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# Time Reversal in Solids (Linear and Nonlinear Elasticity)

## Summary

For a non-specialist description of time reversal, read Reversing Time to Find Wave Sources. For an in-depth overview, read our PDF from Acoustics Today.

Time Reversal (TR) is analogous to filming a pebble dropped into a pond and the resultant ripples emanating outward. Stop the movie and run it backwards. All of the waves propagate backwards-in-time, coalescing on the pebble impact as illustrated in the figure below.

Snapshots of TR back-propagation observed in experiment on a metal plate, showing out-of-plane displacements. Time steps a-b show the wave displacement field up to the focus at c. Unlike the movie analogy, in actual TR the waves at focal time pass through each other and are then rebroadcast as see in steps d-e.

More rigorously, in any linear, time-invariant process, wave propagation may be described as a linear system with different impulse responses. If a point source at r0 emits a Dirac pulse, the δ(t) the jth transducer will record the impulse response hj(t) that corresponds to the Green's function...

Moreover, due to the principle of reciprocity, hj(t) is also the impulse response describing the propagation of a pulse from the jth transducer to the source. TR theory shows that it is the wave phase that makes TR focus. Phase is all we have for locating long duration, noise-like (tremor) signals because discrete arrivals don't exist.

In practice, TR can be carried out in at least three ways...

a) TR can be conducted experimentally by propagating waves forward and backwards in say, a lab sample.

b) In the case where a source is located inside a solid (e.g., Earth), one can take recorded data (re: seismic), TR the data and backpropagate it through an appropriate velocity model.

c) TR can be conducted entirely in a numerical velocity model.

Click image to enlarge.

 Click image to enlarge. We are exploring the possibility of applying TR to reconstruct a spatially and temporally complex earthquake source, locating Earth tremor, as well as exploring the use of TR in NonDestructive Evaluation (NDE) applications. This work is supported by institutional support at Los Alamos (LDRD) and by commercial support.

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA