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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "pipes.mws   resonance in p
ipes - theory and MJM data" }}{PARA 0 "" 0 "" {TEXT -1 86 "2/16/99  re
vised 3/11/99 for length as a function of freq, and for 1/2\" conduit \+
option" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "restart;with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "#D_outer:=0.0178; D_inner:=0.0157; # 1/2\" conduit" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "D_outer:=0.0235; D_inner:=
0.0209;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(D_outerG$\"$N#!\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(D_innerG$\"$4#!\"%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a:=(D_inner+D_outer)/4; # a is effe
ctively the 'radius'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"++++5
6!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "kappa:=(D_inner^2+D
_outer^2)^(1/2)/4; # area rotational inertia" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&kappaG$\"+!3JB'y!#7" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "k:=1.00375*Pi*3/2/L;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"kG,$*&%#PiG\"\"\"%\"LG!\"\"$\"++]i0:!\"*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 45 "V:=5105; # from longitudinal pipe frequencies
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG\"%0^" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 74 "c is a correction factor, related to complex behavio
r of elastic mechanics" }}{PARA 0 "" 0 "" {TEXT -1 77 "(the factor mul
tiplying (ka)^2 is empirical, from trial and error on my part)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "c:=(1+2.4*(k*a)^2)^(1/2);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG*$,&\"\"\"F'*&%#PiG\"\"#%\"LG!
\"#$\"+8OL.n!#8#F'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f:=
V*k^2*kappa/(2*Pi)/c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,$*(%#P
iG\"\"\"%\"LG!\"#,&F(F(*&F'\"\"#F)F*$\"+8OL.n!#8#!\"\"F-$\"+zUO\\X!\")
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eq0:= frq=V*k^2*kappa/(
2*Pi)/c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq0G/%$frqG,$*(%#PiG\"
\"\"%\"LG!\"#,&F*F**&F)\"\"#F+F,$\"+8OL.n!#8#!\"\"F/$\"+zUO\\X!\")" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sol:=solve(eq0,L);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG,$-%'RootOfG6#,(*&%#_ZG\"\"%%$f
rqG\"\"#\".++++++\"*(%#PiGF.F-F.F+F.\",IhLLq'*$F1F.!5T)*=TIM:np?$\"+++
++5!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "s:=allvalues(sol)
;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"sG6&,$*&%$frqG!\"\",&*&%#PiG
\"\"#F(F-!,l!om^L*$,&*&F,\"\"%F(F2\"5pPjvr_\"oM\\%F+\"?+++++k$fZ;s8'oy
#)#\"\"\"F-\"\"&F5$\"+++++5!#;,$F'$!+++++5F:,$*&F(F),&F+F.F/!\"&F5F8,$
F?F<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "len:=evalf(subs(frq
=250,s[1])); # length for given frequency " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$lenG$\"+;\\<Rv!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "evalf(subs(L=0.599,f));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+g=6ZR!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eval
f(subs(L=0.567,f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#[#f+W!\"(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalf(subs(L=0.505,f));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+:_%H`&!\"(" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 17 "x1:=0.4: x2:=0.6:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 46 "j:=plot(f,L=x1..x2,labels=[`L (m)`,`f (Hz)`]):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "MJMData:=[[0.599,395.4],[
0.5205,524.5],[0.4,874.5]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "h:=p
lot(MJMData,x=x1..x2,style=point,symbol=circle,color=blue);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%(MJMDataG7%7$$\"$*f!\"$$\"%aR!\"\"7$$\"%0_
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