Super Position

waves[x_,t_]:=2 Sin[2x-3 t]+3 Sin[3 x - 2 t+0.5]

waves[x,1]

Plot[waves[x,1],{x,0,10}]

Animate[
Plot[waves[x,t],{x,0,10},PlotRange->{-5,5}]
,{t,0,20}]

Some examples of evaluating Fourier Expansions an testing the approximations for various numbers of terms in the expansion

Square Wave

fx[n_]:=4/(n π) Sin[π n/L x]

L=1

fx[3]

Sum[fx[n],{n,1,15,2}]

Plot[Sum[fx[n],{n,1,100,2}],{x,-1,1}]

Saw Tooth Wave

ft[n_]:=2/(π n) (-1)^(n+1) Sin[2 π n/T t]

Clear[ft]

Sum[ft[n],{n,1,4}]

T=1

Parabola

ft[N_]:= π^2/3+Sum[ 4 (-1)^n/n^2 Cos[n x],{n,1,N}]

ft[40]

Plot[{x^2,ft[100]},{x,-π,π}]