KAHIBARO
Table of Contents
Fixed ends and standing waves on a string
A stretched string can support standing waves when waves travel along it, reflect at the ends, and combine with the incoming waves. For a string fixed at both ends, the ends cannot move, so they must always be nodes. This condition strongly restricts which wave patterns are allowed.
Only certain wavelengths fit on the string. If the string has length $L$, then the allowed patterns are those for which an integer number of half wavelengths fits exactly into the length:
$$
L = n \frac{\lambda_n}{2}, \qquad n = 1,2,3,\dots
$$
This gives the allowed wavelengths
$$
\lambda_n = \frac{2L}{n}
$$
Each allowed pattern is called a normal mode, or harmonic.
For a string fixed at both ends, the allowed wavelengths are
$$
\lambda_n = \frac{2L}{n}, \qquad n = 1,2,3,\dots
$$
Only these wavelengths produce stable standing-wave patterns.
Harmonics on a string
The simplest standing wave is the first harmonic, also called the fundamental. In this mode, the string has nodes at both ends and one antinode in the middle. The string length contains half of a wavelength:
$$
L = \frac{\lambda_1}{2}
\quad \Rightarrow \quad
\lambda_1 = 2L
$$
The second harmonic has one extra node in the middle and two antinodes. The length contains one full wavelength:
$$
L = \lambda_2
\quad \Rightarrow \quad
\lambda_2 = L
$$
The third harmonic has two interior nodes and three antinodes:
$$
L = \frac{3\lambda_3}{2}
\quad \Rightarrow \quad
\lambda_3 = \frac{2L}{3}
$$
This pattern continues for higher harmonics.
Frequencies of a vibrating string
The wave speed on the string is related to frequency and wavelength by
$$
v = f \lambda
$$
Since only certain wavelengths are allowed, only certain frequencies are allowed. Substituting $\lambda_n = 2L/n$ gives
$$
f_n = \frac{v}{\lambda_n} = \frac{nv}{2L}
$$
So the frequencies of the harmonics are integer multiples of the fundamental frequency:
$$
f_n = n f_1
$$
where
$$
f_1 = \frac{v}{2L}
$$
For a string fixed at both ends, the allowed frequencies are
$$
f_n = \frac{nv}{2L}, \qquad n=1,2,3,\dots
$$
and the fundamental frequency is
$$
f_1 = \frac{v}{2L}
$$
Role of tension and mass per unit length
The wave speed on a stretched string depends on the tension $T$ and the linear mass density $\mu$, where $\mu$ is mass per unit length:
$$
\mu = \frac{m}{L}
$$
The speed is
$$
v = \sqrt{\frac{T}{\mu}}
$$
Combining this with the harmonic frequency formula gives
$$
f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}
$$
This formula shows how the sound or vibration changes when the string is adjusted. A shorter string gives a higher frequency. Greater tension gives a higher frequency. A heavier string, meaning larger $\mu$, gives a lower frequency.
For a stretched string fixed at both ends,
$$
f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}
$$
Higher tension raises frequency, larger mass per unit length lowers frequency, and shorter length raises frequency.
Nodes and antinodes on strings
In a standing wave on a string, some points never move. These are nodes. Other points oscillate with maximum amplitude. These are antinodes. For the $n$th harmonic of a string fixed at both ends, there are $n+1$ nodes if the two ends are included, and $n$ antinodes.
The distance between adjacent nodes is half a wavelength:
$$
\text{node to node distance} = \frac{\lambda}{2}
$$
The distance between a node and the nearest antinode is one quarter of a wavelength:
$$
\text{node to antinode distance} = \frac{\lambda}{4}
$$
These distances help identify the harmonic number from an observed pattern.
Musical strings
Stringed instruments use these standing-wave patterns. A guitar, violin, or piano string is stretched between fixed points, so the standing-wave rules apply directly. The fundamental usually determines the perceived pitch, while higher harmonics add richness to the sound.
Pressing a finger on a string shortens the vibrating length $L$, which increases the frequency. Tightening the tuning peg increases the tension $T$, which also increases the frequency. Changing to a thicker string changes $\mu$, usually lowering the pitch.
Summary table
| Quantity | Formula |
|---|---|
| Allowed wavelengths | $\lambda_n = \dfrac{2L}{n}$ |
| Fundamental wavelength | $\lambda_1 = 2L$ |
| Allowed frequencies | $f_n = \dfrac{nv}{2L}$ |
| Fundamental frequency | $f_1 = \dfrac{v}{2L}$ |
| Wave speed on string | $v = \sqrt{\dfrac{T}{\mu}}$ |
| Harmonic frequencies in terms of string properties | $f_n = \dfrac{n}{2L}\sqrt{\dfrac{T}{\mu}}$ |
Physical picture
A standing wave on a string is not a single bump traveling from one end to the other. It is the result of two identical waves moving in opposite directions along the string. Because of interference, fixed points appear at the nodes, while the antinodes swing up and down.
The fixed ends force the pattern to match the boundary conditions. That is why only specific wavelengths and frequencies are possible. This is the key idea behind standing waves on strings.