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P1[t_] := 
 Module[{pts}, 
  pts = Cases[Plot[uwave1[t, 0.5, z]^2, {z, 0, 50}, PlotRange -> All],
     Line[a_] :> a, Infinity];
  RegionCentroid@
   Polygon@
    Join[{{pts[[1, 1, 1]], 0}}, pts[[1]], {{pts[[1, -1, 1]], 0}}]]

Here P1[t] is a point $(a,b)$ in 2D space which varies with time. How shall I plot the value of $a$ against time?

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This can be accomplished using First:

Plot[First[P1[t]], {t, 0, 8 Pi}]
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