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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "piano.mws  8/24/95" }}
{PARA 0 "" 0 "" {TEXT -1 336 "This is a problem of dispersion of sound
 waves on a piano wire.The dispersion relation of the piano wire is: (
omega)^2/(k)^2=T/mu+alpha*(k^2), where omega is the angular frequency,
 k the wave vector, T the tension of the string, and alpha a small con
stant value. Let us take several individual waves and observe the wave
 group behavior." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 84 "The small individual 'phase' peaks move forward through t
he envelope or 'group peak." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 32 "d:=omega^2/k^2=T/(mu)+alpha*k^2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "for n from 0 to 10" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
2 "do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y.n:=sin((k+n*k1)*x-(omega
+n*omega1)*t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "psi:=simplify(convert(\{y.(0..10)\}
,`+`),trig);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sol:=solve(
d,omega);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "v[p]:=simplify
(sol[1]/k);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "vg:=diff(v[p
],k)=omega1/k1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "vgr:=sol
ve(vg,omega1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "vphase_ov
er_vgroup:=v[p]/vgr;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "psi
fin:=subs(\{omega1=solve(vg,omega1),omega=sol[1]\},psi);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "values:=\{k=5,k1=5/10,alpha=2/10,T=
3,mu=1/100\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "m:=subs(va
lues,psifin);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "animate(m
,x=10..20,t=0..0.2,color=blue,frames=20,numpoints=100); # the small pe
aks move forward thru the envelope" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 99 "The individual peaks come up and move through the 'envelope' or
 'group' peak, showing vphase>vgroup" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "vgval:=evalf(subs(values,lhs(vg)));" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "vpval:=evalf(subs(values,v[p]));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "ratio:=vpval/vgval;" }}}}{MARK "0 0 0" 
18 }{VIEWOPTS 1 1 0 1 1 1803 }