The so-called beam equation is, see [1], chapter 11, $\frac{\partial^2 {\phi}}{\partial t^2} + \gamma^2\frac{\partial^4 {\phi}}{\partial t^4}=0.$
When substituting a trial function $\phi = e^{\mathfrak{j}(\omega t - \kappa x)}$ you arrive at the so-called dispersion relationship between the temporal and spatial frequencies, $\omega, \kappa$, resp., $$\omega ^2 - \gamma ^2 \kappa ^4=0.$$
If you have other partial derivatives, but with still linear terms, you get, in general, a relationship $$\omega = W(\kappa)$$ where the real roots of this equation for each $\omega$ represents a mode of propagation. For the beam equation you have two solutions $\omega = \gamma \kappa^2$ and $\omega = -\gamma \kappa^2$, and you may keep the former for $\gamma > 0.$
For real $k$ the heat equation $\frac{\partial {u}}{\partial t} + \mathcal {k} \frac{\partial^2 {u}}{\partial t^2}=0$ is not really a wave equation because it leads to the dispersion relationship $\omega = -\mathfrak{j} \kappa^2$ implying an exponentionally varying amplitude, not an unchanging propagating wave.
But if instead of the 3rd order time derivative you have
$$\frac{\partial {\phi}}{\partial t} + \alpha \frac{\partial {\phi}}{\partial x} + \beta \frac{\partial^3 {\phi}}{\partial t^3}=0$$ then you get its dispersion as $$\omega =\alpha \kappa - \beta \kappa^3.$$ This 3rd order equation is the linearized Kortweg-deVries equation and it represents the long-wave propagation in a shallow water.
In general, to get real propagating solutions with real coefficients the time and spatial derivatives must be all even or all odd. The reason the Schroedinger equation leads to propagating waves is because the 1st order time derivative has an imaginary factor that cancels so the dispersion relationship has only real coefficients.
[1]: Whitham: LINEAR AND NONLINEAR WAVES
