Question

I am looking for clarification on a paper that I found which describes the wave equation. The paper can be found here:
http://www.math.tau.ac.il/~turkel/notes/wave.pdf

I am not sure how the author came up with his substitutions. I am either missing something or perhaps there is a typo that is throwing me off. Anyone have an idea?

Answer 1

I think he meant to start with
Define
\[ \phi = \frac{\partial u}{\partial x} \\
\psi = \frac{\partial u}{\partial t}
\]

Answer 2

I am not exactly sure but I dont think that substitution works. I think he may be trying to use stream functions and velocity potential.