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BEATS
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BRIEF ABSTRACT
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This exercise deals with the relationship between a beat's frequency and the frequency of the lower tone.  (A beat (a wa-wa tone) is a superposition of two tones vibrating at close frequencies.)  This makes for a good trigonometry review -- no calculus is required for this problem.
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GENERAL INFORMATION
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FileName:   BEATS

Full title:  What's the Beat?

Last Update: 6/3/96

Developers: 
	Aaron Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA.
	Rosaline Secrest, Physics Teacher, South Vigo High School, Terre Haute, IN 47802 USA.
	Susan Clements, Mathematics Department, South Vigo High School, Terre Haute, IN 47802  USA

Contact:  
Aaron Klebanoff, Department of Mathematics
Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA 
Phone:  812-877-8151.  Email:  Aaron.Klebanoff@rose-hulman.edu.  
FAX:  812-877-3198.

Support:  
The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849:  Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. 
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STATEMENT OF PROBLEM 
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Preliminaries
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What's a beat?
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If two harmonic waves have similar frequencies, a phenomena known as beats, that is, a repetitive variation of  the sound intensity, occurs.  For example, when a musical instrument is being tuned to match the pitch of another instrument, the characteristic "wa-wa" sound is often produced as the two frequencies approach each other.
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Frequency
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Mathematically, we can represent oscillations with the trigonometric functions sine or cosine.  For example, the cosine function
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has a frequency of 1 Hz or 1 vibration per second.  If we want to increase the number of vibrations per second, we increase the frequency f in the expression Cos(2 Pi f t).  For example, the wave
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undergoes precisely 4.2 oscillations per second.
:[font = subsubsection; inactive; preserveAspect; startGroup]
Period and Amplitude
:[font = text; inactive; preserveAspect]
The period of oscillation is the length of time it takes to complete one full cycle (vibration.)  The period of Cos[2 Pi f / t] is inversely related to the frequency by the relation Period = 1 / f.
:[font = text; inactive; preserveAspect]
The amplitude of a wave is the distance from the average value to the peak value of the oscillation.  In all of the oscillations above, the amplitude has a constant value of 1.  However, some functions oscillate with varying amplitudes.  The curve below has a decreasing amplitude starting at 1 and decaying to zero.
:[font = input; preserveAspect; startGroup]
Plot[ Exp[-t/2] Cos[2 Pi t], {t, 0, 5},
 PlotPoints -> 50]
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Graphics["<<>>"]
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Its worth remarking that functions such as the one above are not periodic, and yet we still have a feeling for how long it takes to oscillate through a full cycle.  In this case, we would speak of the quasi-period instead of period.
:[font = subsection; inactive; preserveAspect; startGroup]
The Problems.  For each graph of a wave, play the tone as well if you have Mathematica or some other utility that allows you to play tones oscillating at prescribed frequencies.
;[s]
3:0,0;75,1;86,2;177,-1;
3:1,12,9,Times,1,14,0,0,0;1,13,10,Times,3,14,0,0,0;1,12,9,Times,1,14,0,0,0;
:[font = subsubsection; inactive; preserveAspect]
1. Graph a wave oscillating at 120Hz for 0 < t < 0.1 second.
:[font = subsubsection; inactive; preserveAspect]
2. Graph a wave oscillating at 125Hz for 0 < t < 0.1 second.	
:[font = subsubsection; inactive; preserveAspect]
3. Graph the resultant of the two harmonic oscillations by plotting the sum of the two wave functions over the time interval 0 < t < 1 second.  Also graph parts of the wave to get a feel for what it looks like over smaller time intervals.
:[font = subsubsection; inactive; preserveAspect]
4. Use the graph produced in (3) to determine
(a) the maximum beat amplitude and when the maxima occur
(b) the beat period
(c) the beat frequency  (i.e., the number of beats per second)
:[font = subsubsection; inactive; preserveAspect]
5. Repeat 1 - 4 above if the two frequencies are 150 Hz and 155 Hz.
:[font = subsubsection; inactive; preserveAspect; endGroup; endGroup]
6. Give a conjecture for how the beat frequency relates to the frequency of the base (lower) tone?  Test your conjecture with another example.
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
KEYWORDS  
:[font = subsection; inactive; preserveAspect; endGroup]
Beats, sound, trigonometric functions, plotting, amplitude, frequency, period, cosine function, oscillations.
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
TEACHER NOTES
:[font = subsection; inactive; preserveAspect; startGroup]
ISSUES RELATED TO THE PROBLEM 
:[font = subsubsection; inactive; preserveAspect; endGroup]
Mathematica's Play command makes this problem a lot more interesting, but the problem was written so that Mathematica isn't a necessity.
;[s]
4:0,0;11,1;106,2;117,3;136,-1;
4:1,10,8,Times,3,12,0,0,0;1,10,8,Times,1,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,1,12,0,0,0;
:[font = subsection; inactive; preserveAspect; startGroup]
Prerequisites
:[font = subsubsection; inactive; preserveAspect; endGroup]
Students must have a working knowledge of amplitude, frequency, and period as they relate to the cosine function.   
:[font = subsection; inactive; preserveAspect; startGroup]
Time allotment - time management
:[font = subsubsection; inactive; preserveAspect; endGroup]
Less than one class period is required to complete this problem.
:[font = subsection; inactive; preserveAspect]
Expectations
:[font = subsection; inactive; preserveAspect]
Students will be able to recognize the quantitative characteristics of the resultant sound.
:[font = subsection; inactive; preserveAspect; startGroup]
Future payoffs
:[font = subsubsection; inactive; preserveAspect; endGroup]
Students will gain a greater understanding of the graphical analysis of a trigonometric functions  and of the relationship between mathematical functions and physical phenomena. 
:[font = subsection; inactive; preserveAspect; startGroup]
Extensions
:[font = subsubsection; inactive; preserveAspect; endGroup]
Sound frequencies could be sampled using CBL equipment.
:[font = subsection; inactive; preserveAspect; startGroup]
References and Sources
:[font = subsubsection; inactive; preserveAspect; endGroup; endGroup]
Daniel Stump, "Using the Graphing Calculator in Sample Physics Problems, The Physics Teacher,  May 1993, pp.317-8
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
POSSIBLE SOLUTION(S)
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
1) the 120Hz tone
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wave1 = Cos[120 2 Pi t]
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Cos[240*Pi*t]
;[o]
Cos[240 Pi t]
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Plot[wave1, {t, 0, 0.1}]
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