Волновое уравнение

[ShMaxG]

А вторая замена вполне естественна, ибо смотрите что получается:

$\[\frac{{{\partial ^2}u}} {{\partial {r^2}}} + \frac{2} {r}\frac{{\partial u}} {{\partial r}} = \frac{1} {r}\left( {r\frac{{{\partial ^2}u}} {{\partial {r^2}}} + 2\frac{{\partial u}} {{\partial r}}} \right) = \frac{1} {r}\left( {r\frac{{{\partial ^2}u}} {{\partial {r^2}}} + 2\frac{{\partial r}} {{\partial r}}\frac{{\partial u}} {{\partial r}} + \frac{{{\partial ^2}r}} {{\partial {r^2}}}u} \right) = \frac{1} {r}\frac{{{\partial ^2}}} {{\partial {r^2}}}\left( {ru} \right)\]$


Значит, $\[{u_{tt}} = \frac{1} {r}\frac{{{\partial ^2}}} {{\partial {r^2}}}\left( {ru} \right) \Leftrightarrow {\left( {ru} \right)_{tt}} = \frac{{{\partial ^2}}} {{\partial {r^2}}}\left( {ru} \right)\]$.

Ну и вот.