λ=1;v=1;

?Animate

Animate[Plot[Sin[2π/λ (x-v t)],{x,0,λ},ImageSize->Large],{t,0,10,1/1000}]

Animate[Plot[Sin[2π/λ (x+v t)],{x,0,λ},ImageSize->Large],{t,0,10}]

Animate[Plot[{Sin[2π/λ (x+v t)]+Sin[2π/λ (x-v t)]},{x,0,λ},PlotRange->{-2,2},ImageSize->Large],{t,0,10}]

Animate[Plot[{Sin[2π/λ (x+v t)],Sin[2π/λ (x-v t)],Sin[2π/λ (x+v t)]+Sin[2π/λ (x-v t)]},{x,0,λ},PlotRange->{-2,2},ImageSize->Large],{t,0,10}]

Animate[Plot[E^(-1/2 (-5+x+v t)^2/.3)+E^(-1/2 (-3+x+v t)^2/.8)+E^(-1/2 (-7+x+v t)^2/.8),{x,0,10},ImageSize->Large],{t,0,10}]

λ1=1;λ2=1.1;

Animate[Plot[Sin[2 π/λ1 (x-v t)]+Sin[2 π/λ2 (x-v t)],{x,0,50},PlotPoints->500,ImageSize->Large],{t,0,10}]

L=10;

B[n_,a_]=Integrate[2/L 1/a Sin[n π x/L],{x,0,a}]

ListPlot[Table[B[n,1],{n,1,500}],Joined->True,PlotRange->All,ImageSize->Large]

ListPlot[Table[B[n,1/10],{n,1,500}],Joined->True,PlotRange->All,ImageSize->Large]

b=1;

Animate[Plot[b^3/(b^2+(x-v t)^2),{x,-10,10},PlotRange->{0,2},ImageSize->Large],{t,0,10}]

Animate[Plot[{2 v b^3(x-v t)/(b^2+(x-v t)^2)^2,b^3/(b^2+(x-v t)^2)},{x,-10,10},PlotRange->{-2,2},ImageSize->Large],{t,0,10}]