# Wave equation

Simulations of the linear, hyperbolic wave equation on various domains, obtained by a finite difference scheme.

## Waves with dodecahedral symmetry on a sphere

Solution of the wave equation on a sphere. Reflecting discs have been placed around the vertices of a regular dodecahedron. The initial state is a set of circular waves concentrated at the centers of the faces of the dodecahedron. The image shows the wave height.

## Average energy of waves with dodecahedral symmetry on a sphere

Solution of the wave equation on a sphere. Reflecting discs have been placed around the vertices of a regular dodecahedron. The initial state is a set of circular waves concentrated at the centers of the faces of the dodecahedron. The image shows the energy of the wave, averaged over time.

## Average energy of a branched flow

Time-averaged energy of a solution to the wave equation in a random environment

## Waves with icosahedral symmetry on the sphere

Solution of the wave equation on a sphere. Reflecting discs have been placed around the vertices of a regular icosahedron. The initial state is a set of circular waves concentrated at the centers of the faces of the icosahedron. The image shows the wave height.

## Average energy of waves with icosahedral symmetry on the sphere

Solution of the wave equation on a sphere. Reflecting discs have been placed around the vertices of a regular icosahedron. The initial state is a set of circular waves concentrated at the centers of the faces of the icosahedron. The image shows the wave energy, averaged over a time interval.

## Energy flux of waves with dodecahedral symmetry on a sphere

Solution of the wave equation on a sphere. Reflecting discs have been placed around the vertices of a regular dodecahedron. The initial state is a set of circular waves concentrated at the centers of the faces of the dodecahedron. The image shows the energy flux: radial coordinate and luminosity depend on the norm of the flux, and color hue depends on its direction.

## Waves in a Julia set on the Riemann sphere

A solution of the wave equation in a domain on the Riemann sphere, which is given by an approximation of a Julia set with parameter 0.37468 + 0.21115 i. The initial state is given by two circular waves, with opposite longitudes and positive latitude, and opposite sign. The colors and radial coordinate show the wave height.

## Waves in a Julia set on the Riemann sphere

A solution of the wave equation in a domain on the Riemann sphere, which is given by an approximation of a Julia set with parameter 0.37468 + 0.21115 i. The initial state is given by two circular waves, with opposite longitudes and positive latitude, and opposite sign. The colors and radial coordinate show the energy averaged over a time interval.

## Waves with octahedral symmetry on the sphere

Solution of the wave equation on a sphere. Reflecting discs have been placed around the vertices of a regular octahedron. The initial state is a set of circular waves concentrated at the centers of the faces of the octahedron. The image shows the wave height.

## Average energy of waves with octahedral symmetry on the sphere

Solution of the wave equation on a sphere. Reflecting discs have been placed around the vertices of a regular octahedron. The initial state is a set of circular waves concentrated at the centers of the faces of the octahedron. The image shows the wave energy averaged over a time interval.

## Waves outside a disconnected Julia set on the Riemann sphere

A solution of the wave equation in a domain on the Riemann sphere, which is given by the complement of an approximation of a Julia set with parameter -0.77145 -0.10295 i. The initial state is given by two circular waves, with opposite longitudes and positive latitude, and opposite sign. The colors and radial coordinate show the energy averaged over a time interval.

## Waves outside a disconnected Julia set on the Riemann sphere

A solution of the wave equation in a domain on the Riemann sphere, which is given by the complement of an approximation of a Julia set with parameter -0.77145 -0.10295 i. The initial state is given by two circular waves, with opposite longitudes and positive latitude, and opposite sign. The colors and radial coordinate show the wave height.

## Waves in a Julia set on the Riemann sphere, equirectangular projection

A solution of the wave equation in a domain on the Riemann sphere, which is given by an approximation of a Julia set with parameter 0.37468 + 0.21115 i. The initial state is given by two circular waves, with opposite longitudes and positive latitude, and opposite sign. The image shows an equirectangular projection of the sphere.

## Waves with cubic symmetry on the sphere

A solution of the wave equation on a sphere, obtained by a finite difference scheme. Reflecting obstacles of constant radius have been placed on the vertices of a cube. The initial state is a set of circular waves concentrated near the centers of the faces of the cube, which form a regular octahedron, and at the midpoints of the cube’s edges.

## Energy flux of waves with cubic symmetry on the sphere

A solution of the wave equation on a sphere, obtained by a finite difference scheme. Reflecting obstacles of constant radius have been placed on the vertices of a cube. The initial state is a set of circular waves concentrated near the centers of the faces of the cube, which form a regular octahedron, and at the midpoints of the cube’s edges.The radial coordinate and luminosity depend on the intensity of the energy flux, while the color hue depends on its direction.

## Waves with icosahedral symmetry on the sphere, 2d projection

Solution of the wave equation on a sphere. Reflecting discs have been placed around the vertices of a regular icosahedron. The initial state is a set of circular waves concentrated at the centers of the faces of the icosahedron. The image shows an equirectangular projection of the wave height.

## Average energy of waves with icosahedral symmetry on the sphere, 2d projection

Solution of the wave equation on a sphere. Reflecting discs have been placed around the vertices of a regular icosahedron. The initial state is a set of circular waves concentrated at the centers of the faces of the icosahedron. The image shows an equirectangular projection of the wave energy averaged over a time interval.

## Waves with dodecahedral symmetry on the sphere, 2d projection

Solution of the wave equation on a sphere. Reflecting discs have been placed around the vertices of a regular dodecahedron. The initial state is a set of circular waves concentrated at the centers of the faces of the dodecahedron. The image shows an equirectangular projection of the wave height.

## Average energy of waves with dodecahedral symmetry on the sphere, 2d projection

Solution of the wave equation on a sphere. Reflecting discs have been placed around the vertices of a regular dodecahedron. The initial state is a set of circular waves concentrated at the centers of the faces of the dodecahedron. The image shows an equirectangular projection of the energy averaged over a time interval.

## Waves with octahedral symmetry on the sphere, 2d projection

Solution of the wave equation on a sphere. Reflecting discs have been placed around the vertices of a regular octahedron. The initial state is a set of circular waves concentrated at the centers of the faces of the octahedron. The image shows an equirectangular projection of the energy averaged over a time interval.

## Waves with cubic symmetry on the sphere, 2d projection

Solution of the wave equation on a sphere. Reflecting discs have been placed around the vertices of a cube. The initial state is a set of circular waves concentrated at the centers of the faces and edges of the cube. The image shows an equirectangular projection of the wave height.