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 ± 12% of 343 m/s.
In this project, the reflection factor r and pressure amplitude p_{0}
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 p_{0 }will be calculated from the
maximum and minimum pressure.
Instruments used in this experiment are listed
in the table 1.
Instruments 
Type 
Kundt´s tube with
microphone and speaker 

Signal generator 
Agilent 33220A 
Oscilloscope 
Tektronix DPO2002B 
Coaxial cables 

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 xaxis the position with error is presented,
while on the yaxis 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 p_{0 }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 p_{0 }is given by the formula:
The error of the pressure amplitude p_{0}_{
}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 p_{0} with
error is p_{0} = 21,0 ± 0,5mV.
Now the speed of sound can be calculated. The
positions of two minimum, a_{1}_{ }and a_{2},
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 a_{1} and a_{2 }is measured six times
for three different halfwavelengths n. The average for these points is
calculated for further calculations. Position a_{1 }and a_{2},
average of a_{1 }and a_{2}, resonance frequency
f and total half wavelengths n are presented in table 2.
Table 2: Table presents three measurements for
resonance frequency f, a_{1}, a_{2}, average for a_{1}
and a_{2}, and total half wavelengths n.
Errors in the average of a_{1}_{
}and a_{2}_{ }are
calculated as a combination of errors in the measurements and the errors in the
instruments. The total errors for u(a_{1}) and u(a_{2})
for each of the measurements is:
Results of calculations for the total errors for
u(a_{1}) and u(a_{2}) are presented in table 3.
Average of a_{1 }and a_{2} 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 f_{1 }=
985Hz:
Speed of sound calculated with resonance
frequency f_{1} = 985Hz is v_{1} = 344,6 ± 0,4 m/s.
For resonance frequency f_{2} =
1251Hz:
Speed of sound calculated with resonance
frequency f_{2 }= 1251Hz is v_{2} = 345,7 ± 0,6 m/s.
For resonance frequency f_{3 }=
1521Hz:
Speed of sound calculated with resonance
frequency f_{3} = 1521Hz is v_{3} = 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 a_{1} and a_{2}, errors for average a_{1} and a_{2},
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 ± 12% 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 p_{0
}was calculated from the maximum and minimum pressure measured with
the oscilloscope. Pressure amplitude p_{0} with error is p_{0}
= 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 xaxis
the position with error was presented, while on the yaxis 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.