LAB 2: SPEED OF SOUND
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Lab 2: Speed of Sound
XXX
Lab Partner: XXX
XXX
Date of Experiment: 12 February 2024
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Abstract
This experiment aims to experimentally measure the speed of sound using an acoustic
waveguide setup, then to determine the accuracy of the experimental results through error
analysis and uncertainty analysis. The general procedure involved using the materials provided
to take measurements of the time delay between two signals from microphones, use the time
delay and the set frequency to approximate the experimental phase angle, and finally measure the
speed of sound based on the measured distance between the two microphones and the slope from
the least squares fit line of the frequency and phase angle plot. Sources of error arose during the
experiment such as unexpected negative slope in the data, and a less than ideal correlation
coefficient, but adjustments were made to ensure accuracy. The experiment resulted in the
experimentally measured speed of sound to be 𝑐𝑒𝑥𝑝 = 323. 48 𝑚/𝑠, compared against a
theoretical value of 𝑐𝑎𝑖𝑟 = 344 𝑚/𝑠 yields a 5.96% error. The uncertainty analysis revealed that
the calculated uncertainty was higher than expected, yet after finding the confidence interval of
the experimental value, it was confirmed that the true value for the speed of sound lies within the
confidence interval, indicating a successful uncertainty analysis. This experiment was important
because being able to determine the speed of sound is crucial to many fields, such as the
hypersonics industry, and by being able to experimentally determine the confidence interval for
the true value allows general calculations to be used when deciding design characteristics.
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Table of Contents
Abstract...........................................................................................................................................2
Table of Contents........................................................................................................................... 3
Nomenclature................................................................................................................................. 4
Experiment 2: Speed of Sound..................................................................................................... 5
Required Equations....................................................................................................................... 5
Equations: Measuring Speed of Sound...................................................................................... 5
Equations: Uncertainty Analysis................................................................................................6
Experimental Apparatus and Procedure.....................................................................................7
Measuring Speed of Sound:....................................................................................................... 8
Uncertainty analysis.................................................................................................................10
Results and Discussion.................................................................................................................10
Results: Measuring the Speed of Sound.................................................................................. 10
Discussion: Measuring the Speed of Sound.............................................................................13
Results: Uncertainty Analysis..................................................................................................14
Discussion: Uncertainty Analysis............................................................................................ 16
Conclusion.................................................................................................................................... 18
References..................................................................................................................................... 20
Appendix.......................................................................................................................................21
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Nomenclature
c
Speed of Sound (General)
τ
Time Delay (General)
ϕ
Phase Difference/Phase Angle (General)
𝑓
Frequency
x
Measured Distance Between the Microphones
𝑆ϕ𝑓
Standard Error of the Fit
𝑡𝑣,95%
Student t-distribution
𝑢𝑚
Statistical Uncertainty from the Slope
𝑆𝑚
Precision Estimate
N (K)
Number of Measurements
v
Number of Measurements - 2
𝑢𝑥
Uncertainty of Distance Between Microphones
𝑢𝑐
Uncertainty in the Speed of Sound
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Experiment 2: Speed of Sound
The primary objective behind this lab is to evaluate the acoustic waveguide as a way of
measuring the speed of sound in the air, in other words, to measure the speed of sound from the
phase delay between two microphones. In addition to measuring the speed of sound, based on
the results from the experiment, an uncertainty analysis is to be conducted to determine the
confidence interval of the least squares fit, statistical uncertainty from the slope of the fitted
curve, and relative uncertainty in the measurement of the speed of sound. Per the lab manual
provided during the experiment, a speaker is driven by a function generator through a power
amplifier, the speaker then produces a sound wave that travels through the acoustic waveguide
assembly at the speed of sound. Then, two microphones that the experiment uses measure the
acoustic wavelength from the sound, and by finding the time delay between the two
microphones, a phase difference between the signals is found, allowing the speed of sound to be
approximated.
Required Equations
Equations: Measuring Speed of Sound
This experiment is split into two parts: the estimate of the speed of sound, c, and then the
uncertainty of analysis. The main value that was measured from this lab was τ, known as the
time delay between the signals, which is measured in µ𝑠 (with the signals being the two
microphones), and the value that the experimenter set was the frequency, which is measured in
Hz. The time delay produces a phase difference, ϕ, between the two signals, and this phase
difference is used to approximate c, which is the speed of sound.
Using the definition of the time delay, τ:
τ =
𝑥
, where x is measured as the distance between the two microphones.
𝑐
The definition of phase difference, ϕ, is defined as :
(1) ϕ =
2πτ
𝑇
= 2π𝑓τ, where T is the period, so 1/T = the frequency,
The definition for the phase difference can be rewritten as:
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ϕ=
2π𝑥𝑓
𝑐
By isolating the speed of sound and noting that the phase angle is measured at different
frequencies, the speed of sound can be experimentally determined as:
2π𝑥
(2) c = 𝑑ϕ/𝑑𝑓 where 𝑑ϕ/𝑑𝑓 is the slope of the frequency vs the phase angle, which
can be found by plotting the linear line of best fit of the experimentally collected
values.
Equations: Uncertainty Analysis
The confidence interval of the least squares fit using the standard error of the fit, 𝑆ϕ𝑓, is defined
as:
(3) ϕ = (ϕ𝑜 + 𝑚 · 𝑓) ± 𝑡𝑣,95% · 𝑆ϕ𝑓(95%), where (ϕ𝑜 + 𝑚 · 𝑓) is found from the
fitted line from the experimental data, 𝑡𝑣,95% is the student t distribution value
found in the appendix (A-3), ν is defined as the number of data points (N) - 2,
± 𝑡𝑣,95% · 𝑆ϕ𝑓(95%) is the confidence interval,
𝑆ϕ𝑓 is the standard error of the fit, defined as:
𝐾
(4) 𝑆ϕ𝑓 =
2
∑ [ϕ𝑖−(ϕ𝑜+𝑚·𝑓𝑖)]
𝑖=1
𝑣
, using the same definitions as (3).
Secondly, the statistical uncertainty from the slope of the least squares fit, (the linear line of best
fit from the experimental data), 𝑢𝑚, defined from the precision estimate, 𝑆𝑚, is:
(5) 𝑢𝑚 = 𝑡𝑣,95 · 𝑆𝑚, where 𝑡𝑣,95 is the same as the calculated value from the
confidence interval analysis.
NOTE: K is just an index to account for each test, K=N)
𝑆𝑚 used in (5) is the precision estimate of the least squares fit, defined as:
(6) 𝑆
= 𝑆ϕ𝑓 ·
𝑚
𝑁
𝐾
2
𝐾
2
𝑁 ∑ 𝑓𝑖 −[ ∑ 𝑓𝑖]
𝑖=1
𝑖=1
, which use the same definitions (3)
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Lastly, the relative uncertainty of the calculated speed of sound, 𝑢𝑐, is calculated by using the
uncertainty in the slope, 𝑢𝑚, and the uncertainty in the measurement between the two
microphones, 𝑢𝑥. Since the speed of sound is dependent upon two independent factors from each
other, x, the distance between the two microphones, and 𝑑ϕ/𝑑𝑓, which is the slope of the fitted
line, the uncertainty of the speed of sound would only be dependent on the uncertainty in the
slope and the uncertainty of the measurement between the two microphones. Relative
uncertainty, by definition, is defined as the calculated uncertainty divided by the measured value.
It is important to note that 𝑢𝑥 is found from the actual equipment, which is from a caliper that
was used. The particular caliper that was used in this experiment had an uncertainty of
± 0. 001𝑚𝑚. By using the root-sum-squared method, and the relative uncertainty of both 𝑢𝑚
and 𝑢𝑥, defined as
𝑢𝑚
𝑢
and 𝑥𝑥 , respectively, the relative uncertainty of the speed of sound is
𝑚
found by:
𝑢
(7) 𝑐𝑐 =
𝑢
𝑢
2
2
[ 𝑥𝑥 ] + [ 𝑚𝑚 ] , where all values 𝑢𝑐 are previously
computed/experimentally calculated. To solve for the uncertainty in the
measurement of the speed of sound, simplify equation (7) to:
(8) 𝑢 = 𝑐 ·
𝑐
𝑢
2
𝑢
2
[ 𝑥𝑥 ] + [ 𝑚𝑚 ]
Experimental Apparatus and Procedure
In addition to the Lab Manual, the following experiment was used in this lab:
Acoustic Wave Guide
1 - Caliper
Oscilloscope-Hitachi VC-6020
2 - microphones
Function Generator-HP 33120A
1 - feed through terminator 50Ω
Power Amplifier
The final experimental set up is shown below:
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Figure 1: Final Experimental Setup
Measuring Speed of Sound:
The following steps were taken to experimentally measure the speed of sound, c.
1) Before installing the two microphones, measure the distance between the two microphone
ports on the waveguide.
a) After obtaining the distance, the uncertainty of this measurement was found using
the caliper's specifications.
2) Insert the microphones into the waveguide.
3) As seen in Figure 1, connect the function generator to the TAPE/TUNER input of the
power amplifier, ensuring that the 50Ω feed through terminator is connected in-line
between the function generator and the power amplifier. Then, connect the speaker on
the waveguide to the SPEAKERS output of the power amplifier. Lastly, connect the
microphones to the oscilloscope.
a) Verify that the speaker is on the channel to which the function generator is
attached
b) The microphone closest to the speaker should be connected to CH1, and the
microphone closest to the damper should be connected to CH2
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4) On the function generator, first select a 1-kHz, 100mVPP sine wave and take note of the
microphone signals on the oscilloscope, click auto-range to automatically scale the
waves.
5) This next step requires the time delay to be measured. On the oscilloscope, click the
measure button, and then select time delay. Then, vary the frequency from 1kHz to
3.0kHz in increments of 10Hz (using the function generator)
a) The result will most likely be in microseconds, so be sure to take note of that and
convert the time delay from microseconds to seconds before doing calculations
6) Measure the time delay at each frequency, and then compute the phase angle for each
frequency using equation (1).
a) As previously mentioned, it is very important to compute the phase angle using
the time delay in SECONDS, not microseconds.
b) Additionally, if the time delay is a negative value, add a period to the time delay
(in seconds). This is done by adding 1/f, or 1/frequency (Hz), to the time delay (s)
7) Input all of the collected data into Excel and plot the phase angle versus the frequency
(with the phase angle on the y-axis and the frequency on the x-axis)
a) Whatever spreadsheet software that can create plots and provide data from the
plots can be used, however, the following steps use Excel.
b) Ideally, the data should fall on a straight line of best fit, in reality, this most likely
is not the case.
8) On the plot, select the “+” sign, click Trendline > More Options > Linear > Display
Equation on chart > Display R-squared value on chart.
a) The equation that is displayed on the chart for the line of best fit (otherwise
known as the least squares fit of a straight line) must be of the form “y = mx+b”,
which correlates to the (ϕ𝑜 + 𝑚 · 𝑓) portion of equation (3). Where m is the
slope of the least squares fit., which is 𝑑ϕ/𝑑𝑓 for equation (2)
2
b) The 𝑅 value that is displayed is the correlation coefficient between ϕ & 𝑓, the
closer to 1, the the more linear the data is
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9) After step 8, the experiment itself is done. Use the measured value for the distance
between the microphones, x, and the slope of the least squares fit, m, and equation (2) to
calculate the experimental for the speed of sound, c. Compare the experimental value for
c to the theoretical value, which ideally is
Uncertainty analysis
The equations that were listed in the “Equations: Uncertainty Analysis” were all computed for
the uncertainty. NOTE: All of the following calculations can be done in Excel, however, the ()
section will show the order of the equations used during the analysis.
10) To obtain a confidence interval of the least squares fit, the 𝑡𝑣,95% · 𝑆ϕ𝑓(95%) portion of
equation (3), use equation (4) to first find 𝑆ϕ𝑓 and then the student t distribution table in
the appendix as well as v = N-2 = 20-2 = 18 to find 𝑡𝑣,95%.
11) To obtain the statistical uncertainty of the slope using the precision estimate, first use
equation (6) to find the precision estimate, 𝑆 , using the same 𝑆ϕ𝑓 found in 10), and
𝑚
then, use equation (5) using the same 𝑡𝑣,95% found in 10) to find the statistical uncertainty.
a) Plot the precision error on the chart as an error bar. On Excel, this is done by
clicking the chart > “+” sign > Error Bars > More Options > Fixed Value, and
then typing in the found statistical uncertainty.
12) Lastly, to obtain the uncertainty in the measurement of the speed of sound, equation (8) is
used, utilizing the experimentally measured/calculated values of c, m, and x, as well as
the uncertainty value found from 11) and the uncertainty of the caliper (± 0. 001𝑚𝑚)
Results and Discussion
Results: Measuring the Speed of Sound
Starting with the experimental results from Measuring the Speed of Sound, the distance between
the two microphones was first calculated, where:
𝑥 = 139. 025𝑚𝑚 = 0. 139𝑚
Additionally, the caliper used indicated an uncertainty of ± 0. 0010𝑚𝑚, so:
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𝑢𝑥 = 0. 0010mm
Next, the following chart was generated in Excel, showing the experimentally calculated values
for the Frequency in Hz (set by the experimenter), Time Delay in both microseconds and seconds
(measured), and the Phase Angle in radians (calculated from the frequency and the time delay):
Figure 2: Experimentally Collected Values.
In the chart above (Figure (2)), the frequency and time delay in SECONDS were both used to
calculate the phase angle, in radians, using equation (1). The three highlighted rows have a time
delay that has an unexpectedly low value. As mentioned in the procedure, to account for these
errors, the period for that particular frequency is added to the time delay. For example, in Row 1:
τ = τ𝑢𝑛𝑒𝑥𝑝 +
1
𝑓
−6
1
= (− 428 × 10 ) + 1000 = 0. 000572 𝑠
−6
where τ𝑢𝑛𝑒𝑥𝑝 is the unexpected value for the time delay, (− 428 × 10 ) is the conversion of
1
1
the time delay from microseconds to seconds, and 1000 is the the period for this test, 𝑓 .
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The experimental values generated were used to generate the frequency versus the phase angle
graph, seen below:
Figure 3: Phase Angle (Radians) vs. Frequency (Hz). The data points that were collected are
shown by the big blue dots on the solid curved line. The dotted blue line going through the
data is the line of best fit, or the least squares fit, for this data set. In the upper right hand
2
corner, the equation for the line of best fit is shown, as well as the 𝑅 value.
2
In the highlighted box in the above image (Figure(3)), the 𝑅 value is displayed, as well as the
2
line of best fit. The 𝑅 value is the coefficient of correlation between ϕ & 𝑓 , and it was found
that:
2
𝑅 = 0.9908
2
In addition to the 𝑅 value, the equation for the line of best fit is displayed, where:
Fit: 𝑦𝑓𝑖𝑡 =
− 0. 0027𝑥𝑓𝑖𝑡 + 6. 3429
The equation of the line of best fit is also known as the least squares fit of a straight line to the
data. Rewriting the equation in terms of the phase angle and the frequency, the following
equation is generated:
ϕ = (ϕ𝑜 + 𝑚 · 𝑓) => 𝑦𝑓𝑖𝑡 =
− 0. 0027𝑥𝑓𝑖𝑡 + 6. 3429
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=> ϕ𝑒𝑥𝑝 = (6. 3429 + (− 0. 0027) · 𝑓)
Where ϕ𝑒𝑥𝑝 corresponds to 𝑦𝑓𝑖𝑡, 𝑓 corresponds to 𝑥𝑓𝑖𝑡, ϕ𝑜 corresponds to the y-intercept, 6.3429,
and m corresponds to the slope, -0.0027. The slope of ϕ𝑒𝑥𝑝 can be rewritten as 𝑑ϕ/𝑑𝑓, so to
experimentally determine the speed of sound, use equation (2) with the experimental slope and
measured distance between the microphones (0.139m), where:
2π𝑥
2π(0.139)
c = 𝑑ϕ/𝑑𝑓 = |−0.0027| = 323. 48 𝑚/𝑠
This experimental value for the speed of sound can be compared against the theoretical value for
the speed of sound. For a perfect gas c = 344m/s. So the error analysis follows as:
𝑇ℎ𝑒𝑜𝑟𝑦 − 𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 |
| 344−323.48 | × 100 = 5. 96%
% 𝑒𝑟𝑟𝑜𝑟 = ||
| × 100 = |
|
𝑇ℎ𝑒𝑜𝑟𝑦
344
Discussion: Measuring the Speed of Sound
The experimental measurement of the speed of sound was calculated to be 323.48 m/s and was
compared against a theoretical value of c = 344 m/s, yielding a percent error of 5.96%. Across
the duration of this experiment, there were a few indicators that could have been a source of error
during the experiment. First, the acoustic waveguide was not completely insulated inside of a
sound proof box. Since there were other sounds in the lab such as other people talking as well as
other people conducting the experiment, there is a chance that our microphones could have
picked up some unwanted sounds, thus causing an incorrect reading for the time delay, in turn
causing an incorrect calculation for the speed of sound. This incorrect reading may have been a
possible reason why there was a flatter line around a phase angle of 1 radian from 1800 to 2200
Hz.
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Another source of error during the experiment was that the original slope for the line of
best fit was found to be negative, which goes along with the data. This negative sign was not
expected, as the derived equation for the speed of sound, equation (2), only called for the slope
as measured. However, if the measured slope was used in equation (2) as is, the speed of sound
would be calculated as a negative value, which is not possible as speed cannot be negative. As a
result, it was determined that the absolute value of equation (2) would yield the correct
experimental value for the speed of sound, which only required taking the absolute value of the
slope.
A final error that was seen after plotting the data was that the correlation between the
2
phase angle and frequency was not 1, instead being 𝑅 = 0.9908. This value indicates that the
data itself was not completely linear, which is the theoretical trend. Despite this correlation value
not being 1, with a percent error of 0.92%, calculated as:
% 𝑒𝑟𝑟𝑜𝑟 = ||
𝑇ℎ𝑒𝑜𝑟𝑦 − 𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 |
| × 100
𝑇ℎ𝑒𝑜𝑟𝑦
= ||
1−0.9908 |
| × 100
1
= 0. 92%
The line of best fit was still a pretty accurate fit for the nonlinear data values that were
experimentally generated. Overall, the experiment yielded an equation for the phase angle in
terms of frequency with a correlation coefficient proving that the line of best fit is 99.08%
accurate.
Results: Uncertainty Analysis
To obtain the confidence interval from the standard fit, equation (3) was primarily used. It is
important to note that the equation for the phase angle in terms of the frequency is required, but
this was previously calculated as:
ϕ𝑒𝑥𝑝 = (6. 3429 + (− 0. 0027) · 𝑓)
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What first needs to be calculated to use equation (3) is 𝑡𝑣,95%, the student t distribution, which
can be found using the table in the appendix (A-3). With a total of 20 measurements that were
taken, v is now found, where:
ν = 𝑁𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡𝑠 − 2 = 20 − 2 = 18
So taking the value of ν and plugging that into the student t distribution equation, it can be seen
that the value needed from the table and the respective answer is:
𝑡18,95% = 2.101
In addition to the student t distribution, the standard error, 𝑆ϕ𝑓, is required, computed below as:
𝐾
𝑆ϕ𝑓 =
2
∑ [ϕ𝑖−(ϕ𝑜+𝑚·𝑓𝑖)]
𝑖=1
𝑣
=
0.52252
18
= 0. 170
NOTE: Standard Error Analysis computed in Excel is added in the appendix, A-4
Taking the standard error and the student t distribution values, the confidence interval for
equation (3) can be found, as:
ϕ = (ϕ𝑜 + 𝑚 · 𝑓) ± 𝑡𝑣,95% · 𝑆ϕ𝑓(95%)
=> ϕ𝑒𝑥𝑝 = (6. 3429 + (− 0. 0027) · 𝑓) ± 2. 101 · 0. 170
=> ϕ𝑒𝑥𝑝 = (6. 3429 + (− 0. 0027) · 𝑓) ± 0. 35717
In interval format:
− 0. 35717 ≤ ϕ𝑒𝑥𝑝 = (6. 3429 + (− 0. 0027) · 𝑓) ≤ 0. 35717
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To obtain the statistical uncertainty of the slope, equation (5) is primarily used. Before using this
equation, the previously calculated value for the student t distribution, 𝑡18,95% = 2.101, and the
precision estimate for the least squares fit, 𝑆𝑚 is required. Note that 𝑆𝑚 uses the previously
calculated value for the standard error, 𝑆ϕ𝑓= 0.170. The precision error is calculated below:
𝑆𝑚 = 𝑆ϕ𝑓 ·
𝑁
𝐾
𝐾
2
2
= 0. 170 ·
𝑁 ∑ 𝑓𝑖 −[ ∑ 𝑓𝑖]
𝑖=1
−5
20
5
2
20(917*10 )−(42000)
= 9. 086 * 10
𝑖=1
NOTE: Standard Error Analysis computed in Excel is added in the appendix, A-5
Using the precision estimate and the student t distribution, equation (5) yields:
−5
−4
𝑢𝑚 = 𝑡𝑣,95 · 𝑆𝑚 = 2. 101 · (9. 086 * 10 ) = 1. 909 * 10
This uncertainty generally makes sense due to how small the slope is. Finally, to find the total
uncertainty in the measurement of the speed of sound, equation (8) is used with all of the
previously calculated and experimentally collected values, as:
𝑢𝑐 = 𝑐 ·
𝑢 2
𝑢 2
[ 𝑥𝑥 ] + [ 𝑚𝑚 ]
= 323. 48 ·
−4 2
0.0010 2
1.909*10 ⎤
⎡
⎡ 0.139 ⎤ + ⎢ −0.0027 ⎥ = 32. 6 𝑚/𝑠
⎣
⎦
⎣
⎦
NOTE: The relative uncertainty for 𝑢𝑐 using equation (7) is 𝑢𝑐 = 0.100.
Discussion: Uncertainty Analysis
Two more graphs were generated from the uncertainty analysis, both derivations from Figure (3).
First, is the graph of the phase angle versus the frequency showing the confidence interval using
the error bars. The general idea behind adding errors to the data is to show the uncertainty of
each data point, indicating that the true value is within an interval of the experimentally
measured value. ()The error bars are depictions of the confidence interval applied to each data
point, seen below:
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Figure 4: Uncertainty in the Line of Best Fit using Error Bars
Second, the graph of the phase angle versus the frequency shows the confidence interval using
two other linear fit lines, one adding the uncertainty to the line of best fit (the orange/top line)
and one subtracting the uncertainty from the line of best fit (the green/bottom line). The chart
with the data used to graph the uncertainty lines is included in the appendix, A-8, as:
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Figure 5: Uncertainty in the Line of Best Fit using Parallel LInes
It can be seen that the uncertainty lines enclose all of the data, therefore, the correct confidence
interval for the line of best fit was calculated. Despite the correct confidence interval being
calculated, the uncertainty for the speed of sound has an unexpectedly high value. Ideally, the
uncertainty would be smaller, indicating that the experimental value is closer to the theoretical
value, however, since the experimental value is further away, the uncertainty is higher, which
could indicate an incorrectly measured value during the experiment. However, using the idea that
the true value will be within the confidence interval, it can be found that:
𝑐𝑒𝑥𝑝 − 𝑢𝑐 ≤ 𝑐𝑡𝑟𝑢𝑒 ≤ 𝑐𝑒𝑥𝑝 + 𝑢𝑐 => 290. 88 ≤ 344 ≤ 356. 80
Since the above statement is true, it can be seen that the true value for the speed of sound lies
within the confidence interval calculated, indicating that the experimental uncertainty for the
speed of sound is accurate.
Conclusion
This lab was focused on experimentally determining the speed of sound in the air using an
acoustic waveguide setup and then conducting an uncertainty analysis to determine the
confidence intervals of the experimentally collected values. It was found that experimentally, the
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speed of sound was found to be 𝑐𝑒𝑥𝑝 = 323. 48 𝑚/𝑠, and when compared against a theoretical
value of 𝑐𝑎𝑖𝑟 = 344 𝑚/𝑠, the relative percent error was found to be 5.96%, a respectably low
error, indicating that the experiment yielded relatively accurate data. However, some errors were
noticed in the experiment, ranging from needing to adjust equations due to receiving an
impossible measure of speed to having a less-than-ideal value for the measurement of the
correlation coefficient (with a percent error of 0.92%).
Despite these errors, uncertainty analysis proved that the experimentally found that the
−4
uncertainty for the slope was 𝑢𝑚 = 1. 909 * 10 , the uncertainty for the caliper was
𝑢𝑥 = 0. 0010𝑚𝑚, and the uncertainty for the speed of sound was found to be 𝑢𝑐=32.6 m/s.
After analyzing the confidence interval for the speed of sound, 𝑢𝑐, it was found that the true
value for the speed of sound lies within the confidence interval, indicating that the calculated
value is correct. Additionally, it was found that after plotting the line of best fit
ϕ𝑒𝑥𝑝 = (6. 3429 + (− 0. 0027) · 𝑓) ± 0.35717 with the confidence interval (Figure ()) , it
was deemed that with the uncertainty interval, all of the experimentally collected data values
were between the interval, validating the accuracy using uncertainty analysis. This analysis
proves that despite the experimental errors that arose, overall, the experiment was successful.
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References
Figliola, R. S., & Beasley, D. E. (2010). Theory and Design for Mechanical Measurements (5th
ed.). Wiley.
Lab Manual 2: Speed of Sound
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Appendix
A-1 - Figure 1: Final Experimental Setup, setup notes and required measurements to be
collected will be explained in the following sections. Additionally, the “Dampening Material ''
is only added to ensure that the generated sound does not reverberate back towards the
microphones, which would cause incorrect signals to be projected on the oscilloscope, thus
affecting the time delay.
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A-2: Figure 2: Experimentally Collected Values. The Frequency was set by the experimenter
(column 1), the oscilloscope captured the time delay in microseconds (second column), the
time delay was the converted from microseconds to seconds (third column), and the phase
angle was calculated from the frequency and the time delay in seconds by using equation (1)
(fourth column).
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A-3: Student t-distribution table, used to calculate the t value required for the
calculations for the uncertainty
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A-4: Standard Error Analysis performed in Excel. The data values on the left is the
experimental values for the phase angle, and the computations on the left sums
ϕ𝑖 − (ϕ𝑜 + 𝑚 · 𝑓𝑖) for the whole experiment, finds the value for v, and the
computes the standard error using equation (4)
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A-5: Precision Estimate Analysis performed in Excel. The data values on the left are
the frequency values used in the experiment, and the values on the right
complete the rest of the required values to properly use equation (6) such as the
(sum of the frequency)^2, the sum of the (frequency^2), and the standard error.
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A-6: Figure (4): Error bars applied to each data point of Figure 3, depicting the
confidence interval being applied to each data point. The true value of the phase angle
at each set frequency is within the interval shown by the red bars.
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A-7 Figure (5): This image depicts the uncertainty of the line of best fit for the Phase
Angle vs. Frequency data set. The upper line shows the line of best fit with adding the
uncertainty (labeled “Phase Angle + uc”) and the lower line shows the line of best fit
with subtracting the uncertainty (labeled “Phase Angle - uc”). The blue dotted line is
the same line of best fit shown in Figure 3 (labeled Linear (Recorded Values))
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A-8: Data used to graph the uncertainty lines.
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A-9: Data Collected in Lab with TA Signature