5.2 Poisson’s formula by descent
A solution in two dimensions is a solution in three that does not depend on \(x_3\). Hadamard’s method of descent applies Kirchhoff’s formula to such data. The sphere \(|y - x| = ct\) in \(\mathbb R^3\), with \(x\) in the plane \(x_3 = 0\), projects twice onto the disc \(|y - x| < ct\) in the plane, with area element \(dS = ct\,dy/\sqrt {c^2t^2 - |y - x|^2}\), so \(\frac 1{4\pi c^2t}\int _{\text {sphere}}g\,dS = \frac {1}{2\pi c}\int _{\text {disc}}g(y)/\sqrt {c^2t^2 - |y - x|^2}\,dy\).
Theorem 5.2.1 (Poisson). In \(\mathbb R^2\) the solution of \(u_{tt} = c^2\Delta u\), \(u(x, 0) = f\), \(u_t(x, 0) = g\), is \[ u(x, t) = \frac {\partial }{\partial t}\biggl (\frac {1}{2\pi c}\int _{|y - x| < ct}\frac {f(y)\,dy}{\sqrt {c^2t^2 - |y - x|^2}}\biggr ) + \frac {1}{2\pi c}\int _{|y - x| < ct}\frac {g(y)\,dy}{\sqrt {c^2t^2 - |y - x|^2}} . \]
Descending once more, to data depending on \(x_1\) only, and using \(\int _{-h}^h dy_2/\sqrt {h^2 - y_2^2} = \pi \), recovers d’Alembert’s term \(\frac 1{2c}\int _{x - ct}^{x + ct}g\) (Problem 5.6.5).
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