Measuring the speed of sound with Kundt´s tube

PHYS291 Project

By Karolina Berg

 June 2020

 

Introduction

Kundt´s tube is an experimental acoustical apparatus invented in 1866 by German physicist August Kundt for the measurement of the speed of sound in a gas or a solid rod. In this project the goal is to find the speed of sound in air, by using Kundt´s tube. The speed of sound in air is 343 m/s in room temperature (20 °C), and at 40°C the speed is 355 m/s. Since the instruments have errors and the temperature in the room is not exactly 20°C, the expected speed of sound in air in this project is ± 1-2% of 343 m/s. In this project, the reflection factor r and pressure amplitude p₀ will also be calculated. In Kundt´s tube, the sound wave is a harmonic pressure wave propagating in a gas. The sound wave is sent against a wall where it is reflected, and the reflection factor r is expected to be 0 < r < 1. Pressure amplitude p₀ will be calculated from the maximum and minimum pressure.

 

Instruments used in this experiment are listed in the table 1.

 

 Table 1: Instruments used in this experiment.

 

Procedure

Figure 1: Shows how the instruments are connected together.

 

The equipment is connected together as shown in figure 1. To make the measurements, the amplitude of the signal from the signal generator is set to 2 Vpp, and the frequency to 900 Hz. From that, one can hear sound from the speaker. Kundt´s tube is adjusted, so one can hear where the sound is most clear and loudest. Thereafter, the frequency is adjusted so one can see the signal from the oscilloscope decrease. Frequency is then adjusted to see the maximum impact on the screen of the oscilloscope and the resonance frequency f is measured to be 985 Hz.

 

Kundt´s tube is adjusted to a minimum, and the measurements for position and amplitude are made. The amplitude here is measured as the voltage produced by the microphone, and it is measured in the unit volt since there is no calibration done from voltage to pressure change. Kundt´s tube position is adjusted 0,5 cm in the beginning and in the end, so one can achieve a smooth graph in the beginning and in the end of the plot. Rest of the time, the tube is adjusted 1 cm. The adjustment continues until a half wavelength of the signal is achieved in a graph plot. 

 

The error measurement of the position is the measurement error of the meter stick, which is s_x = 0.05 cm. The measurement error of the oscilloscope is 3% of the measurement of the amplitude. Calculation of error to the first amplitude is presented below, while the rest of the results from the measurements and calculations are presented here: https://folk.uib.no/kgr016/table.html

Amplitude (1): 3.0mV * 0.03 = 0.09 mV. 

 

Data from the table are plotted in C++. The code for the program is presented here: https://folk.uib.no/kgr016/rootprogram.c. Graph with errors is presented below and is run from the C++ code by using ROOT. On the x-axis the position with error is presented, while on the y-axis is the amplitude with error. 

 

Figure 2: Shows position and amplitude with errors.

 

Calculations

Next step is to calculate the reflection factor r and the pressure amplitude p₀ using the minimum and maximum values of the amplitude from the graph. The microphone converts sound pressure to voltage, so the voltage is proportional to the sound pressure. One can therefore set p_max = U_max and p_min = U_min. The reflection factor r is given by the formula:

Error in the measurement of reflection factor r is calculated by using partial derivation, so that the combined uncertainties are taken into account. The formula for reflection factor r is partially derived with respect to p_max, and then with respect to p_min.

 

 

Reflection factor r with error is r = 0,857 ± 0,005.

 

The pressure amplitude p₀ is given by the formula:

 

The error of the pressure amplitude p₀ is calculated using partial derivation, so that the combined uncertainties are taken into account. The formula for the pressure amplitude is partially derived with respect to p_max, and then with respect to p_min.

 

 

Pressure amplitude p₀ with error is p₀ = 21,0 ± 0,5mV.

 

Now the speed of sound can be calculated. The positions of two minimum, a₁ and a₂, which are as far apart as possible, are measured. By choosing two positions that are as far apart as possible, the relative uncertainty will be minimized. This happens because two points with a good distance give a larger measurement area and then error becomes smaller in relation to the measurement area. Position a₁ and a₂ is measured six times for three different half-wavelengths n. The average for these points is calculated for further calculations. Position a₁ and a₂, average of a₁ and a₂, resonance frequency f and total half wavelengths n are presented in table 2. 

 

Table 2: Table presents three measurements for resonance frequency f, a₁, a₂, average for a₁ and a₂, and total half wavelengths n.

 

Errors in the average of a₁ and a₂  are calculated as a combination of errors in the measurements and the errors in the instruments. The total errors for u(a₁) and u(a₂) for each of the measurements is:

 

Results of calculations for the total errors for u(a₁) and u(a₂) are presented in table 3. Average of a₁ and a₂ is afterwards used for calculating the speed of sound by using formula:

Errors in the speed of sound from formula above can be calculated by using:

For resonance frequency f₁ = 985Hz:

 

Speed of sound calculated with resonance frequency f₁ = 985Hz is v₁ = 344,6 ± 0,4 m/s.

 

For resonance frequency f₂ = 1251Hz:

 

Speed of sound calculated with resonance frequency f₂ = 1251Hz is v₂ = 345,7 ± 0,6 m/s.

 

For resonance frequency f₃ = 1521Hz:

 

Speed of sound calculated with resonance frequency f₃ = 1521Hz is v₃ = 344,1 ± 0,5 m/s.

 

Calculated speed of sound with its errors from three different resonance frequency f is listed in table 3.

Table 3: Presents measured resonance frequency f, average a₁ and a₂, errors for average a₁ and a₂, and speed v with its errors u.

 

The average for the speed of sound is calculated with data from table 3, by using the formula:

 

Error for the average speed of sound is calculated by using the formula:

 

The speed of sound as an average from the measurements is v = 344,7 ± 0,3 m/s.

 

Conclusion

In this project Kundt´s tube has been used to measure the speed of sound in air. The speed of sound in air is 343 m/s in room temperature (20 °C), and at 40°C the speed is 355 m/s. Since the temperature in the room was closer to 20°C, the expected value was ± 1-2% of 343 m/s. The result of calculating the speed of sound in air was v = 344,7 ± 0,3 m/s. This value is 0,49% higher than the theoretical value 343 m/s, and the conclusion is that the value is very good, and the measurements have been precise.

 

The reflection factor r was expected to be 0 < r < 1, and the calculated value is r = 0,857 ± 0,005. From this value, one can conclude that most of the sound wave is reflected. Pressure amplitude p₀ was calculated from the maximum and minimum pressure measured with the oscilloscope. Pressure amplitude p₀ with error is p₀ = 21,0 ± 0,5mV.

 

Position to Kundt´s tube and amplitude was measured, and a plot was presented with the measurement errors. The plot was plotted in C++ and ROOT was used to run the code. The plot can be seen in figure 1 as a smooth Gauss function. On the x-axis the position with error was presented, while on the y-axis the amplitude with error was presented. 

 

Possible sources of error that could affect the result data in this project were equipment errors, misreading of data, incorrect plotting of data and possible old equipment such as weak coaxial cables.

 

Files in the project can be seen here:

Root Program: https://folk.uib.no/kgr016/rootprogram.c

Data Table: https://folk.uib.no/kgr016/table.html

 

References

[1] R. Maad. PHYS114 - “Oppgave 6. Kundts rør: Lydhastighet og varmekapasitet for gasser. Teoridel” Bergen: Institutt for fysikk og teknologi, UiB: vår 2020.