# Why a classical solution of the wave equation has to be $C^2$?

A (classical) solution of the wave equation $$u_{tt}-c^2u_{xx}=0,\qquad (x,t)\in\mathbb{R}\times\mathbb{R}^*_+,$$ is required to be of class $$C^2$$. Why?

I mean, why one imposes that all second partial derivatives, even $$u_{xt}$$ which does not appear in the PDE, to be continuous?!