Because it is computationally difficult to solve the 3D wave equation on arbitrary space-time, some 2D wave equation examples have been included in this section. The RNPL code used in generating them can be accessed here, and it is basically the 3D code reduced to 2 dimensions.



6.1 A Gaussian Wave on Minkowski Space-Time

This example represents the propagation of an initially Gaussian wave on a 2+1-dimensional Minkowski space-time. The base grid is made out of 16 × 16 cells, and the animations were made at level 6, which is enough to ensure convergence. The phase jump of &Pi as the wave hits the boundary is easily noticeable, as are the interference patters created by the reflected waves.

Phi.flv Mu.flv
Phi.flv Mu.flv


6.2 A Gaussian Wave on Compactified Minkowski Space-Time

This example represents the propagation of an initially Gaussian wave on a 2+1-dimensional compactified Minkowski space-time. The base grid is made out of 16 × 16 cells, and the animations were made at level 6. The slow-down in the wave's speed as it approaches the boundary and its square nature are easily noticeable, as is the fact that the wave thins out considerably as it approaches the boundary.

Phi.flv Mu.flv
Phi.flv Mu.flv

It is possible to uncompactify data files using DV. To do this, load a register that uses a compactified metric into DV (such as the data files used to produce the two animations above), and open the Functions menu. Scroll down to the uncompact(A) function, select the desired register, and set the Mask level equal to the dimension of the register. Clicking Go! will then produce a new register, which contains the uncompactified data.

Note: Do not use the mouse wheel to scroll through DV functions or registers! Doing so can crash DV, in which case all unsaved registers are lost! For this reason, it is a good idea to save all registers immediately after creation.

The two animations below have been made by uncompactifying the level 5 data files corresponding to the animations above. It can be noticed that the shape of the wave function becomes round, and under uncompactification the spatial region gets transformed to a much larger domain, with the wave occupying only a small area in the center.

Phi.flv Mu.flv
Phi.flv Mu.flv


6.3 Two Traveling Waves on Compactified Minkowski Space-Time

This example represents 2 colliding traveling waves on a 2+1-dimensional compactified Minkowski space-time. The base grid is made out of 16 × 16 cells, and the animations were made at level 6. The two traveling waves collide early in the simulation, and the resulting wave spreads throughout most of the space-time.

Phi.flv Mu.flv
Phi.flv Mu.flv


6.4 Interfering Gaussians on Minkowski Space-Time

The animation below shows 5 initially Gaussian waves interfering on a 2+1-dimensional Minkowski space-time. The base grid is made out of 16 × 16 cells, and the animations were made at level 6. The resulting interference pattern is quite intricate, in part because of the reflectivity of the boundaries.

Phi.flv Mu.flv
Phi.flv Mu.flv



Some of the material presented on this web page is based upon work supported in part by the Princeton Applied and Computational Mathematics Program and by the National Science Foundation under Grant No. 0745779.