(*Wave Equation*) 
σ = 1;  
    c = 1;

    f[x_] := ((1/(σ*(Sqrt[2*π]))))*(E^(-((x^2)/(2*(σ^2)))));

    Show[Table[
        Plot[(1/2)*(f[x + c*t] + f[x - c*t]), {x, -2 π, 2  π}, 
          PlotRange -> {0, 0.5}], {t, 0.0, 8, 0.05}]]

σ = 1;  c = 5;
Show[
      Table[Plot[
          (1/2)*(f[x + c*t] + f[x - c*t]), {x, -2 π, 2  π}, 
          PlotRange -> {0, 0.5}], {t, 0.0, 8, 0.05}]]

σ = 2;  c = 1;
Show[
      Table[Plot[
          (1/2)*(f[x + c*t] + f[x - c*t]), {x, -2 π, 2  π}, 
          PlotRange -> {0, 0.5}], {t, 0.0, 8, 0.05}]]

(*Heat Equation*) 
u[
          x_] := ((1/(Sqrt[4*π*k*t])))*(E^(-((x^2)/(4*k*t))));

k = 1;
Show[
      Table[Plot[u[x], {x, -2 π, 2  π}, 
          PlotRange -> {0, 3}], {t, 0.01, 3, 0.01}]]

k = 0.2;
Show[
      Table[Plot[u[x], {x, -2 π, 2  π}, 
          PlotRange -> {0, 3}], {t, 0.01, 8, 0.01}]]

k = 5;
Show[
      Table[Plot[u[x], {x, -2 π, 2  π}, 
          PlotRange -> {0, 3}], {t, 0.01, 8, 0.01}]]