Explain, using a graphical method, how a standing wave is produced by the superposition of two travelling waves, identify the positions of nodes and antinodes, describe the energy distribution, and relate the theory to the three canonical laboratory demonstrations required by Cambridge AS & A Level Physics (9702) syllabus.
When two waves of the same frequency, amplitude and wavelength travel in opposite directions along the same medium, the resultant displacement at any point is the algebraic sum of the two incident displacements.
\[
y_1=A\sin(kx-\omega t),\qquad
y_2=A\sin(kx+\omega t)
\]
\[
\text{Resultant: }y=y1+y2
=2A\cos(\omega t)\,\sin(kx)
\]
The graphical method visualises the addition of the two opposite‑travelling waves at successive instants.
From \(y=2A\cos(\omega t)\sin(kx)\):
\[
\sin(kx)=0\;\Longrightarrow\;kx=n\pi\;(n=0,1,2,\dots)
\]
\[
\boxed{x_n=n\frac{\lambda}{2}}
\]
Nodes are spaced \(\lambda/2\) apart.
\[
|\sin(kx)|=1\;\Longrightarrow\;kx=\Bigl(n+\tfrac12\Bigr)\pi
\]
\[
\boxed{x_a=\Bigl(n+\tfrac12\Bigr)\frac{\lambda}{2}}
\]
Antinodes are also \(\lambda/2\) apart, but offset by \(\lambda/4\) from the nearest node.
For a string of linear mass density \(\mu\) the instantaneous energy per unit length is
\[
E(x,t)=\tfrac12\mu\left(\frac{\partial y}{\partial t}\right)^2
+\tfrac12 T\left(\frac{\partial y}{\partial x}\right)^2,
\]
which reduces to a sinusoidal envelope that vanishes at the nodes and peaks at the antinodes.
Boundary conditions restrict the allowed wavelengths. The integer \(n\) (mode number) counts the number of half‑waves fitting into the length \(L\).
The corresponding frequencies are
\[
f_n=\frac{nv}{2L}\qquad\text{(fixed–fixed / open–open)},
\qquad
f_n=\frac{(2n-1)v}{4L}\qquad\text{(closed–open)},
\]
where \(v\) is the wave speed in the medium. The first three modes for a fixed–fixed string are illustrated below (placeholder for diagram).
| Experiment | Medium & Boundary Conditions | Wavelength Condition | Typical Observation | Safety / Practical Tips |
|---|---|---|---|---|
| Stretched string (fixed–fixed) | String fixed at both ends → nodes at each end | \(L=n\lambda/2\;(n=1,2,3,\dots)\) | Alternating nodes and antinodes; resonant frequencies \(f_n=n v/2L\) | Secure clamps firmly; keep fingers away from the vibrating region; use a driver with adjustable amplitude to avoid excessive tension. |
| Air column – open–open tube | Both ends open → antinodes at each end | \(L=n\lambda/2\;(n=1,2,3,\dots)\) | Resonant standing sound waves; distinct “harmonic” tones heard with a speaker or tuning fork. | Do not over‑inflate the tube; keep the speaker at a safe distance to avoid hearing damage at high amplitudes. |
| Air column – open–closed tube | Closed end → node; open end → antinode | \(L=(2n-1)\lambda/4\;(n=1,2,3,\dots)\) | Only odd harmonics appear; resonance when the length matches a quarter‑wave series. | Use a rubber stopper that fits tightly; never place the open end near a heat source. |
| Microwave cavity (parallel plates) | Electromagnetic wave between two reflecting plates → electric‑field nodes at the plates | \(d=n\lambda/2\) where \(d\) is plate separation | Standing‑wave pattern detected with a probe; intensity minima correspond to nodes. | Never look directly at the microwave source; use appropriate shielding and follow the laboratory’s microwave safety protocol. |
Count‑and‑Measure Method
1. Identify and count the number of nodes (or half‑waves) between two fixed reference points (e.g. the ends of a string or the closed end of a pipe).
2. If \(N\) half‑waves are observed over a length \(L\), then
\[
\lambda=\frac{2L}{N}.
\]
3. Use the measured wavelength with the known wave speed \(v\) to obtain the frequency \(f=v/\lambda\).
This directly satisfies the AO3 requirement “determine wavelength from node/antinode positions”.
Given:
Wave speed on the string:
\[
v=\sqrt{\frac{T}{\mu}}=\sqrt{\frac{80}{0.002}}=200\;\text{m s}^{-1}.
\]
Fundamental wavelength (first mode, \(n=1\)):
\[
\lambda_1=2L=1.50\;\text{m}.
\]
Fundamental frequency:
\[
f1=\frac{v}{\lambda1}= \frac{200}{1.50}=133.3\;\text{Hz}.
\]
Students can verify the result experimentally by adjusting the driver frequency until the first standing‑wave pattern (one antinode between the two nodes) is observed.
| Feature | Mathematical Condition | Position along the medium | Physical Meaning |
|---|---|---|---|
| Node | \(\sin(kx)=0\) | \(x_n=n\frac{\lambda}{2}\) | Zero displacement at all times (waves 180° out of phase). |
| Antinode | \(|\sin(kx)|=1\) | \(x_a=\bigl(n+\tfrac12\bigr)\frac{\lambda}{2}\) | Maximum displacement \(\pm2A\) (waves in phase). |
| Energy density | \(E(x,t)=\tfrac12\mu\dot y^{2}+\tfrac12T y'^{2}\) | Zero at nodes, maximum at antinodes | Shows localisation of kinetic + potential energy. |
| Fundamental mode | \(n=1\) | One half‑wave in the length \(L\) | Lowest possible frequency \(f_1=v/2L\). |
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