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In the new Simons Foundation video Terence Tao - From Rotating Needles to Stability of Waves: Local Smoothing... (November 29, 2023) after 01:15, Tao says:

In mathematics we describe waves by partial differential equations, and there are dozens and dozens of wave equations out there... I'm not going to list them all, but I just want to show you what two of them look like, at least to a mathematician.

  • The free wave equation $$\partial_{tt}u = \Delta u$$ where u : $\mathbf{R} \times \mathbf{R}^d \rightarrow \mathbf{R}$ is a scalar field;
  • The free Schrödinger equation $$\partial_{t}u = i\Delta u$$ where u : $\mathbf{R} \times \mathbf{R}^d \rightarrow \mathbf{C}$ is a complex field.

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Question: Terry Tao didn't list the "dozens and dozens of wave equations out there" in his talk, but is there such a list somewhere?

Presumably these include effects of dimensionality, nonlinearity etc. and are all defined by partial differential equations and perhaps include different definitions of "action" and conserved properties (energy, momentum, probability, etc.)

After 14:12:

.And it turns out that.. in fact all waves obey a principle of least action, it's just that different wave equations have different actions.

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    $\begingroup$ suspect his blog shows several, terrytao.wordpress.com/2009/01/22/… $\endgroup$
    – Will Jagy
    Dec 12, 2023 at 23:23
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    $\begingroup$ @MariusS.L. could be, but when I listen to Tao talk, he seems to always choose his words carefully and try to make every sentence accurate. "Dozens and dozens" doesn't feel cavalier or flippant; it sounds more like 50 to 100 different equations with non-trivial differences. I assume "out there" means discussed in the literature, not simply "possible" as you propose. With that definition of "out there" then perhaps there's been a review, in which case a list exists. $\endgroup$
    – uhoh
    Dec 12, 2023 at 23:44
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    $\begingroup$ @MariusS.L. and " I'm not going to list them all..." suggests to me at least that he feels that he feels listing the ones "out there" is indeed doable, otherwise he'd say something like "We can never list them all..." $\endgroup$
    – uhoh
    Dec 12, 2023 at 23:51
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    $\begingroup$ General relativity alone contributes many, depending on the assumptions on the stress-energy tensor. $\endgroup$
    – Deane
    Dec 13, 2023 at 2:10
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    $\begingroup$ Another possible and more general definition of a wave equation is a PDE that has a well posed initial value problem and its solutions exhibit oscillatory behavior $\endgroup$
    – Deane
    Dec 13, 2023 at 2:19

2 Answers 2

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Well, wave equations are generally second-order in time (although exceptions exist, e.g., the shallow-water equations, Dirac equation, etc.). However, instead of the Laplacian, wave equations can have more complicated spatial operators. For example, waves (with transverse displacement $w$) in an Euler-Bernoulli beam are described by

$$\frac{\partial^2 }{\partial x^2}\left(EI\cfrac{\partial^2 w}{\partial x^2}\right) = - \mu\cfrac{\partial^2 w}{\partial t^2}.$$

In case of a thin plate, the transverse displacement $\zeta$ satisfies the biharmonic equation

$$D\,\nabla^2\nabla^2 \zeta = -2\rho h \, \ddot{\zeta}.$$

The situation is more complicated when curved beams and shells are considered as the curvature couples the transverse and longitudinal displacements, and the general wave equation in classical elastodynamics is typically of the form

$$\frac{\partial^2\Psi}{\partial t^2} = \widehat{H}_\mathbf{x}\Psi.$$

Above, the operator $\widehat{H}_\mathbf{x}$ is usually Hermitian, but there are also classical systems where it is not (see, for instance, Section 3 of this paper). Additionally, $\Psi(\mathbf{x}, t)$ is a vector field (typically composed of displacements) that depends on the spatial coordinate $\mathbf{x}$ and time $t$. In elastodynamics, the form of $\widehat{H}_\mathbf{x}$ is usually dictated by the approximations one makes (e.g., what terms in the various strain expressions are important) while deriving the equations of equilibrium, and many such equations have been written down. For instance, Table 4 of this paper, lists five different equations for waves in a curved rod ("beam" in engineering literature). And this list is far from exhaustive.

Finally, although the above examples are all from elastodynamics, there are wave equations with similar complexity in fluid mechanics, electromagnetism, plasma physics, etc. Many of them are nonlinear as well.

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    $\begingroup$ speaking of thin plates: 1, 2 $\endgroup$
    – uhoh
    Dec 13, 2023 at 0:53
  • $\begingroup$ I'm curious why the operator in front of $\Psi$ should be Hermitian. In quantum mechanics, Hermiticity means the eigenvalues of the operator are real, which causes expectations values to be real. What does it mean in terms of this wave equation? $\endgroup$ Dec 13, 2023 at 13:42
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    $\begingroup$ I should've mentioned that I was describing general wave equations in elastodynamics. In elastodynamics, the operator $\hat{H}$ is almost always Hermitian and it would have real coefficients. This way, after Fourier transforming in time, you get an eigenvalue equation with real eigenvalues $\omega$ (frequency of the normal modes): $\hat{H}\Psi + \omega^2\Psi = 0$. $\endgroup$
    – B215826
    Dec 13, 2023 at 14:19
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I think an interesting categorization is into the following four categories.

The common factor:
partial derivation with respect to spatial coordinates and time coordinate.

  • Wave equation of newtonian mechanics:
    second derivative wrt spatial coordinate
    second derivative wrt time coordinate

  • Schrödinger equation:
    second derivative wrt spatial coordinate
    first derivative wrt time coordinate

  • Klein-Gordon equation:
    second derivative wrt spatial coordinates
    second derivative wrt time coordinate
    (Second derivative twice, as in the wave equation of newtonian mechanics, but the Klein-Gordon equation implements the Minkowski metric.)

  • Dirac equation:
    first derivative wrt spatial coordinates
    first derivative wrt time coordinate
    (implements Minkowski metric)



As far as I am aware of the following permutation does not occur in physics:
first derivative wrt spatial coordinates
second derivative wrt time coordinate

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  • $\begingroup$ This classification is not quite correct. There are thousands of wave equations that do not fit into any of these categories. Also, sometimes the Schrodinger equation is considered to be a diffusion equation rather than a wave equation. $\endgroup$
    – B215826
    Dec 13, 2023 at 0:14
  • $\begingroup$ @B215826 It could help if you concretely mention just one of the thousands. Also, considering the Schrödinger eqation to be a diffusion equation, as Feynman did, is not rigorous. $\endgroup$
    – Kurt G.
    Dec 13, 2023 at 4:38
  • $\begingroup$ See this answer. $\endgroup$
    – Kurt G.
    Dec 13, 2023 at 4:44
  • $\begingroup$ @KurG. Did you see my answer to OP's question? $\endgroup$
    – B215826
    Dec 13, 2023 at 6:24
  • $\begingroup$ @B215826 . I did not realize first that that was your answer. +1 to both. Unfortunately the paper you link is not publicly accessible. Table 4 sounds interesting. On the other hand it is imho opinion based if changing the Hermitian operator leads to a new wave equation. The classification by Cleonis is correct, in particular because the Dirac equation is mentioned. $\endgroup$
    – Kurt G.
    Dec 13, 2023 at 10:29

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