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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 47 "Quelques calculs autour d
e la membrane vibrante" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 32 "Partant de l'equation des ondes," }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "Delta(F) = diff(F(t),t,t);" "6#/-%&DeltaG6#%\"FG-%%diff
G6%-F'6#%\"tGF-F-" }}{PARA 0 "" 0 "" {TEXT -1 3 "ou " }{XPPEDIT 18 0 "
Delta;" "6#%&DeltaG" }{TEXT -1 211 " est le laplacien, il s'agit d'app
rendre a manipuler Maple en vue d'experimenter autour de la \"resoluti
on\" de l'equation des ondes sur une membrane circulaire (un tambour).
 Le principe du calcul est le suivant :" }}{PARA 0 "" 0 "" {TEXT -1 
108 "1.  l'equation se separe en temps, ramenant ainsi l'etude a la re
cherche de fonctions propres du laplacien ;" }}{PARA 0 "" 0 "" {TEXT 
-1 157 "2. la partie en espace se separe en coordonnees polaires, la p
artie angulaire est elementaire, la partie radiale s'exprime a l'aide \+
des fonctions de Bessel ;" }}{PARA 0 "" 0 "" {TEXT -1 99 "3. etant don
nees des conditions initales (F(R)=0 au bord, qu'on prendra en R=1, et
 F(t), ainsi que " }{XPPEDIT 18 0 "diff(F(t),t);" "6#-%%diffG6$-%\"FG6
#%\"tGF)" }{TEXT -1 37 " donnes en t=0 comme fonctions de r, " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 79 ", il reste a develop
per la solution generale sur la base des fonctions propres." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 305 "Le temps ne pe
rmettra pas d'aborder ce dernier point. L'objectif de la seance (qui e
st meme peut-etre un peu optimiste) est de parvenir a afficher l'anima
tion d'un des modes propres. Le but est de decouvrir le systeme \"par \+
immersion\", en alternant manipulations simples et recherches dans l'a
ide en ligne." }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 92 "1. Exprimer l'eq
uation des ondes en coordonnes polaires et effectuer la separation en \+
temps." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "F:=f(r,theta,t);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%\"fG6%%\"rG%&thetaG%\"tG" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "wave_eq:=linalg[laplacian
](F,[r,theta,z],coords=cylindrical)-diff(F,t,t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(wave_eqG,&*&,(-%%diffG6$-%\"fG6%%\"rG%&thetaG%\"tGF.
\"\"\"*&F.F1-F)6$F+-%\"$G6$F.\"\"#F1F1*&-F)6$F+-F66$F/F8\"\"\"F.!\"\"F
1F>F.F?F1-F)6$F+-F66$F0F8!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 38 "eval(subs(F=A(r,theta)*T(t),wave_eq));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&,(*&-%%diffG6$-%\"AG6$%\"rG%&thetaGF-\"\"\"-%\"TG6#
%\"tGF/F/*(F-F/-F(6$F*-%\"$G6$F-\"\"#F/F0\"\"\"F/*&*&-F(6$F*-F86$F.F:F
/F0F;F;F-!\"\"F/F;F-FBF/*&F*F/-F(6$F0-F86$F3F:F/!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "expand(%/A(r,theta)/T(t));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&-%%diffG6$-%\"AG6$%\"rG%&thetaGF+
\"\"\"*&F(\"\"\"F+\"\"\"!\"\"\"\"\"*&-F&6$F(-%\"$G6$F+\"\"#F-F(F1F2*&-
F&6$F(-F76$F,F9F-*&F(\"\"\")F+\"\"#F-F1F2*&-F&6$-%\"TG6#%\"tG-F76$FIF9
F-FFF1!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "time_part:=n
umer(select(has,%,t)-lambda^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*
time_partG,&-%%diffG6$-%\"TG6#%\"tG-%\"$G6$F,\"\"#!\"\"*&)%'lambdaGF0
\"\"\"F)\"\"\"F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "space_p
art:=numer(select(has,%%,r)+lambda^2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%+space_partG,**&-%%diffG6$-%\"AG6$%\"rG%&thetaGF-\"\"\"F-F/F/*
&-F(6$F*-%\"$G6$F-\"\"#F/)F-F6\"\"\"F/-F(6$F*-F46$F.F6F/*()%'lambdaGF6
F8F*F/F7F8F/" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 33 "2. Resoudre la pa
rtie temporelle." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sol_tim
e:=dsolve(time_part,T(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)sol_t
imeG/-%\"TG6#%\"tG,&*&%$_C1G\"\"\"-%$sinG6#*&%'lambdaGF-F)F-F-F-*&%$_C
2GF--%$cosGF0F-F-" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 62 "3. Separer l
a partie spaciale en parties angulaire et radiale." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 50 "eval(subs(A(r,theta)=phi(r)*U(theta),space_
part));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(-%%diffG6$-%$phiG6#%\"r
GF+\"\"\"-%\"UG6#%&thetaGF,F+F,F,*(-F&6$F(-%\"$G6$F+\"\"#F,F-\"\"\")F+
F7F8F,*&F(F,-F&6$F--F56$F0F7F,F,**)%'lambdaGF7F8F(F8F-F8F9F8F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "expand(%/phi(r)/U(theta));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&*&-%%diffG6$-%$phiG6#%\"rGF,\"\"
\"F,F-\"\"\"F)!\"\"F-*&*&-F'6$F)-%\"$G6$F,\"\"#F-)F,F7F.F.F)F/F-*&-F'6
$-%\"UG6#%&thetaG-F56$F?F7F.F<F/F-*&)%'lambdaGF7F.F8F.F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "polar_part:=numer(select(has,%,thet
a)+kappa^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+polar_partG,&-%%dif
fG6$-%\"UG6#%&thetaG-%\"$G6$F,\"\"#\"\"\"*&)%&kappaGF0\"\"\"F)F1F1" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "radius_part:=numer(select(h
as,%%,r)-kappa^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,radius_partG,
**&-%%diffG6$-%$phiG6#%\"rGF-\"\"\"F-F.F.*&-F(6$F*-%\"$G6$F-\"\"#F.)F-
F5\"\"\"F.*()%'lambdaGF5F7F6F7F*F.F.*&)%&kappaGF5F7F*F7!\"\"" }}}
{EXCHG {PARA 5 "" 0 "" {TEXT -1 102 "4. Resoudre la partie angulaire e
t deduire de la physique une contrainte sur les valeurs du parametre.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dsolve(polar_part,U(the
ta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"UG6#%&thetaG,&*&%$_C1G\"
\"\"-%$sinG6#*&%&kappaGF+F'F+F+F+*&%$_C2GF+-%$cosGF.F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "assume(kappa,integer);" }}}{EXCHG 
{PARA 5 "" 0 "" {TEXT -1 122 "5. Resoudre la partie radiale. La physiq
ue impose que la solution soit reguliere a l'origine. Par definition, \+
la fonction " }{XPPEDIT 18 0 "J[nu];" "6#&%\"JG6#%#nuG" }{TEXT -1 42 "
 de Bessel est reguliere a l'origine pour " }{XPPEDIT 18 0 "nu;" "6#%#
nuG" }{TEXT -1 91 " entier. Verifier que c'est la seule solution possi
ble a une constante multiplicative pres." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "dsolve(radius_part,phi(r));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$phiG6#%\"rG,&*&%$_C1G\"\"\"-%(BesselJG6$%'kappa|irG
*(-%%csgnG6#%'lambdaGF+F4F+F'F+F+F+*&%$_C2GF+-%(BesselYGF.F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "dsolve(radius_part,phi(r),se
ries);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$phiG6#%\"rG,&*(%$_C1G\"
\"\")F',$%'kappa|irG!\"\"F+++F'F+\"\"!,$*&*$)%'lambdaG\"\"#\"\"\"F8,&F
.!\"%\"\"%F+!\"\"F/\"\"#*&*$)F6F;F8F8*&F9\"\"\",&F.!\")\"#;F+\"\"\"F<
\"\"%-%\"OG6#F+\"\"'F+F+*(%$_C2GF+)F'F.F+++F'F+F1,$*&*$F5F8F8,&F.F;F;F
+F<F/\"\"#*&*$F@F8F8*&FS\"\"\",&F.\"\")FEF+\"\"\"F<\"\"%FH\"\"'F+F+" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Ce developpement montre que l'es
pace des solutions regulieres a l'origine est de dimension 1. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sol_radius:=eval(subs(Bessel
Y=0,%%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+sol_radiusG/-%$phiG6#%
\"rG*&%$_C1G\"\"\"-%(BesselJG6$%'kappa|irG*(-%%csgnG6#%'lambdaGF,F5F,F
)F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "La fonction csgn a pour \+
effet de renvoyer |" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 
38 "|, il suffit donc de se restreindre a " }{XPPEDIT 18 0 "0 < lambda
;" "6#2\"\"!%'lambdaG" }{TEXT -1 19 ". De meme, changer " }{XPPEDIT 
18 0 "kappa;" "6#%&kappaG" }{TEXT -1 4 " en " }{XPPEDIT 18 0 "-kappa;
" "6#,$%&kappaG!\"\"" }{TEXT -1 62 " change le signe de BesselJ, on pe
ut donc se contenter du cas " }{XPPEDIT 18 0 "kappa;" "6#%&kappaG" }
{TEXT -1 9 " positif." }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 107 "6. Expl
oiter la condition aux limites F(R)=0 en R=1. (Il existe miraculeuseme
nt une fonction BesselJZeros)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 35 "sol_lambda:=BesselJZeros(kappa,nu);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%+sol_lambdaG-%-BesselJZerosG6$%'kappa|irG%#nuG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sol_radius:=BesselJ(kappa,so
l_lambda*r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+sol_radiusG-%(Besse
lJG6$%'kappa|irG*&-%-BesselJZerosG6$F(%#nuG\"\"\"%\"rGF." }}}{EXCHG 
{PARA 5 "" 0 "" {TEXT -1 81 "7. Animer en fonction du temps la partie \+
radiale pour un mode propre particulier." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 121 "plots[animate](subs(kappa=2,nu=3,sol_radius*cos(sol_
lambda*t)),r=0..1,t=0..2*Pi/subs(kappa=2,nu=3,sol_lambda),frames=10);
" }}{PARA 13 "" 1 "" {INLPLOT "6%-%(ANIMATEG6,7#-%'CURVESG6$7S7$\"\"!F
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