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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.

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    $\begingroup$ What is this operator $P$? $\endgroup$
    – Charlie
    Jun 7, 2021 at 18:18
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    $\begingroup$ Link to abstract page? $\endgroup$
    – Qmechanic
    Jun 7, 2021 at 18:24
  • $\begingroup$ P is an undefined scalar operator. But a usual form is $r\cos{\theta}\alpha$, $\alpha$ being a time-independent scalar function. $\endgroup$ Jun 7, 2021 at 18:52
  • $\begingroup$ It is hard to figure out what you are asking. What is $P$ and what is $\Omega$ $\endgroup$
    – user196418
    Jun 7, 2021 at 18:53
  • $\begingroup$ It is unimportant. They are just scalar functions (which depend on the $r$ and $\theta$ coordinates in the polar spherical coordinates). $\Psi$ is the quantity that should be described by a wave motion. $\endgroup$ Jun 7, 2021 at 18:56

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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.

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  • $\begingroup$ Ah, I'll try that and see if that equation can be further decomposed. Thanks! $\endgroup$ Jun 7, 2021 at 19:09
  • $\begingroup$ @nicoguaro do you know a good reference for this (Elastodynamics)? $\endgroup$ Jul 24, 2021 at 20:14
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    $\begingroup$ @GiorgioPastasciutta, I think that it would be better if you ask a question for reference suggestions (and maybe tag me in the comments). Having said that, I like Auld, B. A. (1973). Acoustic fields and waves in solids. Рипол Классик.. $\endgroup$
    – nicoguaro
    Jul 26, 2021 at 14:53
  • $\begingroup$ I'll definitely do, thank you for suggestion and availability $\endgroup$ Jul 27, 2021 at 16:48

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