Sound Waves and Beats
Aj Klatch
Natira Yefchak
Mr. Edmondson
H. Engineering Sci. II
3/01/02
4/5
2
I.
Abstract
The purpose of this lab was to measure single C4 and C5
frequencies of a flute and then combine the two frequencies in order to
form a beat pattern. This lab was able to be accomplished with the use of
a computer, ULI microphone, logger pro, and a Casio SK-10 keyboard.
The data achieved for this lab is as follows. The C4 frequencies
produced 27 cycles in .05 seconds which gave us a period of .00185s and a
frequency value of 540hz. The C4 key also gave us an amplitude of .560.
The C5 frequency produced 54 cycles in .05 seconds with a period
.000910seconds giving us a resultant frequency of 1098hz.
When combining the C4 and C5 frequencies the number of cycles
in .05seconds was 26. The beat period was .00234 and our calculated
frequency was 427hz. Finally, our percent error compared to our
calculated frequency for the C4 was 0.00% whereas the calculated
frequency for the C5 was 1.67%.
3
Sound Waves and Beats
II.
Materials
The materials used in this lab included a Windows PC, a Universal Lab Interface,
a Vernier microphone, Logger Pro, and an electric keyboard.
III.
Apparatus
The apparatus for the lab consisted of an electric keyboard connected to the
Vernier microphone, which connected to the ULI, which was connected to the computer.
IV.
Procedure
Part I
1.
First connect the Vernier microphone to DIN 1 of the Universal Lab Interface.
2.
Open "Exp 21" from the Physics with Computers experiment files of Logger Pro.
3.
Holding the microphone close to the sound source, produce a sound and collect
the resulting data.
4.
Count and record the number of complete cycles shown after the first peak in your
data.
5.
Click the Examine button, and record the times for the first and last peaks of the
waveform. Divide the difference by the number of cycles to determine the period.
6.
Calculate the frequency and record it in the data table.
7.
Drag mouse across graph and record minimum and maximum y values for an
adjacent peak and trough.
8.
Calculate the amplitude by taking half of the difference between the maximum
and minimum y values. Record your data.
9.
Print your graph.
10.
Save your data by choosing "Store Latest Run" from the Data menu.
11.
Repeat Steps 3 through 9 for the second frequency. Store the latest run.
4
Part II
12.
Simultaneously hold down two keys on the keyboard.
13.
Collect the data while the two tones are sounding.
14.
Count the number of amplitude maxima after the first maximum and record
it in your data table.
15.
Click the Examine button. Record the times for the first and last amplitude
maxima. Divide the difference by the number of cycles to determine the period
of beats. Calculate the beat frequency from the beat period. Record these values in
your data table.
5
V.
Data
a. Tables
i. Part I – Simple Waveforms
Note
Cycles
First Max
(s)
Next Max
(s)
Delta-T
(s)
Period (s)
C4
27
.000820
.00275
.0500
.00185
Calculated
Frequency
(hz)
540
C5
54
.00100
.00191
.0500
.000910
1098
Note
Peak (V)
Trough (V)
Amplitude (V)
C4
3.10
1.98
.560
C5
3.36
1.74
.810
Note
Parameter A (V)
Parameter B (s^-1)
F=B/2pi
C4
.590
3392
540
C5
.810
6785
1080
Error = 1.67%
ii. Part II – Beats
Number of
First Max (s) Last Max (s)
Delta-t (s)
Beat (s)
Cycles
26
Calculated
Beat (hz)
.000360
.00270
.0500
.00234
427
6
VI.
Math
a. Period
i. P=s/cycles
ii. P=.05seconds/27cycles=.00185
b. Frequency
i. F=1/P
ii. F=1/.00185=540hz
c. Amplitude
i. (Peak-Trough)/2 = A
ii. (3.1-1.98)/2 = .56
d. Calculated Frequency
i. B=2pi*F
ii. B=2*pi*540=3392
iii. F=B/2pi
iv. F=3392/2pi=540
e. Percent Error
i. |accepted-achieved|/accepted * 100%
ii. |1080-1098|/1080 * 100% = 1.67%
7
VII.
Conclusion
In closing this lab was done in a very perfunctory manner and
completed in minimal time. This lab was more of a continuation of our
previous sound lab.
Again, the data we achieved for this lab is as follows. The C4
frequencies produced 27 cycles in .05 seconds which gave us a period of
.00185s and a frequency value of 540hz. The C4 key also gave us an
amplitude of .56. The C5 frequency produced 54 cycles in .05 seconds
with a period .00091seconds giving us a resultant frequency of 1098hz.
When combining the C4 and C5 frequencies the number of cycles
in .05seconds was 26. The beat period was .00234 and our calculated
frequency was 427hz.
Finally, the only error that we encountered throughout this lab is a
different C5 frequency then expected. Our C5 frequency ended up to be
1098hz whereas the calculated frequency would have been 540hz. This
gave us a percent of error of 1.67%.
8
VIII.
1.
Questions
Since B corresponds to 2 f in the curve fit, use the curve fit information to
determine the frequency. Compare this frequency with the frequency calculated earlier. Which would you expect to be more accurate? Why?
The earlier calculated frequency would be more accurate.
2.
Compare the parameter A to the amplitude of the waveform.
They are the same.
3.
The trigonometric identity sin x + sin y = 2sin(x+y/2)  cos(x - y/2)
is useful in modeling beats. Show how the beat frequency you measured
above can be predicted using two sinusoidal waves of frequency f and f ,
whose ressure variations are described by sin(2 f t) and sin(2 f t).
These are derived from half angle trig identities. We can make a simple
proof of this identitiy by subbing in the following values and achieving the same
answer on both parts of the equation, thus, this shows that the identity is valid.
This can also be further reduced to get a numerical value for 2ft.
f = 540
f = 1080
2ft+2ft
2sin( f t + f t)  cos( f t - f t)