Page 8 - Winter 2008

Fig. 2. Spatial map plots showing the progression of the creation of a time reversed focus. The color scale shows the amplitude distribution of the wavefield at discrete moments in time. Note that the image dis- plays the surface velocity (out of plane).
signals would then be reversed in time and the receivers ideally act as highly directional sources to generate the reversed version of the ripples. In practice, Cassereau et al. showed that if the receivers in a TRM surround a source that the spacing need not be less than a half wavelength;8 however, for a perfect reversed reconstruction of the movie (including at the nearfield scale) an infinite set of receivers is necessary. If only a few receivers are used (imagine a group of receivers on only one side of the source), the incoming ripples would not be circular and one would expect only a partial reconstruction of the impulse generated by the pebble drop. As a final requirement, when the rip- ples arrive at the drop location, a peb- ble would have to emerge from the water at the same time that the ripples converge. Without this final step the converging ripples would simply pass through each other and thereby create outward propagating ripples again. The latter result is similar to what happens in TR acoustic and elastic wave experiments. The incoming waves coalesce at the source position, pass through each other and continue to propagate outward, duplicating the original forward propagation. Source re-emission is one of the main differ- ences between an actual and an ideal TR experiment.
The TR of a pebble drop in a pond is an example of TR in free space. It is nearly equivalent to creating a phased array, which can also be used to focus energy at a specific location. Phased arrays focus energy at a point in space by introducing appropriate delays to each transducer such that the energy arrives at the desired location at the same time. One of the major advan- tages of using TR over phased arrays in free space is that one must accurately calculate the delays in applying phased arrays, while the TR process requires no such calculation. With TR, the forward signals are just flipped in time and the proper delays are naturally encoded in the forward propagation signals; however, the proper encoding takes place only if all received, reversed, and re-emitted signals
are properly synced in time.
Now let us look at results from an experiment to illus-
trate the differences between ideal and actual TR. The fol-
lowing experiment was conducted in a thin, rectangular alu- minum plate, with a source transducer placed near the cen- ter. This transducer emits a pulse consisting of a few cycles of a sine wave. A number of transducers are placed at various other locations on the plate. They are the TRM, detecting the first arrival of the pulse as well as the reflections from the boundaries. The signals detected by the receiving transducers are then time reversed and rebroadcast from these same transducers. Figure 2 displays experimentally-obtained data from the back-propagation experiment, showing only the wavefield relatively close to the source location. A laser vibrometer detects out-of-plane velocity signals of the wave- field at progressively later snapshots in time. Notice that well before the focal time, the field is quite diffuse. Later in time,
Fig. 3. Illustration of the concept of image (virtual) sources, and how they enhance Time Reversal (TR) focusing in a closed cavity. S, R, and I represent the Source, Receiver, and Image sources respectively. Subplots (a) and (d) depict a spatial illus- tration of the closed cavity TR process, while subplots (b) and (c) depict the tempo- ral signal recorded at R from the forward propagation and its time reversed version, respectively.
6 Acoustics Today, January 2008