A standing wave on a string forms when two waves of the same frequency and amplitude travel in opposite directions and interfere. For a string fixed at both ends, the ends must always remain nodes, so only certain wavelengths are allowed. This is why the vibration pattern is quantized into harmonics.
For a fixed-end string of length \(L\), the allowed standing-wave wavelengths are
\[
\lambda_n = \frac{2L}{n},
\qquad n=1,2,3,\dots
\]
The integer \(n\) is the harmonic number. It tells you how many half-wavelengths fit into the string length. For example, \(n=1\) is the fundamental mode, \(n=2\) is the second harmonic, and \(n=3\) is the third harmonic.
Once the wavelength is known, the frequency follows from the wave relation
\[
v=f\lambda.
\]
Substituting the allowed wavelength gives
\[
f_n = \frac{v}{\lambda_n} = \frac{nv}{2L}.
\]
This means the harmonic frequencies are integer multiples of the fundamental frequency
\[
f_1 = \frac{v}{2L}.
\]
The nodes are points that never move. For the \(n\)-th harmonic, their positions are
\[
x_m = m\frac{\lambda_n}{2} = m\frac{L}{n},
\qquad m=0,1,2,\dots,n.
\]
So there are \(n+1\) nodes, including the two fixed ends. The antinodes lie halfway between adjacent nodes and are the points of maximum oscillation.
Consider the sample values
\[
L=1\text{ m},\qquad v=200\text{ m/s},\qquad n=3.
\]
The wavelength is
\[
\lambda_3 = \frac{2L}{3} = \frac{2(1)}{3} = \frac{2}{3}\text{ m}.
\]
The frequency is
\[
f_3 = \frac{3(200)}{2(1)} = 300\text{ Hz}.
\]
This matches the expected result for the third harmonic.
The node positions for this mode are
\[
x_m = m\frac{L}{3},
\]
so the nodes are at
\[
0,\ \frac13,\ \frac23,\ 1\text{ m}.
\]
The antinodes are halfway between these values.
Standing waves on strings are essential in musical acoustics. A violin, guitar, or piano string supports harmonics, and these harmonics determine the tone color or timbre of the instrument. The higher the harmonic number, the shorter the wavelength and the higher the frequency.
At more advanced level, one can compare fixed-end conditions with free-end conditions, include damping, or study variable-density strings. But the fixed-end model used here is the standard starting point because it clearly shows how allowed wavelengths, frequencies, nodes, and antinodes arise from boundary conditions.
This calculator uses that standard model. It computes the wavelength, frequency, node positions, and antinode positions, while the interactive plot and contained animation make the harmonic structure easy to visualize.
