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

Question

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.

Answer 1

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.

Answer 2

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