Construct two waves:  one moving L → R, the other moving R→L

y1[x_,t_]:= a Sin[k x - ω t];
y2[x_,t_]:= a Sin[k x + ω t];

Traveling Wave

travmovie1=Animate[Plot[y1[x,t] /. {a->1,k->1, ω->1},{x,0, 2 π},PlotRange->{-2.,2.}],{t,0.,2 π, 0.2},AnimationRunning->False]

Standing wave constructed from two traveling waves moving in opposite directions.

travmovie2=Animate[Plot[(y1[x,t]+y2[x,t]) /. {a->1,k->1, ω->1},{x,0, 2 π},PlotRange->{-2.,2.}],{t,0.,2 π, 0.2},AnimationRunning->False]

(* Export["travmovie.gif",travmovie] *)

Standing Wave in more detail--due to the superposition of two traveling waves

travmovie3=Animate[Plot[Evaluate[{y1[x,t], y2[x,t],y1[x,t]+ y2[x,t]} /. {a->1,k->1, ω->1}],{x,0, 2 π},PlotRange->{-2,2},PlotStyle->{{Dashing[{0.02}],RGBColor[1,0,0]},{Dashing[{0.02}],RGBColor[1,0,0]},{Thickness[0.01],RGBColor[0,0,1]}}, Epilog->{{PointSize[0.02],Point[{((π/2) +t),1}]},{PointSize[0.02],Point[{(2 π+(π/2) -t),1}]} ,{PointSize[0.02],Point[{(-2 π+(π/2) +t),1}]},{PointSize[0.02],Point[{((π/2) -t),1}]},{Line[{{(2 π+(π/2) -t),1},{(2 π+(π/2) -t),0} } ]},{Line[{{(-2 π+(π/2) +t),1},{(-2 π+(π/2) +t),0} } ]},{Line[{{((π/2) +t),1},{((π/2) +t),0} } ]},{Line[{{((π/2) -t),1},{((π/2) -t),0} } ]}
} ],{t,0.,2 π - 2 π/40, 2 π/40},AnimationRunning->True]

(* Export["travmovie3.gif",travmovie] *)