PHYS-1500 PHYSICAL MODELING FALL 2006
Class 25: Beats NAME _________________________________
A well known acoustic phenomenon, called “beats”, occurs whenever two sounds of slightly different frequency and nearly the same amplitude are heard at the same time. The listener hears a single tone whose loudness varies regularly with time. It is the regular, beat-like increase and decrease of loudness from which the phenomenon gets its name. The cause of the beats is the interference of the two sound waves that arrive at the ear at the same time. This can be modeled quite easily with the spreadsheet below. It creates two sine waves and then adds them.
Construct the spreadsheet shown below. Below the picture of the spreadsheet are the contents of the cells. For the cells above the actual program, i.e. rows 1 through 10, only the contents of the cells containing formulae have been given. For rows 12 and 13, the contents of all cells have been given. When row 13 is complete, the entire row should be copied to rows 14 through 312.
Then, use the program to answer the questions on the other side of the page.
B6: =B5-B4
E4: =2*PI()*B4
E5: =2*PI()*B5
A12: 0 B12: =$B$7*SIN($E$4*A12)
A13: =A12+$B$3 B13: =$B$7*SIN($E$4*A13)
D12: =B12+C12
D13: =B13+C13
H4: =1/B4
H5: =1/B5
H6: =1/B6
C12: =$B$8*SIN($E$5*A12)
C13: =$B$8*SIN($E$5*A13)
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Create two graphs from your model. The first should be a graph of x
1
+ x
2
as a function of time, and the second should contain separate plots of x
1
and of x
2
vs. time on the same graph. Start with all the values as they are given on the printout of the spreadsheet on the other side of the paper.
1. Your graph of x
1
+ x
2
vs. t should show the amplitude varying with time. How long does one
“beat” last? How many beats per second would be heard? That is, what is the beat frequency?
(Hint: frequency equals 1/period.)
T beat
= __________________
units f beat
= __________________
units
2. Now change f
2
to 550 Hz, but leave everything else the same. How long does one “beat” last?
What is the beat frequency?
T beat
= __________________
units f beat
= __________________
units
3. Now change f
2
to 570 Hz, but leave everything else the same. How long does one “beat” last?
What is the beat frequency?
T beat
= __________________
units f beat
= __________________
units
4. Now change f
2
to 640 Hz, but leave everything else the same. How long does one “beat” last?
What is the beat frequency?
T beat
= __________________
units f beat
= __________________
units
5. From your data, what relationship can you deduce between the beat frequency and the frequencies of the two component signals?
2

6. Now, set f
2
equal to 550 Hz again, and change
 t to 0.0001 s. Examine the graph that shows both x
1
and of x
2
vs. time. What is the relationship between x
1
and x
2
when x
1
+ x
2
is a maximum?. What is the relationship between x
1
and x
2
when x
1
+ x
2
is a minimum?
7. Set A
2
= 2, and leave A
1
= 1. Change nothing else. Does this set of conditions still produce beats of some sort? What is different about the graph of x
1
+ x
2
vs. time?
8. Open the spreadsheet “SHMDRIVN.XLS”, and go to sheet 2, “LCR Circuit”. Enter the values shown below.
 t = 1.00E-05 s
C = 1.00E-06 F
1/ C = 1000000 F -1 q
0
= 0.00E+00 C i
0
= 0.00E+00 A

 = 60000 rad/s
E
1
= 0 V
E
2
= 10 V
L = 4.00E-04 H
R = 0.1 

= 50000 rad/s i (max) = 2.7987555 A
T = 0.0001257 s f = 7957.7472 Hz
The graph of q vs. t should show a “beat” pattern. What two signals are beating? Does the beat frequency obey the rule that you deduced in part 5?
3