A geometric approach to an equation of J. D' Alembert

By using a geometric framework of PDE's we prove that the set of solutions of the D'Alembert equation $ (*)\hskip 3pt {{\partial ^2\lg f}\over{\partial x\partial y}}=0$ is larger than the set of smooth functions of two variables $ f(x,y)$ of the form $(**)\hskip 3pt f(x,y)=h(x).g(y)$. The set of $ 2$-dimensional integral manifolds of PDE $ (*)$ properly contains the ones representable by graphs of $ 2$-jet-derivatives of functions $ f(x,y)$ expressed in the form $ (**)$.} A generalization of this result to functions of more than two variables is sketched too.