The origin of spurious high frequency waves in numerical weather prediction
I don't have a mathematical illustration at the moment (though I might edit this answer if I come across something in my notes). But the basic problem is sound waves. If you make a numerical model of the full Navier-Stokes equations, that model has to allow sound waves to propagate. That implies that the time step used has to be incredibly short -- under 5 minutes for sure for a grid size of 100 km, and preferably much shorter than that. So sound waves are fast-moving, high frequency, and don't really contribute to weather. That's why weather prediction models tend to use the primitive equations: these equations are very similar to the Navier-Stokes, but sound waves aren't solutions to these equations.

Hopefully that's a start. Feel free to ask for a clarification on how this filtering is done in practice. Like I said, I'll look through my notes for something, I remember doing a problem set involving this.



Edit: Here are some notes from David Randall on the anelastic and Boussinesq approximations. These notes are in more of a classroom format, but at the end they derive the link between the continuity equation and sound waves.

Briefly, sound waves arise from variations in density that are independent of variations in enthalpy/temperature. The continuity equation in general is $ \frac{\partial \rho}{\partial t} + \nabla\cdot (\rho \vec{u}) = 0 $ in which density variations can be accompanied by contraction and expansion of the fluid. The anelastic equations filter out sound waves by replacing this dynamic equation with a diagnostic one, removing the time derivative: $ \nabla\cdot(\rho_0 \vec{u}) = 0 $ where $\rho_0$ is a background density profile. No more time derivative for density means no more expansion and contraction except under changes in enthalpy.

This is how sound waves are filtered out. I realize that I didn't read the question closely enough, though. If you're working with an LES, at that fine of a spatial resolution, you're probably 1) already using the anelastic equations, and 2) properly resolving a lot of the high-frequency oscillations anyway. As @milancurcic mentioned, gravity waves become an issue as the next-highest frequency waves. These notes go through the problem of gravity waves in a shallow-water model, and this article (behind a paywall) shows how a good choice of initialization can help eliminate those waves.