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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 40 "Module 26: Different Boundary C
onditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
196 "We examine a variety of boundary conditions. We will provide illu
strations for how these boundary conditions would look in a vibrating \+
string. We will need the plots package to provide animations." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 20 "(1) Fixed endpoints
." }{TEXT -1 176 " This is the situation we have considered to this po
int in these notes. Here is a graph of a function that has this bound
ary condition. Mathematically, the conditions would be" }}{PARA 0 ""
0 "" {TEXT -1 38 " u(t, 0) = 0 and u(t, L) = 0." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u:=(t,x)->2*sin(x)*cos(t);" }}}
{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);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21
"plot(u(0,x),x=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a
nimate(u(t,x),x=0..Pi,t=0..8);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 67 "plot3d(u(t,x),x=0..Pi,t=0..2*Pi,axes=NORMAL,orientation=[-160,
60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 23 "(2) Elastic attachment
." }{TEXT -1 199 " This situation assumes we have a spring, or other e
lastic device attached to the string. Such an arrangement would tend t
o bring the string back to the displacement. Such conditions could be \+
written" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "di
ff(u,x);" "6#-%%diffG6$%\"uG%\"xG" }{TEXT -1 27 "(t, 0) = k u(t, 0) \+
and " }{XPPEDIT 18 0 "diff(u,x);" "6#-%%diffG6$%\"uG%\"xG" }{TEXT
-1 21 "(t, L) = - k u(t, L)." }}{PARA 0 "" 0 "" {TEXT -1 81 "Such cond
itions are reminiscent of radiation cooling for heat diffusion problem
s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "u:=(t,x)->cos(t)*sin(x
+Pi/4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "diff(u(t,x),t,t)
-diff(u(t,x),x,x);\nD[2](u)(t,0)-k*u(t,0);\nD[2](u)(t,Pi/2)-k*u(t,Pi/2
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(u(0,x),x=0..Pi/2
,y=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "animate(u(t,x)
,x=0..Pi/2,t=0..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot
3d(u(t,x),x=0..Pi/2,t=0..Pi,axes=NORMAL,orientation=[25,60]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 24 "(3) Frictionless sleeve." }
{TEXT -1 233 " This models having the string attached to carts which h
old the string with a horizontal tangent at the ends, but allows the s
tring to move up and down as the cart runs along a frictionless 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
$%\"uG%\"xG" }{TEXT -1 12 "(t, L) = 0. " }}{PARA 0 "" 0 "" {TEXT -1
78 "In this illustration, one end is fixed, the other is on a friction
less sleeve." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "u:=(t,x)->si
n(t+Pi/2)*cos(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "diff(u
(t,x),t,t)-diff(u(t,x),x,x);\nD[2](u)(t,0);\nu(t,Pi/2);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(u(0,x),x=0..Pi/2);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "animate(u(t,x),x=0..Pi/2,t=0..8);"
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "plot3d(u(t,x),x=0..Pi/2,t
=0..2*Pi,axes=NORMAL,orientation=[-70,55]);" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 259 31 "(4) Changing Boundary Condition" }{TEXT -1 362 ". We s
uppose we have a half infinite string in this model and we change the \+
boundary condition with time. Take the initial conditions to be zero, \+
and take U(t, 0) = b(t). We expect to see a signal move down the strin
g. Here is an analysis of the problem. With no appeal to initial condi
tions or boundary conditions, we found that solutions should have the \+
form " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 12 "(x + c t) + " }
{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 98 "(x - c t). Using that th
e initial conditions are zero for x > 0, recall that we can conclude t
hat " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 12 "(x) = 0 and " }
{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 30 "(x) = 0 for x > 0. The \+
term " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 70 "(x + c t) will \+
be zero since c > 0 and t > 0. Thus, what ever happens," }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " u(t, x) = " }
{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 10 "(x - c t)," }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "and this is zero a
s long as x > c t. We ask how to extend " }{XPPEDIT 18 0 "phi;" "6#%$
phiG" }{TEXT -1 145 " to the negative numbers. The answer must lie in \+
the boundary condition. Recall that it did earlier, also. Well, we kno
w that u(t, 0) = b(t), so " }}{PARA 0 "" 0 "" {TEXT -1 10 " \+
" }{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 34 "(n) = b(-n/c). H
ere is an example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "b:=x->
sin(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "u:=(t,x)->piecew
ise(x " 0 "" {MPLTEXT 1 0 33 "ani
mate(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 260 11 "Assignment:" }{TEXT -1
97 " Send a signal down a half infinite string that consists of only o
ne period of the sine function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "We can no
t leave the subject of vibrating strings without saying a word or two \+
about music and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 ""
0 "" {TEXT -1 14 "Standing waves" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 67 "Here we say a bit about standing waves. L
ook first at these graphs." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
64 "plot([sin(x),sin(2*x),sin(3*x)],x=0..Pi,color=[BLACK,RED,BLUE]);"
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 ""
{TEXT -1 164 "With these as initial functions, we know what are the so
lutions for the wave equation.\n sin( x ) cos( t ), or sin( 2 x ) \+
cos( 2 t), or sin( 3 x ) cos( 3 t )." }}{PARA 0 "" 0 "" {TEXT -1
81 "To see what happens to any other initial distribution, we make the
Fourier series" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 12 " f(x) = " }{XPPEDIT 18 0 "sum(a[n]*sin(n*x),n);" "6#-
%$sumG6$*&&%\"aG6#%\"nG\"\"\"-%$sinG6#*&F*F+%\"xGF+F+F*" }{TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 6 "to get" }}{PARA 0 "" 0 "" {TEXT -1 23
" u( t, x ) = " }{XPPEDIT 18 0 "sum(a[n]*sin(n*x)*cos(n*t),n
);" "6#-%$sumG6$*(&%\"aG6#%\"nG\"\"\"-%$sinG6#*&F*F+%\"xGF+F+-%$cosG6#
*&F*F+%\"tGF+F+F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 18 "Here's an example." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 32 "f:=x->piecewise(x " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=0..Pi);" }}}{PARA 0 ""
0 "" {TEXT -1 36 "Fourier coefficients can be found by" }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "for n from 1 to 5 do\n a[n]:=in
t(f(x)*sin(n*x),x=0..Pi)/int(sin(n*x)^2,x=0..Pi);\nod;\nn:='n';" }}}
{PARA 0 "" 0 "" {TEXT -1 51 "Here is a check that this is a good repre
sentation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot([f(x),sum
(a[n]*sin(n*x),n=1..5)],x=0..Pi,color=[BLACK,RED]);" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 42 "Here are
the standing waves that are used." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 64 "plot([sin(x),sin(3*x),sin(5*x)],x=0..Pi,color=[BLACK,
RED,BLUE]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0
"" 0 "" {TEXT -1 70 "Here are the standing waves multiplied by the app
ropriate coefficient." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "plo
t([a[1]*sin(x),a[3]*sin(3*x),a[5]*sin(5*x)],x=0..Pi,\n color=[BLAC
K,RED,BLUE]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 398 "What sounds go
od? Even if you are not a musician, you probably know that the C-major
chord is C - E -G. This sounds good. Why it sounds good is a combinat
ion of mathematics and biology. What to note here is the good approxim
ation for the plucked string could resemble C, together with faint ove
rtones of E and G. Let me remind you that if C = 256 vibrations per se
cond, then E = ~323 and G = ~ 384." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "256,256*3/2,256*5/4;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 283 "This plucked string would be a
bout C if the vibrations were 256 vibrations per second. There are tho
se other terms. I have become deaf to high pitched signals, especially
as faint as the fourth, fifth, and higher terms would be. Thus, I wou
ld hear only the first three terms anyway. " }}{PARA 0 "" 0 "" {TEXT
-1 209 " Other initial distributions than this string plucked in t
he middle might not have this same distribution. there might be a prom
inent C# overtone. C and C# played together sounds BAD. Try it on your
piano." }}{PARA 0 "" 0 "" {TEXT -1 90 " The mathematics of music \+
would be an interesting direction to go from here. We won't." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 1 " " }}}{MARK "0 0" 40 }{VIEWOPTS 1 1 0 1 1 1803
1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }