# Lorentz transformation, the one-dimensional wave equation

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 $$D$$.

Question:

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 $$B=\pm\sinh\varphi,\,C=B,\,D=A$$.