I am stuck deriving the familiar d'Alembert formula using the Method of Descent, going from 2D to 1D. After using Kirchhoff's Formula, writing the solution independently of $y$ and some manipulations, I have obtained the following result. $$\frac{1}{2\pi c} \int_{x-ct}^{x+ct} \frac {g(x)} {({(ct)^2} - x^2)^{(1/2)}}dx $$

Where the initial conditions are $f(x) \equiv 0$ and $u_t = g(x)$.

How do I further reduce the integral to the familiar equation: $${u(x,t) = \frac{1}{2c} \int_{x-ct}^{x+ct}g(\overline{x}) d\overline{x}} $$

It is very close to the desired result, but the next step escapes me at this moment. Should I consider Polar coordinates or a change of variables?

Thanks in advance.


I am unsure if this answers your question, but maybe it might still help. The proof of the wave equation i know uses the descend method only for higher dimensions. The derivation of the 2d wave equation i know (following evans) is rather obtained by a spliting method, rewriting the wave equation into a product of transport equations, that are solved first for the homogenous transport equation and then for the inhomogenous transport equation.

  • $\begingroup$ I would do something similar, but that's not the homework problem.... I've looked at integrating it, polar coordinates, but to no avail. $\endgroup$ – Bob Dobalina Apr 7 '14 at 1:58

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