Page 17 - Winter 2010

WAVE PROPAGATION IN ARRAYS OF SCATTERERS TUTORIAL: PART 2
J. D. Maynard
Department of Physics, The Pennsylvania State University University Park, Pennsylvania 16802
“The problem of wave propagation in disordered systems remains open, but some very important general results have been established.”
This is the second part of a tutorial culty in high dimensions, some very
series on wave propagation in
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A periodic array of scatterers is relatively easy to treat, because the periodic array is formed with a “unit cell” repeat- ed to fill space, and it is only necessary to solve a scattering problem for one unit cell. The results are based on rigorous theorems: Floquet’s theorem4 for one dimension and Bloch’s theorem5 for two and three dimensions. Wave propagation in a disordered array is considerably more difficult, because now it is necessary to solve the full problem, satisfying boundary conditions on each of a large number of scatterers. The problem dates back to the era of Lord Rayleigh— 1869–1919—(with interest in sound propagation through forests, etc.), and since that era a significant number of dis- tinguished scientists have worked on the problem. There was some progress, but nothing like a complete understanding.
More recent interest in waves in disordered systems arose in condensed matter physics, with the development of sub-micron sized components for integrated circuits for computers, etc. The connecting wires in integrated circuits are non-crystalline metals, and within metals electrons move as waves (as prescribed by quantum mechanics) scattered by the disordered array of ions. Ordinarily, thermal motion of the ions would wash out the wave character of the electrons, so the problem could be treated relatively easily and accu- rately with a diffusion equation. However, if a wire were suf- ficiently small, then an electron could maintain its wave nature, and it would then be necessary to understand wave propagation in a disordered array of scatterers. This area of condensed matter research, referred to as “mesoscopic physics” or “long-range phase coherence” led to significant
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arrays of scatterers. The first part of the tutorial discussed wave propaga- tion in arrays of scatterers arranged periodically; this part will cover waves propagating in disordered (non-peri- odic) arrays. A later part will treat waves propagating in a relatively new type of array: the “Penrose tile” or “quasicrystal.”2,3
important general results have been established. These results, together with the solved problem in one dimension, are the subjects of this article.
One-dimensional disordered systems
The starting point for the discus- sion of waves in disordered systems will, of course, be the solved one-dimension- al case. An essential first step will be to
review the results for waves in one-dimensional periodic sys- tems, as presented in the first part of this tutorial series. Some of those results may be summarized1 as follows:
1. The model used for illustrating the results is shown in Fig. 1a; it is a string, with some mass per unit length, stretched to a constant tension. The waves are transverse waves which travel with the characteristic speed for the string. The scatterers are point masses separated with a lattice constant, a. One mass and one section of string, of length a, comprise a unit cell.
2. The wave field solutions (which may be referred to as eigenfunctions, normal modes, wave functions, etc.) for peri- odic systems are “extended;” that is, the modulus of the eigenfunction is the same in every unit cell. This is illustrat- ed in Fig. 1b.
3. A simple “balls-and-springs” understanding of the extended eigenfunctions is that each section of string between pairs of masses acts as a “local oscillator,” and for the periodic system all of the local oscillators are identical and have identical “local oscillator frequencies.” When the system is excited near a local oscillator frequency, any energy (how-
Fig. 1. Review of the model system and results for wave propagation in a one- dimensional periodic array of scatterers. (a) Illustration of the model system, con- sisting of a string with masses (scatterers) spaced periodically. (b) The modulus of the wave field solution for a periodic system; since the modulus is the same in every unit cell, the wave field is said to be “extended.” (c) The wave field itself is more com- plicated due to interference effects.
advances in understanding waves in disordered systems. The first rigorous theorem (analogous to Floquet’s theo- rem for periodic arrays) applicable to wave propagation in a disordered array was found by the mathematician Hillel Furstenberg in 1963.7 Furstenberg’s theorem is relatively recent on a time scale which dates back to Lord Rayleigh, so wave propagation in disordered arrays has been a long and difficult problem. But Furstenberg’s theorem treats only one- dimensional systems, and unlike Floquet’s theorem, it cannot be extended into two or three dimensions. So the problem of waves in disordered systems remains open. Despite the diffi-
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