entropy – Rarefaction wave is an entropic solution!!

Let $f: mathbb{R} rightarrow mathbb{R}$ be a strictly convex function.

And $u:mathbb{R}×(0,+infty( rightarrow mathbb{R}$

where $u(x,0)=u_l$ if $x lt 0$
and $u(x,0)=u_r$ if $x gt 0$

And consider the conservation law equation:

$frac{partial u}{partial t}+frac{partial f(u)}{partial x}$, $x in mathbb{R}$, $t gt 0$.

We know that if u does not satisfy the entropy condition $u_r lt u_l$, then the rarefaction wave occur.

So my question is why are rarefaction waves considered as

1)entropic solutions

Although the case of rarefaction waves came after the function being non-entropic!!