<TeXmacs|1.0.6>
<style|<tuple|generic|maxima|axiom>>
<\body>
<section|Cylindrical TM Maxwell Cavity Mode>
<with|prog-language|axiom|prog-session|default|<\session>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
)clear all
</input>
<\output>
\ \ \ All user variables and function definitions have been cleared.
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
)library )dir "/home/andreas/axiom"
</input>
<\output>
\ \ \ TexFormat is already explicitly exposed in frame initial\
\ \ \ TexFormat will be automatically loaded when needed from\
\ \ \ \ \ \ /home/andreas/axiom/TEX.NRLIB/code
\ \ \ TexFormat1 is already explicitly exposed in frame initial\
\ \ \ TexFormat1 will be automatically loaded when needed from\
\ \ \ \ \ \ /home/andreas/axiom/TEX1.NRLIB/code
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
J:=operator 'J
</input>
<\output>
<with|mode|math|math-display|true|J<leqno>(1)>
<axiomtype|BasicOperator >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
psi(rho,phi) == J(gamma*rho)*exp(PP*%i*m*phi)
</input>
<\output>
<axiomtype|Void >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
psiglob:=psi(sqrt(x^2+y^2),atan(y/x))
</input>
<\output>
\ \ \ Compiling function psi with type (Expression Integer,Expression\
\ \ \ \ \ \ Integer) -\<gtr\> Expression Complex Integer\
<with|mode|math|math-display|true|J<left|(>\<gamma\><sqrt|y<rsup|2>+x<rsup|2>><right|)>e<rsup|<left|(>i*P*P*m*arctan
<left|(><frac|y|x><right|)><right|)>><leqno>(3)>
<axiomtype|Expression Complex Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
D(psi(rho,phi),rho)
</input>
<\output>
\ \ \ Compiling function psi with type (Variable rho,Variable phi)
-\<gtr\>\
\ \ \ \ \ \ Expression Complex Integer\
<with|mode|math|math-display|true|\<gamma\>e<rsup|<left|(>i*P*P*m\<phi\><right|)>>J<rsub|
><rsup|,><left|(>\<gamma\>\<rho\><right|)><leqno>(5)>
<axiomtype|Expression Complex Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
cross(vector [0,0,1], vector [x,y,0])
</input>
<\output>
<with|mode|math|math-display|true|<left|[>-y,<space|0.5spc>x,<space|0.5spc>0<right|]><leqno>(7)>
<axiomtype|Vector Polynomial Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
\;
</input>
</session>>
<section|Rectangular Cavity Mode>
According to Jackson, p. 357, (8.17), we need to solve the Helmholtz
equation
<\eqnarray*>
<tformat|<cwith|1|1|3|3|cell-halign|l>|<table|<row|<cell|(\<nabla\><rsup|2>+\<mu\>\<varepsilon\>\<omega\><rsup|2>)<matrix|<tformat|<table|<row|<cell|\<b-E\>>>|<row|<cell|\<b-B\>>>>>>>|<cell|=>|<cell|\<b-0\>,>>>>
</eqnarray*>
subject to <with|mode|math|n\<times\>\<b-E\>=0> and
<with|mode|math|n\<cdot\>\<b-B\>=0>. The ansatz is
<\equation*>
\<b-E\>=<matrix|<tformat|<table|<row|<cell|E<rsub|x,x>(x)E<rsub|x,y>(y)E<rsub|x,z>(z)>>|<row|<cell|E<rsub|y,x>(x)E<rsub|y,y>(y)E<rsub|y,z>(z)>>|<row|<cell|E<rsub|z,x>(x)E<rsub|z,y>(y)E<rsub|z,z>(z)>>>>>
</equation*>
and likewise for <with|mode|math|\<b-B\>>. The boundary conditions are
<\eqnarray*>
<tformat|<table|<row|<cell|E<rsub|x>(x,<with|math-level|1|<tabular|<tformat|<table|<row|<cell|0>>|<row|<cell|b>>>>>>,z)>|<cell|=>|<cell|0,>>|<row|<cell|E<rsub|x>(x,y,<with|math-level|1|<tabular|<tformat|<table|<row|<cell|0>>|<row|<cell|c>>>>>>)>|<cell|=>|<cell|0,>>>>
</eqnarray*>
and so on, as well as
<\eqnarray*>
<tformat|<table|<row|<cell|H<rsub|x>(<with|math-level|1|<tabular|<tformat|<table|<row|<cell|0>>|<row|<cell|a>>>>>>,y,z)>|<cell|=>|<cell|0.>>>>
</eqnarray*>
So
<\equation*>
E<rsub|x>=\<alpha\><rsub|x>exp(i\<beta\><rsub|x>x)sin<left|(><frac|n\<pi\>y|b><right|)>sin<left|(><frac|o\<pi\>z|c><right|)>exp(-i\<omega\>t)=\<alpha\><rsub|x>e<rsub|x>s<rsub|y>s<rsub|z>
</equation*>
and analogous terms for <with|mode|math|E<rsub|y>> and
<with|mode|math|E<rsub|z>> satisfy the first batch of boundary conditions.
Because of the Helmholtz equation, we find that
<with|mode|math|\<beta\><rsub|x>=m\<pi\>/a>; otherwise, not all vector
components would share the same eigenvalue, which would not solve the
equation.
<with|prog-language|axiom|prog-session|default|<\session>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
)clear all
</input>
<\output>
\ \ \ All user variables and function definitions have been cleared.
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
)library )dir "/home/andreas/axiom"
</input>
<\output>
\ \ \ TexFormat is already explicitly exposed in frame initial\
\ \ \ TexFormat will be automatically loaded when needed from\
\ \ \ \ \ \ /home/andreas/axiom/TEX.NRLIB/code
\ \ \ TexFormat1 is already explicitly exposed in frame initial\
\ \ \ TexFormat1 will be automatically loaded when needed from\
\ \ \ \ \ \ /home/andreas/axiom/TEX1.NRLIB/code
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
factors:=[f,g,h];
</input>
<\output>
<axiomtype|List OrderedVariableList [f,g,h] >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
coord := [x,y,z];
</input>
<\output>
<axiomtype|List OrderedVariableList [x,y,z] >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
curl(v)== vector [D(v.3,y)-D(v.2,z),D(v.1,z)-D(v.3,x),D(v.2,x)-D(v.1,y)];
</input>
<\output>
<axiomtype|Void >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
c:=1/sqrt(epsilon*mu);
</input>
<\output>
<axiomtype|Expression Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
sines(i) == sin(factors.i*coord.i);
</input>
<\output>
<axiomtype|Void >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
cosines(i) == cos(factors.i*coord.i);
</input>
<\output>
<axiomtype|Void >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
k:=sqrt(f^2+g^2+h^2);omega:=k*c;
</input>
<\output>
<axiomtype|Expression Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
zdep1:=exp(%i*h*z); zdep2:=exp(-%i*h*z);
</input>
<\output>
<axiomtype|Expression Complex Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
zf1:=1; zf2:=-1;
</input>
<\output>
<axiomtype|Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
zdep:=zf1*zdep1 + zf2*zdep2;
</input>
<\output>
<axiomtype|Expression Complex Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
C:=%i/(f^2+g^2);
</input>
<\output>
<axiomtype|Fraction Polynomial Complex Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
efield := vector [
C*f*h*cosines(1)* \ sines(2)*(zf1*zdep1-zf2*zdep2),
C*g*h* \ sines(1)*cosines(2)*(zf1*zdep1-zf2*zdep2),
\ \ \ \ \ \ \ \ sines(1)* \ sines(2)*zdep];
</input>
<\output>
<axiomtype|Vector Expression Complex Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
hfield:=1/(-%i*omega*mu)*(-curl efield);
</input>
<\output>
<axiomtype|Vector Expression Complex Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
efield2:=1/(-%i*omega*epsilon)*(curl hfield);
</input>
<\output>
<axiomtype|Vector Expression Complex Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
efield2-efield
</input>
<\output>
<with|mode|math|math-display|true|<left|[>0,<space|0.5spc>0,<space|0.5spc>0<right|]><leqno>(71)>
<axiomtype|Vector Expression Complex Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
hfield
</input>
<\output>
<with|mode|math|math-display|true|<left|[><frac|<left|(><left|(>-i*g*h<rsup|2>-i*g<rsup|3>-i*f<rsup|2>g<right|)>cos
<left|(>g*y<right|)>e<rsup|<left|(>i*h*z<right|)>>+<left|(>i*g*h<rsup|2>+i*g<rsup|3>+i*f<rsup|2>g<right|)>cos
<left|(>g*y<right|)>e<rsup|<left|(>-i*h*z<right|)>><right|)>sin
<left|(>f*x<right|)><sqrt|\<epsilon\>\<mu\>>|<left|(>g<rsup|2>+f<rsup|2><right|)>\<mu\><sqrt|h<rsup|2>+g<rsup|2>+f<rsup|2>>>,<space|0.5spc><frac|<left|(><left|(>i*f*h<rsup|2>+i*f*g<rsup|2>+i*f<rsup|3><right|)>cos
<left|(>f*x<right|)>e<rsup|<left|(>i*h*z<right|)>>+<left|(>-i*f*h<rsup|2>-i*f*g<rsup|2>-i*f<rsup|3><right|)>cos
<left|(>f*x<right|)>e<rsup|<left|(>-i*h*z<right|)>><right|)>sin
<left|(>g*y<right|)><sqrt|\<epsilon\>\<mu\>>|<left|(>g<rsup|2>+f<rsup|2><right|)>\<mu\><sqrt|h<rsup|2>+g<rsup|2>+f<rsup|2>>>,<space|0.5spc>0<right|]><leqno>(72)>
<axiomtype|Vector Expression Complex Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
hfield2:=vector [
-%i*g/(f^2+g^2)*epsilon*omega*sines(1)*cosines(2)*(zf1*zdep1+zf2*zdep2),
\ %i*f/(f^2+g^2)*epsilon*omega*cosines(1)*sines(2)*(zf1*zdep1+zf2*zdep2),
0]
</input>
<\output>
<with|mode|math|math-display|true|<left|[><frac|<left|(>-i\<epsilon\>g*cos
<left|(>g*y<right|)>e<rsup|<left|(>i*h*z<right|)>>+i\<epsilon\>g*cos
<left|(>g*y<right|)>e<rsup|<left|(>-i*h*z<right|)>><right|)>sin
<left|(>f*x<right|)><sqrt|h<rsup|2>+g<rsup|2>+f<rsup|2>>|<left|(>g<rsup|2>+f<rsup|2><right|)><sqrt|\<epsilon\>\<mu\>>>,<space|0.5spc><frac|<left|(>i\<epsilon\>f*cos
<left|(>f*x<right|)>e<rsup|<left|(>i*h*z<right|)>>-i\<epsilon\>f*cos
<left|(>f*x<right|)>e<rsup|<left|(>-i*h*z<right|)>><right|)>sin
<left|(>g*y<right|)><sqrt|h<rsup|2>+g<rsup|2>+f<rsup|2>>|<left|(>g<rsup|2>+f<rsup|2><right|)><sqrt|\<epsilon\>\<mu\>>>,<space|0.5spc>0<right|]><leqno>(73)>
<axiomtype|Vector Expression Complex Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
hfield-hfield2
</input>
<\output>
<with|mode|math|math-display|true|<left|[>0,<space|0.5spc>0,<space|0.5spc>0<right|]><leqno>(74)>
<axiomtype|Vector Expression Complex Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
bcs:=[
eval(efield.1, z=0),
eval(efield.2, z=0),
eval(efield.3, z=0),
eval(hfield.1, x=0),
eval(hfield.2, y=0),
eval(hfield.3, z=0)
]
</input>
<\output>
<with|mode|math|math-display|true|<left|[>0,<space|0.5spc><frac|2i*g*h*cos
<left|(>g*y<right|)>sin <left|(>f*x<right|)>|g<rsup|2>+f<rsup|2>>,<space|0.5spc><frac|2i*f*h*cos
<left|(>f*x<right|)>sin <left|(>g*y<right|)>|g<rsup|2>+f<rsup|2>>,<space|0.5spc>0,<space|0.5spc>0,<space|0.5spc>0<right|]><leqno>(76)>
<axiomtype|List Expression Complex Integer >
</output>
<\input|<with|color|red|<with|mode|math|\<rightarrow\>> >>
\;
</input>
</session>>
\;
</body>
<\initial>
<\collection>
<associate|page-type|letter>
</collection>
</initial>
<\references>
<\collection>
<associate|auto-1|<tuple|1|1>>
<associate|auto-2|<tuple|2|1>>
<associate|auto-3|<tuple|3|?>>
</collection>
</references>
<\auxiliary>
<\collection>
<\associate|toc>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|Cylindrical
TM Maxwell Cavity Mode> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-1><vspace|0.5fn>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|Rectangular
Cavity Mode> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-2><vspace|0.5fn>
</associate>
</collection>
</auxiliary>