Derivation of D'Alembert's Solution Could anyone provide references wherein D'Alembert's solution to the one-dimensional wave equation is derived from basic properties?  I keep running into the approach where D'Alembert's solution is mysteriously presented out thin air and then shown to satisfy the wave equation.  I am looking for references that show how D'Alembert's solution is derived.
$$\;$$
**Giuseppe: Either of your references is a good first start.  The next step, which is what I'm looking for, would be to derive D'Alembert's solution:
$$y(x,t) = \frac{1}{2}\left[F\left(x-ct\right)+F\left(x+ct\right)\right]+\frac{1}{2c}{\int\limits_{x-ct}^{x+ct}}G\left(s\right)ds$$
where $F$ and $G$ are odd periodic extensions of $f$ and $g$ respectively, and where the initial conditions are
$$y\left(x,0\right)=f\left(x\right)\;,\;0{\lt}x{\lt}L,$$
$${y_t}\left(x,0\right)=g\left(x\right)\;,\;0{\lt}x{\lt}L\;.$$
D'Alembert's is more general in the sense that $f$ and $g$ do not have to be restricted to well behaved functions like the trigonometric functions.

For the derivation of D'Alembert's solution to the one-dimensional wave equation you may follow the following references:
$(1)~~$"Linear Partial Differential Equations for Scientists and Engineers" by Tyn Myint-U & Lokenath Debnath (Chapter $5$, section $5.3$)
$(2)~~$d’Alembert’s solution of the wave equation / energy
$(3)~~$Wikipedia
$(4)~~$WolframMathWorld
