[acchow]

The infinite implied speed is of course because the Euler equations are just a rough approximation of what is happening, right? Much like Newtonian physics was an approximation?

There’s a real causality bound of the speed of light

[wwalker3]

If you use the theory of nonlinear partial differential equations to analyze the behavior of compressible materials, you can find what are called the "characteristic" speeds, which are the speeds that various types of waves propagate at.

Compressible materials tend two have two different characteristic speeds, one for sound waves and one for shock waves.

The speed of sound basically works out to speed = sqrt(stiffness / density). So as a material gets stiffer, the speed of sound goes up. An infinitely stiff (i.e. incompressible) material by implication would have an infinite speed of sound, though this can't happen in any real material.

Shock waves travel faster than sound, at a speed related to the pressure difference across the shock. The greater the pressure difference, the faster the shock travels. So if you had an infinite pressure difference, you could have an infinitely fast shock wave, but again this can't happen in the real world.

However, sound and shock speeds only apply to pressure waves in a material. Other influences like gravity and electromagnetism travel at the speed of light. So for example, if you're doing fluid dynamics for plasma, then you'll have a third characteristic speed, the speed of light, because of the charged nature of the material attracting and repelling itself.

There are also more exotic characteristics, like the speed of a propagating combustion front in a flammable material. But when you get to this level you're no longer just solving one simple set of differential equations.