I would like to know, is it possible to make a linear change of dependent and independent variables such that Laplace's Equation $u_{xx}+u_{yy}$ transforms to the Wave Equation $\bar{u}_{\bar{x}\bar{x}}-\bar{u}_{\bar{y}\bar{y}}=0$. I believe the answer is no, and I have a rather involved method to calculate it, but, I'm curious how the community would approach this problem.
Thanks in advance for your insights.
Let $\xi = ax+by$, $\eta = cx+dy$ be the most general possible linear substitution. We want to find out what $$ \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) u = 0 $$ is in terms of $\xi$ and $\eta$. (Remark: there is actually no point in transforming both variables, but we may as well be as general as possible, just to be convincing.) (Further, this is rather poor notation, but it will do)
We have: $$ \frac{\partial f}{\partial x} = \frac{\partial \xi}{\partial x} \frac{\partial f}{\partial \xi} + \frac{\partial \eta}{\partial x} \frac{\partial f}{\partial \eta} = a \frac{\partial f}{\partial \xi} + c \frac{\partial f}{\partial \eta}, \\ \frac{\partial f}{\partial y} = \frac{\partial \xi}{\partial y} \frac{\partial f}{\partial \xi} + \frac{\partial \eta}{\partial y} \frac{\partial f}{\partial \eta} = b \frac{\partial f}{\partial \xi} + d \frac{\partial f}{\partial \eta}, $$ and hence $$ \frac{\partial^2 u}{\partial x^2} = a \frac{\partial}{\partial \xi}\left( a \frac{\partial u}{\partial \xi} + c \frac{\partial u}{\partial \eta}\right) + c \frac{\partial}{\partial \eta} \left( a \frac{\partial u}{\partial \xi} + c \frac{\partial u}{\partial \eta} \right) = a^2 u_{\xi\xi} + 2ac u_{\xi\eta} + c^2 u_{\eta\eta}, $$ and similarly $$ \frac{\partial^2 u}{\partial y^2} = b^2 u_{\xi\xi} + 2bd u_{\xi\eta} + d^2 u_{\eta\eta}, $$ so we have $$ (a^2+b^2)u_{\xi\xi} + 2(ac+bd)u_{\xi\eta} + (c^2+d^2)u_{\eta\eta} = 0 $$
Now, we want this to be equivalent to the wave equation, $u_{\xi\xi}-u_{\eta\eta}=0$, so we must have $$ ac+bd = 0 \\ a^2+b^2 = -(c^2+d^2), $$ or $$ a^2+b^2+c^2+d^2=0 $$ Well, it's fairly clear what the problem is going to be here if we have a real change of variables: that last inequality can only hold if $a,b,c,d=0$, which is not an allowed substitution since you want $ad-bc \neq 0$ to have independence of the new variables, among other problems.
Therefore you can't do it with a real change of variables, but can with a complex one. The best exposition I've seen of this is here, which goes into far more generality than I have.