Page 20 - Fall 2008

Fig. 13. A graphic rendition of the tuning-fork, Penrose-tile, two-dimensional qua- sicrystal. This particular Penrose-tile is formed with a pattern of two unit cells—a skinny rhombus and a fat rhombus.
function ψ(x), with some subscripts added, must be of the form
(1)
where k is an important parameter called the Bloch wavenumber. In Eq. 1, n is an integer indexing different bands (e.g., in Fig. 6, the circles have n = 1, the squares are for n = 2, etc.), and Uk,n(x) is a function that is periodic in x with the period equal to the lattice constant a, so that Uk,n(x+a)=Uk,n(x). Solutions as in Eq. 1 are called Bloch wavefunctions.
The Bloch wavenumber must lie within the range -π/a to π/a, and it must have discrete values given by positive or neg- ative integer multiples of π/l. With l/a = N, k may have 2N+1 possible values. However, k = - π/a and k = + π/a generate exactly the same mode, so that there are effectively only 2N values for k. In the example illustrated in Figs. 6 and 7, l/a = 4, so that k may have eight possible values. It should now be noted from Fig. 6 that each band has eight distinct modes: “a” through “h” for the first (n = 1) band, and “i” through “p” for the second (n = 2) band. Thus the Bloch wavenumber is an index to different modes within each band. Now the band structure diagram can be described—it is a plot with the Bloch wavenumber indexing the possible modes on the hor- izontal axis (with 2N discrete values between -π/a to π/a), and the frequencies of the modes, one band after another, on the vertical axis. For the example with N = l/a = 4, the band structure diagram is shown in Fig. 8. The letters refer to the modes in Fig. 7, and the frequencies and their symbols are taken from Fig. 6.
It should be noted that in Fig. 8, the symbols for modes “h” and “i” could be placed on either side of the plot, because -π/a and π/a are equivalent. These modes were plotted where they are in Fig. 8 simply to maintain the back-and-forth pattern of the modes in the figure. It should also be noted that if N = l/a is increased, then the curves in Fig. 8 remain the same, but the number of modes in each band increases, and the density of symbols on the curves for each band increases. For a “large” system (l → ∞), the band structure is given by the continuous curves in Fig. 8; in this case the vertical extremes of the con- tinuous curves are referred to as “band edges.”
Another important feature of wave propagation in peri- odic media is that there is non-trivial dispersion; that is, the frequency is a nonlinear function of the Bloch wavenumber. For the plain string with no scatterers, the Bloch wavenum- ber becomes k=2π/λ, and the dispersion is linear with f = ck/2π. For a string with scatterers, the frequency as a function of Bloch wavenumber is given by the nonlinear curves of Fig. 8. The non-trivial dispersion of waves in periodic systems
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referred to as the “Brillouin zone boundary.”1,4 In two and three dimensions, the nature of band structure and Brillouin zone boundaries is very interesting; a systematic approach to
periodic wave systems in higher dimensions was one of the
may be used to advantage in many devices.
The limits of the Bloch wavevector at -π/a and π/a is
major contributions of Leon Brillouin.
1
that all have a “sharp” frequency of 440 Hz. The local oscilla- tors are coupled together with arcs of steel wire spot-welded at the ends of the tines of the tuning forks, as shown. The coupling causes the sharp frequency to broaden out into a band. While the system is fairly complicated, the periodic (square lattice) nature and the effects of the periodicity on the natural frequencies for the system are quite definite. This system was used to see if the array of tuning forks could cor- rectly produce the predictions of Floquet’s theorem.
The square lattice of tuning forks pictured in Fig. 12 was actually a test case for extending the research to a two- dimensional quasicrystalline system. In this case, the square periodic lattice was replaced with a quasicrystalline lattice known as a “Penrose tile.”13 One way of describing a periodic system is to specify its construction as “taking a unit cell and repeating it to fill space.” To construct a Penrose tile, one is allowed to use more than one unit cell to fill space.
A graphical rendition of an actual tuning fork Penrose tile is shown in Fig. 13. As may be observed, this particular Penrose tile is formed with a pattern of two unit cells—a skinny rhombus and a fat rhombus. The tuning fork array experiment was used to provide the first evidence of the effects on wave propagation in a two-dimensional Penrose
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articles on wave propagation in disordered and quasicrys- talline arrays of scatterers.
Appendix
To understand a band structure diagram, it is helpful to examine the mathematical form of the modes ψ(x) of a peri- odic system. From Floquet’s theorem,5 it is found that the
tile quasicrystal.
The author is planning to prepare additional tutorial
Wave Propagation in Scatterers 19