(*Define variables*)
v:=π/6(*m/s, arbitrary speed*)
ω1:=π/2 (*rad/s, our reference angular frequency*)
ω2:=1.00*π/2  (*Edit this angular frequency*)
Y:=1

y1[x_,t_]:=Y*Sin[(ω1/v)x-ω1*t]
y2[x_,t_]:=Y*Sin[(ω2/v)x-ω2*t]

Animate[
Show[
Plot[{y1[x,t],y2[x,t],y1[x,t]+y2[x,t]},
{x,0,2π},
PlotRange-> {-2,2},
AxesLabel->{"x (m)","y (m)"},
PlotStyle->{{Blue,Dashed,Thickness[.001]},{Orange,Dashed,Thickness[.001]},Black},
ImageSize->450
]
],
{t,0,48},AnimationRate->1
]

Plot[
{y1[0,t],y2[0,t],y1[0,t]+y2[0,t]},
{t,0,36π},
AxesLabel->{"t (s)","y (m)"},
PlotStyle->{{Blue,Dashed,Thickness[.001]},{Orange,Dashed,Thickness[.001]},Black},
ImageSize->450
]