Required Practicals: Waves

The AQA A-Level Physics specification includes several required practicals related to waves. You must understand the experimental methods, be able to describe the apparatus, explain sources of uncertainty, and perform calculations from experimental data. These practicals are assessed in Paper 3 and the Practical Endorsement.

Spec mapping: This lesson consolidates AQA 7408 Required Practicals 3 (investigation into the variation of the frequency of stationary waves on a string with length, tension and mass per unit length, and resonance in standing waves in air columns) and 4 (interference effects using Young's double slit and a diffraction grating). It also covers the practical-skills strand: planning, hazard identification, uncertainty estimation, graphical analysis, and improvements to reduce systematic and random error. (Refer to the official AQA specification document for exact wording.)

Synoptic links:

• Section 3.3.1 (stationary waves): Required Practical 3 is the experimental partner of the stationary-waves theory lesson; the harmonic-frequency formula fₙ = nv/(2L) is the prediction tested.
• Section 3.3.2 (Young's double slits, gratings): Required Practical 4 is the experimental partner of the double-slit and grating lessons; the relations w = λD/s and d sin θ = nλ are the predictions tested.
• Section 3.1 (measurements and their errors): percentage uncertainty, combining uncertainties, identifying random vs systematic error, and constructing a graph with error bars are foundational skills examined here on Paper 3.

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Required Practical: Measuring Wavelength Using Young's Double Slits

Aim

To measure the wavelength of laser light using a double slit interference pattern.

Apparatus

• Laser (e.g., red He-Ne laser, λ ≈ 633 nm)
• Double slit (known slit separation s, typically 0.1–0.5 mm)
• Screen (white card or wall)
• Metre ruler or measuring tape
• Ruler with mm divisions or travelling microscope

Method

1. Set up the laser to shine through the double slit onto the screen.
2. Measure the distance D from the double slit to the screen using a metre ruler. Use a large D (1–2 m) to produce widely spaced fringes.
3. Observe the interference pattern on the screen — evenly spaced bright and dark fringes.
4. Measure the distance across several fringes (e.g., measure across 10 bright fringes).
5. Calculate the fringe spacing: w = total distance / number of fringe spacings.
6. Calculate the wavelength using w = λD/s, so λ = ws/D.

Safety

• Never look directly into the laser beam.
• Avoid directing the beam towards other people.
• Place a "Laser On" warning sign on the door.
• Use a screen to terminate the beam.
• Avoid reflective surfaces that could redirect the beam.

Sources of Uncertainty

• Slit separation s: the main source of uncertainty, as it is very small and may not be accurately known. Measure using a travelling microscope if possible.
• Screen distance D: measure from the slits to the screen, ensuring the ruler is parallel to the beam. Uncertainty ≈ ±1 mm.
• Fringe spacing w: measure across many fringes and divide to reduce percentage uncertainty.

Worked Example 1 — A student measures across 12 bright fringes and obtains a distance of 42.0 mm. The slit separation is 0.40 mm and the screen distance is 2.80 m. Calculate the wavelength.

Number of fringe spacings = 12 − 1 = 11

w = 42.0/11 = 3.818 mm = 3.818 × 10⁻³ m

λ = ws/D = (3.818 × 10⁻³ × 4.0 × 10⁻⁴)/2.80

λ = (1.527 × 10⁻⁶)/2.80 = 5.45 × 10⁻⁷ m = 545 nm

This is consistent with green laser light.

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Required Practical: Measuring Wavelength Using a Diffraction Grating

Aim

To measure the wavelength of light using a diffraction grating of known line spacing.

Apparatus

• Light source (laser or discharge lamp with colour filter)
• Diffraction grating (known number of lines per mm)
• Screen or spectrometer with angular scale
• Metre ruler
• Protractor or goniometer (for angle measurement)

Method

1. Set up the light source to shine through the diffraction grating.
2. Observe the diffraction pattern — sharp maxima at specific angles.
3. Measure the angle θ from the central maximum (zero order) to the first-order maximum.
4. For a screen: measure the distance x from the central maximum to the first-order maximum, and the distance D from the grating to the screen. Then tan θ = x/D, so θ = arctan(x/D).
5. Calculate: d sin θ = nλ, so λ = d sin θ/n.

Worked Example 2 — A grating with 300 lines per mm is used. The first-order maximum is observed at 10.2° from the central maximum. Calculate the wavelength.

d = 1/(300 × 10³) = 1/(3.00 × 10⁵) = 3.33 × 10⁻⁶ m

λ = d sin θ/n = (3.33 × 10⁻⁶ × sin 10.2°)/1

sin 10.2° = 0.1771

λ = 3.33 × 10⁻⁶ × 0.1771 = 5.90 × 10⁻⁷ m = 590 nm

This is consistent with the sodium yellow doublet.

Advantages of a Grating Over a Double Slit

• Sharper, more defined maxima — easier to measure angles precisely.
• Multiple orders available — can take measurements at n = 1, 2, 3 and average.
• Higher angular dispersion — better separation of closely spaced wavelengths.

Worked Example 3 — The same grating gives a second-order maximum at 21.0°. Calculate the wavelength and compare with the first-order result.

λ = d sin θ/n = (3.33 × 10⁻⁶ × sin 21.0°)/2

sin 21.0° = 0.3584

λ = (3.33 × 10⁻⁶ × 0.3584)/2 = (1.193 × 10⁻⁶)/2 = 5.97 × 10⁻⁷ m = 597 nm

Average of both measurements: (590 + 597)/2 = 594 nm

Using two orders improves the reliability of the result.

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Required Practical: Measuring the Speed of Sound

Aim

To determine the speed of sound in air using stationary waves.

Method 1: Resonance Tube (Closed Pipe)

Apparatus: Glass tube partially submerged in water, tuning fork of known frequency f.

1. Strike the tuning fork and hold it above the open end of the tube.
2. Raise the tube out of the water slowly, increasing the length of the air column.
3. Listen for the first resonance — a loud increase in sound volume. Record the air column length L₁.
4. Continue raising the tube to find the second resonance at length L₂.
5. The distance between successive resonances corresponds to half a wavelength:

λ/2 = L₂ − L₁, so λ = 2(L₂ − L₁)

6. Calculate the speed of sound: v = fλ

Worked Example 4 — A tuning fork of frequency 512 Hz produces resonance in a tube at air column lengths of 16.0 cm and 49.2 cm. Calculate the speed of sound.

λ/2 = L₂ − L₁ = 49.2 − 16.0 = 33.2 cm = 0.332 m

λ = 2 × 0.332 = 0.664 m

v = fλ = 512 × 0.664 = 340 m s⁻¹

Why Use the Difference Method?

The first resonance occurs at approximately λ/4, but there is an "end correction" e at the open end (the antinode is slightly beyond the open end). By using the difference L₂ − L₁, the end correction cancels out:

L₁ = λ/4 + e L₂ = 3λ/4 + e L₂ − L₁ = 3λ/4 − λ/4 = λ/2 (the end correction e cancels)

Method 2: Two-Microphone Method

Apparatus: Signal generator, loudspeaker, two microphones connected to a dual-trace oscilloscope.

1. Place both microphones in front of the loudspeaker, both the same distance away.
2. The oscilloscope shows two in-phase traces.
3. Move one microphone away from the speaker until the traces are next in phase again.
4. The distance moved equals one wavelength λ.
5. Read the frequency f from the signal generator.
6. Calculate: v = fλ.

This method avoids the end correction problem entirely.

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Required Practical: Vibrations on a String (Melde's Experiment)

Aim

To investigate the factors affecting the frequency of stationary waves on a string.

Apparatus

• Vibration generator connected to a signal generator
• String (e.g., nylon or cotton)
• Pulley and hanging masses (to vary tension)
• Metre ruler
• Balance (to weigh the string and calculate μ)

Method

1. Attach one end of the string to the vibration generator. Pass the string over a pulley and hang masses from the other end to provide tension T.
2. Set the signal generator to a low frequency and slowly increase it.
3. At certain frequencies, standing wave patterns appear. The lowest frequency that produces a standing wave is the fundamental.
4. Record the frequency f and count the number of loops (antinodes) to determine the harmonic number n.
5. Measure the vibrating length L of the string.
6. Calculate the wavelength: λ = 2L/n.
7. Investigate the effect of changing L, T, or μ on the resonant frequency.

Investigation 1: Varying Length

Keep tension T and mass per unit length μ constant. Vary L by changing the position of the bridge or pulley.

f₁ = v/(2L) → f₁ ∝ 1/L

A graph of f₁ vs 1/L should be a straight line through the origin with gradient v/2.

Investigation 2: Varying Tension

Keep L and μ constant. Vary T by changing the hanging mass.

f₁ = (1/2L)√(T/μ) → f₁ ∝ √T

A graph of f₁ vs √T should be a straight line through the origin.

Worked Example 5 — A string of length 0.80 m and mass per unit length 2.0 × 10⁻³ kg m⁻¹ is under a tension of 5.0 N. The string vibrates in its fundamental mode. Calculate the expected frequency.

v = √(T/μ) = √(5.0/(2.0 × 10⁻³)) = √(2500) = 50.0 m s⁻¹

f₁ = v/(2L) = 50.0/(2 × 0.80) = 50.0/1.60 = 31.3 Hz

Worked Example 6 — The same string is observed vibrating with 4 antinodes at a frequency of 125 Hz. Calculate the wave speed and verify consistency with the tension.

4 antinodes → 4th harmonic → λ₄ = 2L/4 = 2 × 0.80/4 = 0.40 m

v = fλ = 125 × 0.40 = 50.0 m s⁻¹ ✓

This matches the expected wave speed from v = √(T/μ), confirming the result.

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Practical Skills and Uncertainty

Reducing Uncertainty in Fringe Measurements

• Measure across as many fringes as possible (e.g., 10–20).
• Use a travelling microscope for sub-millimetre measurements.
• Repeat measurements and calculate a mean.
• Calculate percentage uncertainty: % uncertainty = (absolute uncertainty / measured value) × 100%.

Worked Example 7 — A student measures across 10 fringes as 35 ± 1 mm. Calculate the fringe spacing and its absolute uncertainty.

Number of spacings = 10 − 1 = 9

w = 35/9 = 3.89 mm

% uncertainty in total distance = (1/35) × 100% = 2.86%

Absolute uncertainty in w = 2.86% × 3.89 = 0.11 mm

w = 3.9 ± 0.1 mm

Reducing Systematic Errors

• Ensure the screen is perpendicular to the laser beam.
• Measure D from the slits to the screen, not from the laser.
• Account for parallax when reading rulers.
• In the resonance tube experiment, use the difference method to eliminate the end correction.

Exam Tip: In Paper 3 questions about required practicals, you must be able to describe the method clearly, identify key sources of uncertainty, and suggest improvements. Always link your improvements to specific sources of error.

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Combining Uncertainties: The Standard Rules

For calculations involving multiple measured quantities, percentage uncertainties combine according to standard rules:

Operation	Rule for combining uncertainties
Addition: z = a + b	Δz = Δa + Δb (absolute uncertainties add)
Subtraction: z = a − b	Δz = Δa + Δb (absolute uncertainties still add — worst case)
Multiplication: z = a × b	Δz/z = Δa/a + Δb/b (% uncertainties add)
Division: z = a/b	Δz/z = Δa/a + Δb/b (% uncertainties add)
Power: z = aⁿ	Δz/z =
Product/Quotient: z = aᵐ × bⁿ / cᵖ	Δz/z =

Worked Example 8 — A pendulum of length L = (0.800 ± 0.005) m has a measured period T = (1.795 ± 0.015) s. The acceleration due to gravity is calculated using g = 4π²L/T². Compute g and its percentage uncertainty.

g = 4π² × 0.800 / (1.795)² = (4π² × 0.800) / 3.222 = 31.58 / 3.222 = 9.80 m s⁻²

Percentage uncertainty in L: ΔL/L = 0.005/0.800 = 0.625%

Percentage uncertainty in T²: 2 × ΔT/T = 2 × (0.015/1.795) = 2 × 0.836% = 1.67%

Total percentage uncertainty in g: 0.625% + 1.67% = 2.30%

Absolute uncertainty in g: 0.023 × 9.80 = 0.23 m s⁻². So g = (9.80 ± 0.23) m s⁻², consistent with the accepted value of 9.81 m s⁻² to within the experimental uncertainty.

Worked Example 9 — In a Young's double-slit experiment, the dominant systematic error is identified as an under-reported slit separation s = (0.30 ± 0.02) mm. With D = (1.50 ± 0.005) m and w = (3.50 ± 0.05) mm, calculate (a) the wavelength and its percentage uncertainty, (b) the most effective single improvement.

(a) λ = ws/D = (3.50 × 10⁻³)(0.30 × 10⁻³)/(1.50) = 7.00 × 10⁻⁷ m = 700 nm

% uncertainty in w: 0.05/3.50 = 1.43% % uncertainty in s: 0.02/0.30 = 6.67% ← dominant % uncertainty in D: 0.005/1.50 = 0.33%

Total: 8.43% → absolute uncertainty = 0.0843 × 700 = 59 nm

Final: λ = (700 ± 59) nm

(b) The slit separation s dominates at 6.67%. The single most effective improvement is to measure s directly using a calibrated travelling microscope (reducing Δs from ±0.02 mm to ±0.002 mm), which would drop the dominant percentage uncertainty from 6.67% to 0.67% — a tenfold improvement.

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Graphical Analysis: Linearisation

In practical-skills questions, the underlying physics often requires linearising a non-linear relation before plotting. A linear graph allows you to read the gradient as a known physical combination of measured quantities and verify the relation.

Example: f₁ vs Tension in Melde's Experiment