show an understanding of experiments that demonstrate two-source interference using water waves in a ripple tank, sound, light and microwaves

Interference and Diffraction – Cambridge AS & A Level Physics (9702)

0. AS‑Level Physics Overview (for context)

This hand‑out concentrates on Topic 8 (Interference & Diffraction) but it is useful to see where it fits in the whole syllabus.

1. Principle of Superposition

• When two or more waves occupy the same region of space, the resultant displacement is the vector sum of the individual displacements.
• This leads to interference – a systematic pattern of alternating maxima (constructive) and minima (destructive).

2. Coherence – Conditions for a Stable Interference Pattern

For observable, stationary fringes the sources must be coherent. In practice this means:

1. Same (or very narrow‑band) frequency – otherwise the relative phase changes too rapidly.
2. Fixed phase relationship – the phase difference remains constant (e.g., two points on the same wavefront).
3. Path‑difference < coherence length – the difference in optical/acoustic path must be smaller than the coherence length of the source.
4. Point‑like source size compared with the wavelength, to avoid averaging over many phases.

Remember: a phase change of π occurs on reflection from a medium of higher acoustic or optical impedance (e.g., light reflecting from a denser medium). This must be included when predicting nodal/antinodal positions.

3. Two‑Source Interference – Geometry and Equations

Two coherent sources S₁ and S₂ are separated by a distance \(d\). At a point P on a screen a distance \(D\) (≫ \(d\)) from the sources, the path‑difference is

\[

\Delta r = d\sin\theta \;\approx\; \frac{d\,y}{D},

\]

where \(y\) is the lateral displacement of P from the centre line and \(\theta\) the angle to that line.

• Constructive interference: \(\displaystyle \Delta r = m\lambda\quad(m=0,1,2,\dots)\)
• Destructive interference: \(\displaystyle \Delta r = \left(m+\tfrac12\right)\lambda\)

Hence the fringe spacing (distance between adjacent bright or dark lines) is

\[

\boxed{\Delta y = \frac{\lambda D}{d}} \tag{1}

\]

3.1 Intensity Distribution

If the two waves have amplitudes \(A1, A2\) (or intensities \(I1, I2\)) the resultant intensity is

\[

I = I1 + I2 + 2\sqrt{I1 I2}\,\cos\Delta\phi,

\qquad

\Delta\phi = \frac{2\pi\Delta r}{\lambda}.

\]

For identical sources (\(I1=I2=I_0\)) this reduces to the familiar form

\[

\boxed{I = 4I_0\cos^{2}\!\left(\frac{\Delta\phi}{2}\right)} \tag{2}

\]

4. Diffraction – The Envelope that Modulates Interference

• Single‑slit diffraction: minima when \(a\sin\theta = m\lambda\) ( \(m=\pm1,\pm2,\dots\) ), where \(a\) is the slit width.
• The intensity envelope from diffraction multiplies the interference term (2). The observable pattern is therefore a series of bright‑dark fringes whose overall brightness follows the single‑slit envelope.

5. Diffraction Grating (many‑slit interference)

For a grating with spacing \(d\) (inverse of line density) the condition for principal maxima is

\[

\boxed{d\sin\theta = n\lambda}\qquad (n = 0, \pm1, \pm2,\dots) \tag{3}

\]

Equation (3) is the basis for quantitative wavelength determination and for many Cambridge exam questions.

6. Experimental Demonstrations of Two‑Source Interference

6.1 Water Waves – Ripple‑Tank Experiment

• Apparatus: Ripple tank with transparent base, two identical dippers (or a single dipper with a split barrier), variable‑frequency oscillator, stroboscopic light, overhead projector or white screen.
• Procedure:

  1. Place the dippers a fixed separation \(d\) (typically 5–10 cm). Connect both to the same oscillator so they oscillate at the same frequency.
  2. Adjust the frequency until a clear, regular pattern appears. Measure the wavelength \(\lambda\) on the tank surface (ruler or calibrated grid).
  3. Turn on the strobe to freeze the pattern and project it onto the screen.
  4. Measure the distance \(D\) from the source plane to the screen and the fringe spacing \(\Delta y\) between successive nodal lines.
  5. Use equation (1) to predict \(\Delta y\) and compare with the measured value; calculate the percentage error.

• Key Observations: Straight, equally spaced nodal (destructive) and antinodal (constructive) lines radiating from the midpoint. The pattern disappears if the two dippers are driven at different frequencies (loss of coherence).
• Practical Tips:

  • Level the tank to avoid sloping waves.
  • Ensure the dippers are as close to point sources as possible (diameter ≪ \(\lambda\)).
  • Record several fringe spacings and average to reduce random error.

6.2 Sound Waves – Double‑Speaker (Young’s) Experiment

• Apparatus: Two identical loudspeakers driven from the same audio source, rigid mounting frame, long straight rail (or a wall) for the microphone track, calibrated microphone, sound‑level meter or oscilloscope.
• Procedure:

  1. Mount the speakers a distance \(d\) apart (e.g. 0.30 m) and point them toward the rail.
  2. Select a pure tone (e.g. 500 Hz). The wavelength in air is \(\lambda = v{\text{sound}}/f \approx 0.68\) m (using \(v{\text{sound}} = 340\) m s⁻¹).
  3. Place the microphone at a fixed distance \(D\) (≈ 1.5 m) from the speaker plane.
  4. Slide the microphone along the rail, recording the intensity at regular intervals (e.g. every 2 cm).
  5. Identify positions of maxima and minima; compute \(\theta\) from \(\tan\theta \approx y/D\) and verify \(d\sin\theta = m\lambda\).

• Typical Result: Alternating “loud” and “soft” zones whose spacing agrees with equation (1). The pattern vanishes if the speakers are powered from independent amplifiers (coherence lost).
• Practical Tips:

  • Use acoustic foam around the set‑up to minimise reflections.
  • Check that the microphone is positioned on the same height as the speaker centres.
  • Average several readings at each position to reduce background noise.

6.3 Light – Double‑Slit Laser Experiment

• Apparatus: Low‑power monochromatic laser (e.g. He‑Ne, \(\lambda = 632.8\) nm), double‑slit slide (known slit separation \(d\) and width \(a\)), screen, metre ruler or vernier calipers, optional narrow‑band filter.
• Procedure:

  1. Align the laser so the beam is normal to the slit plane; secure the slide on a stable mount.
  2. Measure the distance \(D\) from the slits to the screen (typically 1.0–2.0 m).
  3. Record the positions of several bright fringes on the screen; compute the average fringe spacing \(\Delta y\).
  4. Calculate the wavelength from equation (1) and compare with the known laser value; discuss any discrepancy.
  5. Optional: Measure the width of the single‑slit envelope and verify \(a\sin\theta = m\lambda\).

• Observations: A series of equally spaced bright and dark bands; the intensity of outer bands follows the single‑slit diffraction envelope. With white light the pattern is washed out unless a narrow‑band filter is inserted.
• Practical Tips:

  • Secure the laser to avoid drift during measurement.
  • Use a screen with a fine grid to improve positional accuracy.
  • Measure \(\Delta y\) over several fringes (e.g., 5‑10) and divide by the number of intervals to minimise random error.

6.4 Microwaves – Double‑Slit (or Michelson) Interferometer

• Apparatus: Microwave transmitter (≈ 10 GHz, \(\lambda = 3.0\) cm), double‑slit plate (slit separation \(d\) ≈ 6 cm), parabolic reflector to collimate the beam, movable microwave receiver connected to a millimetre‑wave power meter, ruler for lateral displacement.
• Procedure:

  1. Emit a continuous‑wave microwave beam; place the double‑slit plate in the beam path.
  2. Position the receiver at a distance \(D\) (≈ 1 m) behind the slits.
  3. Scan the receiver laterally across the beam, recording received power at regular intervals (e.g., every 1 cm).
  4. Determine the fringe spacing \(\Delta y\) from the recorded maxima; verify \(\Delta y = \lambda D/d\).
  5. Repeat with a different slit separation to illustrate the inverse dependence on \(d\).

• Result: A sinusoidal variation of received intensity, with maxima and minima matching equations (1) and (2). The centimetre‑scale wavelength makes the pattern easy to measure with a ruler.
• Practical Tips:

  • Align the transmitter, slits and receiver carefully to avoid angular mis‑alignment.
  • Shield the set‑up from metallic objects that could cause unwanted reflections.
  • Allow the transmitter to warm up for a few minutes to achieve stable output.

7. Practical Skills Checklist (AO3 – Planning, Conducting, Evaluating)

• Safety: eye protection for lasers; avoid electric shock with speakers and microwave equipment; keep water away from electrical components.
• Calibration: verify the distance \(D\) with a measuring tape or laser rangefinder; check the source separation \(d\) with calipers.
• Uncertainty analysis:

  • Estimate uncertainties in \(d\), \(D\) and \(\Delta y\) (e.g., ±0.5 mm for ruler measurements).
  • Propagate errors to obtain the uncertainty in the calculated wavelength using \(\lambda = \Delta y\,d/D\).

• Systematic errors to consider:

  • Non‑point‑like sources (finite dipper or speaker size).
  • Wave‑front curvature when \(D\) is not ≫ \(d\).
  • Reflections from tank walls, room walls or microwave cabinet.

• Data handling: take at least five measurements of fringe spacing, average, and calculate standard deviation.
• Evaluation: discuss how well the experimental results agree with theory, identify dominant error sources, and suggest improvements (e.g., using a larger \(D\) to improve the small‑angle approximation).

8. Comparison of the Four Experiments

9. Worked Example – Determining the Wavelength from a Ripple‑Tank Pattern

1. Source separation: \(d = 8.0\;\text{cm}\).
2. Screen distance: \(D = 50.0\;\text{cm}\).
3. Measured fringe spacing: \(\Delta y = 2.5\;\text{cm}\).
4. Using equation (1):\[

   \lambda = \frac{\Delta y\,d}{D}

   = \frac{2.5\;\text{cm}\times 8.0\;\text{cm}}{50.0\;\text{cm}}

   = 0.40\;\text{cm} = 4.0\;\text{mm}.

   \]

5. Verification: The oscillator frequency gave a wavelength of ≈ 4 mm on the tank surface (measured independently), confirming coherence and the validity of the experiment.

10. Further Reading / A‑Level Preview

• Thin‑film interference: constructive/destructive conditions for light reflected from a film of thickness \(t\) (phase change on one surface only).
• Fabry‑Pérot interferometer: multiple‑beam interference, finesse, and applications in spectroscopy.
• Michelson interferometer: path‑difference measurement, determination of the speed of light, and modern applications (e.g., LIGO).
• Coherence length and spectral width: quantitative relation \(L_c = \lambda^2/\Delta\lambda\) and its relevance to lasers and radio transmitters.
• Diffraction of X‑rays and electron beams: Bragg’s law \(2d\sin\theta = n\lambda\) as a natural extension of the grating equation.

11. Summary

• Interference results from the superposition of coherent waves; coherence requires identical (or narrow‑band) frequency, a fixed phase relationship, and a path‑difference smaller than the coherence length.
• For two point sources the conditions for bright and dark fringes are \(d\sin\theta = m\lambda\) and \(d\sin\theta = (m+\tfrac12)\lambda\); fringe spacing follows \(\Delta y = \lambda D/d\).
• The intensity pattern is given by \(I = I1+I2+2\sqrt{I1I2}\cos\Delta\phi\); for equal sources this simplifies to \(I = 4I_0\cos^{2}(\Delta\phi/2)\).
• Diffraction of each aperture produces an envelope that modulates the interference fringes; a diffraction grating yields sharp principal maxima described by \(d\sin\theta = n\lambda\).
• Four classic laboratory demonstrations (water, sound, light, microwaves) illustrate the same physics across vastly different wavelength regimes and provide excellent opportunities for quantitative analysis, error estimation, and development of practical experimental skills.