Science labs should promote reasoning that resembles the work that scientists do. However, this is often not the case. We present a lab in which students strive to find out which of two models best describes a physics experiment. The quantification of measurement uncertainties—another topic that is often neglected in high school curricula—determines the quality of the data, and will play a central role in the final decision between these models and is a vital skill in the modern world.

Science labs should promote reasoning that resembles the work that scientists do.^{1} However, this is often not the case.^{2} We present a lab in which students strive to find out which of two models best describes a physics experiment. The quantification of measurement uncertainties—another topic that is often neglected in high school curricula^{3}—determines the quality of the data, and will play a central role in the final decision between these models and is a vital skill in the modern world.^{4}

In this low-cost lab, students blow on a PVC tube that is closed at one end to create a tone (like you do on a bottle). The tone’s frequency can be measured using the free smartphone app phyphox (available for Android and iOS, www.phyphox.org),^{5} after which the wavelengths can be calculated.

The wavelength of the fundamental frequency is often modeled as if exactly one-quarter wavelength fits inside the tube. However careful consideration shows that the antinode of the standing wave lies somewhat outside the tube itself (Fig. 1). An end correction has to be taken into account for accurate acoustic measurements.^{6–8} Nevertheless, some college-level textbooks do not remark on this effect^{9,10}; others make a qualitative remark^{11–13} but neglect the effect for practical purposes, and some more scientifically oriented textbooks quantify the effect in a footnote.^{14,15}

This lab does not aim to measure the end correction or its dependency on the radius. Instead, taking the correction into account or not gives rise to two mathematical models that describe what fraction of a wavelength fits into the tube. The goal for this lab is for students to decide which of these models best fits their data and to refine their experiment successively. This activity is similar to what is described in the NGSS (Appendix F, p. 52),^{16} but also something that students have difficulty with.^{17} The agreement between model and data will be determined using measurement uncertainties, which shall be estimated at an introductory level.

## Two models

In the “simple model,” the tube length *L* of the tube fits exactly one-quarter of a wavelength for the fundamental frequency [see Fig. 1(a)]. Or, the other way around, the wavelength θ is four times the length, resulting in our simple model (of the form *y* = *ax*):

In the other model, which we shall call “extended model,” the antinode lies a distance Δ*L* outside the tube [see Fig. 1(b)]. This end correction can be calculated as^{18}:

where *r* is the radius of the tube. The effective length *L*′ = *L* + Δ*L* now fits a quarter of a wavelength. Filling this in and solving for the wavelength gives the extended model (of the form *y* = *ax*+*b*):

where the intercept is

which is appropriate for the lengths and wavelengths used in this lab.^{19}

## Experiment

In our experiment, we have used four PVC electrical conduit tubes, of lengths 7, 10, 13, and 16 cm, with an inner radius of 0.70 cm. The tubes are closed off at the bottom.

To measure the frequencies we use phyphox’s option *Audio Spectrum*, which creates a Fourier transformed frequency spectrum and indicates the *Peak-Frequency* (see Fig. 2). In settings, we have set phyphox so that it takes 4096 samples, in 85.33 ms (*Period used*), to create the Fourier transform. This results in a frequency binning of

With this, we assume a (crude) measurement uncertainty for the frequency of 12 Hz.

More samples, resulting in smaller frequency bins, could have been chosen. However, in our experience, this setting works best for students, as the spectrum stabilizes quickly. This makes recording the spectrum fast, which helps because some students have difficulty in getting a tone.

Using these settings, we have measured the following frequencies for our tubes: *f*_{7} = 1125 Hz, *f*_{10} = 820 Hz, *f*_{13} = 633 Hz, and *f*_{16} = 516 Hz. The temperature of the air in the tube determines the speed of sound. This is calculated using the standard equation for the speed of sound in air:

where *T* is the temperature of the air in the tube in kelvin. The temperature in the room during our measurements was 24 °C, and the temperature of the air we blew into the tube was 34 °C. For that, we have assumed a temperature of (29 ± 5) °C = (302 ± 5) K, resulting in a speed of sound of *v* = (348 ± 3) m/s.

The wavelengths are then calculated using:

The uncertainty of the wavelength is determined by calculating the largest and smallest possible values by taking the largest and smallest possible values of the speed of sound and the frequency. This crude determination of the uncertainty overestimates the uncertainty of the wavelengths. However, as most students do not have experience with the determination of measurement uncertainties, we have found this a good first step.^{3} Although the uncertainties are overestimated, they still allow distinction between the models later on.

## Results

To determine which of the two competing models best describes the data, we have made plots of the wavelength as a function of the length of the tube. Then we have fitted a least-squares fit through the data, either of the form *y* = *ax*, resembling the simple model, or of the form *y* = *ax*+*b*, resembling the extended model. We have chosen the least-squares method, as most software available to students makes use of this. The downside to the least-squares fit is that all measurements have equal weights, i.e., measurements with small uncertainties have the same weight as measurements with large uncertainties. The next subsections describe the results of the simple and extended model.

### Simple model

Figure 3 shows the results of our measurements (blue dots) and the fit (green line) compared to the simple model (black dashed line), with the residuals underneath. The residual plot shows all the data with, in this case, the simple model subtracted (placing it horizontally at zero). The residual plot shows the difference between the measurements, the fit, and the model in centimeters.

The fit does not describe the data very well, as it does not go through the uncertainty of the smallest tube. The slope of the curve, 4.248, deviates almost 6% from the slope of the simple model. The measurements show no agreement with the simple model, as none of the measurement uncertainties overlap with the simple model.

Additional statistical analysis of the least-squares fit confirms this. The uncertainty of the fit’s slope, 4.248 ± 0.013, indicates no correspondence to the simple model. This additional analysis of the uncertainties is not required for students to come to a solid conclusion. It does, however, prove that the conclusion, based on the lack of overlapping uncertainties, is well founded.

### Extended model

Figure 4 shows the same measurements (blue dots) but now compared to the extended model (black dashed line). The fit (red line) shows much better agreement with the measurements, as the line goes through all four uncertainty intervals. The slope of this fit, 4.08, deviates 2% from the theoretical slope, and the intercept, 2.1 cm, deviates 18%. Although improved as compared with the simple model, the uncertainties of the measurements show no overall agreement with the theoretical extended model, as the extended model only goes through two of the four uncertainty intervals.

Additional statistical analysis of the least-squares fit gives uncertainties of the slope and the intercept of the fit: 4.08 ± 0.06 for the slope and 2.1 ± 0.7 cm for the intercept. This is almost in agreement with the extended model, that predicts a slope of 4 and an intercept of 1.72 cm [the constant in Eq. (4)].

The residual plot shows a systematic ∼1-cm shift above the extended model. This is an opportunity to discuss systematic uncertainties and experimental refinement, exemplary for a scientific process, with students. Our hypothesis for this shift is that the presence of the mouth and lips extends the length of the tube somewhat. This would result in longer wavelengths and smaller frequencies.

When comparing the results of the fits resembling the simple and the extended model, one has to conclude that the extended model describes the data better than the simple model, but there is no overall agreement with the theory. This conclusion is also supported by looking at the uncertainties of the slopes and intercept, resulting from statistical analysis of the least-squares fit.

## Refining the experiment

To test our hypothesis that the presence of the mouth and lips extend the tube somewhat, we have changed the setup of our experiment. Instead of blowing on the tubes with our mouth, we now use a compressor to blow air over the tube (see Fig. 5). This is a variation on the experiment before but requires students to work with a compressor, which might not be available in every physics classroom. This variation, however, allows for more robust uncertainty analysis which we shall discuss.

One advantage of this setup is that the temperature can easily be measured by blowing on a thermometer for a while. This gives *T* = (23 ± 1.5) °C = (296 ± 1.5) K, resulting in *v* = (345.0 ± 0.9) m/s. The second advantage is that we can make a time-series of the frequency, using phyphox’s *Frequency History* function. This way we were able to measure the peak frequency hundreds of times in a few seconds. The frequencies do not seem to depend on the flow rate of the air or the angle with which we blow on the tube. The frequency measurements show a normal distribution, which allows us to calculate an accurate mean value and standard deviation: *f*_{pres,7} = (1164 ± 12) Hz, *f*_{pres,10} = (827 ± 9) Hz, *f*_{pres,13} = (641 ± 5) Hz, and *f*_{pres,16} = (522 ± 4) Hz.

Figure 6 shows the results of the measurements using compressed air (blue squares). Because our measurements have a standard deviation, we can use a chi-squared fit (red line). This fit has the advantage of giving measurements with smaller uncertainties a larger weight in the fit, resulting in a better fit. The measurements show a clear overlap with the fit as well as the extended model (black dashed line) and confirm the extended model. The slope of the fit, 4.01, deviates 0.3% from the theoretical slope, and the intercept, 1.8 cm, deviates 5% from the theoretical value.

Additional statistical analysis shows that the value for the chi-squared is *χ*^{2} (2) = 0.39, which is small but acceptable. The slope of the fit is 4.01 ± 0.02 and the intercept is (1.8 ± 0.2) cm, indicating that these measurements are—as predicted by the overlap of measurement uncertainties with the theory—in very strong agreement with the extended model.

We also see that the frequencies for the compressed air are systematically higher than the frequencies we had measured before, resulting in the disappearance of the systematic upward shift in wavelengths. This would support our hypothesis of the extension of the tube by the lips. Another indication supporting our hypothesis is that we have been able to reproduce the effect qualitatively, by holding the tube by the rim (see Fig. 7). Holding the tube like this simulates the presence of the lips near the rim, extending the tube. This resulted in lower frequencies, depending on where we held the tube.

## Conclusion

We have presented and successfully tested an inexpensive, hands- and minds-on high school lab in which students have to decide which of two models best describes their data.

The determination of measurement uncertainties is kept at an introductory level and the decision between models is based on the (lack of) overlap between uncertainty intervals and the theoretical models. The reduced complexity of this analysis makes the lab accessible for high school students. Nevertheless, the conclusions are well-founded as the result of this analysis is confirmed by the result of more advanced statistical analysis.

As such, this lab offers an opportunity for students to engage in an authentic scientific procedure by comparing measurement data with models, discussing (systematic) measurement uncertainties, and evaluating and refining an experiment. All of this can be done without having to rely on advanced statistical analysis.

## References

*AP Edition*), 3rd ed. (

**Karel Kok** and **Franz Boczianowski** *are researchers in the physics education group at Humboldt-Universität zu Berlin, Germany. Their fields of research are experimenting, measurement uncertainties, and modeling in class. The experiment of the article originated from the time Karel was teaching high school and is now also used in the first semester lab for students in Berlin.*