Problem in question can be found here: https://i.stack.imgur.com/aXEZB.jpg
I'm currently reviewing for a PDE exam and was looking over some of the wave equation notes found on the Stanford website. In particular, I was looking at a problem that derives the solution formula to the wave equation in $\mathbb{R}^3$. I was able to find the steps needed to derive it, but am confused by some of them.
On the third line, where it breaks into two double integrals with $x+tz$ terms, how do we get to the fourth line? More specifically how does it change to $\Phi(y)* (y-x/t) ds(y)$ in the fourth line? Just an explanation of this one step is needed. Thank you!
They perform a substitution $y = x + t z$. Solving for $z$, we have $\frac{y-x}{t} = z$, which explains the term on the right in the fourth line.