Derivation of D'Alembert's Solution

Question

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.

Answer 1

• 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 $\square$.

Answer 2

So you are looking for a physical derivation of the wave equation. This is the one I like the most in the context of mechanics.

The wave equation appears also in electromagnetism, where it is much more fundamental, since it does not involve any approximation and descends directly from the equations of Maxwell; see here.

Answer 3

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