Decay of Waves IV: Numerical Interlude
by Willie Wong
I offer two videos. In both videos the same colour scheme is used: we have four waves in red, green, blue, and magenta. The four represent the amplitudes of spherically symmetric free waves on four different types of spatial geometries: 1 dimension flat space, 2 dimensional flat space, 3 dimensional flat space, and a 3 dimensional asymptotically flat manifold with “trapping” (has closed geodesics). Can you tell which is which? (Answer below the fold.)
First we recall that the flat geometries in odd spatial dimensions are privileged with the strong Huygens’ principle, which was discussed in this post from last year. In particular, we expect the solution to remain quiescence away from the wave zone. Now, for all of the plots, we see that the weak Huygens’ principle (also just called finite speed of propagation) holds: that “outside” of bubble that is the wave front, the solutions remain fixed at zero. However, only the blue and red curves converge quickly to zero in the interior of the bubble. The green and magenta ones clearly show some late-time tails. So we can conclude that the blue and the red ones correspond to odd spatial dimensions with flat geometry.
To distinguish between one and three dimensions, we recall that the one dimensional wave does not decay, whereas the three dimensional wave should have amplitude decaying like . So based on the rapid decay of the solution, we conclude that the red curve is the flat three dimensional case, and the blue curve is the flat one dimensional case.
How do we tell the remaining two? One way is to use asymptotic flatness. Inside the wave zone on an asymptotically flat manifold, we expect the amplitude decay to be well approximated by that of the flat case, which is of the order where is the number of spatial dimensions. So we see that if we follow the typical amplitude near the wave front, we can conclude that the green curve which decays slower is the 2 dimensional wave while the magenta one is the three dimensional one with trapping.
Looking at the green solution we also see a feature typical of even dimensional wave equations: that the tail inside the interior region decays at more-or-less the same rate as the wave front (and not faster).
Lastly, about the spikes shown in the green and red curves near the beginning of the videos: that is due to the “incoming” component of the initial data focussing near the origin, concentrating their energy, which causes a temporary spike in amplitude that quickly goes away after some short time. We don’t see it in the magenta curve because to impose trapping, the point corresponding to 0 along the horizontal axis is not the origin, but the “throat” of a two-ended asymptotically flat manifold, which makes it look not dissimilar to the popular conception of a wormhole.