show an understanding of experiments that demonstrate stationary waves using microwaves, stretched strings and air columns (it will be assumed that end corrections are negligible; knowledge of the concept of end corrections is not required)

Stationary (Standing) Waves – Cambridge A‑Level Physics 9702

1. From Progressive to Stationary Waves

• Progressive waves travel through a medium with speed \(v\), frequency \(f\) and wavelength \(\lambda\) ( \(v = f\lambda\) ).
• They may be transverse (e.g. a string, light) or longitudinal (e.g. sound in air).
• Principle of superposition (Syllabus 8.1):

  When two coherent waves of the same frequency travel in opposite directions, the resultant displacement at any point is the algebraic sum of the two individual displacements. The sum can be a stationary (standing) wave.

• Key properties of a stationary wave:

  • Nodes – points of zero displacement (or zero pressure variation for sound).
  • Antinodes – points of maximum displacement (or pressure).
  • Distance between successive nodes (or antinodes) = \(\lambda/2\).
  • Only discrete frequencies (harmonics) satisfy the boundary conditions of a bounded medium.

2. Syllabus 8.1 – 8.4: Core Concepts

2.1 Superposition – Principle of Stationary Waves (8.1)

Adding a forward‑travelling wave \(y1=A\sin(kx-\omega t)\) to a backward‑travelling wave \(y2=A\sin(kx+\omega t)\) gives

\[

y = 2A\cos(\omega t)\sin(kx) .

\]

The spatial factor \(\sin(kx)\) fixes nodes at \(kx = 0,\pi,2\pi,\dots\); the temporal factor \(\cos(\omega t)\) makes the antinodes oscillate in time.

2.2 Interference (8.3)

Two coherent sources of the same frequency produce a pattern of constructive and destructive interference. For two point sources separated by distance \(d\) the condition for bright (constructive) fringes on a distant screen is

\[

d\sin\theta = n\lambda \qquad (n = 0,\pm1,\pm2,\dots)

\]

and for dark (destructive) fringes

\[

d\sin\theta = \left(n+\tfrac12\right)\lambda .

\]

These relations are the basis of Young’s double‑slit experiment, a typical AO3 illustration of interference.

2.3 Diffraction (8.2)

• Single‑slit diffraction: a wave incident on a narrow slit of width \(a\) spreads out; minima occur when \(a\sin\theta = m\lambda\) (\(m = 1,2,3,\dots\)). The central maximum is the brightest and widest.
• Diffraction grating: for a grating with slit spacing \(d\) the condition is the same as for interference,

  \[

  d\sin\theta = n\lambda \qquad (n = 1,2,3,\dots)

  \]

  but many orders are observable because many slits contribute.

2.4 Stationary Waves in Bounded Media (8.4)

For a medium of length \(L\) the allowed standing‑wave patterns are set by the boundary conditions at each end.

End corrections are explicitly neglected as required by the syllabus; students should be aware that in a real apparatus the effective length may differ by a small amount (≈0.6 × diameter for tubes).

3. Experiments Demonstrating Stationary Waves

Each experiment follows the Cambridge AO3 format: aim, apparatus, safety, method, data‑recording, analysis (AO2) and systematic‑error checklist.

3.1 Microwaves in a Rectangular Waveguide

Aim: Observe the standing‑wave pattern of microwaves and determine the wavelength (hence the speed) inside the guide.

Method (condensed)

1. Secure the waveguide on a bench; connect the transmitter to a stable power supply.
2. Place the metal plate at the far end and lock it in position.
3. Insert the detector probe at the near end, set the voltmeter to read microwave intensity.
4. Move the probe in 1 mm increments, recording the voltage \(V\) at each position \(x\).
5. Identify successive minima (or maxima) in the voltage trace; note their positions \(x1, x2, …\).
6. Repeat the whole scan three times to assess repeatability.

Data analysis (AO2)

• Spacing between adjacent minima (or maxima): \(\Delta x = x{i+1}-xi\).

  The average \(\overline{\Delta x}\) corresponds to \(\lambda/2\); therefore \(\lambda = 2\overline{\Delta x}\).

• Uncertainty in \(\lambda\):

  \[

  \sigma\lambda = 2\sqrt{\frac{\sum{i=1}^{N}(\Delta x_i-\overline{\Delta x})^2}{N(N-1)}} .

  \]

• Calculate the speed \(c = f\lambda\) using the known source frequency \(f\) (provided by the transmitter specifications). Compare with the accepted value \(c = 3.00\times10^8\;\text{m s}^{-1}\) and comment on any discrepancy.

Systematic‑error checklist

• Mis‑alignment of the reflector – introduces an unwanted phase shift.
• Finite size of the detector probe – averages over a small region, reducing contrast between nodes and antinodes.
• Temperature drift of the microwave source power.
• Neglected end correction – the electrical length of the guide differs slightly from its physical length.

3.2 Stretched String Fixed at Both Ends

Aim: Verify the relationship \(f_n = \dfrac{n}{2L}\sqrt{T/\mu}\) and visualise node‑antinode patterns.

Method (condensed)

1. Clamp the string, measure its length \(L\) with a metre rule (±1 mm).
2. Attach a known set of masses to the free end; compute the tension \(T = mg\).
3. Turn on the driver and sweep the frequency upward from ~5 Hz.
4. Observe the pattern with the strobe; when a clear standing‑wave pattern appears, note the frequency \(f\) from the generator and count the number of antinodes \(n\).
5. Repeat for at least three different tensions to test the \(\sqrt{T}\) dependence.
6. For each tension, record several harmonics (e.g., \(n = 1\) to \(4\)).

Data analysis (AO2)

• Plot \(f_n\) versus \(n\). The gradient should be \(\displaystyle \frac{1}{2L}\sqrt{\frac{T}{\mu}}\).
• Perform a linear regression; compare the experimental gradient with the theoretical value.
• Propagate uncertainties from:

  • \(f\) (±0.5 Hz from the generator),
  • \(L\) (±1 mm),
  • \(\mu\) (from mass/length measurement),
  • \(T\) (±0.01 N from mass balance).

Systematic‑error checklist

• End correction – clamps may not be perfect nodes; a small effective length change is possible.
• Air currents causing damping of the vibration.
• Non‑uniform tension if the string is not perfectly horizontal.
• Finite width of the strobe illumination – may blur the exact node positions.

3.3 Air‑Column Resonance (Open–Open & Closed–Open Tubes)

Aim: Observe longitudinal standing waves in air, determine the speed of sound, and compare the open–open with the closed–open configurations.

Method (condensed)

1. Configure the tube in the open–open arrangement (both ends open). Place the speaker at one end, the microphone near the opposite end.
2. Set the piston to a convenient length \(L\) (e.g., 0.50 m). Sweep the generator frequency slowly and watch the oscilloscope trace.
3. When a sharp increase in amplitude occurs, record the frequency \(f\) – this is a resonance (antinode at both ends).
4. Record at least the first three resonances; assign harmonic numbers \(n = 1,2,3\).
5. Repeat the whole procedure with the tube closed at the piston end (closed–open configuration) and record the odd‑harmonic resonances (\(n = 1,3,5,\dots\)).
6. Measure the ambient temperature \(T{\text{room}}\) and, if desired, use \(v = 331 + 0.6\,T{\text{room}}\) m s\(^{-1}\) for a refined sound‑speed value.

Data analysis (AO2)

• Open–open: Verify \(L = n\lambda/2\) by calculating \(\lambda = 2L/n\) for each resonance and then \(v = f\lambda\). The three (or more) values of \(v\) should cluster around the accepted speed of sound.
• Closed–open: Use \(\lambda = 4L/(2n-1)\) (odd \(n\)). Plot \(f\) versus \((2n-1)\); the slope should equal \(v/4L\).
• Uncertainty propagation: combine errors in \(L\) (±1 mm), frequency (±1 Hz), and temperature (±1 °C) to obtain \(\sigma_v\).

Systematic‑error checklist

• End correction – the antinode is slightly outside the open end (≈0.6 × tube diameter); the syllabus permits neglect, but students should comment on its typical size.
• Leakage of sound at tube joints.
• Non‑planar wavefronts from the speaker, especially at low frequencies.
• Background noise affecting detection of weak resonances.

4. Comparative Summary of the Three Classic Experiments

5. Key Take‑aways for the Exam

• Stationary waves arise from the superposition of two opposite‑travelling waves of the same frequency.
• Boundary conditions dictate the allowed wavelengths: \(\lambdan = 2L/n\) for node‑node or antinode‑antinode, \(\lambdan = 4L/(2n-1)\) for node‑antinode.
• For each experimental system the fundamental frequency follows directly from the wave speed and the appropriate \(\lambda_1\).
• When answering AO3 questions, always comment on the neglected end corrections, describe at least two systematic errors, and suggest how they could be minimised.
• Typical exam calculations: (i) determine \(\lambda\) from measured node spacing, (ii) compute \(v\) or \(f\) using \(v=f\lambda\), (iii) compare with accepted values and discuss uncertainties.