AcademyMechanical Waves
Academy
Standing Waves on Strings
Level 1 - Physics topic page in Mechanical Waves.
Principle
Standing waves form when equal waves of the same frequency travel in opposite directions.
On a string, reflections at the ends can supply the oppositely traveling wave and lock the pattern into fixed nodes and antinodes.
Notation
\(A\)
amplitude of each traveling wave
\(\mathrm{m}\)
\(k\)
wavenumber
\(\mathrm{rad\,m^{-1}}\)
\(\omega\)
angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(L\)
string length
\(\mathrm{m}\)
\(f_1\)
fundamental frequency
\(\mathrm{Hz}\)
\(v_w\)
wave speed
\(\mathrm{m\,s^{-1}}\)
Method
Derivation 1: Add two oppositely traveling waves
Take two waves with equal amplitude and frequency, one moving right and one moving left.
Right-moving wave
\[y_1=A\sin(kx-\omega t)\]
Left-moving wave
\[y_2=A\sin(kx+\omega t)\]
Add them
\[y=y_1+y_2=2A\sin(kx)\cos(\omega t)\]
This is no longer a traveling profile. The spatial and temporal parts separate.
Derivation 2: Identify nodes and antinodes
The factor \(\\sin(kx)\) controls how large the oscillation is at each position.
Nodes
\[\sin(kx)=0\]
Node positions
\[x=\frac{m\pi}{k}=m\frac{\lambda}{2}\]
Antinodes
\[|\sin(kx)|=1\]
The standing-wave snapshot below has fixed nodes at the ends and one antinode in the middle.
Derivation 3: Read the fundamental for a string with fixed ends
For the lowest standing mode, the string holds half a wavelength between its ends.
Fundamental geometry
\[L=\frac{\lambda_1}{2}\]
Fundamental wavelength
\[\lambda_1=2L\]
Fundamental frequency
\[f_1=\frac{v_w}{2L}\]
Rules
These are the compact results from the derivation above.
Standing-wave form
\[y=2A\sin(kx)\cos(\omega t)\]
Node positions
\[x=m\frac{\lambda}{2}\]
Fundamental wavelength
\[\lambda_1=2L\]
Fundamental frequency
\[f_1=\frac{v_w}{2L}\]
Examples
Question
A string of length
\[0.80\,\mathrm{m}\]
carries waves at \[120\,\mathrm{m\,s^{-1}}\]
Find its fundamental frequency.Answer
For the fundamental,
\[f_1=\frac{v_w}{2L}=\frac{120}{2(0.80)}=75\,\mathrm{Hz}\]
Checks
- Nodes do not move; antinodes have the largest oscillation amplitude.
- A standing wave is not a single traveling crest moving along the string.
- The spacing between neighboring nodes is \(\\lambda/2\).
- The fundamental mode is the lowest-frequency mode with nodes at both ends.