This notebook shows how to produce an animation, and save it as an animated GIF file.
The figures are cut out of this notebook to make it smaller.
by Jim Swift

$TextStyle  = {FontSize  ->  12};

Plot[BesselJ[0, x],  {x,  0,  10}];

Plot[BesselY[0, x],  {x,  0,  10}];

nFrames  =  24;
Clear[t];
rMin  =   .01;
rMax  =  20;
f[r_,  t_]  = 
        2 ( BesselJ[0, r] Cos[2  π t/nFrames]  +  
              BesselY[0,  r] Sin[
                  2  π t/nFrames]);
amp  = 
        2 (Sqrt[BesselJ[0, rMin]^2  +  BesselY[0,  rMin]^2]);
frame3D[t_]  :=  
        ParametricPlot3D[{r Cos[θ],  r Sin[θ],  
            f[r, t]},  {r,  rMin,  rMax},  {θ,  0,  2  π},  
          PlotRange  -> {-amp,  amp},  
          ImageSize  ->  500];

frame3D[18];

movie3D  =  
        Table[frame3D[t],  {t, 0,  nFrames  - 1 }];

Show[movie3D[[19]] ] ;

The Help Browser in Mathematica gives information about Export.  Be sure to look at "Additional Information."
You will have to change the file name. I suggest you use "Input" " Get File Path..."
These Export commands are commented out, since you would get error messages unless your file structure is very similar to mine.

That  cell took about 2.5 minutes on my 933 MHz computer.