(*^ ::[ 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. 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(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. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsubsection; inactive; preserveAspect; endGroup] 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. :[font = section; inactive; Cclosed; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Preliminaries :[font = subsubsection; inactive; preserveAspect; startGroup] What's a beat? :[font = text; inactive; preserveAspect; endGroup] 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. :[font = subsubsection; inactive; preserveAspect; startGroup] Frequency :[font = text; inactive; preserveAspect] Mathematically, we can represent oscillations with the trigonometric functions sine or cosine. 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.59766 L .69469 .59201 L .69593 .58446 L .69841 .56389 L .70089 .53652 L .70337 .50306 L .70833 .42164 L .71825 .23285 L .72321 .14551 L .72569 .10783 L .72817 .07553 L .73065 .04947 L .73189 .039 L .73313 .03033 L .73438 .02353 L .73562 .01865 L .73686 .0157 L .7381 .01472 L .73934 .0157 L .74058 .01865 L .74182 .02353 L .74306 .03033 L .74554 .04947 L .74802 .07553 L .75298 .14551 L .75794 .23285 L .76786 .42164 L .77282 .50306 L .7753 .53652 L .77778 .56389 L .77902 .57506 L .78026 .58446 L .7815 .59201 L .78274 .59766 L .78398 .60139 L .78522 .60316 L .78646 .60296 L .7877 .6008 L .78894 .59669 L .79018 .59065 L .79142 .58272 L .79266 .57297 L .79762 .51712 L .80258 .43918 L .80754 .34743 L .8125 .2516 L .81746 .16187 L Mistroke .81994 .12231 L .82242 .08775 L .8249 .0591 L .82738 .03712 L .82862 .02882 L .82986 .0224 L .8311 .0179 L .83234 .01535 L .83358 .01475 L .83482 .01613 L .83606 .01947 L .8373 .02474 L .85714 .30902 L .8621 .40362 L .86706 .48818 L .87202 .55372 L .87326 .56626 L .8745 .57709 L .87574 .58612 L .87698 .59329 L .87822 .59856 L .87946 .6019 L .8807 .60328 L .88194 .60269 L .88318 .60013 L .88442 .59563 L .8869 .58092 L .88814 .5708 L .88938 .55894 L .89187 .53028 L .89683 .45617 L .90675 .2706 L .91171 .17885 L .91667 .10091 L .91915 .06979 L .92163 .04507 L .92287 .03531 L .92411 .02739 L .92535 .02135 L .92659 .01723 L .92783 .01507 L .92907 .01487 L .93031 .01664 L .93155 .02037 L .93279 .02603 L .93403 .03358 L .93651 .05414 L .93899 .08152 L .94147 .11497 L .94643 .19639 L Mistroke .95635 .38519 L .96131 .47252 L .96379 .5102 L .96627 .5425 L .96875 .56857 L .96999 .57904 L .97123 .5877 L .97247 .5945 L .97371 .59939 L .97495 .60233 L .97619 .60332 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect] 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 :[font = input; preserveAspect; startGroup] Plot[Cos[2 Pi 4.2 t], {t, 0, 1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 0.309017 0.294302 [ [(0.2)] .21429 .30902 0 2 Msboxa [(0.4)] .40476 .30902 0 2 Msboxa [(0.6)] .59524 .30902 0 2 Msboxa [(0.8)] .78571 .30902 0 2 Msboxa [(1)] .97619 .30902 0 2 Msboxa [(-1)] .01131 .01472 1 0 Msboxa [(-0.5)] .01131 .16187 1 0 Msboxa [(0.5)] .01131 .45617 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .30902 m .21429 .31527 L s P [(0.2)] .21429 .30902 0 2 Mshowa p .002 w .40476 .30902 m .40476 .31527 L s P [(0.4)] .40476 .30902 0 2 Mshowa p .002 w .59524 .30902 m .59524 .31527 L s P [(0.6)] .59524 .30902 0 2 Mshowa p .002 w .78571 .30902 m .78571 .31527 L s P [(0.8)] .78571 .30902 0 2 Mshowa p .002 w .97619 .30902 m .97619 .31527 L s P [(1)] .97619 .30902 0 2 Mshowa p .001 w .0619 .30902 m .0619 .31277 L s P p .001 w .1 .30902 m .1 .31277 L s P p .001 w .1381 .30902 m .1381 .31277 L s P p .001 w .17619 .30902 m .17619 .31277 L s P p .001 w .25238 .30902 m .25238 .31277 L s P p .001 w .29048 .30902 m .29048 .31277 L s P p .001 w .32857 .30902 m .32857 .31277 L s P p .001 w .36667 .30902 m .36667 .31277 L s P p .001 w .44286 .30902 m .44286 .31277 L s P p .001 w .48095 .30902 m .48095 .31277 L s P p .001 w .51905 .30902 m .51905 .31277 L s P p .001 w .55714 .30902 m .55714 .31277 L s P p .001 w .63333 .30902 m .63333 .31277 L s P p .001 w .67143 .30902 m .67143 .31277 L s P p .001 w .70952 .30902 m .70952 .31277 L s P p .001 w .74762 .30902 m .74762 .31277 L s P p .001 w .82381 .30902 m .82381 .31277 L s P p .001 w .8619 .30902 m .8619 .31277 L s P p .001 w .9 .30902 m .9 .31277 L s P p .001 w .9381 .30902 m .9381 .31277 L s P p .002 w 0 .30902 m 1 .30902 L s P p .002 w .02381 .01472 m .03006 .01472 L s P [(-1)] .01131 .01472 1 0 Mshowa p .002 w .02381 .16187 m .03006 .16187 L s P [(-0.5)] .01131 .16187 1 0 Mshowa p .002 w .02381 .45617 m .03006 .45617 L s P [(0.5)] .01131 .45617 1 0 Mshowa p .002 w .02381 .60332 m .03006 .60332 L s P [(1)] .01131 .60332 1 0 Mshowa p .001 w .02381 .04415 m .02756 .04415 L s P p .001 w .02381 .07358 m .02756 .07358 L s P p .001 w .02381 .10301 m .02756 .10301 L s P p .001 w .02381 .13244 m .02756 .13244 L s P p .001 w .02381 .1913 m .02756 .1913 L s P p .001 w .02381 .22073 m .02756 .22073 L s P p .001 w .02381 .25016 m .02756 .25016 L s P p .001 w .02381 .27959 m .02756 .27959 L s P p .001 w .02381 .33845 m .02756 .33845 L s P p .001 w .02381 .36788 m .02756 .36788 L s P p .001 w .02381 .39731 m .02756 .39731 L s P p .001 w .02381 .42674 m .02756 .42674 L s P p .001 w .02381 .4856 m .02756 .4856 L s P p .001 w .02381 .51503 m .02756 .51503 L s P p .001 w .02381 .54446 m .02756 .54446 L s P p .001 w .02381 .57389 m .02756 .57389 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .60332 m .02505 .60315 L .02629 .60262 L .02753 .60176 L .02877 .60054 L .03001 .59899 L .03125 .59709 L .03373 .59227 L .03869 .57865 L .04365 .55995 L .05357 .50879 L .06349 .44263 L .08333 .28593 L .09325 .20715 L .10317 .13603 L .10813 .10504 L .1131 .0779 L .11806 .05511 L .12302 .03712 L .1255 .03003 L .12798 .02425 L .13046 .01982 L .1317 .01812 L .13294 .01676 L .13418 .01574 L .13542 .01507 L .13666 .01475 L .1379 .01477 L .13914 .01514 L .14038 .01586 L .14162 .01693 L .14286 .01834 L .14534 .02219 L .14782 .02739 L .15278 .04175 L .15774 .06115 L .1627 .08523 L .18254 .21807 L .20238 .37772 L .2123 .45282 L .22222 .51712 L .22718 .54367 L .23214 .56579 L .2371 .58308 L .23958 .58979 L .24206 .59519 L .24454 .59923 L .24578 .60074 L .24702 .6019 L .24826 .60272 L Mistroke .2495 .60319 L .25074 .60332 L .25198 .60309 L .25322 .60252 L .25446 .6016 L .2557 .60034 L .25694 .59874 L .25942 .5945 L .2619 .58891 L .26687 .57381 L .27183 .55372 L .28175 .50015 L .30159 .35506 L .32143 .19639 L .33135 .12682 L .33631 .09687 L .34127 .07092 L .34623 .04947 L .34871 .04055 L .35119 .03291 L .35367 .02656 L .35615 .02155 L .35739 .01956 L .35863 .0179 L .35987 .01659 L .36111 .01562 L .36235 .015 L .36359 .01473 L .36483 .0148 L .36607 .01523 L .36731 .01599 L .36855 .01711 L .36979 .01857 L .37103 .02037 L .37599 .03096 L .37847 .03824 L .38095 .04679 L .39087 .0929 L .40079 .15524 L .42063 .30902 L .44048 .46279 L .4504 .52513 L .45536 .55046 L .46032 .57124 L .46528 .58708 L .46776 .59304 L .47024 .59766 L .47148 .59947 L .47272 .60093 L .47396 .60204 L Mistroke .4752 .60281 L .47644 .60323 L .47768 .6033 L .47892 .60303 L .48016 .60241 L .4814 .60145 L .48264 .60013 L .48512 .59648 L .4876 .59147 L .49008 .58513 L .49504 .56857 L .5 .54711 L .50992 .49122 L .51984 .42164 L .53968 .26298 L .5496 .1858 L .55952 .11788 L .56448 .08902 L .56944 .06431 L .5744 .04422 L .57937 .02912 L .58185 .02353 L .58309 .02125 L .58433 .0193 L .58557 .01769 L .58681 .01643 L .58805 .01551 L .58929 .01494 L .59053 .01472 L .59177 .01484 L .59301 .01531 L .59425 .01613 L .59549 .0173 L .59673 .0188 L .59921 .02285 L .60169 .02824 L .60417 .03496 L .60913 .05224 L .61905 .10091 L .62897 .16521 L .63889 .24031 L .65873 .39996 L .66865 .47252 L .67857 .53281 L .68353 .55688 L .68849 .57629 L .69097 .58412 L .69345 .59065 L .69593 .59585 L .69841 .5997 L Mistroke .69965 .60111 L .70089 .60217 L .70213 .60289 L .70337 .60326 L .70461 .60329 L .70585 .60296 L .70709 .60229 L .70833 .60128 L .70957 .59992 L .71081 .59821 L .71329 .59378 L .71577 .58801 L .71825 .58092 L .72321 .56292 L .72817 .54014 L .7381 .482 L .75794 .33211 L .77778 .17541 L .7877 .10924 L .79266 .08152 L .79762 .05808 L .80258 .03938 L .80506 .03192 L .80754 .02576 L .81002 .02095 L .81126 .01905 L .8125 .01749 L .81374 .01628 L .81498 .01541 L .81622 .01489 L .81746 .01472 L .8187 .01489 L .81994 .01541 L .82118 .01628 L .82242 .01749 L .8249 .02095 L .82738 .02576 L .83234 .03938 L .8373 .05808 L .84722 .10924 L .85714 .17541 L .87698 .33211 L .8869 .41088 L .89683 .482 L .90675 .54014 L .91171 .56292 L .91667 .58092 L .91915 .58801 L .92163 .59378 L .92287 .59617 L Mistroke .92411 .59821 L .92535 .59992 L .92659 .60128 L .92783 .60229 L .92907 .60296 L .93031 .60329 L .93155 .60326 L .93279 .60289 L .93403 .60217 L .93651 .5997 L .93775 .59794 L .93899 .59585 L .94147 .59065 L .94643 .57629 L .95139 .55688 L .95635 .53281 L .97619 .39996 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect; endGroup] 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] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.190476 0.272877 0.330442 [ [(1)] .21429 .27288 0 2 Msboxa [(2)] .40476 .27288 0 2 Msboxa [(3)] .59524 .27288 0 2 Msboxa [(4)] .78571 .27288 0 2 Msboxa [(5)] .97619 .27288 0 2 Msboxa [(-0.75)] .01131 .02505 1 0 Msboxa [(-0.5)] .01131 .10766 1 0 Msboxa [(-0.25)] .01131 .19027 1 0 Msboxa [(0.25)] .01131 .35549 1 0 Msboxa [(0.5)] .01131 .4381 1 0 Msboxa [(0.75)] .01131 .52071 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .27288 m .21429 .27913 L s P [(1)] .21429 .27288 0 2 Mshowa p .002 w .40476 .27288 m .40476 .27913 L s P [(2)] .40476 .27288 0 2 Mshowa p .002 w .59524 .27288 m .59524 .27913 L s P [(3)] .59524 .27288 0 2 Mshowa p .002 w .78571 .27288 m .78571 .27913 L s P [(4)] .78571 .27288 0 2 Mshowa p .002 w .97619 .27288 m .97619 .27913 L s P [(5)] .97619 .27288 0 2 Mshowa p .001 w .0619 .27288 m .0619 .27663 L s P p .001 w .1 .27288 m .1 .27663 L s P p .001 w .1381 .27288 m .1381 .27663 L s P p .001 w .17619 .27288 m .17619 .27663 L s P p .001 w .25238 .27288 m .25238 .27663 L s P p .001 w .29048 .27288 m .29048 .27663 L s P p .001 w .32857 .27288 m .32857 .27663 L s P p .001 w .36667 .27288 m .36667 .27663 L s P p .001 w .44286 .27288 m .44286 .27663 L s P p .001 w .48095 .27288 m .48095 .27663 L s P p .001 w .51905 .27288 m .51905 .27663 L s P p .001 w .55714 .27288 m .55714 .27663 L s P p .001 w .63333 .27288 m .63333 .27663 L s P p .001 w .67143 .27288 m .67143 .27663 L s P p .001 w .70952 .27288 m .70952 .27663 L s P p .001 w .74762 .27288 m .74762 .27663 L s P p .001 w .82381 .27288 m .82381 .27663 L s P p .001 w .8619 .27288 m .8619 .27663 L s P p .001 w .9 .27288 m .9 .27663 L s P p .001 w .9381 .27288 m .9381 .27663 L s P p .002 w 0 .27288 m 1 .27288 L s P p .002 w .02381 .02505 m .03006 .02505 L s P [(-0.75)] .01131 .02505 1 0 Mshowa p .002 w .02381 .10766 m .03006 .10766 L s P [(-0.5)] .01131 .10766 1 0 Mshowa p .002 w .02381 .19027 m .03006 .19027 L s P [(-0.25)] .01131 .19027 1 0 Mshowa p .002 w .02381 .35549 m .03006 .35549 L s P [(0.25)] .01131 .35549 1 0 Mshowa p .002 w .02381 .4381 m .03006 .4381 L s P [(0.5)] .01131 .4381 1 0 Mshowa p .002 w .02381 .52071 m .03006 .52071 L s P [(0.75)] .01131 .52071 1 0 Mshowa p .002 w .02381 .60332 m .03006 .60332 L s P [(1)] .01131 .60332 1 0 Mshowa p .001 w .02381 .04157 m .02756 .04157 L s P p .001 w .02381 .05809 m .02756 .05809 L s P p .001 w .02381 .07461 m .02756 .07461 L s P p .001 w .02381 .09113 m .02756 .09113 L s P p .001 w .02381 .12418 m .02756 .12418 L s P p .001 w .02381 .1407 m .02756 .1407 L s P p .001 w .02381 .15722 m .02756 .15722 L s P p .001 w .02381 .17374 m .02756 .17374 L s P p .001 w .02381 .20679 m .02756 .20679 L s P p .001 w .02381 .22331 m .02756 .22331 L s P p .001 w .02381 .23983 m .02756 .23983 L s P p .001 w .02381 .25636 m .02756 .25636 L s P p .001 w .02381 .2894 m .02756 .2894 L s P p .001 w .02381 .30592 m .02756 .30592 L s P p .001 w .02381 .32244 m .02756 .32244 L s P p .001 w .02381 .33897 m .02756 .33897 L s P p .001 w .02381 .37201 m .02756 .37201 L s P p .001 w .02381 .38853 m .02756 .38853 L s P p .001 w .02381 .40505 m .02756 .40505 L s P p .001 w .02381 .42158 m .02756 .42158 L s P p .001 w .02381 .45462 m .02756 .45462 L s P p .001 w .02381 .47114 m .02756 .47114 L s P p .001 w .02381 .48766 m .02756 .48766 L s P p .001 w .02381 .50419 m .02756 .50419 L s P p .001 w .02381 .53723 m .02756 .53723 L s P p .001 w .02381 .55375 m .02756 .55375 L s P p .001 w .02381 .57027 m .02756 .57027 L s P p .001 w .02381 .5868 m .02756 .5868 L s P p .001 w .02381 .00852 m .02756 .00852 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .60332 m .02502 .602 L .02624 .60016 L .02867 .59495 L .0311 .58773 L .03353 .57859 L .03839 .55485 L .04325 .52453 L .06268 .35778 L .0724 .26355 L .08212 .17495 L .09184 .10054 L .0967 .07086 L .10155 .04706 L .10641 .02962 L .10884 .02338 L .11006 .02088 L .11127 .01881 L .11249 .01716 L .1131 .0165 L .1137 .01594 L .11431 .01548 L .11492 .01513 L .11552 .01489 L .11613 .01475 L .11674 .01472 L .11735 .01478 L .11795 .01496 L .11856 .01523 L .11917 .01561 L .11978 .0161 L .12099 .01736 L .12221 .01904 L .12342 .02111 L .12585 .02642 L .12828 .03325 L .13071 .04153 L .14043 .08762 L .15015 .14993 L .15986 .22143 L .16958 .29452 L .1793 .36181 L .18902 .41686 L .19388 .4382 L .19631 .44711 L .19874 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L .97012 .29989 L .97072 .29995 L .97133 .3 L .97194 .30004 L Mistroke .97255 .30006 L .97315 .30008 L .97376 .30009 L .97437 .30008 L .97498 .30007 L .97558 .30004 L .97619 .3 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = text; inactive; preserveAspect; endGroup; endGroup] 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 :[font = input; preserveAspect; startGroup] wave1 = Cos[120 2 Pi t] :[font = output; output; inactive; preserveAspect; endGroup] Cos[240*Pi*t] ;[o] Cos[240 Pi t] :[font = input; preserveAspect; startGroup] Plot[wave1, {t, 0, 0.1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 9.52381 0.309017 0.294302 [ [(0.02)] .21429 .30902 0 2 Msboxa [(0.04)] .40476 .30902 0 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preserveAspect; startGroup] wave2 = Cos[125 2 Pi t] :[font = output; output; inactive; preserveAspect; endGroup] Cos[250*Pi*t] ;[o] Cos[250 Pi t] :[font = input; preserveAspect; startGroup] Plot[wave2, {t, 0, 0.1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 9.52381 0.309017 0.294302 [ [(0.02)] .21429 .30902 0 2 Msboxa [(0.04)] .40476 .30902 0 2 Msboxa [(0.06)] .59524 .30902 0 2 Msboxa [(0.08)] .78571 .30902 0 2 Msboxa [(0.1)] .97619 .30902 0 2 Msboxa [(-1)] .01131 .01472 1 0 Msboxa [(-0.5)] .01131 .16187 1 0 Msboxa [(0.5)] .01131 .45617 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .30902 m .21429 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Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Play[wave2, {t, 0, 1}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Sound["<<>>"] ;[o] -Sound- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 3) the resultant of the two harmonic oscillations :[font = input; preserveAspect; startGroup] Plot[wave1 + wave2, {t, 0, 0.1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 9.52381 0.308632 0.147343 [ [(0.02)] .21429 .30863 0 2 Msboxa [(0.04)] .40476 .30863 0 2 Msboxa [(0.06)] .59524 .30863 0 2 Msboxa [(0.08)] .78571 .30863 0 2 Msboxa [(0.1)] .97619 .30863 0 2 Msboxa [(-2)] .01131 .01395 1 0 Msboxa [(-1)] .01131 .16129 1 0 Msboxa [(1)] .01131 .45598 1 0 Msboxa [(2)] .01131 .60332 1 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L .21808 .01544 L .21823 .01509 L .21838 .01903 L .21853 .02719 L .21868 .03946 L .21883 .05566 L .22003 .2879 L .22063 .42616 L .22093 .48691 L .22123 .53723 L .22137 .55754 L .22152 .57421 L .22167 .58699 L .22182 .59571 L .22197 .60023 L .22212 .60049 L .22227 .5965 L .22242 .58831 L .22257 .57604 L .22272 .55988 L .22302 .5169 L .22362 .39802 L .22422 .25906 L .22452 .19204 L .22481 .13192 L .22511 .08217 L .22526 .06212 L .22541 .04569 L Mistroke .22556 .03311 L .22571 .02456 L .22586 .02016 L .22601 .01998 L .22616 .02401 L .22631 .0322 L .22646 .04441 L .22661 .06046 L .22691 .10309 L .22721 .15759 L .22781 .28889 L .2284 .42413 L .2287 .48338 L .229 .53234 L .22915 .55205 L .2293 .56819 L .22945 .58052 L .2296 .58887 L .22975 .59312 L .2299 .59321 L .23005 .58916 L .2302 .58102 L .2305 .55304 L .23065 .53361 L .2308 .51093 L .23199 .25997 L .23259 .1369 L .23289 .08892 L .23304 .06963 L .23319 .05386 L .23334 .04183 L .23349 .03372 L .23364 .02963 L .23379 .02962 L .23394 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.24994 .09856 L .25024 .13545 L .25054 .18215 L Mistroke .25114 .29345 L .25173 .40672 L .25203 .45584 L .25233 .49608 L .25248 .51214 L .25263 .52517 L .25278 .535 L .25293 .5415 L .25308 .54458 L .25323 .54421 L .25338 .5404 L .25353 .53324 L .25368 .52283 L .25383 .50933 L .25413 .47397 L .25473 .37798 L .25532 .26769 L .25562 .21517 L .25592 .16848 L .25622 .1303 L .25637 .1151 L .25652 .10278 L .25667 .09351 L .25682 .08742 L .25697 .08458 L .25712 .08501 L .25727 .08871 L .25742 .09559 L .25757 .10556 L .25772 .11844 L .25802 .15212 L .25832 .19458 L .25951 .3974 L .25981 .44146 L .26011 .47742 L .26026 .49171 L .26041 .50327 L .26056 .51194 L .26071 .51761 L .2631 .27185 L .2634 .22528 L .2637 .18403 L .264 .15042 L .26415 .1371 L .2643 .12634 L .26445 .1183 L .2646 .11308 L .26475 .11074 L .2649 .11129 L .26505 .11472 L Mistroke .2652 .12096 L .26534 .1299 L .26549 .14139 L .26669 .2974 L .26729 .38653 L .26759 .42481 L .26789 .45592 L .26804 .46823 L .26819 .47814 L .26834 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.28359 .41364 L .28374 .41982 L .28389 .42432 L .28404 .42708 L .28419 .42809 L .28434 .42734 L .28449 .42488 L .28464 .42074 L .28479 .41503 L .28494 .40782 L .28524 .38947 L .28583 .34144 L .28643 .2883 L .28673 .2637 L .28703 .24231 L .28718 .23319 L .28733 .22529 L .28748 .2187 L .28763 .21352 L .28778 .20978 L .28793 .20755 L .28808 .20681 L .28823 .20757 L .28838 .2098 L .28853 .21344 L .28883 .22466 L .28912 .24042 L .28942 .25968 L Mistroke .29002 .30375 L .29062 .3464 L .29092 .36411 L .29107 .37161 L .29122 .37807 L .29137 .38342 L .29152 .3876 L .29167 .39055 L .29182 .39226 L .29197 .39272 L .29212 .39195 L .29226 .38998 L .29241 .38684 L .29271 .3774 L .29301 .3643 L .29361 .33087 L .29421 .29485 L .29451 .27853 L .29481 .26458 L .29496 .25873 L .29511 .25373 L .29526 .24963 L .29541 .24649 L .29556 .24432 L .2957 .24314 L .29585 .24294 L .296 .24372 L .29615 .24543 L .2963 .24804 L .2966 .25567 L .2969 .26604 L .2972 .27841 L .2978 .30581 L .2981 .31916 L .2984 .33123 L .2987 .34137 L .29885 .34555 L .29899 .34908 L .29914 .35191 L .29929 .35402 L .29944 .3554 L .29959 .35605 L .29974 .35597 L .29989 .3552 L .30004 .35377 L .30019 .3517 L .30049 .34591 L .30079 .33832 L .30139 .32006 L .30199 .30176 L Mistroke .30229 .29399 L .30258 .28769 L .30273 .28519 L .30288 .28316 L .30303 .28161 L .30318 .28054 L .30333 .27995 L .30348 .27983 L .30363 .28016 L .30378 .28092 L .30393 .28207 L .30408 .28358 L .30438 .28749 L .30498 .2975 L .30558 .30777 L .30587 .3121 L .30602 .31395 L .30617 .31556 L .30632 .31691 L .30647 .31798 L .30662 .31879 L .30677 .31933 L .30692 .31961 L .30707 .31963 L .30722 .31943 L .30737 .31903 L .30767 .3177 L .30797 .3159 L .30827 .3139 L .30857 .31196 L .30872 .3111 L .30887 .31034 L .30902 .3097 L .30916 .30921 L .30931 .30889 L .30946 .30873 L .30961 .30875 L .30976 .30894 L .30991 .3093 L .31006 .30982 L .31036 .31126 L .31096 .31511 L .31126 .31702 L .31141 .31786 L .31156 .31857 L .31171 .31912 L .31186 .31949 L .31201 .31965 L .31216 .31957 L .31231 .31924 L Mistroke .31246 .31865 L .3126 .31779 L .31275 .31666 L .31305 .3136 L .31335 .30959 L .31395 .29964 L .31455 .28932 L .31485 .28501 L .315 .28325 L .31515 .28181 L .3153 .28074 L .31545 .28006 L .3156 .27982 L .31575 .28003 L .31589 .28071 L .31604 .28188 L .31619 .28353 L .31634 .28566 L .31664 .29127 L .31694 .2985 L .31754 .31625 L .31814 .33491 L .31844 .34306 L .31859 .34658 L .31874 .34964 L .31889 .35216 L .31904 .3541 L .31919 .35541 L .31933 .35604 L .31948 .35597 L .31963 .35518 L .31978 .35365 L .31993 .3514 L .32023 .34477 L .32053 .33555 L .32173 .28373 L .32203 .27079 L .32233 .25953 L .32248 .25478 L .32263 .25072 L .32277 .24745 L .32292 .24502 L .32307 .24349 L .32322 .2429 L .32337 .24329 L .32352 .24467 L .32367 .24704 L .32382 .25038 L .32412 .25984 L .32442 .27262 L Mistroke .32472 .28809 L .32532 .32359 L .32592 .35824 L .32621 .37255 L .32636 .37852 L .32651 .38355 L .32666 .38756 L .32681 .39046 L .32696 .3922 L .32711 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L .37422 .54089 L .37437 .52591 L Mistroke .37467 .48658 L .37497 .43666 L .37557 .31696 L .37617 .19411 L .37647 .1405 L .37676 .09644 L .37691 .07884 L .37706 .06455 L .37721 .05379 L .37736 .04674 L .37751 .04349 L .37766 .04412 L .37781 .04862 L .37796 .05693 L .37916 .23644 L .37976 .36528 L .38006 .42707 L .38035 .48221 L .38065 .52748 L .3808 .54554 L .38095 .56018 L .3811 .57117 L .38125 .57834 L .3814 .5816 L .38155 .58087 L .3817 .57616 L .38185 .56753 L .382 .55511 L .38215 .53906 L .38245 .49705 L .38275 .44389 L .38335 .31689 L .38394 .18708 L .38424 .13062 L .38454 .08435 L .38469 .06591 L .38484 .05099 L .38499 .03981 L .38514 .03254 L .38529 .02929 L .38544 .03012 L .38559 .03502 L .38574 .04393 L .38589 .05672 L .38604 .07322 L .38634 .11632 L .38693 .23341 L .38753 .36853 L .38783 .43313 L .38813 .49067 L Mistroke .38843 .53777 L .38858 .5565 L .38873 .57165 L .38888 .58297 L .38903 .5903 L .38918 .59353 L .38933 .5926 L .38948 .58752 L .38963 .57837 L .38978 .56526 L .38993 .54839 L .39112 .31665 L .39172 .18201 L .39202 .12363 L .39232 .07591 L .39247 .05695 L .39262 .04165 L .39277 .03023 L .39292 .02287 L .39307 .01968 L .39322 .0207 L .39337 .02594 L .39352 .0353 L .39366 .04867 L .39381 .06585 L .39411 .1106 L .39471 .23167 L .39531 .37083 L .39561 .43719 L .39591 .49616 L .39621 .54431 L .39636 .56342 L .39651 .57881 L .39666 .59028 L .39681 .59764 L .39696 .60078 L .3971 .59966 L .39725 .5943 L .3974 .58475 L .39755 .57118 L .3977 .55376 L .398 .50847 L .3983 .45149 L .3989 .31624 L .3995 .17899 L .3998 .11966 L .4001 .07128 L .40025 .05212 L .40039 .03669 L .40054 .02523 L Mistroke .40069 .0179 L .40084 .01482 L .40099 .01602 L .40114 .0215 L .40129 .03118 L .40159 .06249 L .40174 .08368 L .40189 .10815 L .40668 .31569 L .40727 .17809 L .40757 .11878 L .40787 .07055 L .40802 .05149 L .40817 .03619 L .40832 .02488 L .40847 .01771 L .40862 .01478 L .40877 .01615 L .40892 .02179 L .40907 .03162 L .40922 .04548 L .40937 .06319 L .40967 .10901 L .41027 .23203 L .41086 .37241 L .41116 .43898 L .41146 .49791 L .41176 .54577 L .41191 .56466 L .41206 .5798 L .41221 .59097 L .41236 .59802 L .41251 .60085 L .41266 .5994 L .41281 .59372 L .41296 .58388 L .41311 .57003 L .41326 .55238 L .41356 .50676 L .41385 .44969 L .41445 .31503 L .41505 .17934 L .41535 .12104 L .41565 .07374 L .4158 .0551 L .41595 .04018 L .4161 .02919 L .41625 .0223 L .4164 .01958 L .41655 .02109 L Mistroke .4167 .0268 L .41685 .03661 L .417 .05038 L .41715 .06791 L .41744 .11314 L .41804 .2341 L .41864 .37161 L .41894 .43665 L .41924 .4941 L .41954 .54064 L .41969 .55895 L .41984 .57359 L .41999 .58434 L .42014 .59106 L .42029 .59366 L .42044 .59209 L .42059 .58639 L .42073 .57665 L .42088 .56301 L .42103 .54568 L .42133 .50102 L .42163 .44532 L .42223 .31429 L .42283 .18275 L .42313 .12639 L .42343 .0808 L .42358 .06289 L .42373 .04859 L .42388 .03811 L .42402 .0316 L .42417 .02913 L .42432 .03075 L .42447 .03642 L .42462 .04606 L .42477 .05951 L .42492 .07657 L .42522 .12046 L .42582 .23737 L .42642 .36975 L .42672 .43219 L .42702 .48722 L .42732 .53167 L .42746 .54911 L .42761 .56301 L .42776 .57317 L .42791 .57946 L .42806 .58178 L .42821 .58012 L .42836 .5745 L .42851 .56502 L Mistroke .42866 .55182 L .42881 .5351 L .42911 .49215 L .42941 .43874 L .43001 .3135 L .43061 .18826 L .4309 .13477 L .43105 .11174 L .4312 .09163 L .43135 .07474 L .4315 .06129 L .43165 .05149 L .4318 .04546 L .43195 .04328 L .4321 .04498 L .43225 .05052 L .4324 .05981 L .43255 .07271 L .4327 .08902 L .433 .13084 L .43419 .36684 L .43449 .42565 L .43479 .47736 L .43494 .49961 L .43509 .51901 L .43524 .5353 L .43539 .54823 L .43554 .55764 L .43569 .5634 L .43584 .56542 L .43599 .5637 L .43614 .55826 L .43629 .5492 L .43644 .53666 L .43659 .52083 L .43689 .48031 L .43719 .43007 L .43778 .31269 L .43838 .1958 L .43868 .14607 L .43883 .12469 L .43898 .10607 L .43913 .09046 L .43928 .07809 L .43943 .06911 L .43958 .06365 L .43973 .06179 L .43988 .06353 L .44003 .06884 L .44018 .07763 L Mistroke .44033 .08976 L .44048 .10504 L .44078 .14409 L .44137 .24718 L .44197 .3629 L .44227 .41712 L .44257 .46468 L .44272 .48509 L .44287 .50285 L .44302 .51772 L .44317 .5295 L .44332 .53801 L .44347 .54314 L .44362 .54484 L .44377 .54309 L .44392 .53794 L .44407 .52945 L .44422 .51779 L .44436 .50312 L .44466 .46571 L .44496 .41948 L .44556 .3119 L .44616 .20527 L .44646 .16009 L .44661 .14073 L .44676 .12389 L .44691 .10981 L .44706 .0987 L .44721 .09069 L .44736 .08589 L .44751 .08436 L .44765 .0861 L .4478 .09108 L .44795 .09922 L .44825 .12437 L .44855 .15998 L .44975 .35799 L .45005 .40674 L .45035 .44937 L .4505 .46761 L .45065 .48345 L .4508 .49667 L .45095 .50709 L .45109 .51458 L .45124 .51902 L .45139 .52038 L .45154 .51865 L .45169 .51386 L .45184 .50611 L .45214 .48226 L Mistroke .45244 .44859 L .45274 .40714 L .45334 .31114 L .45394 .21653 L .45424 .17663 L .45439 .15958 L .45453 .1448 L .45468 .13248 L .45483 .12279 L .45498 .11587 L .45513 .1118 L .45528 .11061 L .45543 .11232 L .45558 .11688 L .45573 .12422 L .45603 .14667 L .45633 .17825 L .45693 .2607 L .45753 .35217 L .45782 .39466 L .45812 .43167 L .45827 .44745 L .45842 .46112 L .45857 .47249 L .45872 .4814 L .45887 .48774 L .45902 .49143 L .45917 .49244 L .45932 .49076 L .45947 .48644 L .45962 .47956 L .45992 .45861 L .46022 .42925 L .46052 .39328 L .46112 .31045 L .46171 .22939 L .46201 .19542 L .46216 .18096 L .46231 .16846 L .46246 .15809 L .46261 .14999 L .46276 .14425 L .46291 .14096 L .46306 .14013 L .46321 .14176 L .46336 .14581 L .46351 .15221 L .46381 .17157 L .46411 .19858 L .4653 .34553 L Mistroke .4659 .41186 L .4662 .43622 L .46635 .44555 L .4665 .45282 L .46665 .45793 L .4668 .46082 L .46695 .46147 L .4671 .45989 L .46725 .45613 L .4674 .45024 L .4677 .43256 L .46799 .40801 L .46829 .37814 L .46889 .30986 L .46949 .24367 L .46979 .21616 L .46994 .20452 L .47009 .1945 L .47024 .18624 L .47039 .17983 L .47054 .17537 L .47069 .17289 L .47084 .17241 L .47099 .17393 L .47114 .17739 L .47128 .18273 L .47158 .19865 L .47188 .22063 L .47248 .27693 L .47308 .33816 L .47338 .36616 L .47368 .39027 L .47383 .40043 L .47398 .40915 L .47413 .41631 L .47428 .42183 L .47443 .42563 L .47458 .42769 L .47472 .428 L .47487 .42656 L .47502 .42342 L .47517 .41864 L .47547 .40455 L .47577 .38525 L .47607 .36196 L .47727 .25913 L .47757 .23852 L .47772 .22988 L .47787 .22249 L .47802 .21646 L Mistroke .47816 .21185 L .47831 .20871 L .47846 .20707 L .47861 .20694 L .47876 .20829 L .47891 .21109 L .47906 .21528 L .47936 .22747 L .47966 .24403 L .48026 .2857 L .48086 .33017 L .48116 .3502 L .48145 .36723 L .4816 .37432 L .48175 .38035 L .4819 .38524 L .48205 .38893 L .4822 .39139 L .48235 .3926 L .4825 .39256 L .48265 .3913 L .4828 .38886 L .48295 .38528 L .48325 .37504 L .48355 .36132 L .48385 .34504 L .48445 .30901 L .48504 .27553 L .48534 .26214 L .48549 .25662 L .48564 .25198 L .48579 .24826 L .48594 .2455 L .48609 .24373 L .48624 .24294 L .48639 .24314 L .48654 .24429 L .48669 .24637 L .48684 .24931 L .48714 .25755 L .48744 .26838 L .48804 .29472 L .48863 .3217 L .48893 .33343 L .48923 .34312 L .48938 .34705 L .48953 .35029 L .48968 .35284 L .48983 .35466 L .48998 .35574 L Mistroke .49013 .3561 L .49028 .35575 L .49043 .35471 L .49162 .32764 L .49222 .30879 L .49252 .30011 L .49282 .2926 L .49297 .28941 L .49312 .28664 L .49327 .28432 L .49342 .28248 L .49357 .28112 L .49372 .28024 L .49387 .27985 L .49402 .27991 L .49417 .28042 L .49432 .28134 L .49462 .28427 L .49492 .28839 L .49521 .29329 L .49581 .30383 L .49611 .3087 L .49641 .31287 L .49656 .31462 L .49671 .31613 L .49686 .31737 L .49701 .31834 L .49716 .31904 L .49731 .31947 L .49746 .31965 L .49761 .31958 L .4988 .3135 L .4991 .31161 L .49925 .31078 L .4994 .31007 L .49955 .30949 L .4997 .30906 L .49985 .3088 L .5 .30871 L .50015 .3088 L .5003 .30906 L .50045 .30949 L .5006 .31007 L .5009 .31161 L .5012 .3135 L .5015 .31551 L .50179 .31737 L .50194 .31816 L .50209 .31881 L .50224 .31929 L Mistroke .50239 .31958 L .50254 .31965 L .50269 .31947 L .50284 .31904 L .50299 .31834 L .50314 .31737 L .50329 .31613 L .50359 .31287 L .50419 .30383 L .50479 .29329 L .50508 .28839 L .50538 .28427 L .50553 .28263 L .50568 .28134 L .50583 .28042 L .50598 .27991 L .50613 .27985 L .50628 .28024 L .50643 .28112 L .50658 .28248 L .50688 .28664 L .50703 .28941 L .50718 .2926 L .50838 .32764 L .50867 .33664 L .50897 .34452 L .50912 .34786 L .50927 .35072 L .50942 .35301 L .50957 .35471 L .50972 .35575 L .50987 .3561 L .51002 .35574 L .51017 .35466 L .51032 .35284 L .51047 .35029 L .51077 .34312 L .51107 .33343 L .51137 .3217 L .51196 .29472 L .51256 .26838 L .51286 .25755 L .51301 .25307 L .51316 .24931 L .51331 .24637 L .51346 .24429 L .51361 .24314 L .51376 .24294 L .51391 .24373 L .51406 .2455 L Mistroke .51421 .24826 L .51436 .25198 L .51466 .26214 L .51496 .27553 L .51555 .30901 L .51615 .34504 L .51645 .36132 L .51675 .37504 L .5169 .38065 L .51705 .38528 L .5172 .38886 L .51735 .3913 L .5175 .39256 L .51765 .3926 L .5178 .39139 L .51795 .38893 L .5181 .38524 L .51825 .38035 L .51855 .36723 L .51914 .33017 L .51974 .2857 L .52004 .26385 L .52034 .24403 L .52064 .22747 L .52079 .22077 L .52094 .21528 L .52109 .21109 L .52124 .20829 L .52139 .20694 L .52154 .20707 L .52169 .20871 L .52184 .21185 L .52198 .21646 L .52213 .22249 L .52243 .23852 L .52273 .25913 L .52333 .30937 L .52393 .36196 L .52423 .38525 L .52453 .40455 L .52468 .41232 L .52483 .41864 L .52498 .42342 L .52513 .42656 L .52528 .428 L .52542 .42769 L .52557 .42563 L .52572 .42183 L .52587 .41631 L .52602 .40915 L Mistroke .52632 .39027 L .52752 .27693 L .52782 .24727 L .52812 .22063 L .52842 .19865 L .52857 .18986 L .52872 .18273 L .52886 .17739 L .52901 .17393 L .52916 .17241 L .52931 .17289 L .52946 .17537 L .52961 .17983 L .52976 .18624 L .52991 .1945 L .53021 .21616 L .53051 .24367 L .53111 .30986 L .53171 .37814 L .53201 .40801 L .5323 .43256 L .53245 .44234 L .5326 .45024 L .53275 .45613 L .5329 .45989 L .53305 .46147 L .5332 .46082 L .53335 .45793 L .5335 .45282 L .53365 .44555 L .5338 .43622 L .5341 .41186 L .5347 .34553 L .5353 .26854 L .53559 .23157 L .53589 .19858 L .53619 .17157 L .53634 .16084 L .53649 .15221 L .53664 .14581 L .53679 .14176 L .53694 .14013 L .53709 .14096 L .53724 .14425 L .53739 .14999 L .53754 .15809 L .53769 .16846 L .53799 .19542 L .53829 .22939 L .53948 .39328 L Mistroke .53978 .42925 L .54008 .45861 L .54023 .47023 L .54038 .47956 L .54053 .48644 L .54068 .49076 L .54083 .49244 L .54098 .49143 L .54113 .48774 L .54128 .4814 L .54143 .47249 L .54158 .46112 L .54188 .43167 L .54247 .35217 L .54307 .2607 L .54337 .21704 L .54367 .17825 L .54397 .14667 L .54412 .1342 L .54427 .12422 L .54442 .11688 L .54457 .11232 L .54472 .11061 L .54487 .1118 L .54502 .11587 L .54517 .12279 L .54532 .13248 L .54547 .1448 L .54576 .17663 L .54606 .21653 L .54666 .31114 L .54726 .40714 L .54756 .44859 L .54786 .48226 L .54801 .49552 L .54816 .50611 L .54831 .51386 L .54846 .51865 L .54861 .52038 L .54876 .51902 L .54891 .51458 L .54905 .50709 L .5492 .49667 L .54935 .48345 L .54965 .44937 L .55025 .35799 L .55085 .25354 L .55115 .20392 L .55145 .15998 L .55175 .12437 L Mistroke .5519 .11037 L .55205 .09922 L .5522 .09108 L .55235 .0861 L .55249 .08436 L .55264 .08589 L .55279 .09069 L .55294 .0987 L .55309 .10981 L .55324 .12389 L .55354 .16009 L .55384 .20527 L .55504 .41948 L .55534 .46571 L .55549 .48567 L .55564 .50312 L .55578 .51779 L .55593 .52945 L .55608 .53794 L .55623 .54309 L .55638 .54484 L .55653 .54314 L .55668 .53801 L .55683 .5295 L .55698 .51772 L .55713 .50285 L .55743 .46468 L .55863 .24718 L .55893 .19243 L .55922 .14409 L .55937 .12324 L .55952 .10504 L .55967 .08976 L .55982 .07763 L .55997 .06884 L .56012 .06353 L .56027 .06179 L .56042 .06365 L .56057 .06911 L .56072 .07809 L .56087 .09046 L .56102 .10607 L .56132 .14607 L .56162 .1958 L .56222 .31269 L .56281 .43007 L .56311 .48031 L .56326 .50195 L .56341 .52083 L .56356 .53666 L Mistroke .56371 .5492 L .56386 .55826 L .56401 .5637 L .56416 .56542 L .56431 .5634 L .56446 .55764 L .56461 .54823 L .56581 .36684 L .5664 .24176 L .5667 .18277 L .567 .13084 L .56715 .10849 L .5673 .08902 L .56745 .07271 L .5676 .05981 L .56775 .05052 L .5679 .04498 L .56805 .04328 L .5682 .04546 L .56835 .05149 L .5685 .06129 L .56865 .07474 L .5688 .09163 L .5691 .13477 L .56939 .18826 L .56999 .3135 L .57059 .43874 L .57089 .49215 L .57104 .51511 L .57119 .5351 L .57134 .55182 L .57149 .56502 L .57164 .5745 L .57179 .58012 L .57194 .58178 L .57209 .57946 L .57224 .57317 L .57239 .56301 L .57268 .53167 L .57298 .48722 L .57418 .23737 L .57448 .17513 L .57478 .12046 L .57493 .09699 L .57508 .07657 L .57523 .05951 L .57538 .04606 L .57553 .03642 L .57568 .03075 L .57583 .02913 L Mistroke .57598 .0316 L .57612 .03811 L .57627 .04859 L .57657 .0808 L .57687 .12639 L .57717 .18275 L .57777 .31429 L .57837 .44532 L .57867 .50102 L .57897 .54568 L .57912 .56301 L .57927 .57665 L .57941 .58639 L .57956 .59209 L .57971 .59366 L .57986 .59106 L .58001 .58434 L .58016 .57359 L .58046 .54064 L .58076 .4941 L .58136 .37161 L .58196 .2341 L .58226 .16964 L .58256 .11314 L .58271 .08893 L .58285 .06791 L .583 .05038 L .58315 .03661 L .5833 .0268 L .58345 .02109 L .5836 .01958 L .58375 .0223 L .5839 .02919 L .58405 .04018 L .58435 .07374 L .58465 .12104 L .58495 .17934 L .58555 .31503 L .58615 .44969 L .58644 .50676 L .58659 .53119 L .58674 .55238 L .58689 .57003 L .58704 .58388 L .58719 .59372 L .58734 .5994 L .58749 .60085 L .58764 .59802 L .58779 .59097 L .58794 .5798 L Mistroke .58824 .54577 L .58854 .49791 L .58973 .23203 L .59003 .16641 L .59033 .10901 L .59048 .08446 L .59063 .06319 L .59078 .04548 L .59093 .03162 L .59108 .02179 L .59123 .01615 L .59138 .01478 L .59153 .01771 L .59168 .02488 L .59183 .03619 L .59213 .07055 L .59243 .11878 L .59273 .17809 L .59332 .31569 L .59392 .45177 L .59422 .50926 L .59437 .53382 L .59452 .5551 L .59467 .57278 L .59482 .5866 L .59497 .59637 L .59512 .60194 L .59527 .60323 L .59542 .60022 L .59557 .59296 L .59572 .58155 L .59602 .547 L .59631 .4986 L .59691 .37214 L .59751 .23121 L .59781 .1655 L .59811 .10815 L .59826 .08368 L .59841 .06249 L .59856 .04491 L .59871 .03118 L .59886 .0215 L .59901 .01602 L .59916 .01482 L .59931 .0179 L .59946 .02523 L .59961 .03669 L .5999 .07128 L .6002 .11966 L .6005 .17899 L Mistroke .6011 .31624 L .6017 .45149 L .602 .50847 L .60215 .53276 L .6023 .55376 L .60245 .57118 L .6026 .58475 L .60275 .5943 L .6029 .59966 L .60304 .60078 L .60319 .59764 L .60334 .59028 L .60349 .57881 L .60379 .54431 L .60409 .49616 L .60529 .23167 L .60559 .16696 L .60589 .1106 L .60604 .08659 L .60619 .06585 L .60634 .04867 L .60648 .0353 L .60663 .02594 L .60678 .0207 L .60693 .01968 L .60708 .02287 L .60723 .03023 L .60738 .04165 L .60768 .07591 L .60798 .12363 L .60828 .18201 L .60888 .31665 L .60948 .44886 L .60978 .50438 L .60992 .528 L .61007 .54839 L .61022 .56526 L .61037 .57837 L .61052 .58752 L .61067 .5926 L .61082 .59353 L .61097 .5903 L .61112 .58297 L .61127 .57165 L .61157 .53777 L .61187 .49067 L .61247 .36853 L .61307 .23341 L .61336 .17076 L .61366 .11632 L Mistroke .61381 .09318 L .61396 .07322 L .61411 .05672 L .61426 .04393 L .61441 .03502 L .61456 .03012 L .61471 .02929 L .61486 .03254 L .61501 .03981 L .61516 .05099 L .61546 .08435 L .61576 .13062 L .61606 .18708 L .61665 .31689 L .61725 .44389 L .61755 .49705 L .6177 .51961 L .61785 .53906 L .618 .55511 L .61815 .56753 L .6183 .57616 L .61845 .58087 L .6186 .5816 L .61875 .57834 L .6189 .57117 L .61905 .56018 L .61935 .52748 L .61965 .48221 L .62084 .23644 L .62144 .12523 L .62174 .08448 L .62189 .06893 L .62204 .05693 L .62219 .04862 L .62234 .04412 L .62249 .04349 L .62264 .04674 L .62279 .05379 L .62294 .06455 L .62324 .09644 L .62353 .1405 L .62383 .19411 L .62443 .31696 L .62503 .43666 L .62533 .48658 L .62548 .50773 L .62563 .52591 L .62578 .54089 L .62593 .55243 L .62608 .56039 L Mistroke .62623 .56466 L .62638 .56518 L .62653 .56197 L .62682 .54459 L .62802 .36116 L .62862 .24071 L .62892 .1852 L .62922 .13721 L .62937 .1169 L .62952 .09945 L .62967 .08511 L .62982 .07408 L .62997 .0665 L .63011 .06247 L .63026 .06205 L .63041 .06523 L .63056 .07194 L .63071 .08209 L .63101 .11199 L .63131 .1531 L .63161 .20296 L .63221 .31681 L .63281 .42726 L .63311 .47314 L .63326 .49253 L .63341 .50916 L .63355 .52281 L .6337 .5333 L .63385 .54047 L .634 .54424 L .63415 .54456 L .6343 .54144 L .63445 .53494 L .6346 .52516 L .6349 .49646 L .6352 .45708 L .6364 .24617 L .6367 .19563 L .63699 .15206 L .63714 .13367 L .63729 .11792 L .63744 .105 L .63759 .09511 L .63774 .08838 L .63789 .08488 L .63804 .08466 L .63819 .08771 L .63834 .09397 L .63849 .10333 L .63879 .13071 L Mistroke .63909 .16818 L .63939 .21347 L .63999 .31645 L .64058 .41584 L .64088 .45694 L .64103 .47425 L .64118 .48907 L .64133 .50119 L .64148 .51045 L .64163 .51672 L .64178 .51994 L .64193 .52006 L .64208 .5171 L .64223 .51111 L .64238 .50221 L .64268 .47625 L .64298 .44085 L .64358 .35066 L .64417 .25275 L .64447 .208 L .64477 .16955 L .64492 .15338 L .64507 .13956 L .64522 .12828 L .64537 .11969 L .64552 .1139 L .64567 .11098 L .64582 .11095 L .64597 .11381 L .64612 .11949 L .64627 .1279 L .64657 .15231 L .64687 .1855 L .64716 .22546 L .64776 .31587 L .64836 .40257 L .64866 .43822 L .64881 .45319 L .64896 .46595 L .64911 .47635 L .64926 .48425 L .64941 .48953 L .64956 .49215 L .64971 .49209 L .64986 .48935 L .65001 .48399 L .65016 .47612 L .65045 .45336 L .65075 .42252 L .65135 .34448 L Mistroke .65195 .26036 L .65225 .22212 L .65255 .18941 L .6527 .17571 L .65285 .16405 L .653 .15457 L .65315 .14741 L .6533 .14264 L .65345 .14033 L .6536 .14049 L .65374 .14309 L .65389 .1481 L .65404 .1554 L .65434 .17641 L .65464 .20476 L .65494 .23872 L .65554 .31505 L .65614 .38766 L .65644 .4173 L .65659 .42968 L .65674 .44019 L .65689 .44871 L .65704 .45512 L .65718 .45935 L .65733 .46135 L .65748 .46111 L .65763 .45865 L .65778 .45402 L .65793 .44732 L .65823 .42816 L .65853 .40241 L .65913 .33783 L .65973 .26887 L .66003 .23777 L .66033 .21132 L .66047 .20031 L .66062 .19098 L .66077 .18345 L .66092 .17781 L .66107 .17414 L .66122 .17246 L .66137 .17278 L .66152 .17508 L .66167 .17931 L .66182 .18538 L .66212 .20261 L .66242 .22563 L .66272 .25301 L .66332 .314 L .66391 .37134 L Mistroke .66421 .3945 L .66436 .4041 L .66451 .41221 L .66466 .41872 L .66481 .42356 L .66496 .42667 L .66511 .42803 L .66526 .42763 L .66541 .4255 L .66556 .4217 L .66571 .41629 L .66601 .40107 L .66631 .38086 L .66691 .33082 L .6675 .27816 L .6678 .25469 L .6681 .23492 L .66825 .22677 L .6684 .21991 L .66855 .21444 L .6687 .21041 L .66885 .20787 L .669 .20684 L .66915 .2073 L .6693 .20924 L .66945 .21261 L .6696 .21733 L .66975 .22331 L .6699 .23047 L .67109 .31273 L .67139 .33437 L .67169 .35388 L .67199 .37019 L .67214 .37687 L .67229 .38245 L .67244 .38686 L .67259 .39006 L 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.94987 .10748 L .95017 .15509 L .95047 .21187 L .95106 .33951 L .95166 .46117 L .95196 .51075 L .95211 .53135 L .95226 .54875 L .95241 .56267 L .95256 .57291 L .95271 .57931 L .95286 .58177 L .95301 .58024 L .95316 .57474 L .95331 .56535 L .95346 .55218 L .95376 .51534 L .95406 .46631 L .95465 .34351 L .95525 .21179 L .95555 .15216 L .95585 .10145 L .956 .08039 L .95615 .06264 L .9563 .04845 L .95645 .03804 L .9566 .03156 L .95675 .02913 L .9569 .03078 L .95705 .03649 L .9572 .04618 L .95735 .05973 L .95765 .09756 L .95794 .14781 L .95824 .20759 L .95884 .34155 L Mistroke .95944 .46873 L .95974 .52037 L .95989 .54179 L .96004 .55983 L .96019 .57422 L .96034 .58476 L .96049 .59128 L .96064 .59368 L .96079 .59191 L .96094 .58601 L .96109 .57606 L .96123 .56218 L .96153 .52352 L .96183 .47226 L .96243 .34431 L .96303 .2076 L .96333 .14589 L .96363 .09353 L .96378 .07184 L .96393 .05359 L .96408 .03905 L .96423 .02843 L .96438 .0219 L .96453 .01955 L .96467 .02142 L .96482 .02748 L .96497 .03766 L .96512 .05181 L .96542 .09114 L .96572 .14321 L .96602 .20501 L .96662 .34308 L .96722 .47369 L .96752 .52655 L .96767 .54842 L .96782 .5668 L .96796 .58143 L .96811 .59208 L .96826 .5986 L .96841 .6009 L .96856 .59893 L .96871 .59272 L .96886 .58236 L .96901 .56799 L .96931 .52815 L .96961 .47549 L .97021 .34449 L .97081 .20503 L .97111 .14226 L .9714 .08912 L Mistroke .97155 .06716 L .9717 .04872 L .97185 .03407 L .972 .02343 L .97215 .01694 L .9723 .01472 L .97245 .01678 L .9726 .02311 L .97275 .03361 L .9729 .04812 L .9732 .08832 L .9735 .14136 L .9738 .20416 L .97619 .60332 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Play[wave1 + wave2, {t, 0, 1}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Sound["<<>>"] ;[o] -Sound- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 4) Use the graph produced in (3) to determine :[font = subsubsection; inactive; preserveAspect; startGroup] (a) the maximum beat amplitude and when the maxima occur :[font = text; inactive; preserveAspect] From sight, we can estimate the maximum amplitude as approximately equal to 2 and it occurs near t=0, 0.2, 0.4, 0.6, 0.8, and 1. To be sure, we should plot the graphs near our candidate maximum values. :[font = input; preserveAspect; startGroup] Plot[wave1 + wave2, {t, 0, 0.01}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 95.238095 0.308714 0.147303 [ [(0.002)] .21429 .30871 0 2 Msboxa [(0.004)] .40476 .30871 0 2 Msboxa [(0.006)] .59524 .30871 0 2 Msboxa [(0.008)] .78571 .30871 0 2 Msboxa [(0.01)] .97619 .30871 0 2 Msboxa [(-2)] .01131 .01411 1 0 Msboxa [(-1)] .01131 .16141 1 0 Msboxa [(1)] .01131 .45602 1 0 Msboxa [(2)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .30871 m .21429 .31496 L s P [(0.002)] .21429 .30871 0 2 Mshowa p .002 w .40476 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.02756 .3971 L s P p .001 w .02381 .42656 m .02756 .42656 L s P p .001 w .02381 .48548 m .02756 .48548 L s P p .001 w .02381 .51494 m .02756 .51494 L s P p .001 w .02381 .5444 m .02756 .5444 L s P p .001 w .02381 .57386 m .02756 .57386 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .60332 m .02505 .6033 L .02629 .60326 L .02753 .60319 L .02877 .60308 L .03125 .60279 L .03373 .60237 L .03621 .60184 L .03869 .60119 L .04365 .59954 L .04861 .59742 L .05357 .59483 L .06349 .58829 L .08333 .56987 L .10317 .54475 L .14286 .47713 L .18254 .39236 L .22222 .29908 L .2619 .20682 L .30159 .12501 L .32143 .09069 L .34127 .06198 L .35119 .04996 L .36111 .0396 L .37103 .03098 L .38095 .02415 L .38591 .02141 L .39087 .01914 L .39583 .01734 L .39831 .01661 L .40079 .016 L .40327 .01551 L .40451 .01531 L .40575 .01513 L .40699 .01499 L .40823 .01488 L .40947 .01479 L .41071 .01474 L .41195 .01472 L .41319 .01472 L .41443 .01475 L .41567 .01482 L .41691 .01491 L .41815 .01503 L .42063 .01537 L .42312 .01582 L .4256 .01639 L .43056 .01788 L .43552 .01984 L .44048 .02226 L Mistroke .4504 .02849 L .46032 .03651 L .48016 .05773 L .5 .08538 L .53968 .15698 L .57937 .24397 L .61905 .33745 L .65873 .42788 L .69841 .50604 L .71825 .53796 L .7381 .56397 L .74802 .57455 L .75794 .58342 L .76786 .59051 L .77282 .59338 L .77778 .59579 L .78274 .59774 L .78522 .59854 L .7877 .59923 L .79018 .5998 L .79266 .60025 L .79514 .60058 L .79638 .6007 L .79762 .6008 L .79886 .60086 L .8001 .60089 L .80134 .6009 L .80258 .60088 L .80382 .60082 L .80506 .60074 L .8063 .60063 L .80754 .60049 L .81002 .60011 L .8125 .59963 L .81746 .5983 L .82242 .59651 L .82738 .59425 L .8373 .58838 L .84722 .5807 L .85714 .57128 L .87698 .54747 L .89683 .51756 L .93651 .44264 L .97619 .35423 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] wave1 + wave2 /. t -> 0 :[font = output; output; inactive; preserveAspect; endGroup] 2 ;[o] 2 :[font = input; preserveAspect; startGroup] Plot[wave1 + wave2, {t, 0.15, 0.25}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -1.404762 9.52381 0.308632 0.147343 [ [(0.16)] .11905 .30863 0 2 Msboxa [(0.18)] .30952 .30863 0 2 Msboxa [(0.22)] .69048 .30863 0 2 Msboxa [(0.24)] .88095 .30863 0 2 Msboxa [(-2)] .4875 .01395 1 0 Msboxa [(-1)] .4875 .16129 1 0 Msboxa [(1)] .4875 .45598 1 0 Msboxa [(2)] .4875 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .11905 .30863 m .11905 .31488 L s P [(0.16)] .11905 .30863 0 2 Mshowa p .002 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.04286 .30863 m .04286 .31238 L s P p .001 w .00476 .30863 m .00476 .31238 L s P p .001 w .91905 .30863 m .91905 .31238 L s P p .001 w .95714 .30863 m .95714 .31238 L s P p .001 w .99524 .30863 m .99524 .31238 L s P p .002 w 0 .30863 m 1 .30863 L s P p .002 w .5 .01395 m .50625 .01395 L s P [(-2)] .4875 .01395 1 0 Mshowa p .002 w .5 .16129 m .50625 .16129 L s P [(-1)] .4875 .16129 1 0 Mshowa p .002 w .5 .45598 m .50625 .45598 L s P [(1)] .4875 .45598 1 0 Mshowa p .002 w .5 .60332 m .50625 .60332 L s P [(2)] .4875 .60332 1 0 Mshowa p .001 w .5 .04341 m .50375 .04341 L s P p .001 w .5 .07288 m .50375 .07288 L s P p .001 w .5 .10235 m .50375 .10235 L s P p .001 w .5 .13182 m .50375 .13182 L s P p .001 w .5 .19076 m .50375 .19076 L s P p .001 w .5 .22023 m .50375 .22023 L s P p .001 w .5 .24969 m .50375 .24969 L s P p .001 w .5 .27916 m .50375 .27916 L s P p .001 w .5 .3381 m .50375 .3381 L s P p .001 w .5 .36757 m .50375 .36757 L s P p .001 w .5 .39704 m .50375 .39704 L s P p .001 w .5 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Cos[48.*Pi] + Cos[50.*Pi] ;[o] Cos[48. Pi] + Cos[50. Pi] :[font = input; preserveAspect; startGroup] N[%] :[font = output; output; inactive; preserveAspect; endGroup] 2. ;[o] 2. :[font = text; inactive; preserveAspect; endGroup] Due to the periodicity of the waves, if 0 and 0.2 are maxima, so are the remaining values 0.4, 0.6, 0.8, and 1. :[font = subsubsection; inactive; preserveAspect] (b) based on the result from (a), the beat period is 2/10 seconds. Remark: The period is the length of time it takes before the wave starts over again. It is easy to confused her because the beats do not look like function if graphed over the unit interval because the oscillations are so frequent. :[font = subsubsection; inactive; preserveAspect; endGroup] (c) the beat frequency is 5 beats/second. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 5a) the 150 Hz tone :[font = input; preserveAspect; startGroup] wave1 = Cos[150 2 Pi t] :[font = output; output; inactive; preserveAspect; endGroup] Cos[300*Pi*t] ;[o] Cos[300 Pi t] :[font = input; preserveAspect; startGroup] Plot[wave1, {t, 0, 0.1}, PlotPoints -> 50] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 9.52381 0.309017 0.294302 [ [(0.02)] .21429 .30902 0 2 Msboxa [(0.04)] .40476 .30902 0 2 Msboxa [(0.06)] .59524 .30902 0 2 Msboxa [(0.08)] .78571 .30902 0 2 Msboxa [(0.1)] .97619 .30902 0 2 Msboxa [(-1)] .01131 .01472 1 0 Msboxa [(-0.5)] .01131 .16187 1 0 Msboxa [(0.5)] .01131 .45617 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .30902 m .21429 .31527 L s P [(0.02)] .21429 .30902 0 2 Mshowa p .002 w .40476 .30902 m .40476 .31527 L s P [(0.04)] .40476 .30902 0 2 Mshowa p .002 w .59524 .30902 m .59524 .31527 L s P [(0.06)] .59524 .30902 0 2 Mshowa p .002 w .78571 .30902 m .78571 .31527 L s P [(0.08)] .78571 .30902 0 2 Mshowa p .002 w .97619 .30902 m .97619 .31527 L s P [(0.1)] .97619 .30902 0 2 Mshowa p .001 w .0619 .30902 m .0619 .31277 L s P p .001 w .1 .30902 m .1 .31277 L s P p .001 w .1381 .30902 m .1381 .31277 L s P p .001 w .17619 .30902 m .17619 .31277 L s P p .001 w .25238 .30902 m .25238 .31277 L s P p .001 w .29048 .30902 m .29048 .31277 L s P p .001 w 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.91424 .59991 L .91484 .59671 L .91545 .59246 L .91667 .58092 L .91788 .56545 L .92031 .52368 L .92274 .46957 L .9276 .33728 L .93246 .19858 L .93489 .13679 L .93732 .08492 L .93853 .06366 L .93975 .04594 L .94096 .03202 L .94157 .02655 L .94218 .02209 L .94278 .01868 L .94339 .01631 L .944 .015 L .94461 .01475 L .94521 .01557 L .94582 .01744 L .94643 .02037 L .94704 .02434 L Mistroke .95675 .20738 L .97619 .60332 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Play[wave1, {t, 0, 1}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Sound["<<>>"] ;[o] -Sound- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 5b) the 156 Hz tone :[font = input; preserveAspect; startGroup] wave2 = Cos[156 2 Pi t] :[font = output; output; inactive; preserveAspect; endGroup] Cos[312*Pi*t] ;[o] Cos[312 Pi t] :[font = input; preserveAspect; startGroup] Plot[wave2, {t, 0, 0.1}, PlotPoints -> 50] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 9.52381 0.309016 0.294303 [ [(0.02)] .21429 .30902 0 2 Msboxa [(0.04)] .40476 .30902 0 2 Msboxa [(0.06)] .59524 .30902 0 2 Msboxa [(0.08)] .78571 .30902 0 2 Msboxa [(0.1)] .97619 .30902 0 2 Msboxa [(-1)] .01131 .01471 1 0 Msboxa [(-0.5)] .01131 .16186 1 0 Msboxa [(0.5)] .01131 .45617 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .30902 m .21429 .31527 L s P [(0.02)] .21429 .30902 0 2 Mshowa p .002 w .40476 .30902 m .40476 .31527 L s P [(0.04)] .40476 .30902 0 2 Mshowa p .002 w .59524 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preserveAspect; startGroup] Plot[wave1 + wave2, {t, 0, 0.1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 9.52381 0.308729 0.147295 [ [(0.02)] .21429 .30873 0 2 Msboxa [(0.04)] .40476 .30873 0 2 Msboxa [(0.06)] .59524 .30873 0 2 Msboxa [(0.08)] .78571 .30873 0 2 Msboxa [(0.1)] .97619 .30873 0 2 Msboxa [(-2)] .01131 .01414 1 0 Msboxa [(-1)] .01131 .16143 1 0 Msboxa [(1)] .01131 .45602 1 0 Msboxa [(2)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .30873 m .21429 .31498 L s P [(0.02)] .21429 .30873 0 2 Mshowa p .002 w .40476 .30873 m .40476 .31498 L s P [(0.04)] .40476 .30873 0 2 Mshowa p .002 w .59524 .30873 m .59524 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L .10023 .30576 L .10038 .30353 L .10053 .30167 L .10068 .3002 L .10083 .29914 L .10098 .29848 L .10113 .29821 L .10128 .2983 L .10143 .29872 L .10158 .29941 L .10188 .30141 L .10218 .30379 L .10248 .30607 L .10263 .30703 L .10278 .3078 L .10293 .30834 L .10307 .30865 L .10322 .30869 L .10337 .30847 L .10397 .30535 L .10457 .30067 L .10472 .2997 L .10487 .29892 L .10502 .29841 L .10517 .2982 L .10532 .29834 L .10547 .29887 L .10562 .2998 L .10577 .30114 L .10607 .30498 L .10637 .31017 L .10696 .32263 L .10726 .32854 L .10741 .33109 L .10756 .33323 L .10771 .33489 L .10786 .33599 L .10801 .33646 L .10816 .33626 L .10831 .33535 L .10846 .33371 L .10861 .33135 L .10876 .3283 L Mistroke .10906 .32029 L .10936 .31027 L .10995 .28776 L .11025 .27746 L .1104 .27306 L .11055 .26933 L .1107 .2664 L .11085 .26436 L .111 .2633 L .11115 .26329 L .1113 .26436 L .11145 .26653 L .1116 .26979 L .11175 .27408 L .11205 .2855 L .11235 .29993 L .11295 .33273 L .11324 .34804 L .11339 .35473 L .11354 .36054 L .11369 .3653 L .11384 .36887 L .11399 .37112 L .11414 .37198 L .11429 .37137 L .11444 .36929 L .11459 .36573 L .11474 .36074 L .11504 .34685 L .11534 .32867 L .11594 .28584 L .11624 .26509 L .11654 .24746 L .11668 .2404 L .11683 .23478 L .11698 .23074 L .11713 .22844 L .11728 .22794 L .11743 .22931 L .11758 .23255 L .11773 .23763 L .11833 .27397 L .11893 .32608 L .11923 .35245 L .11953 .37574 L .11968 .38549 L .11983 .39364 L .11997 .39997 L .12012 .4043 L .12027 .4065 L Mistroke .12042 .40648 L .12057 .4042 L .12072 .39968 L .12102 .38424 L .12117 .37361 L .12132 .36131 L .12192 .30113 L .12222 .2692 L .12252 .23995 L .12267 .22721 L .12282 .21617 L .12297 .20709 L .12312 .20022 L .12327 .19576 L .12341 .19382 L .12356 .19451 L .12371 .19783 L .12386 .20375 L .12401 .21216 L .12416 .22291 L .12431 .23578 L .12551 .37485 L .12581 .40495 L .12596 .41712 L .12611 .42695 L .12626 .43418 L .12641 .43862 L .12656 .44013 L .1267 .43864 L .12685 .43415 L .127 .42672 L .12715 .41651 L .1273 .40369 L .1279 .33264 L .1285 .24967 L .1288 .21306 L .12895 .19755 L .1291 .18443 L .12925 .17402 L .1294 .16659 L .12955 .16234 L .1297 .1614 L .12985 .16383 L .13 .1696 L .13014 .17862 L .13029 .19071 L .13059 .22306 L .13089 .26389 L .13149 .35616 L .13179 .39923 L Mistroke .13194 .4182 L .13209 .43483 L .13224 .44873 L .13239 .45955 L .13254 .46702 L .13269 .47093 L .13284 .47115 L .13299 .46766 L .13314 .4605 L .13329 .44981 L .13388 .37706 L .13448 .27715 L .13478 .22791 L .13508 .18524 L .13523 .16766 L .13538 .15316 L .13553 .1421 L .13568 .13476 L .13583 .13135 L .13598 .13197 L .13613 .13663 L .13628 .14527 L .13747 .32038 L .13777 .37523 L .13807 .42481 L .13822 .44616 L .13837 .46452 L .13852 .47944 L .13867 .49057 L .13882 .49762 L .13897 .5004 L .13912 .49881 L .13927 .49287 L .13942 .48268 L .13957 .46845 L .13987 .42912 L .14046 .32044 L .14076 .26075 L .14106 .20467 L .14136 .15736 L .14151 .13841 L .14166 .12324 L .14181 .11221 L .14196 .1056 L .14211 .10359 L .14226 .10625 L .14241 .11355 L .14256 .12536 L .14271 .14142 L .14286 .1614 L Mistroke .14346 .27065 L .14405 .39609 L .14435 .45076 L .1445 .47375 L .14465 .4931 L .1448 .50835 L .14495 .51913 L .1451 .52516 L .14525 .52628 L .1454 .52244 L .14555 .51369 L .1457 .50023 L .14585 .48232 L .14615 .4348 L .14645 .37531 L .14704 .24227 L .14734 .18076 L .14749 .15382 L .14764 .13028 L .14779 .1107 L .14794 .09555 L .14809 .08518 L .14824 .07987 L .14839 .07975 L .14854 .08485 L .14869 .09509 L .14884 .11025 L .14914 .15395 L .14944 .21212 L .15004 .35035 L .15034 .41794 L .15063 .47619 L .15078 .50009 L .15093 .51972 L .15108 .53462 L .15123 .54443 L .15138 .54891 L .15153 .54793 L .15168 .54148 L .15183 .5297 L .15198 .51283 L .15213 .49123 L .15243 .43586 L .15303 .29513 L .15333 .22244 L .15363 .15703 L .15377 .12897 L .15392 .1049 L .15407 .08539 L .15422 .07089 L Mistroke .15437 .06177 L .15452 .05824 L .15467 .06042 L .15482 .06826 L .15497 .08161 L .15512 .10019 L .15542 .1513 L .15602 .29161 L .15632 .36823 L .15662 .43996 L .15692 .50025 L .15707 .52431 L .15721 .54354 L .15736 .55747 L .15751 .56578 L .15766 .56826 L .15781 .56483 L .15796 .55554 L .15811 .54061 L .15841 .49518 L .15871 .43254 L .15901 .35828 L .15961 .20209 L .15991 .13434 L .16006 .10592 L .16021 .08204 L .16036 .06325 L .1605 .05001 L .16065 .04263 L .1608 .04129 L .16095 .04604 L .1611 .05679 L .16125 .07331 L .1614 .09524 L .1617 .15326 L .162 .2257 L .1626 .38701 L .1629 .46129 L .16305 .49375 L .1632 .52208 L .16335 .54561 L .1635 .56381 L .16365 .57623 L .1638 .58258 L .16394 .58271 L .16409 .5766 L .16424 .56437 L .16439 .54629 L .16469 .49429 L .16499 .42523 L Mistroke .16559 .26172 L .16589 .18206 L .16619 .11355 L .16634 .0855 L .16649 .06247 L .16664 .04501 L .16679 .03353 L .16694 .02829 L .16709 .02942 L .16723 .03692 L .16738 .05063 L .16858 .32213 L .16888 .40587 L .16918 .48108 L .16933 .5133 L .16948 .5409 L .16963 .56327 L .16978 .57987 L .16993 .59032 L .17008 .59437 L .17023 .59193 L .17038 .58303 L .17053 .56787 L .17067 .54679 L .17082 .52025 L .17097 .48886 L .17157 .33026 L .17187 .24389 L .17217 .16318 L .17247 .09545 L .17262 .06845 L .17277 .04689 L .17292 .03126 L .17307 .02194 L .17322 .01913 L .17337 .02291 L .17456 .25166 L .17516 .42396 L .17546 .49852 L .17561 .52977 L .17576 .55601 L .17591 .57664 L .17606 .59119 L .17621 .59932 L .17636 .60084 L .17651 .59571 L .17666 .58405 L .17681 .56611 L .17696 .5423 L .17755 .40084 L Mistroke .17815 .22642 L .17845 .14629 L .1786 .1113 L .17875 .08078 L .1789 .05544 L .17905 .03586 L .1792 .02247 L .17935 .0156 L .1795 .01539 L .17965 .02186 L .1798 .03486 L .17995 .05409 L .18025 .1094 L .18055 .18278 L .18114 .35615 L .18144 .44045 L .18174 .51285 L .18189 .54248 L .18204 .56679 L .18219 .58522 L .18234 .59737 L .18249 .60295 L .18264 .60183 L .18279 .59404 L .18294 .57976 L .18413 .29709 L .18443 .21014 L .18473 .13215 L .18488 .09877 L .18503 .07017 L .18518 .04701 L .18533 .02982 L .18548 .01897 L .18563 .01472 L .18578 .01715 L .18593 .02622 L .18608 .04172 L .18623 .06328 L .18638 .09041 L .18653 .1225 L .18713 .28442 L .18743 .37243 L .18772 .45457 L .18802 .52342 L .18817 .55086 L .18832 .57276 L .18847 .58865 L .18862 .59816 L .18877 .60107 L .18892 .59734 L Mistroke .19012 .36807 L .19072 .19585 L .19101 .12146 L .19116 .0903 L .19131 .06414 L .19146 .04357 L .19161 .02906 L .19176 .02092 L .19191 .01933 L .19206 .02433 L .19221 .03579 L .19236 .05344 L .19251 .07688 L .19281 .13883 L .19311 .21598 L .19371 .38711 L .19401 .46559 L .19416 .49983 L .1943 .52967 L .19445 .55445 L .1946 .5736 L .19475 .5867 L .1949 .59347 L .1961 .43225 L .1967 .2653 L .197 .18429 L .1973 .11479 L .19745 .08638 L .1976 .06307 L .19774 .04539 L .19789 .03372 L .19804 .02833 L .19819 .02933 L .19834 .03667 L .19849 .05019 L .19864 .06955 L .19879 .09432 L .19909 .15764 L .19969 .31748 L .20029 .47289 L .20059 .53117 L .20074 .55294 L .20089 .56911 L .20103 .57933 L .20118 .58337 L .20133 .58115 L .20148 .57275 L .20163 .55836 L .20178 .53834 L .20208 .48336 L Mistroke .20328 .17611 L .20358 .11263 L .20373 .08736 L .20388 .06718 L .20403 .05255 L .20418 .04378 L .20433 .04106 L .20447 .04443 L .20567 .25281 L .20627 .40876 L .20657 .47596 L .20672 .50407 L .20687 .52766 L .20702 .5462 L .20717 .55928 L .20732 .56663 L .20747 .56809 L .20762 .56365 L .20777 .55344 L .20791 .5377 L .20806 .5168 L .20836 .46163 L .20866 .39302 L .20926 .24126 L .20956 .17186 L .20971 .14162 L .20986 .11529 L .21001 .09344 L .21016 .07656 L .21031 .06502 L .21046 .05906 L .21061 .05879 L .21076 .0642 L .21091 .07514 L .21106 .09135 L .21135 .13791 L .21165 .19952 L .21225 .34437 L .21255 .41445 L .21285 .47444 L .213 .49893 L .21315 .519 L .2133 .53421 L .21345 .54423 L .2136 .54886 L .21375 .54802 L .2139 .54175 L .21405 .5302 L .2142 .51368 L .21435 .49257 L Mistroke .21464 .43868 L .21524 .30314 L .21554 .23386 L .21584 .17196 L .21599 .14554 L .21614 .12294 L .21629 .10467 L .21644 .09112 L .21659 .08258 L .21674 .07921 L .21689 .08107 L .21704 .08809 L .21719 .10008 L .21734 .11676 L .21764 .16245 L .21823 .28663 L .21853 .35371 L .21883 .41606 L .21913 .46809 L .21928 .48876 L .21943 .50523 L .21958 .51714 L .21973 .52427 L .21988 .52645 L .22003 .52369 L .22018 .51605 L .22033 .50375 L .22063 .46646 L .22093 .41538 L .22123 .35527 L .22182 .23028 L .22212 .17669 L .22227 .15433 L .22242 .13561 L .22257 .12093 L .22272 .11061 L .22287 .10484 L .22302 .10374 L .22317 .1073 L .22332 .11541 L .22347 .12785 L .22362 .14433 L .22392 .18769 L .22422 .2414 L .22481 .35954 L .22511 .41325 L .22526 .43658 L .22541 .45685 L .22556 .47363 L .22571 .48654 L Mistroke .22586 .49533 L .22601 .49983 L .22616 .49995 L .22631 .49573 L .22646 .4873 L .22661 .47487 L .22691 .43935 L .22721 .39259 L .22781 .28336 L .2281 .23087 L .2284 .18617 L .22855 .16799 L .2287 .15315 L .22885 .14195 L .229 .13463 L .22915 .13131 L .2293 .13204 L .22945 .13678 L .2296 .14538 L .22975 .15762 L .2299 .1732 L .2302 .21275 L .2308 .31139 L .2311 .36141 L .2314 .40586 L .23154 .42473 L .23169 .44081 L .23184 .45375 L .23199 .46329 L .23214 .46924 L .23229 .47149 L .23244 .47003 L .23259 .46492 L .23274 .45631 L .23289 .44443 L .23304 .42957 L .23319 .41212 L .23379 .32526 L .23409 .27879 L .23439 .23589 L .23469 .20036 L .23483 .18635 L .23498 .17526 L .23513 .16731 L .23528 .16264 L .23543 .16134 L .23558 .16339 L .23678 .27682 L .23738 .35905 L .23768 .39384 L Mistroke .23783 .40824 L .23798 .42021 L .23813 .42952 L .23827 .43598 L .23842 .43947 L .23857 .43996 L .23872 .43746 L .23887 .43207 L .23902 .42395 L .23917 .41331 L .23977 .35171 L .24037 .27827 L .24067 .24541 L .24082 .23127 L .24097 .21908 L .24112 .20909 L .24127 .20149 L .24142 .19643 L .24156 .19399 L .24171 .19418 L .24186 .19697 L .24201 .20225 L .24216 .20988 L .24246 .23128 L .24276 .259 L .24336 .32244 L .24366 .35231 L .24396 .37735 L .24411 .38737 L .24426 .39545 L .24441 .40143 L .24456 .40521 L .24471 .40674 L .24486 .40601 L .245 .40309 L .24515 .39807 L .24635 .308 L .24665 .282 L .24695 .25935 L .2471 .24991 L .24725 .24199 L .2474 .23574 L .24755 .23127 L .2477 .22865 L .24785 .2279 L .248 .22901 L .24815 .2319 L .24844 .24261 L .24859 .2501 L .24874 .25877 L Mistroke .24934 .30043 L .24964 .3219 L .24994 .3412 L .25009 .3495 L .25024 .35666 L .25039 .36255 L .25054 .36707 L .25069 .37015 L .25084 .37174 L .25099 .37185 L .25114 .37052 L .25129 .36782 L .25144 .36384 L .25173 .35259 L .25233 .32172 L .25263 .30521 L .25293 .29003 L .25323 .27749 L .25338 .27254 L .25353 .26858 L .25368 .26569 L .25383 .26388 L .25398 .26317 L .25413 .26354 L .25428 .26493 L .25443 .26728 L .25458 .27049 L .25473 .27446 L .25532 .29526 L .25562 .30662 L .25592 .31714 L .25622 .3259 L .25637 .32939 L .25652 .33222 L .25667 .33434 L .25682 .33573 L .25697 .3364 L .25712 .33638 L .25832 .31839 L .25861 .31213 L .25891 .30658 L .25906 .30424 L .25921 .30225 L .25936 .30064 L .25951 .29945 L .25966 .29865 L .25981 .29825 L .25996 .29823 L .26011 .29855 L .26041 .3 L Mistroke .26056 .30104 L .26071 .30219 L .26101 .30459 L .26116 .30572 L .26131 .30673 L .26146 .30756 L .26161 .30819 L .26176 .30858 L .2619 .30871 L .26205 .30858 L .2622 .30819 L .26235 .30756 L .2625 .30673 L .2628 .30459 L .2631 .30219 L .26325 .30104 L .2634 .3 L .26355 .29915 L .2637 .29855 L .26385 .29823 L .264 .29825 L .26415 .29865 L .2643 .29945 L .26445 .30064 L .2646 .30225 L .2649 .30658 L .26549 .31839 L .26579 .32469 L .26609 .33028 L .26624 .33257 L .26639 .3344 L .26654 .33569 L .26669 .33638 L .26684 .3364 L .26699 .33573 L .26714 .33434 L .26729 .33222 L .26759 .3259 L .26789 .31714 L .26849 .29526 L .26878 .28415 L .26908 .27446 L .26923 .27049 L .26938 .26728 L .26953 .26493 L .26968 .26354 L .26983 .26317 L .26998 .26388 L .27013 .26569 L .27028 .26858 L Mistroke .27148 .32172 L .27178 .33805 L .27207 .35259 L .27222 .35872 L .27237 .36384 L .27252 .36782 L .27267 .37052 L .27282 .37185 L .27297 .37174 L .27312 .37015 L .27327 .36707 L .27342 .36255 L .27357 .35666 L .27387 .3412 L .27447 .30043 L .27477 .2787 L .27507 .25877 L .27522 .2501 L .27536 .24261 L .27551 .23648 L .27566 .2319 L .27581 .22901 L .27596 .2279 L .27611 .22865 L .27626 .23127 L .27641 .23574 L .27656 .24199 L .27686 .25935 L .27746 .308 L .27806 .36068 L .27836 .3824 L .27851 .39112 L .27866 .39807 L .2788 .40309 L .27895 .40601 L .2791 .40674 L .27925 .40521 L .2794 .40143 L .27955 .39545 L .27985 .37735 L .28105 .259 L .28135 .23128 L .2815 .21964 L .28165 .20988 L .2818 .20225 L .28195 .19697 L .28209 .19418 L .28224 .19399 L .28239 .19643 L .28254 .20149 L Mistroke .28269 .20909 L .28284 .21908 L .28314 .24541 L .28344 .27827 L .28404 .35171 L .28434 .38562 L .28449 .40042 L .28464 .41331 L .28479 .42395 L .28494 .43207 L .28509 .43746 L .28524 .43996 L .28539 .43947 L .28553 .43598 L .28568 .42952 L .28583 .42021 L .28613 .39384 L .28643 .35905 L .28703 .27682 L .28733 .23678 L .28763 .20247 L .28778 .18852 L .28793 .17717 L .28808 .16872 L .28823 .16339 L .28838 .16134 L .28853 .16264 L .28868 .16731 L .28883 .17526 L .28912 .20036 L .28942 .23589 L .29002 .32526 L .29032 .37111 L .29062 .41212 L .29077 .42957 L .29092 .44443 L .29107 .45631 L .29122 .46492 L .29137 .47003 L .29152 .47149 L .29167 .46924 L .29182 .46329 L .29197 .45375 L .29212 .44081 L .29241 .40586 L .29301 .31139 L .29361 .21275 L .29391 .1732 L .29406 .15762 L .29421 .14538 L Mistroke .29436 .13678 L .29451 .13204 L .29466 .13131 L .29481 .13463 L .29496 .14195 L .29511 .15315 L .29541 .18617 L .2957 .23087 L .296 .28336 L .2966 .39259 L .2969 .43935 L .29705 .45876 L .2972 .47487 L .29735 .4873 L .2975 .49573 L .29765 .49995 L .2978 .49983 L .29795 .49533 L .2981 .48654 L .29825 .47363 L .2984 .45685 L .2987 .41325 L .29899 .35954 L .29959 .2414 L .29989 .18769 L .30004 .16443 L .30019 .14433 L .30034 .12785 L .30049 .11541 L .30064 .1073 L .30079 .10374 L .30094 .10484 L .30109 .11061 L .30124 .12093 L .30139 .13561 L .30258 .35527 L .30288 .41538 L .30318 .46646 L .30333 .48709 L .30348 .50375 L .30363 .51605 L .30378 .52369 L .30393 .52645 L .30408 .52427 L .30423 .51714 L .30438 .50523 L .30468 .46809 L .30498 .41606 L .30558 .28663 L .30587 .22088 L Mistroke .30617 .16245 L .30632 .13772 L .30647 .11676 L .30662 .10008 L .30677 .08809 L .30692 .08107 L .30707 .07921 L .30722 .08258 L .30737 .09112 L .30752 .10467 L .30767 .12294 L .30797 .17196 L .30857 .30314 L .30887 .37355 L .30916 .43868 L .30931 .46737 L .30946 .49257 L .30961 .51368 L .30976 .5302 L .30991 .54175 L .31006 .54802 L .31021 .54886 L .31036 .54423 L .31051 .53421 L .31066 .519 L .31081 .49893 L .31096 .47444 L .31156 .34437 L .31216 .19952 L .31246 .13791 L .3126 .11244 L .31275 .09135 L .3129 .07514 L .31305 .0642 L .3132 .05879 L .31335 .05906 L .3135 .06502 L .31365 .07656 L .3138 .09344 L .31395 .11529 L .31425 .17186 L .31455 .24126 L .31515 .39302 L .31545 .46163 L .3156 .49125 L .31575 .5168 L .31589 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L .43509 .23404 L .43524 .23971 L .43539 .24709 L Mistroke .43569 .2664 L .43599 .2904 L .43659 .34388 L .43689 .36847 L .43704 .37915 L .43719 .3884 L .43734 .39596 L .43749 .40165 L .43763 .40528 L .43778 .40675 L .43793 .40597 L .43808 .40294 L .43823 .39769 L .43838 .39029 L .43868 .36968 L .43898 .34276 L .43958 .27969 L .43988 .24922 L .44018 .22333 L .44033 .21291 L .44048 .20455 L .44063 .19846 L .44078 .19483 L .44092 .19376 L .44107 .19532 L .44122 .19952 L .44137 .20628 L .44167 .22698 L .44197 .25576 L .44257 .32722 L .44287 .36351 L .44317 .39568 L .44332 .40925 L .44347 .42067 L .44362 .42966 L .44377 .43598 L .44392 .43945 L .44407 .43997 L .44422 .43747 L .44436 .43199 L .44451 .42362 L .44466 .41251 L .44496 .38305 L .44556 .30485 L .44586 .26306 L .44616 .22454 L .44631 .20765 L .44646 .1929 L .44661 .18064 L .44676 .1712 L Mistroke .44691 .16481 L .44706 .16165 L .44721 .16183 L .44736 .16538 L .44751 .17225 L .44765 .18232 L .44795 .21118 L .44825 .24954 L .44855 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.46246 .52167 L .46261 .52608 L .46276 .52555 L .46291 .52006 L .46306 .50972 L .46321 .49473 L .46351 .4522 L .46411 .33158 L .4647 .2003 L .465 .14557 L .46515 .12329 L .4653 .10514 L .46545 .09154 L .4656 .08284 L .46575 .07925 L .4659 .08087 L .46605 .08771 L .4662 .09961 L .46635 .11635 L .4665 .13755 L .4671 .25644 L .4677 .39615 L .46799 .45817 L .46814 .4846 L .46829 .50713 L .46844 .52523 L .46859 .53847 L .46874 .54653 L .46889 .54919 L Mistroke .46904 .54638 L .46919 .53814 L .46934 .52463 L .46949 .50613 L .46979 .45592 L .47009 .39192 L .47069 .24619 L .47099 .17764 L .47128 .12047 L .47143 .09786 L .47158 .07998 L .47173 .06724 L .47188 .05997 L .47203 .05833 L .47218 .06241 L .47233 .07211 L .47248 .08724 L .47278 .13238 L .47308 .19385 L .47368 .34289 L .47398 .41701 L .47428 .48179 L .47443 .50877 L .47458 .53128 L .47472 .54879 L .47487 .56088 L .47502 .56726 L .47517 .56777 L .47532 .56237 L .47547 .55118 L .47562 .53442 L .47577 .51247 L .47607 .45501 L .47667 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.50613 .59987 L .50628 .60057 L .50643 .59463 L .50658 .58219 L .50688 .5391 L .50703 .50945 L .50718 .47525 L .50778 .30986 L .50808 .22331 L .50838 .14466 L .50852 .11056 L .50867 .08099 L .50882 .05664 L .50897 .03804 L .50912 .02562 L .50927 .01966 L .50942 .02027 L .50957 .02744 L .50972 .041 L .50987 .06063 L .51017 .11617 L .51077 .27244 L .51137 .4407 L .51167 .5103 L .51182 .53852 L .51196 .56148 L .51211 .57866 L .51226 .58967 L .51241 .59428 L .51256 .59239 L .51271 .58405 L .51286 .56946 L .51316 .52304 L .51346 .4574 L .51376 .37854 L .51436 .21035 L .51466 .13626 L .51481 .10479 L .51496 .07801 L .51511 .05653 L .51525 .04081 L Mistroke .5154 .03122 L .51555 .02795 L .5157 .03108 L .51585 .0405 L .516 .05601 L .51615 .07724 L .51645 .13473 L .51675 .20771 L .51735 .37272 L .51765 .44979 L .51795 .51378 L .5181 .53903 L .51825 .55898 L .5184 .57319 L .51855 .58137 L .51869 .58332 L .51884 .57903 L .51899 .56861 L .51914 .5523 L .51944 .50369 L .51974 .43771 L .52034 .27881 L .52064 .20033 L .52094 .13203 L .52109 .10367 L .52124 .08004 L .52139 .06167 L .52154 .04897 L .52169 .0422 L .52184 .04151 L .52198 .04689 L .52213 .05821 L .52228 .07519 L .52243 .09743 L .52273 .1555 L .52333 .30558 L .52363 .38393 L .52393 .45501 L .52408 .4858 L .52423 .51247 L .52438 .53442 L .52453 .55118 L .52468 .56237 L .52483 .56777 L .52498 .56726 L .52513 .56088 L .52528 .54879 L .52542 .53128 L .52572 .48179 L .52632 .34289 L Mistroke .52692 .19385 L .52722 .13238 L .52737 .10748 L .52752 .08724 L .52767 .07211 L .52782 .06241 L .52797 .05833 L .52812 .05997 L .52827 .06724 L .52842 .07998 L .52872 .12047 L .52991 .39192 L .53021 .45592 L .53036 .48306 L .53051 .50613 L .53066 .52463 L .53081 .53814 L .53096 .54638 L .53111 .54919 L .53126 .54653 L .53141 .53847 L .53156 .52523 L .53171 .50713 L .53201 .45817 L .5323 .39615 L .5329 .25644 L .5332 .19141 L .5335 .13755 L .53365 .11635 L .5338 .09961 L .53395 .08771 L .5341 .08087 L .53425 .07925 L .5344 .08284 L .53455 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.50369 L .9569 .53049 L .95705 .5523 L .9572 .56861 L .95735 .57903 L .9575 .58332 L .95765 .58137 L .95779 .57319 L .95794 .55898 L .95824 .51378 L .95854 .44979 L .95884 .37272 L .95944 .20771 L .95974 .13473 L .95989 .10368 L .96004 .07724 L .96019 .05601 L .96034 .0405 L .96049 .03108 L .96064 .02795 L .96079 .03122 L .96094 .04081 L .96109 .05653 L .96123 .07801 L .96153 .13626 L .96183 .21035 L .96243 .37854 L .96273 .4574 L .96303 .52304 L .96318 .54897 L .96333 .56946 L .96348 .58405 L .96363 .59239 L .96378 .59428 L .96393 .58967 L .96408 .57866 L .96423 .56148 L .96453 .5103 L .96482 .4407 L Mistroke .96542 .27244 L .96602 .11617 L .96632 .06063 L .96647 .041 L .96662 .02744 L .96677 .02027 L .96692 .01966 L .96707 .02562 L .96722 .03804 L .96737 .05664 L .96752 .08099 L .96782 .14466 L .96841 .30986 L .96871 .39645 L .96901 .47525 L .96916 .50945 L .96931 .5391 L .96946 .56353 L .96961 .58219 L .96976 .59463 L .96991 .60057 L .97006 .59987 L .97021 .59254 L .97036 .57875 L .97051 .5588 L .97081 .50236 L .9714 .34335 L .972 .17171 L .9723 .10059 L .97245 .07173 L .9726 .04825 L .97275 .03069 L .9729 .01945 L .97305 .01478 L .9732 .01681 L .97335 .02548 L .9735 .0406 L .9738 .08867 L .97619 .60332 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Play[wave1 + wave2, {t, 0, 1}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Sound["<<>>"] ;[o] -Sound- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 5d) Use the graph produced in (5c) to determine :[font = subsubsection; inactive; preserveAspect; startGroup] (a) the maximum beat amplitude and when the maxima occur :[font = text; inactive; preserveAspect] From sight, we can estimate the maximum amplitude as approximately equal to 2 and it occurs near t=0, 1/6, 2/6, 3/6, 4/6, 5/6, and 1. To be sure, we should plot the graphs near our candidate maximum values. :[font = input; preserveAspect; startGroup] Plot[wave1 + wave2, {t, 0, 0.01}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 95.238095 0.308738 0.147291 [ [(0.002)] .21429 .30874 0 2 Msboxa [(0.004)] .40476 .30874 0 2 Msboxa [(0.006)] .59524 .30874 0 2 Msboxa [(0.008)] .78571 .30874 0 2 Msboxa [(0.01)] .97619 .30874 0 2 Msboxa [(-2)] .01131 .01416 1 0 Msboxa [(-1)] .01131 .16145 1 0 Msboxa [(1)] .01131 .45603 1 0 Msboxa [(2)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .30874 m .21429 .31499 L s P [(0.002)] .21429 .30874 0 2 Mshowa p .002 w .40476 .30874 m .40476 .31499 L s P [(0.004)] .40476 .30874 0 2 Mshowa p .002 w .59524 .30874 m .59524 .31499 L s P [(0.006)] .59524 .30874 0 2 Mshowa p .002 w .78571 .30874 m .78571 .31499 L s P [(0.008)] .78571 .30874 0 2 Mshowa p .002 w .97619 .30874 m .97619 .31499 L s P [(0.01)] .97619 .30874 0 2 Mshowa p .001 w .0619 .30874 m .0619 .31249 L s P p .001 w .1 .30874 m .1 .31249 L s P p .001 w .1381 .30874 m .1381 .31249 L s P p .001 w .17619 .30874 m .17619 .31249 L s P p .001 w .25238 .30874 m .25238 .31249 L s P p .001 w .29048 .30874 m .29048 .31249 L s P p .001 w .32857 .30874 m .32857 .31249 L s P p .001 w .36667 .30874 m .36667 .31249 L s P p .001 w .44286 .30874 m .44286 .31249 L s P p .001 w .48095 .30874 m .48095 .31249 L s P p .001 w .51905 .30874 m .51905 .31249 L s P p .001 w .55714 .30874 m .55714 .31249 L s P p .001 w .63333 .30874 m .63333 .31249 L s P p .001 w .67143 .30874 m .67143 .31249 L s P p .001 w .70952 .30874 m .70952 .31249 L s P p .001 w .74762 .30874 m .74762 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[(-2)] .17004 .01224 1 0 Msboxa [(-1)] .17004 .16001 1 0 Msboxa [(1)] .17004 .45555 1 0 Msboxa [(2)] .17004 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .42063 .30778 m .42063 .31403 L s P [(0.15)] .42063 .30778 0 2 Mshowa p .002 w .65873 .30778 m .65873 .31403 L s P [(0.2)] .65873 .30778 0 2 Mshowa p .002 w .89683 .30778 m .89683 .31403 L s P [(0.25)] .89683 .30778 0 2 Mshowa p .001 w .23016 .30778 m .23016 .31153 L s P p .001 w .27778 .30778 m .27778 .31153 L s P p .001 w .3254 .30778 m .3254 .31153 L s P p .001 w .37302 .30778 m .37302 .31153 L s P p .001 w .46825 .30778 m .46825 .31153 L s P p .001 w .51587 .30778 m .51587 .31153 L s P p .001 w .56349 .30778 m .56349 .31153 L s P p .001 w .61111 .30778 m .61111 .31153 L s P p .001 w .70635 .30778 m .70635 .31153 L s P p .001 w .75397 .30778 m .75397 .31153 L s P p .001 w .80159 .30778 m .80159 .31153 L s P p .001 w .84921 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.8807 .3099 L .88194 .30539 L .88318 .30175 L .88442 .29914 L .88566 .29765 L .8869 .29724 L .88814 .29779 L .88938 .29909 L .89187 .30289 L .89311 .30481 L .89435 .30638 L .89559 .30742 L Mistroke .89683 .30778 L .89807 .30742 L .89931 .30638 L .90055 .30481 L .90179 .30289 L .90303 .30089 L .90427 .29909 L .90551 .29779 L .90675 .29724 L .90799 .29765 L .90923 .29914 L .91047 .30175 L .91171 .30539 L .91419 .31499 L .91667 .32541 L .91791 .32989 L .91915 .3333 L .92039 .33527 L .92163 .3355 L .92287 .33379 L .92411 .33008 L .92659 .31721 L .92907 .29936 L .93155 .28088 L .93279 .27303 L .93403 .26695 L .93527 .26317 L .93651 .26211 L .93775 .26401 L .93899 .26892 L .94023 .27669 L .94147 .28696 L .94395 .31261 L .94643 .33981 L .94767 .35175 L .94891 .36141 L .95015 .36806 L .95139 .3711 L .95263 .37019 L .95387 .36519 L .95511 .35626 L .95635 .34383 L .95883 .3113 L .96131 .27521 L .96255 .25869 L .96379 .24469 L .96503 .23423 L .96627 .22811 L .96751 .22688 L .96875 .23078 L Mistroke .96999 .23974 L .97123 .25335 L .97371 .29135 L .97619 .336 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] wave1 + wave2 /. t -> 1/6 :[font = output; output; inactive; preserveAspect; endGroup] 2 ;[o] 2 :[font = text; inactive; preserveAspect; endGroup] Again, due to the periodicity of the waves, if 0 and 1/6 are maxima, so are the remaining values 2/6, 3/6, 4/6, 5/6, and 1. :[font = subsubsection; inactive; preserveAspect] (b) based on the result from (a), the beat period is 1/6 seconds. :[font = subsubsection; inactive; preserveAspect; endGroup] (c) the beat frequency is 6 beats/second. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 6) The beat frequency is the numerical difference between the frequencies of the two original frequencies. :[font = subsubsection; inactive; preserveAspect; startGroup] An example to support the conjecture: 120 Hz and 123 Hz waves :[font = input; preserveAspect; startGroup] wave1 = Cos[120 2 Pi t] :[font = output; output; inactive; preserveAspect; endGroup] Cos[240*Pi*t] ;[o] Cos[240 Pi t] :[font = input; preserveAspect; startGroup] Plot[wave1, {t, 0, 0.1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 9.52381 0.309017 0.294302 [ [(0.02)] .21429 .30902 0 2 Msboxa [(0.04)] .40476 .30902 0 2 Msboxa [(0.06)] .59524 .30902 0 2 Msboxa [(0.08)] .78571 .30902 0 2 Msboxa [(0.1)] .97619 .30902 0 2 Msboxa [(-1)] .01131 .01472 1 0 Msboxa [(-0.5)] .01131 .16187 1 0 Msboxa [(0.5)] .01131 .45617 1 0 Msboxa [(1)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin 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.92659 .10091 L .92907 .06431 L .93031 .04947 L .93155 .03712 L .93279 .02739 L Mistroke .93403 .02037 L .93527 .01613 L .93651 .01472 L .93775 .01613 L .93899 .02037 L .94023 .02739 L .94147 .03712 L .94395 .06431 L .94643 .10091 L .95635 .30902 L .97619 .60332 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; startGroup] wave2 = Cos[123 2 Pi t] :[font = output; output; inactive; preserveAspect; endGroup] Cos[246*Pi*t] ;[o] Cos[246 Pi t] :[font = input; preserveAspect; startGroup] Plot[wave2, {t, 0, 0.1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 9.52381 0.309017 0.294302 [ [(0.02)] .21429 .30902 0 2 Msboxa [(0.04)] 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.43808 .20852 L .43838 .17956 L .43853 .16785 L .43868 .15821 L .43883 .15077 L .43898 .14563 L Mistroke .43913 .14285 L .43928 .14247 L .43943 .14447 L .43958 .14882 L .43973 .15545 L .43988 .16424 L .44018 .18778 L .44137 .32985 L .44197 .40048 L .44227 .42845 L .44242 .43986 L .44257 .44934 L .44272 .45677 L .44287 .46204 L .44302 .46509 L .44317 .46589 L .44332 .46443 L .44347 .46075 L .44362 .45491 L .44377 .44701 L .44407 .42553 L .44436 .3976 L .44496 .3293 L .44556 .25775 L .44586 .22588 L .44616 .19904 L .44631 .188 L .44646 .17874 L .44661 .1714 L .44676 .16606 L .44691 .16279 L .44706 .16164 L .44721 .1626 L .44736 .16566 L .44751 .17075 L .44765 .1778 L .44795 .19728 L .44825 .22292 L .44855 .25319 L .44915 .32036 L .44975 .38355 L .45005 .40911 L .4502 .41973 L .45035 .42869 L .4505 .43589 L .45065 .44123 L .4508 .44464 L .45095 .44609 L .45109 .44556 L .45124 .44308 L Mistroke .45139 .43869 L .45154 .43246 L .45184 .41492 L .45214 .39155 L .45334 .27062 L .45394 .21814 L 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L .51765 .36005 L .5178 .36005 L .51795 .35931 L .5181 .35781 L .51825 .35558 L .51855 .34899 L .51884 .33984 L .51914 .32858 L .51974 .30225 L .52034 .27585 L .52064 .26459 L .52079 .25977 L .52094 .2556 L .52109 .25214 L .52124 .24946 L .52139 .24761 L .52154 .24662 L .52169 .24653 L .52184 .24734 L .52198 .24906 L .52213 .25167 L .52243 .25945 L .52273 .27032 L .52393 .33137 L .52423 .34669 L .52453 .36019 L .52468 .36599 L .52483 .37103 L .52498 .37523 L .52513 .37852 L .52528 .38083 L .52542 .38213 L .52557 .38238 L .52572 .38156 L .52587 .37969 L .52602 .37676 L .52632 .36791 L .52662 .35542 L .52692 .33996 L .52752 .30358 L .52812 .26685 L .52842 .25105 L .52857 .24424 L .52872 .23829 L Mistroke .52886 .2333 L .52901 .22935 L .52916 .22652 L .52931 .22485 L .52946 .22438 L .52961 .22513 L .52976 .2271 L .52991 .23027 L .53006 .23461 L .53021 .24007 L .53051 .25405 L .53111 .2914 L .53171 .33419 L .53201 .35461 L .5323 .37275 L .5326 .38752 L .53275 .39335 L .5329 .398 L .53305 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.84323 .38106 L .84338 .37883 L .84353 .37556 L .84368 .37129 L .84398 .35995 L .84458 .3284 L .84518 .29094 L .84548 .27263 L .84578 .25603 L .84608 .24215 L .84623 .23651 L .84637 .23186 L .84652 .22828 L .84667 .22583 L .84682 .22456 L .84697 .22449 L .84712 .22565 L .84727 .22802 L .84742 .23159 L .84757 .23631 L .84787 .24897 L .84817 .26534 L .84877 .30552 L .84937 .348 L .84966 .36703 L .84996 .38303 L .85011 .38959 L .85026 .39503 L .85041 .39928 L .85056 .40226 L .85071 .40392 L Mistroke .85086 .40421 L .85101 .40313 L .85116 .40068 L .85131 .39688 L .85146 .39178 L .85161 .38543 L .85176 .37793 L .85654 .29926 L .85714 .35201 L .85744 .37597 L .85774 .3964 L .85789 .4049 L .85804 .41207 L .85819 .4178 L .85834 .42201 L .85849 .4246 L .85864 .42555 L .85879 .42482 L .85894 .42241 L .85909 .41835 L .85924 .41268 L .85954 .39683 L .85983 .37569 L .86013 .35041 L .86073 .29325 L .86133 .23816 L .86163 .21545 L .86178 .20593 L .86193 .19784 L .86208 .19129 L .86223 .1864 L .86238 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.87838 .14351 L .87853 .14708 L .87868 .153 L .87883 .16118 L .87898 .17153 L .87928 .19811 L Mistroke .87988 .26921 L .88047 .35054 L .88077 .38929 L .88107 .42374 L .88137 .45186 L .88152 .46302 L .88167 .472 L .88182 .47867 L .88197 .48292 L .88212 .48468 L .88227 .48392 L .88242 .48064 L .88257 .47488 L .88272 .4667 L .88287 .45622 L .88317 .42894 L .88346 .39456 L .88406 .31245 L .88466 .22826 L .88496 .19146 L .88526 .16113 L .88541 .14897 L .88556 .13907 L .88571 .13157 L .88586 .1266 L .88601 .12423 L .88616 .12451 L .88631 .12744 L .88646 .13299 L .88661 .1411 L .88676 .15165 L .88705 .17948 L .88765 .25605 L .88825 .34573 L .88855 .38917 L .88885 .42831 L .88915 .46085 L .8893 .47404 L .88945 .4849 L .8896 .49326 L .88975 .49899 L .8899 .50202 L .89005 .50227 L .8902 .49974 L .89034 .49446 L .89049 .48649 L .89064 .47594 L .89094 .4477 L .89124 .4113 L .89184 .32253 L Mistroke .89244 .2295 L .89274 .18806 L .89304 .15329 L .89319 .13906 L .89334 .12723 L .89349 .11798 L .89363 .11144 L .89378 .10772 L .89393 .10689 L .89408 .10895 L .89423 .1139 L .89438 .12167 L .89453 .13216 L .89483 .16068 L .89543 .24168 L .89603 .33894 L .89663 .43055 L .89693 .46754 L .89707 .48282 L .89722 .49564 L .89737 .50582 L .89752 .51319 L .89767 .51764 L .89782 .51911 L .89797 .51755 L .89812 .51299 L .89827 .50548 L .89842 .49513 L .89872 .46647 L .89902 .4286 L .89962 .33415 L .90022 .2329 L .90051 .18696 L .90081 .14777 L .90096 .13143 L .90111 .1176 L .90126 .10649 L .90141 .09825 L .90156 .09303 L .90171 .09089 L .90186 .09188 L .90201 .09599 L .90216 .10318 L .90231 .11334 L .90261 .14201 L .90291 .1804 L .90321 .22636 L .9038 .33033 L .9044 .4305 L .9047 .47189 L Mistroke .90485 .48928 L .905 .50412 L .90515 .5162 L .9053 .52532 L .90545 .53136 L .9056 .53421 L .90575 .53383 L .9059 .53021 L .90605 .52341 L .9062 .5135 L .9065 .48496 L .9068 .44619 L .90799 .23837 L .90859 .1446 L .90889 .11029 L .90904 .09724 L .90919 .0872 L .90934 .08033 L .90949 .07673 L .90964 .07645 L .90979 .07952 L .90994 .08589 L .91009 .09548 L .91039 .12376 L .91068 .16279 L .91098 .21037 L .91158 .32007 L .91218 .42822 L .91248 .47387 L .91263 .49337 L .91278 .51026 L .91293 .52429 L .91308 .53526 L .91323 .543 L .91338 .54738 L .91353 .54835 L .91368 .54588 L .91397 .53078 L .91517 .36109 L .91577 .24575 L .91607 .19151 L .91637 .14378 L .91652 .12325 L .91667 .10534 L .91682 .09031 L .91697 .07841 L .91712 .06979 L .91726 .06459 L .91741 .06289 L .91756 .06472 L Mistroke .91771 .07007 L .91786 .07886 L .91801 .09097 L .91816 .10623 L .91846 .14534 L .91876 .19397 L .91936 .3084 L .91996 .42379 L .92026 .47354 L .92056 .51402 L .9207 .53003 L .92085 .54289 L .921 .55241 L .92115 .55845 L .9213 .56092 L .92145 .55976 L .9216 .555 L .92175 .5467 L .9219 .53496 L .92205 .51995 L .92235 .48101 L .92295 .37589 L .92355 .25487 L .92385 .19696 L .92414 .14526 L .92444 .10277 L .92459 .08578 L .92474 .07196 L .92489 .06152 L .92504 .05462 L .92519 .05136 L .92534 .0518 L .92549 .05593 L .92564 .06371 L .92579 .07502 L .92594 .0897 L .92624 .12835 L .92654 .17747 L .92714 .29553 L .92773 .41736 L .92803 .47094 L .92833 .51539 L .92848 .53337 L .92863 .54815 L .92878 .55951 L .92893 .56729 L .92908 .57136 L .92923 .57168 L .92938 .56821 L .92953 .56102 L Mistroke .92968 .55019 L .92983 .53588 L .93013 .49766 L .93073 .39122 L .93132 .26553 L .93162 .20433 L .93192 .14895 L .93222 .10258 L .93237 .08364 L .93252 .06791 L .93267 .05561 L .93282 .04694 L .93297 .04203 L .93312 .04094 L .93327 .04369 L .93342 .05026 L .93357 .06056 L .93372 .07444 L .93402 .11211 L .93431 .16115 L .93491 .28174 L .93551 .40909 L .93581 .46619 L .93611 .51442 L .93626 .53431 L .93641 .55099 L .93656 .56421 L .93671 .57378 L .93686 .57956 L .93701 .58145 L .93716 .57943 L .93731 .57353 L .93746 .56381 L .9376 .55042 L .9379 .51342 L .9382 .46462 L .9385 .40678 L .9391 .2775 L .9397 .15473 L .94 .10471 L .94015 .08389 L .9403 .06627 L .94045 .05212 L .9406 .04165 L .94075 .035 L .94089 .03228 L .94104 .03353 L .94119 .03874 L .94134 .04783 L .94149 .06068 L Mistroke .94179 .09689 L .94209 .14532 L .94269 .26729 L .94329 .39916 L .94359 .45941 L .94389 .51116 L .94404 .53289 L .94419 .55142 L .94433 .56649 L .94448 .57787 L .94463 .5854 L .94478 .58896 L .94493 .5885 L .94508 .58403 L .94523 .5756 L .94538 .56334 L .94568 .52803 L .94598 .4801 L .94628 .42228 L .94688 .29053 L .94748 .16246 L .94777 .10907 L .94792 .08646 L .94807 .06703 L .94822 .05106 L .94837 .03878 L .94852 .03037 L .94867 .02595 L .94882 .02559 L .94897 .02929 L .94912 .03702 L .94927 .04866 L .94957 .08296 L .94987 .13025 L .95106 .38781 L .95136 .45079 L .95166 .50573 L .95181 .52918 L .95196 .54948 L .95211 .56634 L .95226 .57952 L .95241 .58882 L .95256 .5941 L .95271 .59529 L .95286 .59237 L .95301 .58538 L .95316 .57441 L .95346 .54122 L .95376 .49468 L .95406 .43744 L Mistroke .95465 .30435 L .95525 .17194 L .95555 .11553 L .9557 .09128 L .95585 .07013 L .956 .0524 L .95615 .03835 L .9563 .02817 L .95645 .02201 L .9566 .01998 L .95675 .02209 L .9569 .02832 L .95705 .03858 L .95824 .23757 L .95884 .37527 L .95914 .44051 L .95944 .49827 L .95974 .54525 L .95989 .56382 L .96004 .57874 L .96019 .58979 L .96034 .59681 L .96049 .59971 L .96064 .59842 L .96079 .59298 L .96094 .58346 L .96109 .56999 L .96123 .55276 L .96153 .50808 L .96183 .45196 L .96243 .31867 L .96303 .18296 L .96333 .12395 L .96363 .07548 L .96378 .05609 L .96393 .04033 L .96408 .02841 L .96423 .02053 L .96438 .01678 L .96453 .01722 L .96467 .02185 L .96482 .0306 L .96497 .04335 L .96512 .05992 L .96542 .10352 L .96602 .22288 L .96662 .36181 L .96692 .42879 L .96722 .48895 L .96752 .53883 L Mistroke .96767 .55899 L .96782 .57557 L .96796 .58833 L .96811 .59707 L .96826 .60168 L .96841 .60209 L .96856 .59829 L .96871 .59033 L .96886 .57833 L .96901 .56245 L .96931 .52005 L .96961 .46555 L .97021 .33322 L .97081 .1953 L .97111 .13413 L .9714 .08293 L .97155 .06202 L .9717 .04465 L .97185 .03108 L .972 .02149 L .97215 .01602 L .9723 .01476 L .97245 .01773 L .9726 .02487 L .97275 .03609 L .9729 .05123 L .9732 .09234 L .9738 .2087 L .97619 .60332 L Mfstroke P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] As expected there are 3 beats per cycle. :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect; endGroup; endGroup] We used the Sound capability of Mathematica to illustrate the "wa-wa" effect. It should be pointed out that the some of the Play calculations may take a while even on fast machines. ;[s] 3:0,0;32,1;43,2;182,-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; ^*)