Standing Wave Frequencies

Equivalent forms

\lambda_n = \frac{2L}{n}

f_n = n f_1

Discrete musical notes emerge from a continuous string — quantization born from boundary conditions alone.

Symbol	Name	SI unit	Dimension	Range
$f_n$	n-th harmonic frequencyoutput Frequency of the n-th allowed standing-wave mode	Hz	T⁻¹	0 – 5000
$n$	Harmonic number Integer mode index (n=1 is fundamental)	dimensionless	1	1 – 10
$v$	Wave speed Speed of transverse waves on the string	m/s	L·T⁻¹	10 – 1000
$L$	String length Length between fixed ends	m	L	0.1 – 2

Unit systems

Symbol	SI	CGS	Imperial
$f_n$	Hz	Hz	Hz
$v$	m/s	cm/s	ft/s
$L$	m	cm	ft

Where it holds

Valid for ideal string with rigidly fixed ends, small amplitude, uniform linear mass density. Real strings have slight inharmonicity from stiffness.

Dimensional analysis

f_n \to [T^{-1}]

n\cdot v/(2L) \to [1]\cdot [L\cdot T^{-1}]/[L] = T^{-1}

Both sides have units of frequency.

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Discovery

Marin Mersenne · 1636

Mersenne discovered the laws of vibrating strings in 'Harmonie Universelle'. The full mode-quantization picture was formalized by Daniel Bernoulli and Euler in the 1740s.

Try this

Why does a guitar string play one note and not a chord of random tones?

A guitar string of length L = 0.65 m has wave speed v = 285 m/s. Find the first three harmonic frequencies.

Research status: stable

Real-world applications

Musical instruments: guitars, violins, pianos, organ pipes set pitch by these modes.
Microwave resonant cavities: same math, with electromagnetic modes between metal walls.
Laser cavities: longitudinal modes $f_n = n c / (2L)$ define laser line structure.
Acoustic engineering: standing waves cause room modes affecting bass response.

Common misconceptions

Higher harmonics are 'overtones' separate from the fundamental — they coexist as a superposition.
Pluck location doesn't matter — it affects the amplitude of each harmonic (and thus timbre).
All instruments have the same harmonic series — open pipes match strings, but closed pipes have only odd harmonics $(n=1$ ,3,5,...).

Experimental verification

Verified by Mersenne (1636) and confirmed daily by every guitar tuner and violinist. Modern spectrum analyzers show the harmonic ladder with deviations < 0.1% for ideal strings.

Derivation

On a string of length L fixed at both ends, allowed wavelengths satisfy the boundary condition y(0,

t) = y(L

,

t) = 0

.

Sinusoidal solutions sin(kx) require

kL = n\pi for

integer n, so wavelengths

\lambda _n = 2L/n

.

Using

f = v/\lambda

gives

f_n = n v / (2L)

.

The series f_1, 2f_1, 3f_1 produces the natural harmonic spectrum.

Limiting cases

n = 1

⟶

f_1 = v / (2L)

Fundamental — one half-wavelength fits the string.

L \to \infty

⟶

f_n \to 0

Infinitely long string has continuous spectrum (no quantization).

L \to 0

⟶

f_n \to \infty

Very short string has very high pitch (small instruments are bright).

What if…

What if one end is free (open)?

Boundary condition becomes anti-node at the free end; allowed modes $are \lambda _n = 4L/(2n-1)$ , only odd harmonics survive.

What if you increase tension?

Wave speed $v = \sqrt{T/\mu }$ rises, so all harmonics rise together — tightening a string raises its pitch.

What if the string is stiff?

Stiffness adds a dispersive correction making higher harmonics slightly sharp (inharmonicity) — why pianos are tuned with stretched octaves.

1

First three harmonics of a guitar string

Given ·

L:: 0.65
v:: 285

Find · f_1, f_2, f_3

Steps

Compute $f_1 = v/(2L) = 285/1.30 = 219.2\,\mathrm{Hz}$ (close to A3)
$f_2 = 2\,\mathrm{f}_1 = 438.5\,\mathrm{Hz}$ (octave above)
$f_3 = 3\,\mathrm{f}_1 = 657.7\,\mathrm{Hz}$ (octave + perfect fifth)

Result ·

f_1 = 285/(2\cdot 0.65) = 219.2\,\mathrm{Hz}

;

f_2 = 438.5\,\mathrm{Hz}

;

f_3 = 657.7\,\mathrm{Hz}

2

Tune a string to A4 (440 Hz) by changing length

Given ·

v:: 285
f 1:: 440

Find · L

Steps

Rearrange $f_1 = v/(2L)$ for L
$L = v / (2\,\mathrm{f}_1) = 285 / 880\,\mathrm{m}$
$L = 0.324\,\mathrm{m}$

Result ·

L = v/(2\,\mathrm{f}_1) = 285/(2\cdot 440) = 0.324\,\mathrm{m} \approx 32.4\,\mathrm{cm}

Wave Speed Equation→Wave Equation→Doppler Effect→