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Manipulate[Plot[Exp[ - (x-t)^2]+Exp[-(x+t)^2],{x,-20,20},
PlotRange->{0,2}],
{t,0,20,1,.1}]
Manipulate[Plot[Exp[ - (x-t)^2]-Exp[-(x+t)^2],{x,-20,20},
PlotRange->{-1,1}],
{t,0,20,.1}]
F[x_]:=If[Abs[x]<1,1-Abs[x],0];
Manipulate[Plot[F[x-t]+F[x+t],{x,-20,20},
PlotRange->{0,2}],
{t,0,20,.1}]