For a wave to continue spread out, you will need to supply energy continuously.
dK= (1/2) (dm)*v[sup]2[/sup] =(1/2)* e*dx*v[sup]2[/sup]
where e is the density per unit length.
P=dk/dt= (1/2)* e*dx/dt*v[sup]2[/sup]= (1/2)*e*u *v[sup]2[/sup]  (The power need to be supplied)
where u is the speed of the wave.
Because v is a function of time v=vo*sim(w*t), average of v[sup]2[/sup] will give you (1/2)vo[sup]2[/sup] and you will find average power.