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{SECT 0 {PARA 256 "" 0 "" {TEXT 260 42 "Section 5.5: Different Boundar
y Conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "
" {TEXT 280 30 "Maple Packages for Section 5.5" }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 189 "We exami
ne a variety of boundary conditions that can be associated with the wa
ve equation. We will provide illustrations for how these boundary cond
itions would look in a vibrating string. " }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT 256 16 "Fixed endpoints." }{TEXT -1 188 "
Fixed endpoints is the situation we have considered to this point in \+
these notes. Here is a graph of a function that has this boundary con
dition. Mathematically, the conditions have been" }}{PARA 0 "" 0 ""
{TEXT -1 10 " " }{TEXT 266 1 "u" }{TEXT -1 1 "(" }{TEXT 265
1 "t" }{TEXT -1 13 ", 0) = 0 and " }{TEXT 264 1 "u" }{TEXT -1 1 "(" }
{TEXT 263 1 "t" }{TEXT -1 2 ", " }{TEXT 262 1 "L" }{TEXT -1 6 ") = 0.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "L:=Pi;\nu:=(t,x)->2*sin(
x)*cos(t);" }}}{PARA 0 "" 0 "" {TEXT -1 93 "We check that this functio
n satisfies the wave equation and is zero at the boundaries: 0 and " }
{TEXT 261 1 "L" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 50 "diff(u(t,x),t,t)-diff(u(t,x),x,x);\nu(t,0);\nu(t,L);" }}}
{PARA 0 "" 0 "" {TEXT -1 31 "We graph the initial condition." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(u(0,x),x=0..L);" }}}
{PARA 0 "" 0 "" {TEXT -1 126 "Here is an animation of the solution. Re
member that it is what happens at the boundaries that is our interest \+
in this Section." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "animate(
u(t,x),x=0..L,t=0..8);" }}}{PARA 0 "" 0 "" {TEXT -1 49 "Finally, here \+
is a graph of the solution surface." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 66 "plot3d(u(t,x),x=0..L,t=0..2*Pi,axes=NORMAL,orientatio
n=[-160,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 19 "Elastic attach
ment." }{TEXT -1 287 " This situation assumes we have a spring, or oth
er elastic device attached to the string. Such an arrangement would te
nd to bring the string back from any displacement. How hard the elasti
c device pulls the string back depends on how far it is displaced. Suc
h conditions could be written" }}{PARA 0 "" 0 "" {TEXT -1 10 " \+
" }{XPPEDIT 18 0 "diff(u,x);" "6#-%%diffG6$%\"uG%\"xG" }{TEXT -1 11
"(t, 0) = k " }{TEXT 267 1 "u" }{TEXT -1 1 "(" }{TEXT 268 1 "t" }
{TEXT -1 13 ", 0) and " }{XPPEDIT 18 0 "diff(u,x);" "6#-%%diffG6$%
\"uG%\"xG" }{TEXT -1 13 "(t, L) = - k " }{TEXT 269 1 "u" }{TEXT -1 7 "
(t, L)." }}{PARA 0 "" 0 "" {TEXT -1 141 "Such conditions are reminisce
nt of radiation cooling for heat diffusion problems. Here is an illust
ration for how such a system would behave." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 "L:=Pi/2;\nu:=(t,x)->cos(t)*sin(x+Pi/4);" }}}{PARA
0 "" 0 "" {TEXT -1 19 "We check that this " }{TEXT 270 1 "u" }{TEXT
-1 57 " satisfies the wave equation and the boundary conditions." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "diff(u(t,x),t,t)-diff(u(t,x)
,x,x);\nD[2](u)(t,0)-u(t,0);\nD[2](u)(t,Pi/2)+u(t,L);" }}}{PARA 0 ""
0 "" {TEXT -1 33 "Here is the initial displacement." }}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 27 "plot(u(0,x),x=0..L,y=0..1);" }}}{PARA 0 ""
0 "" {TEXT -1 21 "Watch the boundaries." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 33 "animate(u(t,x),x=0..Pi/2,t=0..L);" }}}{PARA 0 "" 0 "
" {TEXT -1 30 "Here is the solutions surface." }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 65 "plot3d(u(t,x),x=0..L,t=0..2*Pi,axes=NORMAL,orien
tation=[-30,70]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 20 "Friction
less sleeve." }{TEXT -1 233 " This models having the string attached t
o carts which hold the string with a horizontal tangent at the ends, b
ut allows the string to move up and down as the cart runs along a fric
tionless track. This situation can be described with" }}{PARA 0 "" 0 "
" {TEXT -1 10 " " }{XPPEDIT 18 0 "diff(u,x);" "6#-%%diffG6$%
\"uG%\"xG" }{TEXT -1 15 "(t, 0) = 0 and " }{XPPEDIT 18 0 "diff(u,x);"
"6#-%%diffG6$%\"uG%\"xG" }{TEXT -1 12 "(t, L) = 0. " }}{PARA 0 "" 0 "
" {TEXT -1 87 "In this illustration, the right end is fixed, the left \+
end is on a frictionless sleeve." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 38 "L:=Pi/2;\nu:=(t,x)->sin(t+Pi/2)*cos(x);" }}}{PARA 0 "" 0 ""
{TEXT -1 45 "We check the PDE and the boundary conditions." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "diff(u(t,x),t,t)-diff(u(t,x),x,x);
\nD[2](u)(t,0);\nu(t,L);" }}}{PARA 0 "" 0 "" {TEXT -1 33 "Here is the \+
initial distribution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plo
t(u(0,x),x=0..L);" }}}{PARA 0 "" 0 "" {TEXT -1 80 "The animation is a \+
good way to get an understanding for the boundary conditions." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "animate(u(t,x),x=0..L,t=0..2
*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot3d(u(t,x),x=0.
.L,t=0..2*Pi,axes=NORMAL,orientation=[-40,55]);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 259 27 "Changing Boundary Condition" }{TEXT -1 377 ". We \+
suppose we have a half infinite string in this model and we change the
boundary condition with time. Take the initial conditions to be zero,
and take U(t, 0) = b(t). We expect to see a signal move down the stri
ng. Here is an analysis of the problem. With no appeal to initial cond
itions or boundary conditions, we found in Section 5.2 that solutions \+
should have the form " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 12
"(x + c t) + " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 98 "(x - c t
). Using that the initial conditions are zero for x > 0, recall that w
e can conclude that " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 12 "(
x) = 0 and " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 30 "(x) = 0 fo
r x > 0. The term " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 69 "(
x + c t) will be zero since c > 0 and t > 0. Thus, whatever happens,"
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " u(
t, x) = " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 10 "(x - c t)," }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "and this
is zero as long as x > c t. We ask how to extend " }{XPPEDIT 18 0 "p
hi;" "6#%$phiG" }{TEXT -1 107 " to the negative numbers. The answer mu
st lie in the boundary condition. Recall that it did earlier, also. "
}}{PARA 0 "" 0 "" {TEXT -1 37 " We know that u(t, 0) = b(t), so "
}}{PARA 0 "" 0 "" {TEXT -1 18 " " }{XPPEDIT 18 0 "phi
;" "6#%$phiG" }{TEXT -1 27 "(0 - c t) = u(t, 0) = b(t)." }}{PARA 0 ""
0 "" {TEXT -1 36 "So, for negative numbers n, we have " }{XPPEDIT 18
0 "phi;" "6#%$phiG" }{TEXT -1 110 "(n) = b(-n/c). Here is an example. \+
Think of standing at the end of a long rope and moving the end up and \+
down." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "b:=x->sin(x);" }}}
{PARA 0 "" 0 "" {TEXT -1 90 "From what came above, here would be the s
olution for the wave equation with this boundary." }}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 40 "u:=(t,x)->piecewise(x " 0 "" {MPLTEXT 1 0 33 "animate(u(t,x),x=0..10, t=0.
.10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot3d(u(t,x),x=0.
.10,t=0..10,axes=NORMAL,orientation=[-115,55]);" }}}{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 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 425 "The intent of this Section was to recall that there ca
n be a variety of boundary conditions that change the character of the
solutions for the wave equation. Some of these were illustrated above
. Before the reader grows weary with the endless variations possible, \+
we move to a new chapter and step up the dimension. We next consider p
artial differential equations associated with the steady state equatio
n in multidimensions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1
{PARA 3 "" 0 "" {TEXT 279 16 "Unassisted Maple" }}{PARA 0 "" 0 ""
{TEXT -1 125 "This moving boundary problem suggests ideas about sendin
g signals. Suppose we know the speed of the wave along the string is \+
" }{TEXT 271 1 "c" }{TEXT -1 86 " = 3. We ask, how long will it take f
or the peak of the wave to first reach the point " }{TEXT 272 1 "x" }
{TEXT -1 166 " = 10? This is a problem that Maple can solve unassisted
. We ask Maple to solve an equation involving the unknown as part of t
he argument of a trigonometric function." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "First we define " }{TEXT 273 2 "u.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "u:=(t,x)->piecewise(x " 0 ""
{MPLTEXT 1 0 25 "solve(sin((t-10)/c)=1,t);" }}}{PARA 0 "" 0 "" {TEXT
-1 64 "To get a visualization of this answer, we choose a particular c
." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "c:=3;" }}}{PARA 0 "" 0 "
" {TEXT -1 12 "Having this " }{TEXT 277 1 "c" }{TEXT -1 36 ", watch to
see that at the value of " }{TEXT 276 1 "t" }{TEXT -1 38 " found abov
e, the wave first peaks at " }{TEXT 278 1 "x" }{TEXT -1 7 " = 10. " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "animate(u(t,x),x=0..10,t=0..
10+Pi*c/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "EMAIL: herod@ma
th.gatech.edu or jherod@tds.net" }}{PARA 0 "" 0 "" {TEXT -1 38 "UR
L: http://www.math.gatech.edu/~herod" }}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}{PARA 257 "" 0 "" {TEXT -1 36 "Copyright \251 2003 by James V. Her
od" }}{PARA 257 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 0
}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }