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{SECT 0 {PARA 17 "" 0 "" {TEXT -1 39 "     Paradigms in Physics: Oscil
lations" }}{PARA 17 "" 0 "" {TEXT -1 0 "" }}{PARA 17 "" 0 "" {TEXT -1 
77 "© 2000 Oregon State University                       version Oct 1
6, 2000 DMc" }}{PARA 17 "" 0 "" {TEXT -1 0 "" }}{PARA 17 "" 0 "" 
{TEXT -1 6 "      " }}{PARA 18 "" 0 "" {TEXT -1 29 "Choosing Fourier C
oefficients" }}{PARA 18 "" 0 "" {TEXT -1 20 "for Triangular Waves" }}
{PARA 19 "" 0 "" {TEXT 264 75 "by Jason Janesky, Catherine Meyer, Cori
nne A. Manogue and Philip J. Siemens" }}{PARA 19 "" 0 "" {TEXT -1 43 "
Physics Department, Oregon State University" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 256 "" 0 "" 
{TEXT 258 22 "Worksheet Introduction" }}{PARA 257 "" 0 "" {TEXT -1 78 
"This worksheet is designed as a first experience in the use of Fourie
r series." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" 
{TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 53 "A Fourier series can be
 used to represent a function " }{XPPEDIT 18 0 "g(t) " "6#-%\"gG6#%\"t
G" }{TEXT -1 48 "which is defined on a finite interval of length " }
{XPPEDIT 18 0 "T" "6#%\"TG" }{TEXT 263 3 ",  " }{TEXT -1 22 "or a peri
odic function" }{TEXT 265 8 "  g(t) =" }{TEXT -1 3 " g(" }{TEXT 262 3 
"t+T" }{TEXT -1 61 ").   The fundamental angular frequency of such a f
unction is " }{XPPEDIT 18 0 "omega = 2*Pi/T" "6#/%&omegaG*(\"\"#\"\"\"
%#PiGF'%\"TG!\"\"" }{TEXT -1 172 " ,   Fourier series analysis involve
s the approximation of such a function as an infinite sum of terms who
se frequencies are integer multiples of the fundamental frequency " }
{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 1 "." }}{PARA 261 "" 0 "
" {TEXT -1 1 " " }}{PARA 262 "" 0 "" {TEXT -1 90 "In the worksheet \"F
ourier Series Approximation\" we introduced the  Sine and Cosine Serie
s:" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "g(t)=a[0]/2 +a[1]*cos(omega t)+
a[2]*cos(2omega t)+a[3]*cos(3omega t)" "6#/-%\"gG6#%\"tG,**&&%\"aG6#\"
\"!\"\"\"\"\"#!\"\"F.*&&F+6#F.F.-%$cosG6#*&%&omegaGF.F'F.F.F.*&&F+6#F/
F.-F56#*(F/F.F8F.F'F.F.F.*&&F+6#\"\"$F.-F56#*(FBF.F8F.F'F.F.F." }
{TEXT -1 3 " + " }{TEXT 257 3 "..." }}{PARA 263 "" 0 "" {TEXT -1 5 "..
.+ " }{XPPEDIT 18 0 "+ b[1]*sin(omega t)+b[2]*sin(2omega t)+b[3]*sin(3
omega t) " "6#,(*&&%\"bG6#\"\"\"F(-%$sinG6#*&%&omegaGF(%\"tGF(F(F(*&&F
&6#\"\"#F(-F*6#*(F2F(F-F(F.F(F(F(*&&F&6#\"\"$F(-F*6#*(F9F(F-F(F.F(F(F(
" }{TEXT -1 6 " + ..." }}{PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "
" 0 "" {TEXT -1 37 "The coefficients in the series - the " }{XPPEDIT 
18 0 "a[n]" "6#&%\"aG6#%\"nG" }{TEXT -1 7 "'s and " }{XPPEDIT 18 0 "b[
n]" "6#&%\"bG6#%\"nG" }{TEXT -1 79 "'s -  can be chosen to provide the
 best approximation to the original function " }{XPPEDIT 18 0 "g(t)" "
6#-%\"gG6#%\"tG" }{TEXT -1 90 ".  We can often obtain a useful approxi
mation by keeping only a few terms of the series.  " }}{PARA 266 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 243 "The present workshee
t begins with a Fourier series approximation of a sawtooth waveform wh
ere you guess the most useful coefficients.  Finally, for homework, yo
u will use a systematic method to find the best possible values of the
 coefficients" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT 259 12 "Triangle Wav" }{TEXT 261 1 "e" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 329 "Now i
t's your turn to try to fit a given function with sines and cosines.  \+
There is a mathematical way to arrive at the coefficients which will b
e presented to you shortly.  However using intuition and educated gues
ses will provide you with a lot of insight into the underlying princip
les.  Try to fit the sawtooth pattern below:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "The Sawtooth wave" }}
{EXCHG {PARA 268 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}
{EXCHG {PARA 269 "> " 0 "" {MPLTEXT 1 0 86 "f(t):=piecewise(t<=-Pi,-3/
Pi*t-9/2,t<0,3/Pi*t+3/2,t<=Pi,-3/Pi*t+3/2, t>Pi,3/Pi*t-9/2);" }}}
{EXCHG {PARA 270 "> " 0 "" {MPLTEXT 1 0 35 "plot(f(t),t=-2*Pi..2*Pi,-1
.5..1.5);" }}}{PARA 271 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" 
{TEXT -1 48 "What is the fundamental period of this waveform?" }}
{PARA 273 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 274 "> " 0 "" {MPLTEXT 
1 0 12 "Period :=  ;" }}}{PARA 275 "" 0 "" {TEXT -1 0 "" }}{PARA 276 "
" 0 "" {TEXT -1 45 "this defines the fundamental frequency to be:" }}
{PARA 277 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 278 "> " 0 "" {MPLTEXT 
1 0 21 "omega := 2*Pi/Period;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "The constant term (n=0):" }}
{PARA 279 "" 0 "" {TEXT -1 27 "This is the constant term, " }{XPPEDIT 
18 0 "a[0]" "6#&%\"aG6#\"\"!" }{TEXT -1 141 ".  It does not depend on \+
sine or cosine.  Remember this guess should get you as close to as man
y points as possible.  Enter your guess below:" }}{PARA 280 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 281 "> " 0 "" {MPLTEXT 1 0 11 "a[0] := 1 \+
;" }}}{PARA 282 "" 0 "" {TEXT -1 0 "" }}{PARA 283 "" 0 "" {TEXT -1 36 
"Your Fourier  series so far is then:" }}{PARA 284 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 285 "> " 0 "" {MPLTEXT 1 0 15 "g(t) := a[0]/2;" }}}
{PARA 286 "" 0 "" {TEXT -1 0 "" }}{PARA 287 "" 0 "" {TEXT -1 45 "And p
lotting that against the sawtooth gives:" }}{PARA 288 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 289 "> " 0 "" {MPLTEXT 1 0 44 "plot([f(t),g(t)]
, t=-2*Pi..2*Pi, -1.5..1.5);" }}}{PARA 290 "" 0 "" {TEXT -1 0 "" }}
{PARA 291 "" 0 "" {TEXT -1 67 "You can plot the difference if you woul
d like using the line below:" }}{PARA 292 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 293 "> " 0 "" {MPLTEXT 1 0 44 "plot([f(t)-g(t)], t=-2*Pi.
.2*Pi, -1.5..1.5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 27 "The fundamental term (n=1):" }}{PARA 294 "" 0 "
" {TEXT -1 37 "Now determine what the coefficients, " }{XPPEDIT 18 0 "
a[1]" "6#&%\"aG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b[1]" "6#
&%\"bG6#\"\"\"" }{TEXT -1 260 ", should be.  A hint is to first determ
ine if the sawtooth is even or odd.  There should be a match between t
his (even or odd) and the choice of sine or cosine terms.  An odd func
tion can't be used to describe an even function.  WHY?  Enter your cho
ices below:" }}{PARA 295 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 296 "> \+
" 0 "" {MPLTEXT 1 0 11 "a[1] :=   ;" }}}{PARA 297 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 298 "> " 0 "" {MPLTEXT 1 0 11 "b[1] :=   ;" }}}{PARA 
299 "" 0 "" {TEXT -1 0 "" }}{PARA 300 "" 0 "" {TEXT -1 19 "The series \+
is then:" }}{PARA 301 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 302 "> " 0 
"" {MPLTEXT 1 0 59 "g(t) := a[0]/2 + a[1]*cos(1*omega*t) + b[1]*sin(1*
omega*t);" }}}{PARA 303 "" 0 "" {TEXT -1 0 "" }}{PARA 304 "" 0 "" 
{TEXT -1 43 "Plotting your guess and the waveform gives:" }}{PARA 305 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 306 "> " 0 "" {MPLTEXT 1 0 44 "pl
ot([f(t),g(t)], t=-2*Pi..2*Pi, -1.5..1.5);" }}}{PARA 307 "" 0 "" 
{TEXT -1 0 "" }}{PARA 308 "" 0 "" {TEXT -1 99 "Adjust your parameters \+
until you are satisfied that you are as close to as many points as pos
sible." }}{PARA 309 "" 0 "" {TEXT -1 98 "You can subtract the two wave
forms to give you clues about the next terms using the command below:
" }}{PARA 310 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 311 "> " 0 "" 
{MPLTEXT 1 0 44 "plot([f(t)-g(t)], t=-2*Pi..2*Pi, -1.5..1.5);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "
Higher order terms:" }}{PARA 312 "" 0 "" {TEXT -1 221 "Keep going incl
uding as many terms as you can.   Adjusting them to give as good a fit
 as possible.  You will have to cut and paste the commands you need or
 create them yourself.  Call an instructor over for any questions." }}
{PARA 313 "" 0 "" {TEXT -1 0 "" }}{PARA 314 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 315 "> " 0 "" {MPLTEXT 1 0 11 "a[0] :=   ;" }}}{EXCHG 
{PARA 316 "> " 0 "" {MPLTEXT 1 0 11 "a[1] :=   ;" }}}{EXCHG {PARA 317 
"> " 0 "" {MPLTEXT 1 0 11 "b[1] :=   ;" }}}{EXCHG {PARA 318 "> " 0 "" 
{MPLTEXT 1 0 11 "a[2] :=   ;" }}}{EXCHG {PARA 319 "> " 0 "" {MPLTEXT 
1 0 11 "b[2] :=   ;" }}}{EXCHG {PARA 320 "> " 0 "" {MPLTEXT 1 0 103 "g
(t) := a[0]/2 + a[1]*cos(1*omega*t) + b[1]*sin(1*omega*t) + a[2]*cos(2
*omega*t) + b[2]*sin(2*omega*t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 321 "> " 0 "" {MPLTEXT 1 0 44 "plot([f(t),g(t)], t=-
2*Pi..2*Pi, -1.5..1.5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{PARA 322 "" 0 "" {TEXT -1 0 "" }}{PARA 323 "" 0 "" {TEXT -1 74 
"For homework you will mathematically calculate the correct coefficien
ts.  " }}{PARA 324 "" 0 "" {TEXT -1 0 "" }}{PARA 325 "" 0 "" {TEXT -1 
111 "Compare these calculated results to your guesses, and then use th
em in the program to see how well they work.  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 100 "WHEN YOU ARE TRYING OUT
 YOUR RESULTS, BE SURE TO CHOOSE INLINE PLOTS IN ORDER TO DOCUMENT YOU
R WORK." }}{PARA 326 "" 0 "" {TEXT -1 0 "" }}{PARA 327 "" 0 "" {TEXT 
-1 98 "After you substitute your homework results, print out the works
heet to document your achievement. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{MARK "12" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }