Standing Waves on Strings

A string of length L fixed at both ends supports standing waves with wavelengths λ_n = 2L ÷ n and frequencies f_n = nv ÷ 2L = (n ÷ 2L)√(T ÷ μ), where n = 1, 2, 3, … is the harmonic number and v = √(T ÷ μ) is the wave speed on the string. In brief: the lowest frequency, f₁, is the fundamental (first harmonic); the harmonics are whole-number multiples of it. The nth harmonic has n antinodes and n + 1 nodes (including the fixed ends). The pitch of a stringed instrument rises with greater tension, shorter length and lower mass per unit length. A real note is a mixture of harmonics, which determines its timbre.

Guitars, violins and pianos produce notes by standing waves on strings. This guide derives the allowed frequencies, explains the harmonics and the effect of tension, length and thickness, compares the first few harmonics in a table, works through a guitar string example and corrects common mistakes.

How do standing waves form on a string?

When a string fixed at both ends is plucked, waves travel along it in both directions and are reflected at the ends, with a phase change of π. Waves of the same frequency traveling in opposite directions superpose to give a standing wave: y(x, t) = 2A sin(kx) cos(ωt). The positions where sin(kx) = 0 are the nodes, and where |sin(kx)| = 1 the antinodes. Since the ends are fixed, they must be nodes. This boundary condition allows only waves for which the length of the string is a whole number of half wavelengths: L = nλ ÷ 2. At other frequencies, the waves reflected from the ends do not reinforce each other, and the vibrations die away quickly. In a real string, damping and the energy radiated as sound are small enough that the allowed modes persist for a time (the “ringing” of the string).

What are the allowed frequencies?

With v = fλ and λ_n = 2L ÷ n:

f_n = n v ÷ 2L = n f₁, with f₁ = (1 ÷ 2L) √(T ÷ μ).

• The fundamental f₁ (n = 1) has one antinode in the middle and a wavelength of 2L.
• The second harmonic (n = 2) has a node at the center, and a frequency of 2f₁ (one octave higher). The third harmonic (n = 3) is 3f₁ (an octave and a fifth).
• The frequency is proportional to √T and inversely proportional to L and to √μ. Quadrupling the tension doubles the frequency; halving the length doubles it.

A string plucked at a point excites a mixture of harmonics, but not those with a node at the plucking point. A string plucked at the center lacks the even harmonics, giving a hollow sound. Touching the string lightly at the midpoint and plucking produces the second harmonic (a “natural harmonic” on guitar).

How do instruments use these frequencies?

On a guitar or violin, the player changes the vibrating length by pressing the string against the fingerboard (the frets on a guitar), and the tension of the strings is adjusted with tuning pegs. The strings are of different thickness (mass per unit length): the bass strings are thicker and heavier, giving a lower pitch for similar length and tension. On a piano, strings of different length, thickness and tension cover seven octaves. The vibrating strings alone move little air, so they are coupled to a sound board or a hollow body, which radiates the sound more effectively and has its own resonances that shape the tone. In the equal-tempered scale, each semitone corresponds to a frequency ratio of 2^1/12 = 1.0595, and the 12th fret of a guitar halves the vibrating length, raising the pitch by an octave.

Harmonics compared

Worked example: a guitar string

A string has a length of 0.65 m between its fixed ends, a tension of 80 N and a linear density of 5.0 × 10⁻⁴ kg/m.

1. Wave speed: v = √(T ÷ μ) = √(80 ÷ 5.0 × 10⁻⁴) = √160,000 = 400 m/s.
2. Fundamental: f₁ = v ÷ 2L = 400 ÷ 1.30 = 308 Hz (near E4, 329.6 Hz, but a bit lower).
3. Higher harmonics: f₂ = 615 Hz, f₃ = 923 Hz, f₄ = 1,231 Hz.
4. Tuning up: to raise f₁ to 329.6 Hz, the tension must increase by (329.6 ÷ 307.7)² = 1.147, to 91.8 N, since f ∝ √T.
5. Fretting: pressing the string at a fret so that the vibrating length is 0.55 m gives f₁ = 400 ÷ 1.10 = 364 Hz.
6. Wavelength of the sound in air: the sound produced at 308 Hz has a wavelength of 343 ÷ 308 = 1.11 m in air, which is different from the wavelength 1.30 m on the string, because the speeds differ. The frequency is the same.

How were string vibrations studied?

Pythagoras and his followers are said to have discovered the relations between the lengths of strings and musical intervals. Marin Mersenne published the laws of the vibrating string in 1636: frequency inversely proportional to length and to the square root of the mass per length, and directly proportional to the square root of tension. Joseph Sauveur observed nodes and antinodes in 1701 and introduced the term “harmonics”. Brook Taylor (1713) and d’Alembert (1747) obtained the mathematical theory, and Daniel Bernoulli showed that a string can vibrate with several harmonics at once.

Which mistakes are common?

Mistake 1: Saying that the wavelength of the fundamental is L

It is 2L: the string length is half a wavelength.

Mistake 2: Confusing the wave speed on the string with the speed of sound

The string wave speed is √(T ÷ μ), which is different. The frequency is the same for the string and the sound it creates.

Mistake 3: Forgetting that frequency scales as the square root of tension

Doubling the tension raises the frequency by a factor of 1.41, not 2.

Mistake 4: Counting nodes incorrectly

The nth harmonic has n + 1 nodes if the two fixed ends are counted, and n − 1 nodes in between.