Propagation of a Wave over a Non-Flat 2D Space Emedded in 3D

We are simulating the wave equation applied to a zero form subject to Neumann boundary condtions

\[\ddot{\phi} = \star \mathbf{d} \star \mathbf{d} \phi\]

We use an explicit time integrator.

First, we observe how the wave propagates on a flat surface.

Second, we observe how the wave propagates on a surface with a small ditch in the middle.

Observe that as the wave passes through the hole it has to transverse an extra distance and thus the wave front is delayed relative to the front outside of the hole.

Let us also observe what happens when the hole gets deeper.

Orthonormal Bases

Given a standard orthornormal reference frame, consisting of the unit vectors along the x and y directions, the video below shows a naive mapping of that frame to the surface. When the surface is distorted, one can see that the frame is no longer orthonormal with respect to the metric of the space in which the surface is embedded.

In the video below we compute an orthonormal frame for each point using the square root method.

In the video below we domonstrate the texture unrolled onto a flat surface.