Core Practical: Investigating the Speed of Sound

• Wave Equation: The fundamental relationship linking the speed of a wave ($v$), its frequency ($f$), and its wavelength ($\lambda$) is given by $v = f\lambda$. This equation forms the basis for calculating the speed of sound once frequency and wavelength are determined experimentally.

• Oscilloscope Functionality: A 2-beam oscilloscope is essential as it allows for the simultaneous display of two electrical signals: the direct output from the signal generator and the signal detected by the microphone. This enables direct visual comparison of their phases.

• Phase Shift and Wavelength Measurement: As the microphone is moved away from the loudspeaker, the sound wave travels a greater distance, causing a phase shift in the detected signal relative to the emitted signal. Observing successive points of identical phase coincidence (e.g., trough-peak alignment) allows for the direct measurement of one or more full wavelengths.

• Setup: Connect a signal generator (with loudspeaker) and a microphone to a 2-beam oscilloscope. Initially, place the microphone approximately $50 \text{ cm}$ from the loudspeaker to establish a baseline signal.

• Initial Settings: Set the signal generator to a suitable frequency, typically around $4 \text{ kHz}$, and adjust the oscilloscope's time base so that about three complete cycles of both the emitted and received signals are clearly visible on the screen.

• Wavelength Determination: Carefully adjust the microphone's position until a specific phase coincidence is observed (e.g., a trough on the upper trace aligns with a peak on the lower trace). Record this initial distance. Then, move the microphone further away, noting each subsequent position where the next identical phase coincidence occurs. The distance between these successive coincidences represents one full wavelength.

• Repeat Measurements: To improve accuracy, repeat the process of moving the microphone and recording distances for multiple phase coincidences. This allows for the calculation of a mean wavelength from several full wavelength increments.

• Frequency Verification: While the signal generator provides a nominal frequency, the actual frequency should be determined from the oscilloscope trace by measuring the period ($T$) of the wave and calculating $f = 1/T$. This ensures accuracy by using the detected signal's properties.

• Varying Frequency: To confirm results or investigate frequency dependence, the experiment can be repeated with a different frequency, such as $2 \text{ kHz}$, following the same steps for wavelength and frequency determination.

• Calculating Wavelength: After recording multiple microphone positions ($d_1, d_2, d_3, \dots$) corresponding to successive phase coincidences, the wavelength ($\lambda$) can be calculated. If $N$ full wavelengths occur between $d_1$ and $d_N$, then $\lambda = (d_N - d_1) / (N-1)$. A mean wavelength should be calculated from all valid measurements.

• Determining Frequency: From the oscilloscope trace, measure the time taken for one complete cycle, which is the period ($T$). Ensure the time base scale (e.g., milliseconds) is correctly interpreted. The frequency ($f$) is then calculated using the formula $f = 1/T$.

• Calculating Speed of Sound: Once the mean wavelength ($\lambda$) and the verified frequency ($f$) are obtained, the speed of sound ($v$) is calculated using the wave equation: $v = f\lambda$.

Key Formulas:

• Wave Speed: $v = f\lambda$
• Frequency from Period: $f = \frac{1}{T}$

• Systematic Errors: These errors consistently affect measurements in one direction. A common systematic error is misinterpreting the oscilloscope's time base scale (e.g., confusing milliseconds with seconds), which would lead to an incorrect frequency. Another is relying on an uncalibrated signal generator dial for frequency; using the oscilloscope trace for frequency measurement mitigates this.

• Random Errors: These are unpredictable variations in measurements. Examples include slight inaccuracies in judging the exact point of phase coincidence or minor errors in reading the distance. These can be significantly reduced by taking multiple readings for microphone positions and calculating a mean wavelength.

• Maximizing Distance: To minimize the percentage error in wavelength measurement, it is crucial to make the distance between the microphone and signal generator as large as practically possible. This allows for the observation of more full wavelengths, averaging out small measurement inaccuracies over a larger total distance.

• Electrical Safety: As with any electrical equipment, ensure that connecting leads are in good condition and handle components with care, although the voltages and currents involved are typically low.

• Hearing Protection: Keep the sound volume at a normal listening level to prevent any potential damage to hearing, especially during prolonged experimental sessions.

• Control Variables: Maintain consistent experimental conditions by using the same location, the same set of microphones, and a constant frequency for each set of wavelength measurements. This ensures that only the intended variables are affecting the results.

• Accuracy of Timing: The use of an oscilloscope for timing is highly accurate, as it eliminates human reaction time errors inherent in manual timing methods, contributing to the overall precision of the experiment.