4 Discussion
4.1 Analytical Results
For the resonance tube method, the speed of sound calculated was 340,01±7,39m/s, with an experimental uncertainty of 2,17% and a relative error of 2,94×10-3% (errors at measuring the distances between nodes - bigger than the ones at determining resonant frequencies).
For the two microphones method, the speed of sound obtained was 345,89±13,62m/s, with an experimental uncertainty of 3,94% and a relative error of 1,73% (errors at reading the time interval between peaks - bigger than the ones determining the distance between microphones).
For the echo method, the speed of sound acquired was 341,89±3,11m/s, with an experimental uncertainty of 0,91% and a relative error of 5,56×10-1% (errors at reading the time interval between peaks - bigger that the ones determining the distance between microphones).
In general, the values obtained for the speed of sound are very satisfactory, having obtained experimental uncertainties lower than 4% and relative errors lower than 2%. Furthermore, in the three cases under study, the experimental error covers the absolute error of the measurement and is approximately of the same order, so the error estimates can be considered credible.
Comparing the values obtained for the speed of sound through the different methods, it is possible to state that the most precise value (smaller experimental uncertainty) was obtained by the echo method, while the most accurate value (close to the real one) was obtained through the resonance tube method.
4.2 Uncertainties
In the case of the resonance tube, the associated uncertainties are related to:
- The use of a frequency slightly different from the resonance frequency which can systematically influence the position of the nodes;
- The difficulty in perceiving the exact positions of the nodes (for really small distances, it is not possible to visualize an amplitude difference) which affects the values randomly;
- Random error at marking the distances between the nodes parallax error, width of a marker)
- The existence of unwanted reflections at the closed end of the tube, which does not allow nodes to be found with great accuracy in this area;
Since frequency divergences are the uncertainty with the greatest impact (the analysis assumes that the wave inside the tube is stationary - wave with a specific frequency), it was important to "spend" time on verifying them. Thus, in the second session, by decreasing the step with which the frequency was changed, it was possible to obtain a more precise value for the frequencies and consequently a more accurate value for the speed of sound.
In addition to this, to decrease the uncertainty, only node positions up to the third order were used in the analysis.
In the cases of the two microphones and the echo, the associated uncertainties are related to:
- The parallax errors, considering that the microphones and the source of the sound pulse are collinear and parallel to the tube (they may have small deviations from each other) which can systematically influence the distance considered;
- The identification of the sound produced by two wooden blocks as a perfect pulse (the sound is equivalent to a sine wave, in which amplitude decreases rapidly);
- Considering all the pulses as "equal" (the way the two wooden blocks collide changes not only the amplitude, but also the frequency of the sound) can lead to random errors.
Since the nature of the errors is mostly random, the easiest way to decrease uncertainty is to repeat the tests, decreasing their influence on the final result
Furthermore, it can be seen in comparison with the echo method, that in the two microphone methods, the uncertainty associated with the time interval between the two pulses has a greater impact on the final uncertainty. This, because the distance covered by the pulse is smaller, which implies a smaller time interval, and the uncertainty will correspond to a larger percentage. Thus, one way to reduce the uncertainty is to increase the distance travelled by the pulse.
In the resonance tube method, the errors are mainly systematic, so it is possible to have greater control and make small adjustments to improve the results. On the other hand, for the other two methods, the errors are mostly random, and it is harder to predict their influence on the results. Thus, it makes sense to have obtained a more accurate value with the resonance tube method.
4.2. Assumptions
Based on the results, it is now possible to comment on the influence (or not) of the initial assumptions on experimental activity.
- Since the position of the nodes did not vary along the tube, the assumption that in the resonant tube method, the sound wave created was a standing wave was correct. If it had not been verified, it would be impossible to remove the exact position of the nodes and therefore impossible to calculate the speed of sound;
- Given that the results obtained for the echo method are consistent, the assumption that reflections along the surface of the tube can be considered negligible except for those occurring at the closed end of it was correct. If this did not happen, the microphone would detect the second pulse earlier and the speed of sound obtained would be higher than the real one;
- Since the velocities obtained are slightly higher, it is possible to state that the temperature would be slightly higher and that it was possibly not constant. However, since the fluctuations are small (example: increasing the temperature by 5°C, the speed of sound becomes 3m/s faster, which corresponds to a relative error smaller than 1%) the results are still valid.
https://phizze.com/tables/speed-of-sound-in-air-vs-temperature.html
5 Conclusion
It was possible to achieve the objectives, having obtained very satisfactory values for the speed of sound (experimental uncertainties lower than 4% and relative errors lower than 2%), using different methods.Therefore, although the resonance tube method presents a higher accuracy and the echo method presents a higher precision, it is recommended to perform all the experiments, since each one allows the study of different characteristics of the sound.
On the other hand, small adjustments can be made to the experiments to reduce inaccuracy. One big improvement would be doing experiments in a more temperature controlled environment (the same temperature in all three experiments).
Noteworthy is the acquisition of a larger amount of data, not only by repeating the tests more times, but also by changing the length of the tube, which in the case of the methods concerning the echo and the two microphones, would allow a data analysis through fitting plot. As you can see from the graphic below, the resoults fit within the uncertainties.
5.1. Reflection
Throughout the experimental activity, we were able to put our theoretical knowledge about sound into practice, as well as deepen it. As a matter of fact, during the first week, when we "ignored" the importance of theoretical knowledge, we ended up not predicting the possible results and not identifying (a posteriori) any errors that could arise and be easily corrected.
From another perspective, we realize that advanced or specialized equipment is not required. All you need is a tube, a microphone, a sound generator (a mobile app will do) and an oscilloscope (or a computer app with the same features)!
As a group of students from different study programmes, we profited the most by dividing work in such a way that everyone used his/her best potential.
6. Sources
- https://www.youtube.com/watch?v=cthCLX_9rRQ&list=LL&index=15&t=210s
- https://www.youtube.com/watch?v=u8cHrD1Zw3U
- https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Book%3A_Sound_-_An_Interactive_eBook_(Forinash_and_Christian)/11%3A_Tubes/11.01%3A_Standing_Waves_in_a_Tube/11.1.01%3A_Tube_Resonance
- https://www.acs.psu.edu/drussell/demos/standingwaves/standingwaves.html
- https://www.physicsclassroom.com/class/sound/Lesson-1/Sound-as-a-Longitudinal-Wave
- https://www.khanacademy.org/science/physics/mechanical-waves-and-sound/sound-topic/v/production-of-sound
- https://www.khanacademy.org/science/physics/mechanical-waves-and-sound/sound-topic/v/sound-properties-amplitude-period-frequency-wavelength
- https://www.khanacademy.org/science/physics/mechanical-waves-and-sound/sound-topic/v/speed-of-sound
- https://www.dkfindout.com/uk/science/sound/echoes/
- https://ocw.mit.edu/
- https://phizze.com/tables/speed-of-sound-in-air-vs-temperature.html
- https://en.wikipedia.org/wiki/Standing_wave#/media/File:Standing.gif
- https://www.compadre.org/osp/EJSS/4492/277.htm
- Images (numbers) are from our gallery and cannot be reproduced without authors’ permission (only for academic purposes).