## Fourier… 3

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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:

$\displaystyle \dfrac{\partial^2 u}{\partial t^2} = c^2 \dfrac{\partial^2 u}{\partial x^2}$.

After presenting the equation, the book immediately declares its solution is given by

$\displaystyle u(x, t) = f(x -ct) + g(x + ct),$

for arbitrary function $f$ and $g$, 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

$\displaystyle \xi = x -ct$,

$\displaystyle \eta = x + ct$,

the PDE becomes

$\displaystyle \dfrac{\partial^2 u}{\partial \xi \partial \eta} = 0$.

Integrating this with respect to $\xi$ gives

$\displaystyle \dfrac{\partial u}{\partial \eta} = G(\eta)$,

(G: arbitrary function of $\eta$, independent of $\xi$, since differentiating G with respect to $\xi$ gives $0$).

Then integrating above with respect to $\eta$ gives

$\displaystyle u = f(\xi) + g(\eta)$,

(f: arbitrary function of $\xi$, and $g(\eta) = \int G(\eta) d \eta$).  By substituting the variables back, we reach the solution

$\displaystyle u(x, t) = f(x -ct) + g(x + ct)$.

Okay.  There is a bit of explanation above.  It is better, only very slightly though, than nothing.  But still $\xi$ and $\eta$ 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

lll