Defining 3rd variable for parametricplot3D of two-variable complex function

Question

I am trying to make a parametricplot3D curve of the function Psi(x,t) shown in the included image. My code is the following:

a = 2;  (* a - length*)

A = 1/Sqrt[2*a];

w = 3;(* frequency rad/s*)

psi[x_, t_] := A*((Sin[(Pi x)/a]*E^((-w I) t) + 
     I Sin[2*Pi/a x]*E^((-4*w I) t))/Sqrt[2*a]);

G[x_, t_] := A*Sin[(Pi*x)/a]*Cos[w*t] + Sin[(2*Pi*x)/a]*Sin[4*w*t];

H[x_, t_] := Sin[(2*Pi*x)/a]*Cos[4*w*t] - A*Sin[(Pi*x)/a]*Sin[w*t]

ParametricPlot3D[{G[x, t], H[x, t], Variable z}, {x, -3, a}, {t, 0, Pi}, 
 PlotRange -> All, Mesh -> 40]

So I have tried several things as my Variable z. I tried using (x+t) as well as repeating G[x,t] or H[x,t] However I continue to get an empty graph. Also I have tried to use:

Arg1 = Simplify[ComplexExpand[Re[psi[x_, t_]], a > 0]];

Arg2  = Simplify[ComplexExpand[Im[psi[x_, t_]], a > 0]];

as values for G[x,t] and H[x,t] but get the same issues. I really think my issue is with my Variable z. If anyone has some insight on how I can determine the value of said variable I would be very appreciative.

Answer 1

Here you can make a different scan along the third coordinate. I will show 3 models

a = 2;(*a-length*)A = 1/Sqrt[2*a];

w = 3;(*frequency rad/s*)
psi[x_, t_] := 
 A*((Sin[(Pi x)/a]*E^((-w I) t) + I Sin[2*Pi/a x]*E^((-4*w I) t))/
    Sqrt[2*a]);

{ParametricPlot3D[{Re[psi[x, t]], Im[psi[x, t]], t/2}, {x, -3, a}, {t,
    0, Pi}, PlotRange -> All, Mesh -> None], 
 ParametricPlot3D[{Re[psi[x, t]], Im[psi[x, t]], x/2}, {x, -3, a}, {t,
    0, Pi}, PlotRange -> All, Mesh -> None], 
 ParametricPlot3D[{Re[psi[x, t]], Im[psi[x, t]], 
   Norm[{x, t}]/2}, {x, -3, a}, {t, 0, Pi}, PlotRange -> All, 
  Mesh -> None]}