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{SECT 0 {PARA 201 "" 0 "" {TEXT 205 54 "Transverse modes of a free bea
m  4/14/2004 for MTH 444" }}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 
8 "restart:" }{MPLTEXT 1 206 69 "\nu:=c1*exp(alpha*x)+c2*exp(-alpha*x)
+c3*cos(alpha*x)+c4*sin(alpha*x);" }{MPLTEXT 1 206 15 "\nu1:=diff(u,x)
;" }{MPLTEXT 1 206 16 "\nu2:=diff(u1,x);" }{MPLTEXT 1 206 16 "\nu3:=di
ff(u2,x);" }{MPLTEXT 1 206 43 "\nde:=simplify(diff(u,x,x,x,x)-alpha^4*
u)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG,**&%#c1G\"\"\"-%$expG6
#*&%&alphaGF(%\"xGF(F(F(*&%#c2GF(-F*6#,$F,!\"\"F(F(*&%#c3GF(-%$cosGF+F
(F(*&%#c4GF(-%$sinGF+F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u1G,**
(%#c1G\"\"\"%&alphaGF(-%$expG6#*&F)F(%\"xGF(F(F(*(%#c2GF(F)F(-F+6#,$F-
!\"\"F(F4*(%#c3GF(-%$sinGF,F(F)F(F4*(%#c4GF(-%$cosGF,F(F)F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u2G,**(%#c1G\"\"\")%&alphaG\"\"#F(-
%$expG6#*&F*F(%\"xGF(F(F(*(%#c2GF(F)F(-F-6#,$F/!\"\"F(F(*(%#c3GF(-%$co
sGF.F(F)F(F6*(%#c4GF(-%$sinGF.F(F)F(F6" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#u3G,**(%#c1G\"\"\")%&alphaG\"\"$F(-%$expG6#*&F*F(%\"xGF(F(F(*
(%#c2GF(F)F(-F-6#,$F/!\"\"F(F6*(%#c3GF(-%$sinGF.F(F)F(F(*(%#c4GF(-%$co
sGF.F(F)F(F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/\"\"!F&" }}}
{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 40 "bc1:=simplify(subs(x=-HL
,u2)/alpha^2)=0;" }{MPLTEXT 1 206 41 "\nbc2:=simplify(subs(x= HL,u2)/a
lpha^2)=0;" }{MPLTEXT 1 206 41 "\nbc3:=simplify(subs(x=-HL,u3)/alpha^3
)=0;" }{MPLTEXT 1 206 41 "\nbc4:=simplify(subs(x= HL,u3)/alpha^3)=0;" 
}{MPLTEXT 1 206 26 "\nevalm([[coeff(bc1,c1)]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$bc1G/,**&%#c1G\"\"\"-%$expG6#,$*&%&alphaGF)%#HLGF)!
\"\"F)F)*&%#c2GF)-F+6#F.F)F)*&%#c3GF)-%$cosGF,F)F1*&%#c4GF)-%$sinGF,F)
F1\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$bc2G/,**&%#c1G\"\"\"-%$e
xpG6#*&%&alphaGF)%#HLGF)F)F)*&%#c2GF)-F+6#,$F-!\"\"F)F)*&%#c3GF)-%$cos
GF,F)F5*&%#c4GF)-%$sinGF,F)F5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%$bc3G/,**&%#c1G\"\"\"-%$expG6#,$*&%&alphaGF)%#HLGF)!\"\"F)F)*&%#c2GF
)-F+6#F.F)F1*&%#c3GF)-%$sinGF,F)F)*&%#c4GF)-%$cosGF,F)F1\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$bc4G/,**&%#c1G\"\"\"-%$expG6#*&%&al
phaGF)%#HLGF)F)F)*&%#c2GF)-F+6#,$F-!\"\"F)F5*&%#c3GF)-%$sinGF,F)F)*&%#
c4GF)-%$cosGF,F)F5\"\"!" }}{PARA 8 "" 1 "" {TEXT -1 31 "Error, unable \+
to compute coeff\n" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 38 "c
c1:=expand(exp(-alpha*HL)*(bc3-bc1));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$cc1G/,,*&\"\"#\"\"\"%#c2GF)!\"\"*(-%$expG6#*&%&alphaGF)%#HLGF)F+
%#c3GF)-%$sinGF/F)F+*(F-F+%#c4GF)-%$cosGF/F)F+*(F-F+F3F)F8F)F)*(F-F+F7
F)F4F)F+\"\"!" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 38 "cc2:=e
xpand(exp( alpha*HL)*(bc4-bc2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
$cc2G/,,*&\"\"#\"\"\"%#c2GF)!\"\"*(-%$expG6#*&%&alphaGF)%#HLGF)F)%#c3G
F)-%$sinGF/F)F)*(F-F)%#c4GF)-%$cosGF/F)F+*(F-F)F3F)F8F)F)*(F-F)F7F)F4F
)F)\"\"!" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 40 "dc1:=simpli
fy((cc1-cc2)*exp( alpha*HL));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$dc
1G/,2*&%#c3G\"\"\"-%$sinG6#*&%&alphaGF)%#HLGF)F)!\"\"*&%#c4GF)-%$cosGF
,F)F0*&F(F)F3F)F)*&F2F)F*F)F0*(-%$expG6#,$*(\"\"#F)F.F)F/F)F)F)F(F)F*F
)F0*(F8F)F2F)F3F)F)*(F8F)F(F)F3F)F0*(F8F)F2F)F*F)F0\"\"!" }}}{EXCHG 
{PARA 201 "> " 0 "" {MPLTEXT 1 206 38 "cc3:=expand(exp( alpha*HL)*(bc3
+bc1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$cc3G/,,*&\"\"#\"\"\"%#c1
GF)F)*(-%$expG6#*&%&alphaGF)%#HLGF)F)%#c3GF)-%$sinGF.F)!\"\"*(F,F)%#c4
GF)-%$cosGF.F)F5*(F,F)F2F)F8F)F5*(F,F)F7F)F3F)F)\"\"!" }}}{EXCHG 
{PARA 201 "> " 0 "" {MPLTEXT 1 206 38 "cc4:=expand(exp(-alpha*HL)*(bc4
+bc2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$cc4G/,,*&\"\"#\"\"\"%#c1
GF)F)*(-%$expG6#*&%&alphaGF)%#HLGF)!\"\"%#c3GF)-%$sinGF.F)F)*(F,F2%#c4
GF)-%$cosGF.F)F2*(F,F2F3F)F8F)F2*(F,F2F7F)F4F)F2\"\"!" }}}{EXCHG 
{PARA 201 "> " 0 "" {MPLTEXT 1 206 40 "dc2:=simplify((cc3-cc4)*exp( al
pha*HL));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$dc2G/,2*(-%$expG6#,$*(
\"\"#\"\"\"%&alphaGF.%#HLGF.F.F.%#c3GF.-%$sinG6#*&F/F.F0F.F.!\"\"*(F(F
.%#c4GF.-%$cosGF4F.F6*(F(F.F1F.F9F.F6*(F(F.F8F.F2F.F.*&F1F.F2F.F6*&F8F
.F9F.F.*&F1F.F9F.F.*&F8F.F2F.F.\"\"!" }}}{EXCHG {PARA 201 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 20 "a:=arra
y(1..2,1..2);" }{MPLTEXT 1 206 34 "\na[1,1]:=lhs(subs(c3=1,c4=0,dc1));
" }{MPLTEXT 1 206 34 "\na[1,2]:=lhs(subs(c3=0,c4=1,dc1));" }{MPLTEXT 
1 206 34 "\na[2,1]:=lhs(subs(c3=1,c4=0,dc2));" }{MPLTEXT 1 206 34 "\na
[2,2]:=lhs(subs(c3=0,c4=1,dc2));" }{MPLTEXT 1 206 10 "\nevalm(a);" }
{MPLTEXT 1 206 41 "\ndeterm:=simplify(linalg[det](evalm(a)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%&arrayG6%;\"\"\"\"\"#F(7\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6$\"\"\"F',*-%$sinG6#*&%&alphaG
F'%#HLGF'!\"\"-%$cosGF+F'*&-%$expG6#,$*(\"\"#F'F-F'F.F'F'F'F)F'F/*&F3F
'F0F'F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6$\"\"\"\"\"#,*-%$co
sG6#*&%&alphaGF'%#HLGF'!\"\"-%$sinGF,F0*&-%$expG6#,$*(F(F'F.F'F/F'F'F'
F*F'F'*&F4F'F1F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6$\"\"#\"
\"\",*-%$sinG6#*&%&alphaGF(%#HLGF(!\"\"-%$cosGF,F(*&-%$expG6#,$*(F'F(F
.F(F/F(F(F(F*F(F0*&F4F(F1F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"
aG6$\"\"#F',**&-%$expG6#,$*(F'\"\"\"%&alphaGF/%#HLGF/F/F/-%$cosG6#*&F0
F/F1F/F/!\"\"*&F*F/-%$sinGF4F/F/F2F/F8F/" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#K%'matrixG6#7$7$,*-%$sinG6#*&%&alphaG\"\"\"%#HLGF.!\"\"-%$cosGF+
F.*&-%$expG6#,$*(\"\"#F.F-F.F/F.F.F.F)F.F0*&F4F.F1F.F0,*F1F0F)F0F:F.F3
F07$F(,*F:F0F3F.F1F.F)F.Q(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%'determG,,*&\"\"%\"\"\"-%$expG6#,$*(\"\"#F(%&alphaGF(%#HLGF(F(F(!
\"\"F.F1*&F'F()-%$cosG6#*&F/F(F0F(F.F(F(*&F.F(-F*6#,$*(F'F(F/F(F0F(F(F
(F1*(F'F(F9F(F3F(F(" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 35 "
determ2:=subs(alpha*HL=ahl,determ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%(determ2G,,*&\"\"%\"\"\"-%$expG6#,$*&\"\"#F(%$ahlGF(F(F(!\"\"F.F0*&
F'F()-%$cosG6#F/F.F(F(*&F.F(-F*6#,$*&F'F(F/F(F(F(F0*(F'F(F7F(F2F(F(" }
}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 38 "determ3:=simplify(dete
rm2/exp(4*ahl));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(determ3G,,*&\"
\"%\"\"\"-%$expG6#,$*&\"\"#F(%$ahlGF(!\"\"F(F0*&F.F(-F*6#,$*&F'F(F/F(F
0F(F0*(F'F(F2F()-%$cosG6#F/F.F(F(F.F0*&F'F(F7F(F(" }}}{EXCHG {PARA 
201 "> " 0 "" {MPLTEXT 1 206 24 "plot(determ3,ahl=0..18);" }}{PARA 13 
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206 31 "\ns[1]:=fsolve(determ3=0,ahl=2);" }{MPLTEXT 1 206 31 "\ns[2]:=
fsolve(determ3=0,ahl=4);" }{MPLTEXT 1 206 31 "\ns[3]:=fsolve(determ3=0
,ahl=6);" }{MPLTEXT 1 206 31 "\ns[4]:=fsolve(determ3=0,ahl=7);" }
{MPLTEXT 1 206 10 "\neps:=0.1:" }{MPLTEXT 1 206 10 "\nnmax:=60;" }
{MPLTEXT 1 206 25 "\nfor i from 5 to nmax do " }{MPLTEXT 1 206 52 "\n \+
 s[i]:=fsolve(determ3=0,ahl=s[i-1]+eps..s[i-1]+2);" }{MPLTEXT 1 206 
19 "\nod: unassign('i'):" }{MPLTEXT 1 206 19 "\nseq(s[i],i=1..40);" }
{MPLTEXT 1 206 79 "\n#plot([seq([s[i],0],i=1..nmax)],style=point,symbo
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{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 47 "frequencies:=seq(evalf(2
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^2/Pi),0],i=1..nmax)],style=point,symbolsize=15,view=[0..2600,-1..1]);
" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
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w:=0.025;\nh:=0.0043;\nII:=w*h^3/12;\narea:=w*h;\nsigma := 8600;\nEE:=
9.1e10;\nrho:=sigma*area;      # density per unit length" }{MPLTEXT 1 
206 20 "\nEI:=EE*II;\nL:=1.33;" }{MPLTEXT 1 206 23 "\nfac:=sqrt(EI/rho
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2*Pi)*hertz;" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" 
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" {MPLTEXT 1 206 69 "semitones_from_A440:=seq(12*log[2](hertz[i]/440),
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te_name:=proc(semitones_from_A440)" }{MPLTEXT 1 206 53 "\n   local not
e_names,offset,octave,soctave,nname,ind;" }{MPLTEXT 1 206 14 "\n   off
set:=5;" }{MPLTEXT 1 206 49 "\n   octave:=floor(offset+semitones_from_
A440/12);" }{MPLTEXT 1 206 36 "\n   soctave:=convert(octave,string);" 
}{MPLTEXT 1 206 71 "\n   note_names:=[\"A\",\"Bb\",\"B\",\"C\",\"Db\",
\"D\",\"Eb\",\"E\",\"F\",\"Gb\",\"G\",\"Ab\"];" }{MPLTEXT 1 206 51 "\n
   ind:=1+(floor(0.5+semitones_from_A440) mod 12);" }{MPLTEXT 1 206 
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ames[ind],soctave);" }{MPLTEXT 1 206 10 "\nend proc:" }{MPLTEXT 1 206 
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_A440]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6hnQ$C-16\"Q#F0F$Q#F1F$Q#D2
F$Q#A2F$Q$Eb3F$Q#G3F$Q#C4F$Q#E4F$Q#G4F$Q$Bb5F$Q$Db5F$Q#E5F$Q$Gb5F$Q#A5
F$Q#B6F$Q$Db6F$Q$Eb6F$Q#F6F$Q$Gb6F$Q$Ab6F$Q$Bb7F$Q#B7F$Q#C7F$Q#D7F$Q$E
b7F$Q#E7F$Q$Gb7F$Q#G7F$Q$Ab7F$Q#A8F$Q$Bb8F$Q#B8F$Q#C8F$Q$Db8F$Q#D8F$Q$
Eb8F$Q#E8F$Q#F8F$Q$Gb8F$Q#G8F$Q$Ab8F$FMQ#A9F$Q$Bb9F$Q#B9F$FPQ#C9F$Q$Db
9F$Q#D9F$FSQ$Eb9F$Q#E9F$FUQ#F9F$FVQ$Gb9F$Q#G9F$FXQ$Ab9F$" }}}{EXCHG 
{PARA 201 "> " 0 "" {MPLTEXT 1 206 48 "libname:=\"/home/ringland/src/M
apleSound/siggen\"," }{MPLTEXT 1 206 46 "\n         \"/home/ringland/s
rc/MapleSound/wav\"," }{MPLTEXT 1 206 18 "\n         libname;" }
{MPLTEXT 1 206 14 "\nwith(SigGen);" }{MPLTEXT 1 206 11 "\nwith(WAV);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(libnameG6%QE/home/ringland/src/Ma
pleSound/siggen6\"QB/home/ringland/src/MapleSound/wavF'Q6/usr/local/ma
ple9/libF'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#73%$ASDG%'AttackG%&Decay
G%*FM_SignalG%(FM_SineG%.Integer2BytesG%$LFOG%(LimiterG%*NormalizeG%)Q
uantizeG%)SawtoothG%'SignalG%%SineG%-ViewEnvelopeG%+WriteSoundG%*word2
byteG%.writeintegersG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%(ReadWAVG%
)WriteWAVG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(2*Pi);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3`=$G'!\"*" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 332 "mySine:=proc(Amp,freq,phase,tau,t)\n     A
mp*evalhf(exp(-t/tau)*sin(2*Pi*freq*t+phase)):\nend proc:\nmySignal:=p
roc(amplitudes,frequencs,decayconst,t)\n#   print(decayconst,t);\n#   \+
print(mySine(amplitudes[1],frequencs[1],0,decayconst,t));\n    add( my
Sine(amplitudes[i],frequencs[i],0,decayconst,t) \n   ,i=1..nops(amplit
udes));\nend proc:" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 14 "l
owestmode:=5;" }{MPLTEXT 1 206 27 "\nhighestmode:=15;\ntau:=4.0;" }
{MPLTEXT 1 206 24 "\nfreqs:=[seq(hertz[i],i=" }{MPLTEXT 1 206 10 "lowe
stmode" }{MPLTEXT 1 206 36 "..highestmode)]; # Enter Frequencies" }
{MPLTEXT 1 206 83 "\namps:=[seq(evalf(exp(-(2*i/nmax))),i=lowestmode..
highestmode)]; # Enter Amplitudes" }{MPLTEXT 1 206 31 "\nDuration:=2.0
: # 2 sec. sample" }{MPLTEXT 1 206 76 "\nSampleFreq:=22050: # For test
ing, and speed, try at half the CD sample rate" }{MPLTEXT 1 206 29 "\n
deltaT:=evalf(1/SampleFreq);" }{MPLTEXT 1 206 37 "\nNpoints:=round(Dur
ation*SampleFreq);" }{MPLTEXT 1 206 167 "\nSigData:=Array([seq(mySigna
l(amps,freqs,tau,i*deltaT),i=1..Npoints)],datatype=float):\n#SigData:=
Array([seq(Signal(amps,freqs,i*deltaT),i=1..Npoints)],datatype=float):
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+lowestmodeG\"\"&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%,highestmodeG\"#:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$tauG$\"#S!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
&freqsG7-$\"+XTl%3\"!\"($\"+d,$\\^\"F($\"+])=p,#F($\"+7-i!f#F($\"+UU.O
KF($\"+Y4;`RF($\"+>.+UZF($\"+eBb-cF($\"+uq\"[`'F($\"+eWzQvF($\"+7X[9')
F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%ampsG7-$\"+\\s\"[Y)!#5$\"+JvI
(=)F($\"+kc*)=zF($\"+$Q$GfwF($\"+2A=3uF($\"+1JJlrF($\"+,iSIpF($\"+g/?.
nF($\"+5MW$['F($\"+`3*3F'F($\"+(f1`1'F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'deltaTG$\"+#RZ^`%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Np
ointsG\"&+T%" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 206 83 "WriteWA
V(\"/home/ringland/src/brassbar_modes_5_to_15.wav\", SigData, SampleFr
eq, 16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%41~channel(s)~found.G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%@44100~points~per~channel~found.G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#Q9Quantizing~to~16~bits...6\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#QXSaving~to~/home/ringland/src/brassba
r_modes_5_to_15.wav6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q%Done6\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "?evalhf" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 3 "eva" }}}{EXCHG {PARA 201 "" 0 "" {TEXT -1 0 
"" }}}{PARA 205 "" 0 "" {TEXT -1 0 "" }}}{MARK "27 0 0" 0 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }