A stretched string fixed at both ends supports standing waves. These arise when waves traveling in opposite directions interfere with each other. Because both ends are fixed, the displacement must always be zero at each end, so the ends are always nodes. This boundary condition allows only certain wavelengths and frequencies.
The wave speed on a string depends on the tension \(T\) and the linear mass density \(\mu\). The standard formula is
\[
v=\sqrt{\frac{T}{\mu}}.
\]
A larger tension makes the wave travel faster, while a heavier string (larger \(\mu\)) makes it travel more slowly. This is why tightening a guitar string raises its pitch, while using a thicker string lowers it.
Since the ends are fixed, the allowed standing-wave patterns must fit an integer number of half-wavelengths into the length \(L\):
\[
L=\frac{n\lambda_n}{2},
\qquad n=1,2,3,\dots
\]
Solving for the wavelength gives
\[
\lambda_n=\frac{2L}{n}.
\]
Each positive integer \(n\) gives one harmonic. The case \(n=1\) is the fundamental, \(n=2\) is the second harmonic, and so on.
Using the general wave relation
\[
f=\frac{v}{\lambda},
\]
the harmonic frequencies become
\[
f_n=\frac{v}{\lambda_n}
=\frac{n v}{2L}.
\]
Substituting the string wave speed gives the useful combined formula
\[
f_n=\frac{n}{2L}\sqrt{\frac{T}{\mu}}.
\]
This is the main formula used in this calculator.
The harmonic number also determines the node and antinode structure. For the \(n\)-th harmonic:
\[
\text{nodes at } x=\frac{mL}{n}, \qquad m=0,1,2,\dots,n,
\]
and
\[
\text{antinodes at } x=\frac{(2m+1)L}{2n}, \qquad m=0,1,2,\dots,n-1.
\]
As \(n\) increases, the number of internal nodes increases and the shape of the string becomes more complex.
For the sample values \(L=0.65\ \text{m}\), \(T=100\ \text{N}\), and \(\mu=0.005\ \text{kg/m}\),
\[
v=\sqrt{\frac{100}{0.005}}=\sqrt{20000}\approx 141.4\ \text{m/s}.
\]
For the second harmonic \(n=2\),
\[
f_2=\frac{2(141.4)}{2(0.65)}\approx 217.6\ \text{Hz}.
\]
The corresponding wavelength is
\[
\lambda_2=\frac{2(0.65)}{2}=0.65\ \text{m}.
\]
This ideal model is widely used for guitars, violins, and laboratory string experiments. At university level, the model can be refined to include stiffness, damping, variable tension, and non-uniform density. Even so, the fixed-end standing-wave model remains the standard starting point for harmonic analysis on strings.
This calculator shows both the numerical harmonic values and the geometry of the mode shape. The plot displays how frequency or wavelength changes with harmonic number, while the animation helps visualize nodes, antinodes, and the standing-wave pattern along the string.
