Standing wave and energy conservation

Consider a one-dimensional standing wave formed on the interval $0 leq x leq L$.

Let’s say that this is an ideal model motivated by a string as a medium with both ends closed.

Assume that there is no energy loss in the process of either transfer or reflection of the waves.

Also say that, though it may be impossible practically, that the standing wave is formed by some initial consecutive pulses given to the same direction (say, to the right so that we can model it as something like $Asin(kx-wt)$).

Now, my question. In this ideal case the conservation of energy should hold, and we can deduce that the pulses will be reflected and keep moving to the right and left without damping, so once the standing wave is formed (at time $t=t_1$) then for the rest of the time ($t in [t_1, infty)$) the standing wave should be kept without being destroyed. Is it right?

And also if you find any primary misunderstanding of concepts from my question please tell me.