Here we superpose two eigenstates of the 1 D string and what the superposition evolve in time.  notice that at certain times the superposition state is entirely in one eigenstate or ther other.

c = 1
l = 1
w1 = Pi
w2 = 2 Pi
c1 = 1
c2 = 1
psi1[x_] = Sin[Pi x]
psi2[x_] = Sin[2 Pi x]

P[x_,t_ ]  = c1 psi1[x] Exp[-I w1 t] + c2 psi2[x] Exp[-I w2 t]

Animate[Plot[Re[P[x,t]],{x,-1,1},PlotRange->{{-1,1},{-2,2}}],{t,0,3,.1}]

Animate[Plot[Abs[P[x,t]],{x,-1,1},PlotRange->{{-1,1},{-2,2}}],{t,0,3,.1}]