Speed of Sound in Air Using a Resonance Tube
1. Aim
To determine the speed of sound in air at room temperature using a resonance tube by finding two resonance positions.
2. Apparatus Used
- Resonance tube apparatus with a reservoir and scale
- Tuning fork set of known frequencies
- Rubber hammer or striker
- Thermometer
- Meter scale
- Water
- Laboratory stand with clamp
- Stopwatch
3. Diagram
Fig. 1: Experimental setup of the resonance tube apparatus showing the tube, water reservoir, and tuning fork position.
4. Theory
The resonance tube is a common method used to determine the speed of sound in air. It consists of a long glass tube partially filled with water, with the water level adjustable by raising or lowering a connected reservoir.
When a vibrating tuning fork is held at the open end of the tube, the air column inside the tube vibrates at the same frequency as the tuning fork. At certain lengths of the air column, resonance occurs, resulting in a loud sound.
In a tube open at one end and closed at the other (by the water surface), resonance occurs when the length of the air column is equal to an odd multiple of quarter wavelength of the sound wave:
Where:
- $L$ is the length of the air column
- $n$ is a positive integer (1, 2, 3, ...)
- $\lambda$ is the wavelength of the sound
For the first resonance position ($n = 1$), we have:
For the second resonance position ($n = 2$), we have:
Therefore:
So, $\lambda = 2(L_2 - L_1)$
The speed of sound ($v$) is related to the frequency ($f$) and wavelength ($\lambda$) by:
There's also an end correction ($e$) needed due to the fact that the antinode of the sound wave doesn't exactly form at the open end of the tube but slightly beyond it:
Subtracting these equations:
Which gives us the same result, showing that the end correction cancels out when using the two-position method.
5. Formula
The speed of sound in air is calculated using the formula:
$$v = 2f(L_2 - L_1)$$Where:
- $v$ is the speed of sound in air (m/s)
- $f$ is the frequency of the tuning fork (Hz)
- $L_1$ is the first resonance position, i.e., the length of the air column for the first resonance (m)
- $L_2$ is the second resonance position, i.e., the length of the air column for the second resonance (m)
The theoretical value of the speed of sound in air at a temperature $T$ (in °C) can be calculated using:
$$v_{theoretical} = 331.4 + 0.6T \text{ m/s}$$6. Procedure
- Set up the resonance tube with the reservoir and ensure the apparatus is stable.
- Fill the reservoir with water so that the water level in the tube is near the bottom.
- Select a tuning fork of known frequency (preferably between 512 Hz and 1024 Hz).
- Strike the tuning fork gently with the rubber hammer to set it vibrating.
- Hold the vibrating tuning fork horizontally over the open end of the tube.
- Adjust the water level by lowering the reservoir slowly until a loud sound is heard, indicating the first resonance position.
- Measure and record the length $L_1$ from the water surface to the open end of the tube.
- Further lower the water level until a second loud sound is heard, indicating the second resonance position.
- Measure and record the length $L_2$ from the water surface to the open end of the tube.
- Repeat steps 4-9 at least three times to get average values.
- Measure the room temperature using the thermometer.
- Calculate the speed of sound using the formula $v = 2f(L_2 - L_1)$.
- Compare with the theoretical value at the measured room temperature.
- Repeat the experiment using tuning forks of different frequencies if available.
7. Observation Table
| Observation No. | Frequency of Tuning Fork (Hz) | First Resonance Length $L_1$ (cm) | Second Resonance Length $L_2$ (cm) | Difference ($L_2 - L_1$) (cm) | Room Temperature (°C) |
|---|---|---|---|---|---|
| 1 | |||||
| 2 | |||||
| 3 | |||||
| 4 | |||||
| 5 | |||||
| Average value of $(L_2 - L_1)$ | |||||
8. Calculations
Step 1: Calculate the average difference between the two resonance positions:
Where $n$ is the number of observations.
Step 2: Convert the length measurements from centimeters to meters:
Step 3: Calculate the speed of sound using the formula:
Step 4: Calculate the theoretical value of the speed of sound at room temperature:
Where $T$ is the room temperature in degrees Celsius.
Step 5: Calculate the percentage error:
Sample calculation:
Let's assume:
- Frequency of tuning fork, $f = 512$ Hz
- Average value of $(L_2 - L_1) = 32.5$ cm $= 0.325$ m
- Room temperature, $T = 25°$C
Experimental speed of sound:
$$v_{experimental} = 2 \times 512 \times 0.325 = 332.8 \text{ m/s}$$Theoretical speed of sound:
$$v_{theoretical} = 331.4 + 0.6 \times 25 = 346.4 \text{ m/s}$$Percentage error:
$$\% \text{ Error} = \left|\frac{346.4 - 332.8}{346.4} \times 100\%\right| = 3.93\%$$9. Result
- The experimental value of the speed of sound in air at room temperature (___ °C) is determined to be _____ m/s.
- The theoretical value of the speed of sound in air at the same temperature is calculated to be _____ m/s.
- The percentage error in the experiment is _____%, which may be attributed to various experimental errors and approximations.
10. Precautions
- Strike the tuning fork gently to avoid overtones, which can lead to incorrect resonance positions.
- Hold the tuning fork horizontally over the tube with the prongs vibrating in a vertical plane.
- Ensure that the tuning fork does not touch the edge of the tube.
- Adjust the water level slowly to precisely determine the resonance positions.
- Take multiple readings to minimize random errors.
- Keep the room quiet during the experiment to clearly hear the resonance.
- Measure the room temperature accurately.
- Ensure the inner surface of the tube is clean and dry above the water level.
- Keep the resonance tube vertical throughout the experiment.
- Read the water level accurately by avoiding parallax error.
11. Sources of Error
- End Correction: The effective length of the air column is slightly more than the measured length due to the end correction. Although it theoretically cancels out in the two-position method, some errors might still persist.
- Temperature Variations: The speed of sound varies with temperature. If the temperature changes during the experiment, it can lead to errors.
- Humidity Effects: The humidity of the air affects the speed of sound slightly, which is not accounted for in the simple formula.
- Resonance Identification: The exact position of maximum resonance can be subjective to the listener's ear.
- Parallax Error: While reading the water level, parallax error can occur if the eye is not at the correct level.
- Tuning Fork Imperfections: The tuning fork might not vibrate exactly at its labeled frequency, especially if it is old or damaged.
- Water Meniscus: The curved surface of water due to capillary action can lead to errors in measuring the exact water level.
- Air Damping: The amplitude of the tuning fork's vibration decreases with time due to air damping, which might affect the clarity of resonance.
- Air Currents: Air movements in the room can affect the resonance.
- Tuning Fork Positioning: The position of the tuning fork over the tube can affect the resonance conditions.
12. Viva Voice Questions
A1: The principle is based on standing waves formed in a column of air. When a tuning fork of a specific frequency is held at the open end of a tube with one end closed, resonance occurs at specific lengths where the air column length is an odd multiple of quarter wavelength of the sound wave.
A2: Using two resonance positions eliminates the need to account for end correction, which is the small additional length beyond the physical end of the tube where the sound wave actually forms its antinode. This makes the calculation more accurate.
A3: The speed of sound in air increases with increasing temperature. The relationship is approximately given by v = 331.4 + 0.6T, where T is the temperature in degrees Celsius.
A4: Standing waves are formed when two waves of the same frequency and amplitude, traveling in opposite directions, interfere with each other. They are characterized by nodes (points of zero displacement) and antinodes (points of maximum displacement) that remain fixed in position.
A5: Water provides a reflecting surface for the sound waves. The closed end creates a node in the standing wave, while the open end forms an antinode. This specific configuration allows for the creation of standing waves with lengths that are odd multiples of quarter wavelength.
A6: End correction accounts for the fact that the antinode of the sound wave doesn't form exactly at the open end of the tube, but slightly beyond it. This correction is necessary for accurate determination of the wavelength when using a single resonance position method.
A7: Theoretically, yes, but practical considerations limit the range. Very low frequencies require excessively long tubes, while very high frequencies may produce resonances too close together to distinguish accurately. Tuning forks in the range of 256 Hz to 1024 Hz are typically used.
A8: Increased humidity slightly increases the speed of sound in air. This is because water vapor molecules are lighter than nitrogen and oxygen molecules, which are the main components of dry air.
A9: In a tube open at both ends, resonance occurs when the length of the air column is an integral multiple of half wavelength (L = nλ/2), as both ends would be antinodes. This is different from the resonance condition in a tube closed at one end.
A10: The first resonance occurs when L₁ = λ/4 and the second when L₂ = 3λ/4. Subtracting these equations: L₂ - L₁ = 3λ/4 - λ/4 = λ/2. Therefore, λ = 2(L₂ - L₁).