A few years ago, I was with some students to read a textbook about audio engineering. I did not like the book in a few ways. One of the reasons is about treatment of wave equation.
1D wave equation looks something like this:
After presenting the equation, the book immediately declares its solution is given by
for arbitrary function and , without any derivation, explanation no nothing. The solution comes down like an oracle. I still don’t like the way it is given.
This solution is usually called “d’Alembert’s Solution“. And in case his name chimes in, we can see a slightly better derivation.
The derivation goes like following. Introducing new variables
the PDE becomes
Integrating this with respect to gives
(G: arbitrary function of , independent of , since differentiating G with respect to gives ).
Then integrating above with respect to gives
(f: arbitrary function of , and ). By substituting the variables back, we reach the solution
Okay. There is a bit of explanation above. It is better, only very slightly though, than nothing. But still and come down out of blue sky. It is unclear how this change of variables invented.
I recently saw Fourier comes for rescue, which is like