Does the PDE $\frac{1}{t}\frac{\partial^{2} u}{\partial t^{2}}-\frac{\partial^{2} u}{\partial x^{2}}=0$ have a name? $$\frac{1}{t}\frac{\partial^{2} u}{\partial t^{2}}-\frac{\partial^{2} u}{\partial x^{2}}=0$$
Does this PDE have a specific name? Is it a wave equation?
Can we transform it into a wave equation?
 A: This is a Tricomi-type equation normally describing some quantity's transition from subsonic flow (elliptic region) to supersonic flow (hyperbolic region)
The general form is:
$$
\partial_{tt} u - t^{2k} \Delta u = f(x,t)
$$
About the transforming it into wave equation equation, I would say no based on my Google-fu, $t$ having different signs makes it impossible to globally describe the behavior of a scaled $u$ or transformed $u$ using simply a wave equation. 

EDIT: If $t>0$, $k=1/2$, $f=0$, space dimension is 1 in above, set $\tau = t^{\alpha}$:
$$
\partial_{tt} u = \alpha^2 t^{2\alpha-2}\partial_{\tau\tau} u + \alpha (\alpha-1) t^{\alpha -2} \partial_{\tau} u
$$
hence by letting $\alpha = 3/2$ we could eliminate time in the second derivative, but one more term appears(? I don't know how to deal with this further)
$$
\frac{9}{4}\partial_{\tau\tau} u - \partial_{xx} u - \frac{4}{3\tau}\partial_{\tau} u = 0
$$
A: In fact this belongs to a Tricomi-type equation according to http://eqworld.ipmnet.ru/en/solutions/lpde/lpde402.pdf.
In fact this PDE cannot transform into a wave equation and get the nice non-kernel form general solution and only get the kernel form general solution by using separation of variables:
Let $u(x,t)=X(x)T(t)$ ,
Then $\dfrac{X(x)T''(t)}{t}-X''(x)T(t)=0$
$\dfrac{X(x)T''(t)}{t}=X''(x)T(t)$
$\dfrac{X''(x)}{X(x)}=\dfrac{T''(t)}{tT(t)}=-(f(s))^6$
$\begin{cases}X''(x)+(f(s))^6X(x)=0\\T''(t)+(f(s))^6tT(t)=0\end{cases}$
$\begin{cases}X(x)=\begin{cases}c_1(s)\sin(x(f(s))^3)+c_2(s)\cos(x(f(s))^3)&\text{when}~f(s)\neq0\\c_1x+c_2&\text{when}~f(s)=0\end{cases}\\T(t)=\begin{cases}c_3(s)\text{Ai}(-t(f(s))^2)+c_4(s)\text{Bi}(-t(f(s))^2)&\text{when}~f(s)\neq0\\c_3t+c_4&\text{when}~f(s)=0\end{cases}\end{cases}$
$\therefore u(x,t)=C_1xt+C_2x+C_3t+C_4+\int_sC_5(s)\sin(x(f(s))^3)\text{Ai}(-t(f(s))^2)~ds+\int_sC_6(s)\sin(x(f(s))^3)\text{Bi}(-t(f(s))^2)~ds+\int_sC_7(s)\cos(x(f(s))^3)\text{Ai}(-t(f(s))^2)~ds+\int_sC_8(s)\cos(x(f(s))^3)\text{Bi}(-t(f(s))^2)~ds$
or $C_1xt+C_2x+C_3t+C_4+\sum\limits_sC_5(s)\sin(x(f(s))^3)\text{Ai}(-t(f(s))^2)+\sum\limits_sC_6(s)\sin(x(f(s))^3)\text{Bi}(-t(f(s))^2)+\sum\limits_sC_7(s)\cos(x(f(s))^3)\text{Ai}(-t(f(s))^2)+\sum\limits_sC_8(s)\cos(x(f(s))^3)\text{Bi}(-t(f(s))^2)$
