w[t_, x_] :=  
      Which[t  <  a, 
        Which[x  <  -a - t, 0, x < -a + t, ((x + a + t)/2), 
          x < a - t, t, x < a + t, ((-x + a + t)/2), True, 0], True, 
        Which[x  <  -a - t, 0, x < a - t, ((x + a + t)/2), 
          x < -a + t, a, x < a + t, ((-x + a + t)/2), True, 
          0]]

u // ClearAll; x =. ; t =. ;

a =  .2; x0 =  .55; 
    u[t_, x_]  := w[t, x - x0]  -  w[t, -x - x0]

\(\(\(\ \)\(\(δt\  = \  .01\)\(;\)\(\ \)\(\(Plot[
          u[#, x], {x, \(-2\), 2}, 
          PlotRange\  -> {{\(-2\), 2}, {\(- .3\),  .3}}, 
          AspectRatio ->  .5, 
          PlotStyle\  -> \ {Thickness[ .004], 
              Hue[ .65]}] &\)\  /@ 
\((	\ Range[0, 
          3, δt])\)\)\(;\)\(
\)\(
\)\(
\)\)\)\)