Propagación de las ondas y propagación de perturbaciones

Clear["Global`*"]
a[x_] := 0 /; x<0 
a[x_] := 
      Sin[x]  /; 0 ≤ x ≤ Pi
a[x_] :=  0 /; Pi < x
b[x_] =  0;

v[x_, t_] = 
      0.5 (a[x + t]) + 
        0.5 (∫_0^(x + t))b[s] \[DifferentialD]s

w[x_, t_] = 
      0.5 (a[x - t]) - 
        0.5 (∫_0^(x - t))b[s] \[DifferentialD]s

u[x_, t_] = 
      0.5 (a[x + t] + a[x - t]) + 
        0.5 (∫_(x - t)^(x + t))b[
                s] \[DifferentialD]s

For[t = 0, t < 3, t = t + 1, 
      Plot[v[x, t], {x, -5, 5}, PlotStyle -> {Thickness[0.010]}, 
        PlotRange -> {-1.5, 1.5}]]

For[t = 0, t < 3, t = t + 1, 
      Plot[w[x, t], {x, -5, 5}, PlotStyle -> {Thickness[0.010]}, 
        PlotRange -> {-1.5, 1.5}]]

For[t = 0, t < 3, t = t + 1, 
        Plot[u[x, t], {x, -5, 5}, PlotStyle -> {Thickness[0.010]}, 
          PlotRange -> {-1.5, 1.5}]];