Alternate forms for the wave equation?

Question

Are there alternate ways to write the usual wave equation?:

$$ \frac{\partial^2u}{\partial t^2}=c^2\nabla^2u. $$

I've come across this equation in a paper (Yoshimura 1975):

$$\frac{\partial^2\Psi}{\partial t^2}=P(\nabla\Omega\times\nabla\Psi)\cdot\hat{\mathbf{e}}_{\phi}\, ,$$

and he claims that it describes a wave (I think he forgot the unit vector in the paper). I just don't understand how... What does it really mean?

EDIT: we are in the polar spherical coordinate system. And for the purpose of this question, $P$ and $\Omega$ are just undefined scalar functions.

Reference

Yoshimura, H. (1975). Solar-cycle dynamo wave propagation. The Astrophysical Journal, 201, 740-748.

Answer 1

I don't think that that equation is an alternate form for the wave equation. It might represent a wave phenomenon, though. But then the main problem translates to define what constitutes a wave phenomenon. But I don't think that discussion so useful, at least in this context.

Let me give you an example I'm familiar with (since magnetohydrodynamics is not my field). In elastodynamics you have the following set of equations

$$(\lambda + 2\mu)\nabla(\nabla\cdot \mathbf{u}) - \mu\nabla\times\nabla\times\mathbf{u} = \rho\frac{\partial^2 \mathbf{u}}{\partial t^2}\, .$$

We can say that this represent "wave phenomena". Actually, it presents longitudinal and transverse waves. But this is not evident from this equation.

If we do a Helmholtz decomposition of the displacement field

$$\mathbf{u} = \nabla \phi + \nabla\times\boldsymbol{\psi}\, , \quad \text{with } \nabla\cdot\boldsymbol{\psi} = 0\, ,$$

we get

\begin{align} &\nabla^2 \phi = \frac{1}{\alpha^2} \frac{\partial^2 \phi}{\partial t^2}\, ,\\ &\nabla^2 \boldsymbol{\psi} = \frac{1}{\beta^2} \frac{\partial^2 \boldsymbol{\psi}}{\partial t^2}\, ,\\ \end{align}

with $\alpha^2 = (\lambda + 2\mu)/\rho$, $\beta^2 = \mu/\rho$. Here it is easy to see that they behave like waves.