Space-time microstructures and field patterns
Field patterns are a new type of wave propagating along orderly patterns of characteristic lines which arise in specific space-time materials, whose geometry in one spatial dimension plus time is somehow commensurate with the slope of the characteristic lines, as in the figures below.
Figure 1. Three examples of space-time microstructures. a) A space-time checkerboard in which the slope of the characteristic lines (in red) is the same for the two phases (white and gray) and related to the dimensions of the squares. b) A space-time checkerboard in which the wave speeds of the three phases (white, gray, dark gray) are in a certain ratio (the wave speed is the slope of the characteristic lines in a space-time diagram). c) Space-time rectangular inclusions (gray) in a space-time matrix (white): the dimensions of the rectangles is related to the wave speeds of the two phases.
Figure 2. Evolution of the wave along the discrete network of characteristic lines for the space-time checkerboard of Figure 1b) when there is an instantaneous point-source: n represents the number of time periods and p the number of points in the discrete network. As time evolves, the wave propagates inside a triangular shape having vertex at the point where the source is located (the point labeled with p = 201).
Figure 3. Dispersion diagrams for the three-phase space–time checkerboard shown in Figure 1b) for different values of the material parameters. The dispersion diagram a) corresponds to a condition of unbroken PT– symmetry (the frequency ω is real), whereas b) to a condition of broken PT–symmetry: Imω takes non-zero values indicative of exponentially growing and decaying modes.
Related publications
G.W. Milton, O. Mattei, 2017. Field patterns: A new mathematical object. Proc. R. Soc. A 20160819.
O. Mattei, G.W. Milton, 2017. Field patterns without blow up. New J. Phys.
19 093022.
O. Mattei, G.W. Milton, 2017. Field patterns: a new type of wave with infinitely degenerate band
structure. Europhys. Lett. 120(5), 54003.
See also the news release by Paul Gabrielsen, "Field patterns" as a new mathematical object, and the SIAM NEWS article New Horizons in the Study of Waves in Space-time Microstructures.