> <\body> <\input|> >> )clear all <\output> \ \ \ All user variables and function definitions have been cleared. <\input|> >> )library )dir "/home/andreas/axiom" <\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 <\input|> >> J:=operator 'J <\output> (1)> <\input|> >> psi(rho,phi) == J(gamma*rho)*exp(PP*%i*m*phi) <\output> <\input|> >> psiglob:=psi(sqrt(x^2+y^2),atan(y/x)) <\output> \ \ \ Compiling function psi with type (Expression Integer,Expression\ \ \ \ \ \ \ Integer) -\ Expression Complex Integer\ \+x>ei*P*P*m*arctan >(3)> <\input|> >> D(psi(rho,phi),rho) <\output> \ \ \ Compiling function psi with type (Variable rho,Variable phi) -\\ \ \ \ \ \ \ Expression Complex Integer\ ei*P*P*m\>J\\(5)> <\input|> >> cross(vector [0,0,1], vector [x,y,0]) <\output> -y,x,0(7)> <\input|> >> \; > According to Jackson, p. 357, (8.17), we need to solve the Helmholtz equation <\eqnarray*> |+\\\)>>|>>>>>>||,>>>> subject to \=0> and \=0>. The ansatz is <\equation*> \=(x)E(y)E(z)>>|(x)E(y)E(z)>>|(x)E(y)E(z)>>>>> and likewise for >. The boundary conditions are <\eqnarray*> (x,>|>>>>>,z)>||>|(x,y,>|>>>>>)>||>>> and so on, as well as <\eqnarray*> (>|>>>>>,y,z)>||>>> So <\equation*> E=\exp(i\x)siny|b>sinz|c>exp(-i\t)=\ess and analogous terms for > and > satisfy the first batch of boundary conditions. Because of the Helmholtz equation, we find that =m\/a>; otherwise, not all vector components would share the same eigenvalue, which would not solve the equation. <\input|> >> )clear all <\output> \ \ \ All user variables and function definitions have been cleared. <\input|> >> )library )dir "/home/andreas/axiom" <\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 <\input|> >> factors:=[f,g,h]; <\output> <\input|> >> coord := [x,y,z]; <\output> <\input|> >> 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)]; <\output> <\input|> >> c:=1/sqrt(epsilon*mu); <\output> <\input|> >> sines(i) == sin(factors.i*coord.i); <\output> <\input|> >> cosines(i) == cos(factors.i*coord.i); <\output> <\input|> >> k:=sqrt(f^2+g^2+h^2);omega:=k*c; <\output> <\input|> >> zdep1:=exp(%i*h*z); zdep2:=exp(-%i*h*z); <\output> <\input|> >> zf1:=1; zf2:=-1; <\output> <\input|> >> zdep:=zf1*zdep1 + zf2*zdep2; <\output> <\input|> >> C:=%i/(f^2+g^2); <\output> <\input|> >> 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]; <\output> <\input|> >> hfield:=1/(-%i*omega*mu)*(-curl efield); <\output> <\input|> >> efield2:=1/(-%i*omega*epsilon)*(curl hfield); <\output> <\input|> >> efield2-efield <\output> 0,0,0(71)> <\input|> >> hfield <\output> -i*g*h-i*g-i*fgcos g*yei*h*z>+i*g*h+i*g+i*fgcos g*ye-i*h*z>sin f*x\>|g+f\+g+f>>,i*f*h+i*f*g+i*fcos f*xei*h*z>+-i*f*h-i*f*g-i*fcos f*xe-i*h*z>sin g*y\>|g+f\+g+f>>,0(72)> <\input|> >> 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] <\output> -i\g*cos g*yei*h*z>+i\g*cos g*ye-i*h*z>sin f*x+g+f>|g+f\>>,i\f*cos f*xei*h*z>-i\f*cos f*xe-i*h*z>sin g*y+g+f>|g+f\>>,0(73)> <\input|> >> hfield-hfield2 <\output> 0,0,0(74)> <\input|> >> 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) ] <\output> 0,g*ysin f*x|g+f>,f*xsin g*y|g+f>,0,0,0(76)> <\input|> >> \; > \; <\initial> <\collection> <\references> <\collection> > > > <\auxiliary> <\collection> <\associate|toc> |math-font-series||Cylindrical TM Maxwell Cavity Mode> |.>>>>|> |math-font-series||Rectangular Cavity Mode> |.>>>>|>