Can you have a compensating wave in steady state?
By definition steady state means the system has settled down after a sufficient amount of time to it's normal state.

There is no memory in this system, so talking about system dynamics is not useful. If there were a capacitor or inductor, you could start to talk about the dynamics of the system and its state.

In this system the state is either on or off, there is no in between.

If you had drawn a real switch instead of an ideal switch, then you could also consider switch bounce (where the inductance of the switch and parasitic capacitance elsewhere in the system causes oscillations).

So to answer your question:


  So my question is can there be a compensating wave in steady state? My
  intuition says no since there is no change to compensate like in the
  circuit with the switch.


In this circuit there are no 'waves', it's either on or off.