I checked the timing to draw one curve. On my computer (2013 rMBPro), it took about 3 seconds. Since there are 80 curves per image and 19 images, the entire image will take about 80 minutes. The code as I copied it still does not work, and I suspect that it is an issue with linebreaks. I had to set the format to Code and redo the linebreaks to draw the single line.
Using the Code Stype (from the Format/Style menu) takes an Input formatted cell and removes all the automatic linebreaks. I then add explicit line breaks to keep the length of the line manageable.
I did this first to make one line, (3.08 seconds), then one image (267.6 seconds). Since the next step -- wrapping ListAnimate[Table[*** code for one image]] around what I did, I am sure that it will work.
Here is the code that works to draw one image. Note that I have renamed the R[] function and I left in place the //Timing command. Just copying nad pasting this as Input text has altered the formatting, so check it out.
R3[n_] := (SeedRandom[n]; RandomReal[])
Show[
Table[
Plot[80 - m + .2*Sin[2 Pi*R3[6*m]
+ Sum[4*Sin[2 Pi*R3[4*m] + t +
R3[2 n*m]*2 Pi]*
Exp[-(.3*x + 30 - 1*100*R3[2 n*m])^2/20], {n, 1, 30, 1}]] +
Sum[3 (1 + R3[3*n*m])*Abs[Sin[t + R3[n*m]*2 Pi]]*
Exp[-(x - 1*100*R3[n*m])^2/20],
{n, 1, 4, 1}],
{x, -50, 150}, PlotStyle -> Directive[White, Thick],
PlotRange -> {{-50, 150}, {0, 85}},
Background -> Black,
Filling -> Axis,
FillingStyle -> Black,
Axes -> False,
AspectRatio -> Full,
ImageSize -> {500, 630}],
{m, 1, 80, 1}]
] // Timing
I need to understand the problem better to understand why SeedRandom[] is used the way it is to draw the curves. Just using RandomReal[] makes a different plot.
george