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Fig. 8. Eigenfunctions (transverse wave amplitude as a function of position, ψ (x)) for different eigenfrequencies. (a) and (b) Eigenfunctions for the periodic sys- tem, verifying extended Bloch waves. (c) through (f) Eigenfunctions for the disor- dered system, showing Anderson localization.
quency corresponds to moving a horizontal line vertically upward in Fig. 2. If the starting frequency is in the gap region, where the modes are exponentially-decaying evanescent waves, then very little response is measured at the far end of the wire. As the frequency (or horizontal line) enters a band, then a sequence of resonances in the pickup is observed as eigenfrequencies (the symbols in the bands in Fig. 2) are tra- versed. After a whole band is traversed, the response of the system drops while the frequency is in the next gap. Just such features were observed experimentally, as shown in Fig. 7a. Between gap regions of very low response, there are about 50 resonances corresponding to the eigenfrequencies of the 50 mass system. The more slowly varying structure in Fig. 7a is due the pickup being at one point, which is sometimes near a node in the Bloch standing wave.
After measuring the eigenfrequency spectrum, the drive frequency was fixed at one of the resonances (eigenfrequen- cies) in the band, and the pickup was moved along the length of the wire to measure the eigenfunction. Two eigenfunctions (plotted as wave amplitude as a function of position along the wire) for two different eigenfrequencies are shown in Fig. 8a and 8b. The band which was measured is the second band, for which the eigenfunctions correspond to fitting approxi- mately one half wavelength between the masses; thus in Fig. 8a and 8b, the places where the amplitude drops to nearly zero are the locations of the masses, and the peaks in- between indicate the half wavelengths. The more slowly vary- ing structure in the eigenfunctions is due to the form of the Bloch standing waves. It is important to note that the large amplitude of these eigenfunctions is maintained from the driven end to the far end of the 50 mass system. This extend- ed nature of waves in a periodic system is remarkable given that the reflection coefficient of each mass is 0.9997. The measured results for this system can be compared with the predictions of Floquet’s theorem, and it is found that the sys- tem of masses and steel wire is an accurate realization of a periodic wave mechanical system.
After measuring the periodic system, the positions of
the masses were moved so as to give the system 2% disor- der; that is, the lengths of wire between the masses were set to a(1 + 0.02r), where r was a random number between -1 and + 1. The measurement procedure used for the periodic system was then repeated for the disordered system. The measured eigenfrequency spectrum is shown in Fig. 7b. Instead of having a regular pattern, the eigenfrequencies for the disordered system clump into an irregular pattern, and some eigenfrequencies appear at positions which would be in the gap for the periodic system. By setting the drive fre- quency at one of the new resonances, an eigenfunction for the disordered system could be measured. The eigenfunc- tions for four different eigenfrequencies are shown in Fig. 8c through 8f. Fig. 8c shows a strongly localized state, and the others are localized to a somewhat lesser degree; this is as predicted because the most localized eigenfunction is the one whose eigenfrequency is well within the gap region (farthest to the left in Fig. 7b).
The experiment described above was undertaken to
study the effects of a time-dependent potential field on
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Anderson localization. It turned out that this acoustics
experiment provided the first direct experimental observa- tion of an Anderson localized eigenfunction (Fig. 7.)
Classical Anderson localization in higher dimensions
Guided by discussions in the preceding section, a scheme for observing classical Anderson localization in two
Fig. 7. Measured frequency spectra showing resonances at eigenfrequencies. (a) Spectrum for the steel wire with masses spaced periodically. (b) Spectrum with the mass spacing disordered by 2%; note the eigenfrequency appearing in the gap on the left.
Wave Propagation in Arrays 19