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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 $A\sin(kx-wt)$).

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.

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Indeed, in the ideal world this wave will continue to exist forever. This is not special to standing waves, but I suppose that it is something in the formulation/shape of the standing wave that makes you doubt validity of the energy conservation?

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    $\begingroup$ Something special about standing wave is that it’ll retain shape. Others will have a dispersion. $\endgroup$ Aug 28 '20 at 8:10
  • $\begingroup$ @SuperfastJellyfish Dispersion is a property of a wave packet, and not for every type of waves (e.g., the electromagnetic waves in vacuum are dispersionless). Also, dispersion is not dissipation. $\endgroup$ Aug 28 '20 at 8:28
  • $\begingroup$ Dispersion is a property of the medium. And with my above comment I meant due to dispersion, arbitrary shape will not be retained. $\endgroup$ Aug 28 '20 at 8:33
  • $\begingroup$ A plane wave will not be affected by dispersion. Also, I do not see how it is relevant to the OP's doubts about the energy conservation. $\endgroup$ Aug 28 '20 at 8:38
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    $\begingroup$ Agreed. I just wanted to note the difference between standing wave and pulses since OP is sending pulses. $\endgroup$ Aug 28 '20 at 8:41

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