smpsy=40;
smpsz=10;

sipky=Table[
{{0,y,z},{Sin[m y π]Cos[k z],y,z}}/.{k->1.5,m->1,z->0},
{y,0,1,1/smpsy}];

Graphics3D[{Arrowheads[Small],Arrow[sipky]}]

Animate[
Show[
Graphics3D[{Arrowheads[Small],Arrow[Table[{{x,0,0},{x,0,1/2Sin[m x π]Cos[3t]}},
{x,0,1,1/20}]]}],
Plot3D[1/2Sin[m π y]Cos[(ω/5)(t-(Sqrt[1-(((((m π)/ω)))^2)])z)]/.{ω->15},{y,0,1},{z,0,5},BoxRatios->Automatic,Axes->None,ColorFunction->"ThermometerColors"],
PlotRange->{-3/4,3/4}
],
{{fake1,0,Style["Elektrické pole ve vlnovodu",Bold,17]},0,0,AppearanceElements->"",AnimationRunning->False},
{{fake2,0,Style["(Subscript[E, x])(y,z,t) = sin(((mπ)/b)) cos(ωt-(Subscript[k, z])z)",15]},0,0,AppearanceElements->"",AnimationRunning->False},
{{fake3,0,Style["disperzní vztah: (ω^2)=(c^2)((((mπ)/b))^2 + ((Subscript[k, z])^2))",15]},0,0,AppearanceElements->"",AnimationRunning->False},
{{t,0,"Čas t"},0,∞},
{{m,1,Dynamic["Mód vlnění - m = "<>ToString[m]]},1,4,1,AppearanceElements->"ProgressSlider",AnimationRunning->False},
ControlPlacement->{Top,Top,Top,Bottom,Bottom}
]

Export

Postupné vlnění

imtable=ReleaseHold[Hold@Table[Show[
{Graphics3D[{Arrowheads[Small],Arrow[Table[{{x,0,0},{x,0,1/2Sin[m x π]Cos[(ω/α) t]}},
{x,0,1,1/20}]]},ImageSize->450],
Plot3D[1/2Sin[m π y]Cos[(ω/α)(t-(Sqrt[1-(((((m π)/ω)))^2)])z)],{y,0,1},{z,0,5},BoxRatios->Automatic,Axes->None,ColorFunction->"ThermometerColors"]},
PlotRange->{-3/4,3/4}
],{t,0,((2π α)/ω)-((2π α)/(50ω)),((2π α)/(50ω))}]/.{ω->15,α->5,m->1}];
Export["vlnovod.gif",imtable,"GIF","DisplayDurations"->{0.04},"AnimationRepetitions"->∞]

Kombinace 1. a 4. módu

imtable2=ReleaseHold[Hold@Table[Show[
{Graphics3D[{Arrowheads[Small],Arrow[Table[{{x,0,0},{x,0,1/2(Sin[x π]Cos[(ω/α) t]+1/2Sin[4x π]Cos[(ω/α) t])}},
{x,0,1,1/20}]]},ImageSize->450],
Plot3D[1/2(Sin[π y]Cos[(ω/α)(t-(Sqrt[1-((((π/ω)))^2)])z)]+1/2Sin[4π y]Cos[(ω/α)(t-(Sqrt[1-(((((4π)/ω)))^2)])z)]),{y,0,1},{z,0,5},BoxRatios->Automatic,Axes->None,ColorFunction->"ThermometerColors"]},
PlotRange->{-3/4,3/4}
],{t,0,((2π α)/ω)-((2π α)/(50ω)),((2π α)/(50ω))}]/.{ω->15,α->5,m->4}];
Export["vlnovodtest.gif",imtable2,"GIF","DisplayDurations"->{0.04},"AnimationRepetitions"->∞]

Tlumené vlny

imtable3=ReleaseHold[Hold@Table[Show[
{Graphics3D[{Arrowheads[Small],Arrow[Table[{{x,0,0},{x,0,1/2Sin[m x π]Cos[(ω/α) t]}},
{x,0,1,1/20}]]},ImageSize->450],
Plot3D[1/2Sin[m π y](E^(-2(Sqrt[(((((m π)/ω)))^2)-1])z))Cos[(ω/α)t],{y,0,1},{z,0,5},BoxRatios->Automatic,Axes->None,ColorFunction->"ThermometerColors"]},
PlotRange->{-3/4,3/4}
],{t,0,((2π α)/ω)-((2π α)/(50ω)),((2π α)/(50ω))}]/.{ω->15,α->5,m->5}];
Export["vlnovodtlum.gif",imtable3,"GIF","DisplayDurations"->{0.04},"AnimationRepetitions"->∞]