AcademyMechanical Waves
Academy
String Normal Modes
Level 1 - Physics topic page in Mechanical Waves.
Principle
Normal modes are the discrete standing-wave patterns allowed by the string's boundary conditions.
The string length is fixed, so only wavelengths that fit an integer number of half-waves can survive as steady standing patterns.
Notation
\(n\)
mode number
1
\(L\)
string length
\(\mathrm{m}\)
\(\lambda_n\)
wavelength of mode \(n\)
\(\mathrm{m}\)
\(f_n\)
frequency of mode \(n\)
\(\mathrm{Hz}\)
\(v_w\)
wave speed on the string
\(\mathrm{m\,s^{-1}}\)
Method
Derivation 1: Count how many half-waves fit in the string
For a string fixed at both ends, each end must be a node. The allowed shapes are therefore made of whole numbers of half-wavelengths.
Mode geometry
\[L=n\frac{\lambda_n}{2}\]
Allowed wavelengths
\[\lambda_n=\frac{2L}{n}\]
The first three modes are shown below. Each higher mode inserts one extra interior node.
Derivation 2: Convert the allowed wavelengths to allowed frequencies
Use the wave-speed relation for a string.
Wave-speed relation
\[v_w=f_n\lambda_n\]
Substitute allowed wavelength
\[f_n=\frac{v_w}{\lambda_n}=\frac{nv_w}{2L}\]
Derivation 3: Interpret the harmonic sequence
The frequencies are integer multiples of the fundamental.
Fundamental
\[f_1=\frac{v_w}{2L}\]
Harmonics
\[f_n=nf_1\]
Rules
These are the compact results from the method above.
Allowed wavelengths
\[\lambda_n=\frac{2L}{n}\]
Allowed frequencies
\[f_n=\frac{nv_w}{2L}\]
Harmonic sequence
\[f_n=nf_1\]
Examples
Question
A string of length
\[0.75\,\mathrm{m}\]
supports waves at \[180\,\mathrm{m\,s^{-1}}\]
Find the first three normal-mode frequencies.Answer
First find the fundamental:
\[f_1=\frac{v_w}{2L}=\frac{180}{2(0.75)}=120\,\mathrm{Hz}\]
Then \[f_2=2f_1=240\,\mathrm{Hz},\qquad f_3=3f_1=360\,\mathrm{Hz}\]
Checks
- Normal modes are discrete because the boundary conditions restrict the wavelengths.
- Higher mode number means shorter wavelength and higher frequency.
- The relation \(f_n=nf_1\) holds for an ideal string fixed at both ends.
- Extra interior nodes appear as the mode number increases.