Hola
Demuestre que la ecuación de onda \( \displaystyle\frac{\partial^2 u}{\partial t^2}-a\frac{\partial^2 u}{\partial x^2}=0 \) toma la forma \( \displaystyle\frac{\partial^2 u}{\partial r\partial s}=0 \) mediante el cambio de variable \( r=x+at \) y \( s=x-at \).
Tienes que:
\( \dfrac{\partial u}{\partial t}=\dfrac{\partial u}{\partial r}\cdot \dfrac{\partial r}{\partial t}+\dfrac{\partial u}{\partial s}\cdot \dfrac{\partial s}{\partial t}=a\dfrac{\partial u}{\partial r}-a\dfrac{\partial u}{\partial s} \)
\( \dfrac{\partial^2 u}{\partial t^2}=a\dfrac{\partial u^2}{\partial r^2}\cdot \dfrac{\partial r}{\partial t}+a\dfrac{\partial u^2}{\partial r \partial s}\cdot \dfrac{\partial s}{\partial t}-a\dfrac{\partial^2 u}{\partial s\partial r}\dfrac{\partial r}{\partial t}-a\dfrac{\partial^2 u}{\partial s^2}\dfrac{\partial s}{\partial t}=\\
=a^2\dfrac{\partial u^2}{\partial r^2}-a^2\dfrac{\partial u^2}{\partial r \partial s}-a^2\dfrac{\partial^2 u}{\partial s\partial r}+a^2\dfrac{\partial^2 u}{\partial s^2}=a^2\dfrac{\partial u^2}{\partial r^2}-2a^2\dfrac{\partial^2 u}{\partial s\partial r}+a^2\dfrac{\partial^2 u}{\partial s^2} \)
Análogamente:
\( \dfrac{\partial^2 u}{\partial x^2}=\ldots=\dfrac{\partial u^2}{\partial r^2}+2\dfrac{\partial^2 u}{\partial s\partial r}+\dfrac{\partial^2 u}{\partial s^2} \)
Termina...
Saludos.
P.D. Como bien apunta después hméndez (¡gracias!), para que se cumpla lo indicado la ecuación de partida debe de ser:
\( \displaystyle\frac{\partial^2 u}{\partial t^2}-\color{blue}a^2\color{red}\frac{\partial^2 u}{\partial x^2}=0 \)
