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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 33 "Module 24: Structure of Solutio
ns" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "In
this module we want to contrast the three situations that we have see
n. We take three examples which are different, but similar." }}{PARA
0 "" 0 "" {TEXT -1 31 " We compare the situations:" }}{PARA 0 ""
0 "" {TEXT -1 54 "(1) x is any number on the real line and call this a
n " }{TEXT 258 15 "infinite string" }{TEXT -1 4 ", or" }}{PARA 0 "" 0
"" {TEXT -1 34 "(2) x is positive and call this a " }{TEXT 259 20 "hal
f infinite string" }{TEXT -1 4 ", or" }}{PARA 0 "" 0 "" {TEXT -1 40 "(
3) x is in an interval and call this a " }{TEXT 260 13 "finite string
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 56 " We begin with a \+
function f which has only one bump." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 44 "f:=x->piecewise(x<=0,0,x<=3,x/3,x<=4,4-x,0);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot('f(x)',x=0..4);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 243 "If we are going to assume we have three strings, one fitting e
ach of the situations described above, that we give the string an inti
ial displacement determined by f, and give the string an initial veloc
ity of zero, then we need to recall that " }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 " U(t, x) = " }{XPPEDIT
18 0 "(F(x+t)+F(x-t))/2;" "6#*&,&-%\"FG6#,&%\"xG\"\"\"%\"tGF*F*-F&6#,&
F)F*F+!\"\"F*F*\"\"#F/" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111
"and that this F is obtained from the original f through some extensio
n. Thus, we will need the extension tools." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Extensions Tools" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "M
aking even and odd extensions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 241 "It's clear that in order to illustrate t
hese ideas, we are going to need to be able to make even and odd exten
sions for functions. Here is a procedure. You must execute these two p
arts in order to be able to work the rest of this work sheet." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "ef:=proc(f,x) \n if
x>0 then f(x);\n else f(-x);\n fi;\n end;" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "of:=proc(f,x) \n if x>0 then f
(x);\n else -f(-x);\n fi;\n end;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1
{PARA 3 "" 0 "" {TEXT -1 52 "Even periodic extensions and odd periodic
extensions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 283 "In order to be able to graph solutions for the finite string, \+
we need to be able to make a different extensions of functions than si
mply even or odd. We need to make odd and even, 2 L periodic extensi
ons of functions defined on the interval [0, L]. Here is a procedure \+
to do this." }}{PARA 0 "" 0 "" {TEXT -1 53 " First, we make an eve
n, 2 L periodic extension." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 116 "To do this we make the extension
to the interval [0, 2 L] aware that the goal is to get an even, perio
dic extension." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "pfe:= pro
c(f,x)\n global L;\n\011\011if 0 <= x and x < L then f(x)\n\011
\011elif L <= x and x <= 2*L then f(2*L-x)\n\011\011fi end;" }}}{PARA
0 "" 0 "" {TEXT -1 52 "Here is where we extend the function to all num
bers." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "epf:= (f,x)->pfe(f,
frac(abs(x)/(2*L))*2*L);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 " Nex
t, we make an odd, 2 L periodic extension." }}{PARA 0 "" 0 "" {TEXT
-1 115 "To do this we make the extension to the interval [0, 2 L] awar
e that the goal is to get an odd, periodic extension." }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 114 "pfo:= proc(f,x)\n global L;\n\011
\011if 0 <= x and x < L then f(x)\n\011\011elif L <= x and x <= 2*L th
en -f(2*L-x)\n\011\011fi end;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 57 "Here is where we make extend the function
to all numbers." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "opf:= (f
,x)->sign(x)*pfo(f,frac(abs(x)/(2*L))*2*L);" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We use b
oth these extensions in what follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 " For Situation 1, we \+
define initialgree with f and to be defined for all numbers." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "F1:=x->piecewise(x<=0,0,x<=3
,x/3,x<=4,4-x,0);" }}}{PARA 0 "" 0 "" {TEXT -1 116 "For Situation 2, w
e define F2 to agree with f(x) for positive x's and to be the odd exte
nsion of f for negative x' s" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
16 "F2:=x->of(F1,x);" }}}{PARA 0 "" 0 "" {TEXT -1 104 "For Situation 3
, we make the odd, periodic extension of f. For this example, the peri
od will be 2 L = 8." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "L:=4;
\nF3:=x->opf(F1,x);" }}}{PARA 0 "" 0 "" {TEXT -1 237 "We draw a graph \+
of each of these. For the purposes of drawing these graphs, which will
have considerable overlap, we not only color them differently, but al
so artificially off-set them. This off-setting is for illustration pur
poses only." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot(['F1(x)'
,'F2(x)'+.015,'F3(x)'-.015],x=-8..8,color=[black,red,green]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We define the solution u for ea
ch example." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "u1:=(t,x)->(F
1(x+t)+F1(x-t))/2:\nu2:=(t,x)->(F2(x+t)+F2(x-t))/2;\nu3:=(t,x)->(F3(x+
t)+F3(x-t))/2;" }}}{PARA 0 "" 0 "" {TEXT -1 62 "We will draw the graph
of each u, and make animations of each." }}{PARA 0 "" 0 "" {TEXT -1
40 "Here is u1 as a solution in Situation 1." }}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 67 "plot3d('u1(t,x)',x=-8..8,t=0..8,axes=NORMAL,orient
ation=[-115,25]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "with(p
lots):\nanimate('u1(t,x)',x=-8..8,t=0..20);" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 65 "plot([u1(0,x),u1(3,x),u1(5,x)],x=-6..10,color=[BLA
CK,RED,GREEN]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 40 "Here is u2 as a solution in Situation 2." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot3d('u2(t,x)',x=0..8,t=0.
.8,axes=NORMAL,orientation=[-115,25]);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 48 "with(plots):\nanimate('u2(t,x)',x=0..10,t=0..20);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plot(['u2(0,x)','u2(3,x)'+.
015,'u2(5,x)'+.025],x=0..10,\n color=[BLACK,RED,GREEN]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40
"Here is u3 as a solution in Situation 3." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 66 "plot3d('u3(t,x)',x=0..4,t=0..8,axes=NORMAL,orientatio
n=[-115,25]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "with(plots
):\nanimate('u3(t,x)',x=0..4,t=0..20);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 85 "plot(['u3(0,x)','u3(1,x)'+.015,'u3(2,x)'+.025],x=0..4
,\n color=[BLACK,RED,GREEN]);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Structure of Solu
tions" }}{PARA 0 "" 0 "" {TEXT -1 137 "Before we leave these examples \+
too far behind, they call our attention to at least four properties of
the wave equation that are central." }}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}{PARA 0 "" 0 "" {TEXT -1 162 "(1) Bumps in the initial distribution \+
are split into two parts each having half the height of the original. \+
One bump moves to the right and one moves to the left." }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 291 "(2) We are willing \+
to talk about functions being solutions to the wave equation that are \+
not even differentiable once, much less twice. This luxury is a result
of having d'Alembert's formulation of solutions. Such an extension of
the idea of solution can be made precise with the concept of " }
{TEXT 256 14 "weak solutions" }{TEXT -1 44 ". We will not pursue this \+
idea further here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 18 "(3) Choose points " }{XPPEDIT 18 0 "[t[0], x[0]];" "6#7$&
%\"tG6#\"\"!&%\"xG6#F'" }{TEXT -1 137 " out in the plane and ask what \+
points of the extended f and g influence the behavior of the solution \+
at this chosen point. We see that u(" }{XPPEDIT 18 0 "t[0],x[0];" "6$&
%\"tG6#\"\"!&%\"xG6#F&" }{TEXT -1 32 ") involves the values of f at \+
" }{XPPEDIT 18 0 "x[0]+c*t[0];" "6#,&&%\"xG6#\"\"!\"\"\"*&%\"cGF(&%\"t
G6#F'F(F(" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "x[0]-c*t[0];" "6#,&&
%\"xG6#\"\"!\"\"\"*&%\"cGF(&%\"tG6#F'F(!\"\"" }{TEXT -1 50 " and invo
lves the values of g over the interval [" }{XPPEDIT 18 0 "x[0]-c*t[0];
" "6#,&&%\"xG6#\"\"!\"\"\"*&%\"cGF(&%\"tG6#F'F(!\"\"" }{TEXT -1 3 " , \+
" }{XPPEDIT 18 0 "x[0]+c*t[0]" "6#,&&%\"xG6#\"\"!\"\"\"*&%\"cGF(&%\"tG
6#F'F(F(" }{TEXT -1 32 " ]. This interval, then, is the " }{TEXT 257
20 "domain of dependence" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "[t[0], x
[0]];" "6#7$&%\"tG6#\"\"!&%\"xG6#F'" }{TEXT -1 150 " . If we change th
e initial conditions outside this interval and leave it the same insid
e, the change will not influence the value of u at this point." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "(4) If f \+
and g are zero outside some interval [ a, b], then u will be zero for \+
" }{XPPEDIT 18 0 "b+c*t <= x;" "6#1,&%\"bG\"\"\"*&%\"cGF&%\"tGF&F&%\"x
G" }{TEXT -1 9 " and for " }{XPPEDIT 18 0 "x <= a-c*t;" "6#1%\"xG,&%\"
aG\"\"\"*&%\"cGF'%\"tGF'!\"\"" }{TEXT -1 80 " . Thus, information trav
els no faster than speed c to the left or to the right." }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 11 "Assignment:" }{TEXT -1 87 " Gi
ve the solution u for all three situations if g = 0 and f = sin on the
interval [0, " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }
{TEXT -1 28 " ] and zero everywhere else." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 369 "The first part of these notes are
intended to summarize the progression of complication for the wave eq
uation: a string infinite in both directions, a string infinite in onl
y one direction, and a finite string. The solutions of d'Alembert for \+
these three cases illustrates the contrasts. What happens next is a re
sult of asking about the vibrations in a viscous medium." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1
1 }{PAGENUMBERS 0 1 2 33 1 1 }