Adding random numbers to the arguments of a sum of cosines

Question

This is probably really a simple question.

I would like to plot a sum of cosines with increased argument and for each cosine in sum I would like to add random value:

Plot[Sum[Cos[n*x + RandomReal[{0, 2 Pi}]], {n, 100}], {x, -10, 10}]

However it seems to me that the program, while making a plot, generates a new random number for each x-point plotted. Therefore, even for single cosine (Plot[Cos[x + RandomReal[{0, 2*Pi}]], {x, -10, 10}] ) I get a completely noisy graph of random cosine values.

I would be very grateful if somebody can tell me what to do to prevent the program from generating new random numbers at each x-point.

Answer 1

J.M. has answered the question in the comments. This is to get that answer (slightly edited) on record in a more visible form. For 10 terms, we get

SeedRandom[42];
  With[{n = 10},
    With[{r = RandomReal[{0, 2 Pi}, n]}, 
      Plot[Total[Cos[Range[n] x + r]], {x, -10, 10}]]]

For a 100 terms, which is what the OP asked for, the plot must be stretched out to see the details better.

SeedRandom[42];
  With[{n = 100},
    With[{r = RandomReal[{0, 2 Pi}, n]}, 
      Plot[Total[Cos[Range[n] x + r]], {x, -10, 10},
        AspectRatio -> 1/5,
        ImageSize -> Full]]]