In section 3.1 of The Wave Equation on a Curved Spacetime, Friedman defines what a characteristic hypersurface for the wave equation $$\Box u+a\cdot\nabla u+bu=f$$ is, and then shows that these are precisely the null hypersurfaces of the Lorentzian manifold. He also talks about tensor wave equations, which are the kind of obvious generalizations of the above, e.g. $$\Box u^a+a^{ab}{}_c\nabla_bu^c+b^{ab}u_b=F^a.$$ He says that the characteristics are in fact "multiple characteristics" and that in the vector case, if a characteristic is a level set of some function $S$, then we have $$\tag{1}(\langle\nabla S,\nabla S\rangle)^n=0,$$ which is...strangely phrased. It might be a typo, I'm not sure. But that looks just like the condition for $S$ to define a null surface, with a random power of $n=\dim M$.
So what exactly are these "multiple characteristics" and what does (1) have to do with anything? I found a few papers talking about multiple characteristics but it wasn't clear what they actually meant.