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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Deepsea.mws   18 Aug 95   \+
(phase velocity is twice the group velocity). " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 378 "This is a problem of dis
persion of waves in deep sea waters. Let us only work with two waves a
nd observe the beat pattern. We should be able to see that the phase v
elocity of the individual waves is much greater than the group velocit
y. The dispersion relation is:omega = sqrt(k*g), where omega is the an
gular frequency, k the wave vector, and g the acceleration due to grav
ity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Re
marks:" }}{PARA 0 "" 0 "" {TEXT -1 52 "Illustrates animation with spec
ific colors assigned." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 "" {TEXT -1 34 "write down the dispersion relation" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "omega:=sqrt(g*k);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 74 "psi:=sin(k*x-omega*t); #general form of a
 wave function for deep sea waves" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "y[1]:=subs(k=k1,psi);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "y[2]:=subs(k=k2,psi);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "vp:=ome
ga/k;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "vg:=diff(omega,k);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "ratio:=vp/vg; # phase v
elocity over group velocity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "evalf(subs(k=10,g=98/10,ratio));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "y_total:=y[1]+y[2];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "values:=\{k1=5,k2=9/2,g=98/10\};" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
86 "h:=animate(subs(values,y_total),x=-2..8,t=0..3,color=red,    frame
s=30,numpoints=100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "g:=
animate(subs(values,y[1]),      x=-2..8,t=0..3,color=black,frames=30,n
umpoints=100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 " i:=anima
te(subs(values,y[2]),       x=-2..8,t=0..3,color=blue,  frames=30,nump
oints=100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "j:=display([
h,g]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display([i,j]);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 197 "Notice that the individual peaks move ahead of the 'gr
oup' peak. By single stepping through the animation you can see a trai
ling peak come in from the left, and overtake the peak of the 'envelop
e'." }}}}{MARK "20 0 0" 118 }{VIEWOPTS 1 1 0 1 1 1803 }