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.
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**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.
Best Answer
One approach makes use of the first-order canonical form $\mathbf{v}_t + \boldsymbol{\Lambda} \mathbf{v}_x = \mathbf{0}$ with $\mathbf{v} = \lbrace u_t \pm c u_x \rbrace^\top$ and $\boldsymbol{\Lambda} = \text{diag}\lbrace \mp c \rbrace$, see e.g. this post. Integration is performed by using the method of characteristics componentwise, and by applying the initial conditions.
Another approach makes use of the second-order canonical form $u_{\xi\eta} = 0$ with $\xi = x-ct$ and $\eta = x+ct$, see e.g. here. This PDE can be integrated as $u = F(\xi) + G(\eta)$, where the functions $F$, $G$ are deduced from the initial conditions.
In a certain way, both methods take benefit of the factorization $$ \square u = u_{tt} - c^2 u_{xx} = (\partial_t - c \partial_x)(\partial_t + c \partial_x) u $$ of the d'Alembert operator $\square$.