ys[n_, z_, t_] = (2/Pi)*
          Re[Sum[((-1)^(i + 1)*Sin[Pi*z*i]*
                  Cos[Pi*i*t]/i), {i, 1, n, 1}]];

Do[Plot[ys[40, a, t], {a, 0.0, 1.0}, AxesOrigin -> {0. , 0. }, 
        PlotRange -> {-1.25, 1.25}, PlotPoints -> 105, 
        PlotDivision -> 110, PlotStyle -> {Thickness[0.005]}], {t, 
        0. , 1.975, 0.025}]