Setting each "curve" on a 3D plot to be of different colour similar to that of its 2D plot

Question

Consider the functions

\[Lambda][n_] := ((n \[Pi])/L)^2

\[ScriptCapitalX][
x_] := \[ScriptCapitalC][1] Cos[Sqrt[\[Lambda][n]] x]

\[ScriptCapitalT][t_] := 50 E^(-\[Lambda] t)

to give the Sum product solution

\[ScriptCapitalU][x_, 
t_, \[ScriptCapitalN]_] := \[ScriptCapitalX][x] \[ScriptCapitalT][t]

Notice that the 2D and 3D plot of the Sum product solution, respective are

Plot[Evaluate@
ReplaceAll[
Table[\[ScriptCapitalX][x] \[ScriptCapitalT][t], {n, 1, 3}, {L, 1, 
 1}], {\[ScriptCapitalC][1] -> 1, \[Lambda] -> 1, t -> 0}], {x, 0,
1}, PlotLegends -> "Expressions", AxesLabel -> {"x", "t"}, 
PlotLabel -> "2D Plot Behaviour for \[ScriptCapitalU](x,t)"]

and

Plot3D[ReplaceAll[
Table[\[ScriptCapitalU][x, t, \[ScriptCapitalN]], {n, 1, 3}, {L, 1, 
1}], {\[ScriptCapitalC][1] -> 1, \[Lambda] -> 1}], {t, 0, 1}, {x, 
0, 1}, AxesLabel -> Automatic, 
PlotLabel -> "3D Plot Behaviour for \[ScriptCapitalU](x,t)"]

If you rotate the 3D point such that the x and t plots points "towards" you, you are able to observe that they corresponds qualitatively to the 2D plot.

In the case of the 3D plot, how do I match the different colours to different "n" for n=1 to n=3?

Any help is appreciated.

Answer 1

Plot3D[Evaluate@
  ReplaceAll[Table[\[ScriptCapitalU][x, t, \[ScriptCapitalN]], {n, 1, 3}, {L, 1, 1}], 
 {\[ScriptCapitalC][1] -> 1, λ -> 1}], {t, 0, 1}, {x, 0, 1}, 
  AxesLabel -> Automatic, 
  PlotLabel -> "3D Plot Behaviour for \[ScriptCapitalU](x,t)"]

Note: Interestingly, the combination of options Evaluated ->True + PlotStyle -> {Red, Green, Blue}, which works in version 9, does not work in version 11.2 (on Wolfram Cloud).

Answer 2

\[Lambda][n_] := ((n \[Pi])/L)^2
\[ScriptCapitalX][x_] := \[ScriptCapitalC][1] Cos[Sqrt[\[Lambda][n]] x]
\[ScriptCapitalT][t_] := 50 E^(-\[Lambda] t)
\[ScriptCapitalU][x_, t_, \[ScriptCapitalN]_] := \[ScriptCapitalX][x] \[ScriptCapitalT][t]

Plot3D[Evaluate@ (* Step 2: enforce the evaluation before configuring the colors *)
  ReplaceAll[Table[\[ScriptCapitalU][x, t, \[ScriptCapitalN]], {n, 1, 3}, {L, 1, 1}], {\[ScriptCapitalC][1] -> 1, \[Lambda] -> 1}], {t, 0, 1}, {x, 0, 1}, 
  AxesLabel -> Automatic, PlotLabel -> "3D Plot Behaviour for \[ScriptCapitalU](x,t)",
  PlotStyle -> {Red, Green, Blue}] (* Step 1: use PlotStyle to set up color for each plot surface *)