Main: The Wave equation

The two dimensional Green's function

We can use the method of descent to deduce the two dimensional Greens function

$\begin{displaymath} G_{2}(x_{1},x_{2},t; y_{1},y_{2},\tau) \end{displaymath}$

for the two dimensional wave equation

$\begin{displaymath} {1\over c^{2}}{{\partial^2 \phi}\over{\partial {t}^2}}-\lef... ...l^2 \phi}\over{\partial {x_{2}}^2}}\right) =f(x_{1},x_{2},t). \end{displaymath}$

We find that

$\begin{displaymath} G_{2}(x_{1},x_{2},t; y_{1},y_{2},\tau) = \int\limits^{\in... ...\hbox{\boldmath $x$}},t; {\hbox{\boldmath $y$}},\tau)\,dy^{3}. \end{displaymath}$

Evaluating this integral is in principle similar to the calculation given above, but more tedious. Suffice it say that the integral can be done and, writing ${\hbox{\boldmath$x$}}=(x_{1},x_{2})$ and ${\hbox{\boldmath$y$}}=(y_{1},y_{2})$ , we have

$\begin{displaymath} G_{2}({\hbox{\boldmath $x$}},t; {\hbox{\boldmath $y$}},\tau... ...oldmath $x$}}-{\hbox{\boldmath $y$}}\right\vert^{2}/c^{2}} }. \end{displaymath}$

(In fact, the method of descent was invented not to find the one dimensional Green's function but rather, as the easiest way of finding the two dimensional Green's function!)

Main: The Wave equation

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