Does a π phase shift from reflection advance or lag the wave?

[LCSphysicist]

Homework Statement

I will post below

Relevant Equations

I will post below

Actually, maybe i am confusing something, or valuing unnecessary things, but this doubt arises and i am not sure how answer it.
The reflection of an incident wave changes (suppose it will change, we don't need to talk when or why to go on the discussion) by π rad, but, this reflection lags the wave or "advance" the phase?
That is, let's do a exercise:
Two waves are travelling:
ψ1 = a*sin(wt-kx)
ψ2 = a*sin(wt-kx+p)
THe difference of phase 2 to 1, clearly, is p.
If the wave two reflect with change of phase, the difference of phase between the waves will now be
p + π (advances) or p - π (lags)
?

 

[TSny]

A phase difference of $\phi+\pi$ is equivalent to a phase difference of $\phi-\pi$.

Show that $\sin(\omega t + kx +\phi +\pi) = \sin(\omega t + kx +\phi - \pi)$.

So, there is no need to be concerned with whether the $\pi$ is an advance or a lag.

[If the wave reflects, $\omega t - kx$ changes to $\omega t + kx$. If only the second wave reflects, then the two waves no longer differ by just a phase constant.]