constructed from sine waves at specific amplitudes and phases—a principle enabling both audio compression and additive synthesis.

Fourier series allow any periodic wave to be constructed as a sum of sinusoidal waves with frequencies that are integer multiples of a fundamental. A square wave at 200 Hz can be synthesized by summing sine waves at 200 Hz, 600 Hz, 1000 Hz, and other odd multiples with amplitudes following a 1/n pattern. Real musical sounds are complex waves that can be analyzed into their harmonic components. The display of harmonic amplitudes reveals why different instruments produce different sounds even when playing the same note.

Jean-Baptiste Fourier discovered that any function can be constructed by adding together sine and cosine waves with appropriate amplitudes. Starting with a single sine wave produces a basic shape, but adding more sine waves with higher frequencies and appropriately smaller amplitudes progressively approximates more complex shapes like square waves. As more terms are added, the approximation becomes increasingly accurate, demonstrating that complex functions can be decomposed into simpler sinusoidal components.

Fourier analysis is a mathematical framework for decomposing and reconstructing signals, where Fourier series applies to periodic phenomena (analyzed using frequency descriptors) and Fourier transform extends to non-periodic phenomena; both involve linear operations of analysis (breaking signals into constituent parts) and synthesis (reassembling them), with a fundamental reciprocal relationship between time/space domains and frequency domains that appears throughout the subject.