A monocromatic 1D wave is described by the formula 
y[x,t]=Sin[k x+ωt], ω=((2π)/λ) is the wave number; ω = angular frequency.
Let (Subscript[k, i])=(Subscript[k, 0])+i Δk, Δk << 1, say 0.001. We assume (Subscript[k, 0])=1.

(* Nk Number of wave to add *)
xmax=200;
Deltak=0.001;
k[0]=1.0;omega=0.1;
k[i_]:=k[0]+i*Deltak;

y[i_,x_,t_]:=Sin[k[i]*x-omega*t];(* y is a monochromatic wave with wavenumber k[i] *)
Y[Nk_,x_,t_]:=Sum[y[i,x,t],{i,0,Nk}]; (*Y is the wavepacket formed by superiposing Nk waves *)

Here we ask Mathematica to show the wave form y[x,t]=Sin[k x+ωt] for k=(Subscript[k, 0])
and k=(Subscript[k, 0])+(Subscript[N, k])Δk = 2.0, (Subscript[N, k])=1000.

j=0;
Manipulate[
Plot[y[j,x,t],{x,-xmax/8,xmax/8},PlotLabel->{"k=",k[j]}],{t,0,100,0.1}
]

kk=1000;
Manipulate[
Plot[y[kk,x,t],{x,-xmax/8,xmax/8},PlotLabel->{"k=",k[kk]}],{t,0,100,0.1}
]

Here we ask Mathematica to show the wave form Y[x,t]=(∑_(i=0)^(i=(Subscript[N, k])))Sin[(Subscript[k, i])x+ωt] at t=0 for (Subscript[N, k])=50,100,200,500. You will observe that the large the number of wave we add to form Y[x,t], the more localise it becomes.

Nk=50;
Plot[Y[Nk,x,0],{x,-xmax,xmax},PlotRange->All,PlotLabel->{"Y[x,t] with Nk = ",Nk}]
Nk=100;
Plot[Y[Nk,x,0],{x,-xmax,xmax},PlotRange->All,PlotLabel->{"Y[x,t] with Nk = ",Nk}]
Nk=200;
Plot[Y[Nk,x,0],{x,-xmax,xmax},PlotRange->All,PlotLabel->{"Y[x,t] with Nk = ",Nk}]
Nk=500;
Plot[Y[Nk,x,0],{x,-xmax,xmax},PlotRange->All,PlotLabel->{"Y[x,t] with Nk = ",Nk}]

In the following, we ask Mathematica to display a vidoe showing the progressive motion of the wave form Y[x,t] for an extensive period of time, from t=0 to t = 100.

Manipulate[Nk=100;
Plot[Y[Nk,x,t],{x,-xmax,xmax},PlotRange->{-160,160},PlotLabel->{"Y[x,t] with Nk = ",Nk}]
,{t,0,100,1}]

Manipulate[Nk=200;
Plot[Y[Nk,x,t],{x,-xmax,xmax},PlotRange->{-360,360},PlotLabel->{"Y[x,t] with Nk = ",Nk}]
,{t,0,100,1}]

Manipulate[Nk=500;
Plot[Y[Nk,x,t],{x,-xmax,xmax},PlotRange->{-460,460},PlotLabel->{"Y[x,t] with Nk = ",Nk}]
,{t,0,100,1}]

xmax