The speed of sound measurement experiment is a physics experiment in which the frequency and wavelength of a sound wave are measured and the speed at which sound travels through air is determined using the fundamental wave equation.
A common method uses a resonance tube closed at one end. Sound waves are introduced into the tube, and the resonance positions where the sound becomes loud are measured. Because the distance between adjacent resonance positions corresponds to half a wavelength, the speed of sound can be determined by combining this distance with the known frequency.
In an experimental report, it is important not only to calculate the speed of sound but also to discuss how factors such as determining the resonance position, open-end correction, room temperature, the tube’s inner diameter, sound-source frequency, and background noise affected the results.
This article assumes an experiment in which several resonance positions are measured using an air-column resonance apparatus, and summarizes the items to record in the results, sample experimental values, calculation methods, sources of error, example result statements, and discussion examples that can be used in a report.
Note:
This article is a study reference for discussing results obtained in physics experiments at universities and similar institutions. The experimental values shown here are examples for explanation and are not actual measurement results.
For the use of resonance tubes, oscillators, speakers, tuning forks, water tanks, and related equipment, as well as safety precautions, always follow the laboratory manual and the instructions of your instructor or teaching assistant. Do not use excessively loud sound for extended periods.
Principle of Measuring the Speed of Sound
If the speed of a sound wave is v, its frequency is f, and its wavelength is λ, the fundamental wave equation is expressed as follows.
v = fλ
Therefore, if the frequency of the sound source is known, the speed of sound can be determined by measuring the wavelength.
Resonance Tube Closed at One End
In a tube with one end closed by a water surface or similar boundary and the other end open, a standing wave is formed in which the closed end is a displacement node and the open end is a displacement antinode.
If the air-column length at the first resonance is L1, at the second resonance is L2, and at the third resonance is L3, the following relationships ideally hold.
L1 + e = λ / 4
L2 + e = 3λ / 4
L3 + e = 5λ / 4
e is the open-end correction. Taking the difference between adjacent resonance positions eliminates the open-end correction.
L2 − L1 = λ / 2
Therefore, the wavelength can be calculated using the following equation.
λ = 2(L2 − L1)
When the first and third resonance positions are used, the relationship is as follows.
λ = L3 − L1
Relationship Between Temperature and the Speed of Sound
The speed of sound in air changes with temperature. Near room temperature, if the temperature is t °C, the theoretical speed of sound can be approximated by the following equation.
v = 331.5 + 0.60t
The speed of sound in air increases as the temperature rises.
Main Items to Record in the Results
- Type of resonance tube used
- Inner diameter of the tube
- Room temperature
- Type of sound source
- Nominal frequency of the sound source
- Reading of the frequency meter or oscillator
- First resonance position
- Second resonance position
- Third resonance position
- Difference between adjacent resonance positions
- Calculated wavelength
- Speed of sound obtained experimentally
- Theoretical speed of sound calculated from temperature
- Difference between experimental and theoretical values
- Relative error
- Average resonance position from repeated measurements
- Open-end correction
- Changes in sound level or waveform at resonance
If the wavelength is calculated from only one resonance position, the result is strongly affected by the open-end correction. Measuring several resonance positions and calculating the wavelength from their differences generally improves accuracy.
Sample Experimental Conditions
| Item | Sample Condition |
|---|---|
| Measurement method | Air-column resonance tube closed at one end |
| Sound source | Oscillator and speaker |
| Frequency | 512 Hz |
| Room temperature | 22.0 °C |
| Tube inner diameter | 0.030 m |
| Number of measurements | 3 measurements at each resonance position |
| Theoretical speed of sound | 331.5 + 0.60 × 22.0 = 344.7 m/s |
Sample Experimental Values
Measured Resonance Positions
| Trial | First resonance position L1 (m) | Second resonance position L2 (m) | Third resonance position L3 (m) |
|---|---|---|---|
| 1 | 0.160 | 0.496 | 0.832 |
| 2 | 0.161 | 0.497 | 0.833 |
| 3 | 0.159 | 0.495 | 0.831 |
Average Values
| Item | Average |
|---|---|
| First resonance position | 0.160 m |
| Second resonance position | 0.496 m |
| Third resonance position | 0.832 m |
| L2 − L1 | 0.336 m |
| L3 − L2 | 0.336 m |
The differences between adjacent resonance positions are both 0.336 m, showing that the half-wavelength is nearly constant.
Sample Values Measured at Several Frequencies
| Frequency f (Hz) | Difference between resonance positions ΔL (m) | Wavelength λ = 2ΔL (m) | Speed of sound v = fλ (m/s) |
|---|---|---|---|
| 384 | 0.449 | 0.898 | 344.8 |
| 512 | 0.336 | 0.672 | 344.1 |
| 640 | 0.269 | 0.538 | 344.3 |
| 768 | 0.224 | 0.448 | 344.1 |
Although the wavelength changes with frequency, the speed of sound calculated from the product of frequency and wavelength remains nearly constant at approximately 344 m/s.
Calculation Methods
Average Resonance Position
The average second resonance position is calculated as follows.
Average L2 = (0.496 + 0.497 + 0.495) / 3
= 0.496 m
Wavelength
Using the average positions of the first and second resonances, the difference between the resonance positions is:
ΔL = L2 − L1
= 0.496 − 0.160
= 0.336 m
Because this difference corresponds to half a wavelength, the wavelength is:
λ = 2ΔL
= 2 × 0.336
= 0.672 m
Speed of Sound
For a frequency of 512 Hz, the speed of sound is:
v = fλ
= 512 × 0.672
= 344.1 m/s
Theoretical Speed of Sound from Room Temperature
At a room temperature of 22.0 °C, the theoretical speed of sound is:
v = 331.5 + 0.60t
= 331.5 + 0.60 × 22.0
= 344.7 m/s
Relative Error
Relative error (%) = |Experimental value − Theoretical value| / Theoretical value × 100
= |344.1 − 344.7| / 344.7 × 100
= approximately 0.17%
Calculating the Open-End Correction from the First Resonance Position
At the first resonance, the effective air-column length is one-quarter of the wavelength.
L1 + e = λ / 4
Therefore, the open-end correction is calculated as follows.
e = λ / 4 − L1
= 0.672 / 4 − 0.160
= 0.168 − 0.160
= 0.008 m
If the tube radius is 0.015 m, the open-end correction is approximately 0.53 times the radius.
Average Speed of Sound from Several Frequencies
Average speed of sound = (344.8 + 344.1 + 344.3 + 344.1) / 4
= 344.3 m/s
Main Sources of Error
| Source of Error | Effect on Measurement | Likely Trend in Results | Improvement / Check |
|---|---|---|---|
| Error in determining the resonance position | It is difficult to determine the exact position of maximum sound level | Resonance positions vary forward and backward | Move the water surface or piston slowly and repeat measurements |
| Ignoring open-end correction | The effective air-column length is longer than the physical tube length | Using only the first resonance tends to underestimate the wavelength | Use the difference between adjacent resonance positions |
| Room-temperature measurement error | The theoretical speed of sound is shifted | A systematic difference appears when compared with the experimental value | Measure the air temperature near the tube |
| Oscillator frequency error | Directly affects the calculated speed of sound | All measured values shift by the same proportion | Check with a frequency meter or oscilloscope |
| Scale-reading error | The measured resonance-position length is shifted | Causes variation in wavelength and sound speed | Keep the eye at the same height as the water surface or indicator |
| Background noise | Makes the change in sound level at resonance harder to detect | The range over which resonance appears to occur becomes wider | Measure in a quiet environment and use a microphone if necessary |
| Movement of the water surface | The position of the closed end is unstable | Low-order resonance positions vary | Take readings after the water surface becomes still |
| Large tube inner diameter | Increases the influence of open-end correction | Values calculated from the first resonance are more likely to deviate | Use differences between multiple resonance positions |
| Misalignment of the speaker and tube | Sound does not enter the tube efficiently | Resonance becomes weak and difficult to identify | Fix the speaker directly in front of the open end |
| Effects of humidity and atmospheric pressure | Slightly change air density and the speed of sound | A difference from the temperature-only approximation remains | Record environmental conditions when necessary |
Why Use the Difference Between Adjacent Resonance Positions?
At an open end, the oscillation of the air extends slightly beyond the tube, so the effective air-column length is somewhat longer than the physical distance to the tube opening. This is called the open-end correction.
If the wavelength is calculated from only the first resonance position, the correction must be estimated accurately. In contrast, when the difference between the second and first resonance positions is taken, the open-end correction contained in both measurements cancels out.
Why Is There a Range in Determining the Resonance Position?
Ideally, the amplitude reaches a maximum at one specific air-column length. In practice, however, air viscosity, losses at the tube wall, the bandwidth of the sound source, background noise, and other factors give the resonance curve a finite width.
Therefore, when resonance is judged by ear, different observers may identify slightly different positions as the loudest. Using a microphone together with a voltmeter or oscilloscope to measure amplitude allows the resonance position to be determined more objectively.
Why Does the Speed of Sound Increase with Temperature?
As temperature rises, the thermal motion of air molecules becomes more vigorous. Pressure changes are transmitted to neighboring air more rapidly, so the speed of sound increases.
Therefore, when comparing an experimental value with a standard value for the speed of sound, the room temperature during the experiment should always be recorded and values should be compared under the same temperature conditions.
Why Are Low Frequencies Difficult to Measure?
The lower the frequency, the longer the wavelength, so the spacing between adjacent resonance positions also becomes larger. If the apparatus is not long enough, the second or third resonance may not appear within the tube.
Conversely, if the frequency is too high, the spacing between resonance positions becomes short and scale-reading error accounts for a larger fraction of the wavelength. A frequency suitable for the apparatus length and scale resolution should therefore be selected.
Example of How to Write the Results
A 512 Hz sound wave was introduced into a resonance tube closed at one end, and the air-column lengths from the first through third resonances were each measured three times. The room temperature was 22.0 °C.
The average first, second, and third resonance positions were 0.160, 0.496, and 0.832 m, respectively. The differences between adjacent resonance positions were both 0.336 m, giving a wavelength of 0.672 m.
The speed of sound calculated from the fundamental wave equation was 344.1 m/s. The relative error compared with the theoretical speed of sound of 344.7 m/s at 22.0 °C was approximately 0.17%, showing good agreement between the two values.
Points to Connect to the Discussion
- Were the differences between adjacent resonance positions constant?
- Can you explain why the difference between resonance positions corresponds to half a wavelength?
- Was the speed of sound calculated from several frequencies constant?
- Did the experimental speed of sound agree with the theoretical value calculated from temperature?
- How did the open-end correction affect the result?
- Was there error from determining resonance positions by ear alone?
- Did the measured room temperature represent the air temperature inside the tube?
- Was the oscillator frequency accurate?
- Could the position of the water surface or piston be read accurately?
- Did background noise or speaker position affect the measurement?
- Was there a difference in measurement accuracy between low and high frequencies?
Discussion Example
In this experiment, a resonance tube closed at one end was used to measure the resonance positions of sound waves with a known frequency and determine the speed of sound in air.
At a frequency of 512 Hz, the average first, second, and third resonance positions were 0.160, 0.496, and 0.832 m, respectively. The differences between adjacent resonance positions were both 0.336 m and were therefore nearly constant.
In a tube closed at one end, a standing wave is formed with a displacement node at the closed end and a displacement antinode at the open end. Between adjacent resonance states, the air-column length increases by half a wavelength, so the 0.336 m difference between resonance positions corresponds to half a wavelength. The wavelength was therefore calculated to be 0.672 m.
The speed of sound calculated using the fundamental wave equation v = fλ was 344.1 m/s. The theoretical speed of sound at a room temperature of 22.0 °C was 344.7 m/s, giving a relative error of approximately 0.17%. Thus, the experimental value agreed well with the theoretical value.
When measurements were made at several frequencies, the wavelength decreased as the frequency increased, while the speed of sound obtained from the product of frequency and wavelength remained nearly constant at approximately 344 m/s. This indicates that, at the same air temperature, the speed of sound is almost independent of frequency.
Possible reasons why the experimental and theoretical values did not agree perfectly include error in determining the resonance position, scale-reading error, oscillator frequency error, differences in room-temperature measurement, and open-end correction.
In particular, when resonance is judged by ear, it is difficult to identify the position of maximum sound level as a single exact point. Resonance has a finite width, and background noise and the observer’s position can also affect the judgment of maximum sound level. Therefore, resonance positions should be measured several times and averaged.
At the open end, the air oscillation extends outside the tube, making the effective air-column length longer than the physical tube length. If only the first resonance is used, this open-end correction cannot be ignored. However, using the difference between adjacent resonance positions cancels the open-end correction contained in both measurements and allows the wavelength to be determined more accurately.
To improve measurement accuracy, it is effective to move the water surface or piston slowly and repeatedly check the region around the position of maximum sound level. In addition, using a microphone and oscilloscope to measure amplitude and measuring the room temperature near the tube can reduce subjectivity in resonance detection and temperature-related error.
These results confirm that the wavelength of a sound wave can be measured using air-column resonance and that the speed of sound in air can be determined from the product of frequency and wavelength. They also show that the speed of sound is nearly independent of frequency under the same temperature conditions and changes with temperature.
Discussion Example When the Experimental Speed of Sound Is Lower Than the Theoretical Value
One possible reason why the experimental speed of sound was lower than the theoretical value is that the difference between resonance positions was measured smaller than its actual value, causing the wavelength to be underestimated. The experimental value could also become smaller if the oscillator’s actual frequency was lower than the displayed value or if the air temperature inside the tube was lower than the measured room temperature.
Discussion Example When the Experimental Speed of Sound Is Higher Than the Theoretical Value
Possible reasons why the experimental speed of sound was higher than the theoretical value include measuring the difference between resonance positions larger than its actual value or the oscillator’s actual frequency being higher than the displayed value. In addition, if the room thermometer indicated a temperature lower than the actual air temperature inside the tube, the theoretical speed of sound would be underestimated, making the experimental value appear relatively high.
Summary
In a speed-of-sound measurement experiment, a standing wave is formed in a resonance tube and the wavelength is determined from the difference between adjacent resonance positions. In a tube closed at one end, the difference between adjacent resonance positions corresponds to half a wavelength.
Once the sound-source frequency and wavelength are known, the speed of sound can be calculated from the fundamental wave equation v = fλ. The speed of sound in air also increases as temperature rises.
In the discussion, it is important to relate factors such as resonance-position determination, open-end correction, room temperature, frequency, scale reading, background noise, and tube inner diameter to the experimental results.
