Describe a method involving a measurement of distance and time for determining the speed of sound in air

3 Waves – Determining the Speed of Sound in Air

Learning Objectives (AO1 & AO2 & AO3)

• Explain how the speed of a longitudinal wave in a gas depends on the properties of the medium.
• State the relationship between speed, frequency and wavelength (v = f λ) and relate it to audible pitch.
• Design and carry out a practical method that uses distance and time measurements to obtain the speed of sound in air.
• Analyse experimental data, calculate random and systematic uncertainties and compare the result with the theoretical value at the measured temperature.
• Evaluate the method and suggest realistic improvements (AO3).

Key Theory (AO1)

1. Wave‑speed basics

The general wave‑speed equation is

\$v=\frac{d}{t}\$

• d – distance travelled by the wave (m)
• t – time taken (s)
• v – wave speed (m s⁻¹)

2. Sound as a longitudinal wave

• Particle displacement is parallel to the direction of propagation.
• Key observable quantities:

  • Amplitude – determines loudness (intensity).
  • Frequency (f) – number of compressions per second; determines pitch.
  • Wavelength (λ) – distance between successive compressions.

• These quantities are linked by the fundamental relation

\$v = f\,\lambda\$

3. Temperature dependence (kinetic‑particle model)

In an ideal gas the speed of sound is

\$v = \sqrt{\frac{\gamma\,R\,T}{M}}\$

where γ is the ratio of specific heats (≈ 1.4 for air), R the universal gas constant, T the absolute temperature (K) and M the molar mass of air.

For practical work this reduces to the linear approximation used in the syllabus:

\$v \approx 331\;\text{m s}^{-1}+0.6\,T\;({^\circ}{\rm C})\$

This equation will be used to calculate the theoretical speed at the measured ambient temperature.

4. Vector vs scalar

• Speed is a scalar (magnitude only).
• When the direction of propagation is required we speak of velocity, a vector quantity.

Apparatus (AO2)

Safety & Practical Tips (AO2)

• Secure the speaker, microphone and circuit components to prevent them from falling.
• Conduct the experiment in a quiet room; background noise can cause false triggers.
• Keep the corridor clear of obstacles and avoid parallel walls that produce early reflections.
• Do not place the speaker nearer than 0.5 m to the microphone – the timer may not resolve such short intervals.
• Handle the temperature probe carefully to avoid breakage.

Experimental Procedure (AO2)

1. Measure and mark three (or more) distances along the corridor: d₁, d₂, d₃ … (e.g., 5 m, 10 m, 15 m). Record each to the nearest 0.01 m.
2. Set up the trigger circuit so that pressing a button simultaneously drives the speaker and sends a “start” pulse to the timer.
3. Connect the microphone output to a comparator that produces a clean “stop” pulse when the signal exceeds a preset voltage (this removes background noise).
4. Place the speaker at the start mark and the microphone at the finish mark for the chosen distance.
5. Press the button to generate a short, sharp pulse (e.g., a balloon pop). The timer starts automatically.
6. When the pulse reaches the microphone the comparator triggers the timer to stop. Record the time t.
7. Repeat steps 4‑6 at least five times for each distance and calculate the mean time ⟨t⟩ for that distance.
8. Measure the ambient temperature T (°C) and note it for the whole set of measurements.
9. After all distances are measured, plot the mean time ⟨t⟩ (y‑axis) against distance d (x‑axis). The gradient of the best‑fit straight line equals the speed of sound.
10. Alternatively, compute the speed for each distance using v = d/⟨t⟩ and obtain an overall mean speed.

Data Table

Data Analysis (AO2)

1. Calculate the speed for each distance: v = d/⟨t⟩. Record the values (already shown in the table).
2. Overall mean speed:

\$\bar v = \frac{\sum{i=1}^{n} vi}{n}\$

3. Random uncertainties (propagation of errors):

   • Distance uncertainty, Δd = ±0.01 m.
   • Time uncertainty, Δt = ±0.001 s (resolution of the timer).
   • Relative uncertainty:

     \$\frac{Δv}{v}= \sqrt{\left(\frac{Δd}{d}\right)^2+\left(\frac{Δt}{t}\right)^2}\$

   • Apply the formula to each measurement and obtain an average Δv (e.g., ±2 m s⁻¹ for the sample data).

4. Systematic errors to consider:

   • Reflection of the sound wave from walls or ceiling – can give an early or delayed trigger.
   • Latency of the microphone‑comparator circuit (typically a few µs, but may become significant for short distances).
   • Temperature gradients along the corridor (the speed varies with local temperature).
   • Non‑linearities in the timer or oscilloscope trigger level.

5. Calculate the theoretical speed at the measured temperature (example T = 22 °C):

   \$v_{\rm th}=331+0.6\times22 = 344.2\ {\rm m\,s^{-1}}\$

6. Compare the experimental mean v̄ with v_th and comment on agreement within the combined uncertainty.

Evaluation (AO3)

Example Result

Using the sample data above:

\$\bar v = \frac{344.5 + 346.8 + 346.5}{3}=345.9\ {\rm m\,s^{-1}}\$

Propagated random uncertainty: ±2.0 m s⁻¹.

Theoretical value at 22 °C: 344.2 m s⁻¹.

The experimental result agrees with theory within the calculated uncertainty, confirming the temperature‑dependence of the speed of sound.

Conclusion (AO1 & AO2 & AO3)

The distance‑and‑time method provides a reliable way to measure the speed of sound in air. By using several distances, an electronic trigger circuit and proper error analysis, the experiment yields a value that is consistent with the theoretical prediction based on the kinetic‑particle model. Systematic errors such as reflections and circuit latency can be reduced with the improvements suggested above.

Link to Cambridge IGCSE Physics Syllabus (0625)