AcademyElectromagnetic Waves
Academy
Standing Electromagnetic Waves
Level 1 - Physics topic page in Electromagnetic Waves.
Principle
Standing electromagnetic waves form when waves of the same frequency travel in opposite directions and interfere. Cavities and transmission lines support only certain resonant patterns.
Notation
\(L\)
cavity length
\(\mathrm{m}\)
\(n\)
mode number
1
\(\lambda_n\)
allowed wavelength
\(\mathrm{m}\)
\(f_n\)
resonant frequency
\(\mathrm{Hz}\)
\(E_0\)
traveling-wave electric amplitude
\(\mathrm{V\,m^{-1}}\)
Method
Derivation 1: Superposition
Add equal waves traveling in opposite directions.
Right-moving wave
\[E_1=E_0\sin(kx-\omega t)\]
Left-moving wave
\[E_2=E_0\sin(kx+\omega t)\]
Standing electric field
\[E=2E_0\sin(kx)\cos(\omega t)\]
Derivation 2: Conducting boundaries
At a perfect conductor, the tangential electric field must be zero.
Electric nodes at both ends
\[E(0,t)=0,\quad E(L,t)=0\]
Allowed lengths
\[L=\frac{n\lambda_n}{2}\]
Derivation 3: Resonant frequencies
Use \(c=f\\lambda\) for waves in vacuum.
Allowed wavelengths
\[\lambda_n=\frac{2L}{n}\]
Allowed frequencies
\[f_n=\frac{nc}{2L}\]
Rules
Standing electric field
\[E=2E_0\sin(kx)\cos(\omega t)\]
Allowed wavelengths
\[\lambda_n=\frac{2L}{n}\]
Resonant frequencies
\[f_n=\frac{nc}{2L}\]
Adjacent node spacing
\[\Delta x=\frac{\lambda}{2}\]
Examples
Question
A vacuum cavity is
\[0.75\,\mathrm{m}\]
long. Find the fundamental frequency.Answer
\[f_1=\frac{c}{2L}=\frac{3.0\times10^8}{2(0.75)}=2.0\times10^8\,\mathrm{Hz}\]
Checks
- Electric-field nodes are fixed positions where the tangential electric field is always zero.
- Magnetic-field nodes and electric-field nodes are shifted by one quarter wavelength in a simple standing EM wave.
- Resonant frequencies are integer multiples of the fundamental for a cavity with nodes at both ends.
- A standing wave stores energy in place rather than transporting net energy steadily in one direction.