I am given the partial differential equation $\rho u_{tt}(x, t) + Ek^2u_{xxxx}(x, t) = 0$ for modelling a beam/rod, where
$0 < x < l$, where $l$ is the length of the rod,
$u(0, t) = a \sin(\omega t)$,
$u_x(0, t) = 0$,
$u_{xx}(l, t) = 0$,
$u_{xxx}(l ,t) = 0$.
I am then provided with the scaled equations
$$x = l \zeta, \ \ \ \tau = \omega t, \ \ \ u = av,$$
and told that we therefore have that
$$v_{\tau \tau} + Jv_{\zeta \zeta \zeta \zeta} = 0,$$
where $J = \dfrac{E k^2}{\rho \omega^2 l^4}$, where $E$ is Young's modulus, $k$ is the radius of gyration, $\rho$ is the density, $\omega$ is the frequency, and $l$ is the length of the rod.
However, there is no explanation for how the scaled equations are used to go from $\rho u_{tt}(x, t) + Ek^2u_{xxxx}(x, t) = 0$ to $v_{\tau \tau} + Jv_{\zeta \zeta \zeta \zeta} = 0$. And this is the first exposure to such material, so there is no basis for expecting someone to know what happened here. I would greatly appreciate it if people would please take the time to explain this.