Wave equation three dimensional elastic medium

ex 1.5 p18. QFT for noobs.

Satisfy the wave equation

use dS/df = 0. the "least action" principle, or it should be 'stationary action' (minimum or maximum of the action).

substitute in S = integral of L over time

evaluate by using, I'm guessing, this:

But we have an x and a t term, so... I guess... make two terms one with d/dx one with d/dt.

Plug in the Kinetic and Potential energies, T - V, and set the integral to zero. Say that the integrand must equal zero 'cause... energies are non-negative, I guess.
Then let v squared equal Tau over roe.

Basically, copy the example 1.4 solved for the two dimensional string vibration. I didn't do anything different except change the spatial derivative into a gradient.

... so I must've done something wrong LOL.

I didn't use functional derivatives. Well, I guess I used a bastardized form (which might be incorrect) of a result we showed previously with functional derivatives.