I am struggling with a solution to this question, I don't really understand what it means about the three conditions on the coefficients $A$, $B$, $C$ and $D$.


The one-dimensional wave equation

$ \frac{\partial^{2}u}{\partial x^{2}} - \frac{\partial^{2}u}{\partial t^{2}} = 0$

is subjected to a change of coordinates $u(x, t) = v(\xi, \tau )$, where new coordinates $\xi, \tau$ are linear functions of the old coordinates,

$\xi = Ax + Bt,$ $\tau = Cx + Dt, $

and $A$, $B$, $C$ and $D$ are positive constants. The transformation is chosen so that the new equation is the same as before, up to the names of the variables, i.e.

$ \frac{\partial^{2}v}{\partial\xi^{2}} - \frac{\partial^{2}v}{\partial\tau^{2}} = 0$

State the three conditions on the coefficients $A$, $B$, $C$ and $D$ that are necessary and sufficient to ensure this property. Let $A = \cosh \varphi$ where $\varphi > 0$. Find $B$, $C$ and $D$ in terms of $\varphi$.


Since $$\partial_x=\partial_x\xi\partial_\xi+\partial_x\tau\partial_\tau=A\partial_\xi+C\partial_\tau,\,\partial_t=B\partial_\xi+D\partial_\tau,$$the wave equation is $$0=(A\partial_\xi+C\partial_\tau)(A\partial_\xi u+C\partial_\tau u)-(B\partial_\xi+D\partial_\tau)(B\partial_\xi u+D\partial_\tau u)\\=((A^2-B^2)\partial_\xi^2+2(AC-BD)\partial_\xi\partial_\tau+(C^2-D^2)\partial_\tau^2)u.$$The desired conditions are $A^2-B^2=1,\,AC-BD=0,\,C^2-D^2=-1$. With $A=\cosh\varphi$ we have $B=\pm\sinh\varphi,\,C=B,\,D=A$.


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