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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 29 "Exercise 5: Waves on a St
ring" }}{PARA 257 "" 0 "" {TEXT -1 57 "Due at 10am on Friday, October \+
1, at the start of studio." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "In class, we deri
ved the \"wave equation\", the differential equation that governs the \+
motion of a stretched string." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "waveqn:=diff(y(x,t),x,x)-(1/s^2)*diff(y(x,t),t,t)=0;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'waveqnG/,&-%%diffG6$-%\"yG6$%\"xG%\"tG-%
\"$G6$F-\"\"#\"\"\"*&-F(6$F*-F06$F.F2\"\"\"*$)%\"sG\"\"#F9!\"\"!\"\"\"
\"!" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 38 "1) Show that any function \+
of the form " }{XPPEDIT 18 0 "f(x-s*t);" "6#-%\"fG6#,&%\"xG\"\"\"*&%\"
sGF(%\"tGF(!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "g(x+st);" "6#-%\"
gG6#,&%\"xG\"\"\"%#stGF(" }{TEXT -1 15 " is a solution " }{XPPEDIT 18 
0 "y(x,t);" "6#-%\"yG6$%\"xG%\"tG" }{TEXT -1 23 " for the wave equatio
n." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 263 "The \"animate\" feature is
 useful for making plots of functions of two variables, particularly w
hen one of them is time. Following is an example that shows how some a
rbitrary functional form (called a \"Gaussian\") propogates to the rig
ht if it used for the function " }{XPPEDIT 18 0 "f(x-st);" "6#-%\"fG6#
,&%\"xG\"\"\"%#stG!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 
"f:=exp(-u^2/(2*sigma^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%
$expG6#,$*&*$)%\"uG\"\"#\"\"\"F.*$)%&sigmaG\"\"#F.!\"\"#!\"\"F-" }}}
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Set the value of the speed " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 
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{TEXT -1 300 "To make the picture move, click on the graph (so the box
 around it appears) and then click on the \"play\" button on the menu \+
bar. You should see the functional form move to the right, covering 10
 units of distance in the two units of time, because the speed is five
 units of distance per unit of time." }}}{EXCHG {PARA 4 "" 0 "" {TEXT 
-1 78 "2) Modify the commands above to show, using animation, that the
 solution form " }{XPPEDIT 18 0 "g(x+s*t);" "6#-%\"gG6#,&%\"xG\"\"\"*&
%\"sGF(%\"tGF(F(" }{TEXT -1 34 " is a wave that moves to the left." }}
}{EXCHG {PARA 4 "" 0 "" {TEXT -1 213 "3) Create and animate a rightwar
d-moving sinusoidal wave with wavelength 3 units of distance, with fre
quency 5 per unit of time, and with amplitude 10. Start with the wave \+
in standard form, which I give you below." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "y:=(x,t)->A*sin(k*x-omega*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"yGR6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF)*&%\"AG\"\"\"-%$sinG6#,&*&
%\"kGF/9$F/F/*&%&omegaGF/9%F/!\"\"F/F)F)F)" }}}{EXCHG {PARA 4 "" 0 "" 
{TEXT -1 60 "4) Create an animation showing the second normal mode (i.
e. " }{XPPEDIT 18 0 "n = 2;" "6#/%\"nG\"\"#" }{TEXT -1 158 ") of a str
etched string with fixed ends. Display the entire length of the string
, and let the animation continue for at least three full periods of vi
bration." }}}}{MARK "18 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }