(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "NeXT Mathematica Notebook Front End Version 2.2"; NeXTStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 12, "Times"; ; fontset = leftheader, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; paletteColors = 128; automaticGrouping; magnification = 125; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] FREQ: Variable Frequencies :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] A string vibrating at a constant frequency can be modeled by A sin(2 Pi w t) where w is the frequency of vibration. However, if the frequency changes over time, then w is not the frequency. This set of exercises explores how to find the true frequency given w as well as how to find w given the desired variable frequency. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsection; inactive; preserveAspect] FileName: FREQ :[font = subsection; inactive; preserveAspect] Full title: Variable Frequencies :[font = subsection; inactive; preserveAspect] Last Update: 8/20/96 :[font = subsection; inactive; preserveAspect] Developer: Aaron D. 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. :[font = subsection; inactive; preserveAspect] Contact: Aaron D. 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. :[font = subsection; inactive; preserveAspect; endGroup] 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. :[font = section; inactive; Cclosed; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; preserveAspect; startGroup] When we first learned about trigonometric functions, we found that sin(t) went through 1 cycle in 2 Pi seconds. Furthermore, sin(2 Pi t) goes through 1 cycle per second and sin(2 Pi w t) goes through w cycles in 1 second -- as long as w is a constant. We call the constant, w, the frequency of oscillation. The following problem set is concerned with what happens if w is not a constant, i.e., w = w(t) is a function of time t. For example, suppose w(t) = t. If we graph y = sin(2 Pi w t) = sin(2 Pi t^2) for 5 seconds, it is clear that the frequency of oscillation increases with t. :[font = input; preserveAspect; startGroup] Plot[Sin[2 Pi t^2], {t, 0, 5}, AspectRatio -> Automatic, PlotPoints -> 50] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; preserveAspect; startGroup] 1) Argue, however, that w(t) = t does NOT count the number of oscillations per second and is thus not the frequency of oscillation as follows: :[font = subsubsection; inactive; preserveAspect] a) Estimate the number, f, of cycles on [2, 3]. :[font = subsubsection; inactive; preserveAspect; endGroup] b) Note that your estimate of the frequency, f, does not satisfy w(2) <= f <= w(3). Explain why this implies that w(t) cannot be the frequency of oscillation. :[font = subsection; inactive; preserveAspect] 2) In fact, if w(t) = t, then the frequency W(t), is W(t) = 2t. Show that, indeed, W(2) <= f <= W(3). :[font = subsection; inactive; preserveAspect] 3) Argue that if w(t) = t^2, then the frequency is W(t) = 3t^2 by estimating the number of cycles sin(2 Pi w(t) t) has on 2 <= t <= 3 and comparing it with the values of W(t) on the same interval. :[font = subsection; inactive; preserveAspect] 4) Argue further that if w(t) = 200 + (100/(2 Pi)) sin(2 Pi t) then the frequency is W(t) = 200 + 100t cos(2 Pi t) + (100/(2 Pi)) sin(2 Pi t) by estimating the number of cycles sin(2 Pi w(t) t) has on 1 <= t <= 1.04, multiplying that number by 25 to get the approximate number of cycles per second and comparing that value with the values of W(t) on the same interval. :[font = subsection; inactive; preserveAspect] 5) Given w(t), how can you determine W(t)? :[font = subsection; inactive; preserveAspect] 6) Suppose you are given W(t). How can you find w(t)? :[font = subsection; inactive; preserveAspect] 7) Construct the function sin(2 Pi w(t) t) whose frequency W(t) increases linearly from 263 to 526 cycles per second over a period of 3 seconds. (A tight string oscillating at the given frequencies would play the middle C octave on a piano.) :[font = subsection; inactive; preserveAspect; endGroup] 8) Use Mathematica's Play[] command to play the function created in #7. You might be able to hear that the tones are weighted towards the top of the scale. This is due to the fact that the frequency is exponentially related to the octave: F(n) = 263 2^n is the frequency n octaves above middle-C. Find the frequency W(t) for t in [0, 3] and the corresponding function sin(2 Pi w(t)t) so that the tones increase at a constant rate over the 3 second interval. ;[s] 3:0,0;7,1;18,2;461,-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 = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Exponential Functions, frequency :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; preserveAspect] ISSUES RELATED TO THE PROBLEM :[font = subsection; inactive; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect] Students should be familiar with frequency of trigonometric functions as well as the antiderivative of a function. :[font = subsubsection; inactive; preserveAspect; endGroup] It's advisable, but not necessary to do at least part of OCTAVE and OCTAVE2 from this WWW site. :[font = subsection; inactive; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect; endGroup] Allow 50 - 60 minutes in a computer lab. :[font = subsection; inactive; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect; endGroup] Some students may be unable to finish, but most should be able to finish at least the first 4. The last four require a good understanding of the derivative. Be prepared to answer questions and/or allow for unfinished work. :[font = subsection; inactive; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect; endGroup] Students gain a more thorough understanding of frequency and trigonometric functions. :[font = subsection; inactive; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect; endGroup] See OCTAVE and OCTAVE2 at this WWW site. :[font = subsection; inactive; preserveAspect; endGroup] References and Sources :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; preserveAspect; startGroup] 1) Argue, however, that w(t) = t does NOT count the number of oscillations per second and is thus not the frequency of oscillation. :[font = input; preserveAspect; startGroup] Plot[Sin[2 Pi t^2], {t, 2, 3}, AspectRatio -> Automatic, PlotPoints -> 50] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect; endGroup] Since there are f = 5 cycles for t in [2, 3] and f=5 does not lie between w(2)=2 and w(3)=3 and w(t) is monotone on [2, 3], w(t) must not be counting the number of oscillations per second. :[font = subsection; inactive; preserveAspect] 2) Indeed, if W(t) = 2t, then [W(2)=4] <= [f=5] <= [6=W(3)]. Furthermore, the average of W(t) on [2, 3] is f = 5 which makes sense since the frequency W is increasing linearly. :[font = subsection; inactive; preserveAspect; startGroup] 3) :[font = input; preserveAspect; startGroup] w = t^2; Plot[Sin[2 Pi w t], {t, 2, 3}, PlotPoints -> 50] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] Counting very carefully, we find f=19 cycles in the one second for t on [2, 3]. If W(t) = 3t^2, then [W(2)=12] <= [f=19] <= [W(3)=27]. Indeed, the average of W(t) on [2, 3] is :[font = input; preserveAspect; startGroup] Integrate[3 t^2, {t, 2, 3}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 19 ;[o] 19 :[font = subsection; inactive; preserveAspect; startGroup] 4) :[font = input; preserveAspect] w = 200 + (100/(2 Pi)) Sin[2 Pi t]; :[font = input; preserveAspect] W[t_] = 200 + 100 t Cos[2 Pi t] + (100/(2 Pi)) Sin[2 Pi t]; :[font = input; preserveAspect; startGroup] Plot[Sin[2 Pi w t], {t, 1, 1.04}, PlotPoints -> 50] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] It looks like about 12 and 1/6 cycles occur for t in [1, 1.04] which means that we are averaging about f=228.125 cycles per second. :[font = input; preserveAspect; startGroup] f = (12 + 1/6)/0.04 :[font = output; output; inactive; preserveAspect; endGroup] 304.1666666666667 ;[o] 304.167 :[font = text; inactive; preserveAspect] which means that we are averaging about f=304.167 cycles per second. :[font = input; preserveAspect; startGroup] leftside = W[1] //N :[font = output; output; inactive; preserveAspect; endGroup] 300. ;[o] 300. :[font = input; preserveAspect; startGroup] rightside = W[1.04] //N :[font = output; output; inactive; preserveAspect; endGroup] 304.6906712413029 ;[o] 304.691 :[font = input; preserveAspect; startGroup] NIntegrate[W[t], {t, 1, 1.04}]/0.04 :[font = output; output; inactive; preserveAspect; endGroup] 302.9085845820563 ;[o] 302.909 :[font = text; inactive; preserveAspect; endGroup] Recognizing that we were only able to make a crude approximation (the 1/6 of a cycle part), we did a pretty good job! As we have seen before, W(t) counts the number of cycles per second at time t. Also, over the interval [1, 1.04], the average number of cycles per second, about 303, is pretty darn close to our estimate of 304! :[font = subsection; inactive; preserveAspect] 5) Given w(t), W(t) = D(t w(t)). :[font = subsection; inactive; preserveAspect] 6) Given W(t), w(t) = Integrate[W(t)]/t. :[font = subsection; inactive; preserveAspect; startGroup] 7) :[font = input; preserveAspect] ClearAll[W, t]; :[font = input; preserveAspect; startGroup] W[t_] = 263 + (526 - 263)/3 t :[font = output; output; inactive; preserveAspect; endGroup] 263 + (263*t)/3 ;[o] 263 t 263 + ----- 3 :[font = input; preserveAspect; startGroup] Plot[W[t], {t, 0, 3}, AxesLabel -> {"seconds", "frequency"}] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] w = Simplify[Integrate[W[t], t]/t] :[font = output; output; inactive; preserveAspect; endGroup] (263*(6 + t))/6 ;[o] 263 (6 + t) ----------- 6 :[font = input; preserveAspect; startGroup] g = Sin[2 Pi w t] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Sin[(263*Pi*t*(6 + t))/3] ;[o] 263 Pi t (6 + t) Sin[----------------] 3 :[font = subsection; inactive; preserveAspect; startGroup] 8) :[font = input; preserveAspect; startGroup] Play[g, {t, 0, 3}] :[font = output; output; inactive; preserveAspect; endGroup] Sound["<<>>"] ;[o] -Sound- :[font = input; preserveAspect] W = 263 2^(t/3); :[font = input; preserveAspect; startGroup] Plot[W, {t, 0, 3}] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] w = Simplify[Integrate[W, t]/t] :[font = output; output; inactive; preserveAspect; endGroup] (789*2^(t/3))/(t*Log[2]) ;[o] t/3 789 2 -------- t Log[2] :[font = input; preserveAspect; startGroup] h = Simplify[Sin[2 Pi w t]] //N :[font = output; output; inactive; preserveAspect; endGroup] Sin[7152.064303803312*2.^(0.3333333333333333*t)] ;[o] 0.333333 t Sin[7152.06 2. ] :[font = input; preserveAspect; startGroup] Play[h, {t, 0, 3}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] Sound["<<>>"] ;[o] -Sound- :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect] Students should already be familiar with Mathematica's Plot[] command before attempting this problem. ;[s] 4:0,0;41,1;55,2;96,3;101,-1; 4:1,12,9,Times,1,14,0,0,0;1,13,10,Times,3,14,0,0,0;1,12,9,Times,1,14,0,0,0;1,12,9,Times,1,14,0,0,0; :[font = subsection; inactive; preserveAspect] Because the Play[] command takes a lot of time, students should be warned to play tones for short time intervals -- say {t, 0, 1}. :[font = subsection; inactive; preserveAspect; endGroup; endGroup] It is difficult to hear the difference between the scales in problems number 7 and 8. Thus, it's important to rely on mathematical analysis using the Play[] command only as a check for reasonableness. ^*)