Page 14 - Fall 2008

Fig. 2. Illustration of a system where the medium is filled with scatterers.
dimensional model wave medium will be a string, having some mass per unit length ρL, stretched to a constant tension. The sound waves will be transverse vibrations of the string (i.e., the displacement of the string will be orthogonal to the length of the string), traveling with a wave speed c. It will be assumed that the string is infinitely long, so that boundary conditions may be ignored. The periodic array of scatterers will be point masses positioned at equal intervals (with spac- ing “a,” called the “lattice constant,”) along the string. Each point scatterer will have the same mass, m. A section of the string with a mass that is repeated periodically will be referred to as a “unit cell.” The time dependence of the motion of the string will be simple harmonic, given by cos(2πft), where f is the frequency of the transverse vibra- tions. With the string extending along the x-axis, the dis- placement of the transverse vibration in the y-direction, ψ, will be a function of its x-position, thus ψ(x). It should be noted that this formulation includes standing waves only; traveling waves may be formed as linear combinations of
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Fig. 4. The properties of waves in periodic arrays of scatterers are well known in solid state physics. In this case the waves are the quantum mechanical nature of the electron and the scatterers are ions in a crystal lattice.
The treatment of the model system involves finding the possible modes of vibration of the string and the natural fre- quency for each mode, for a given value of the “strength” of the scatterers (that is, the size of the mass m). The modes and frequencies may be indexed with subscripts.
Zero masses
The first possibility to consider is what happens when the mass of each scatterer is zero (i.e., m=0). That is the situ- ation for which the string actually has no scatterers at all. In this case, the modes of vibration are proportional to cos(2πx/λ) or sin(2πx/λ), where λ= c/f is the wavelength of the vibration. Because the string is infinite in length, it is not possible to show the mode of vibration for the entire string in a figure. Because the modes are periodic in the position x, with periodicity λ, figures of modes may be made by illus- trating them only over a finite length l (from x = 0 to x = l), with l at least as large as a half-wavelength. It is also not pos- sible to show all of the modes for all possible (i.e., arbitrary) wavelengths λ; however, it will turn out that only certain wavelengths will be important for understanding what hap- pens when the scatterers (with m greater than 0) are placed on the string. Thus, the following conditions are added: (a) examples of the modes of vibration will be limited to those modes that correspond to fitting an integer number of half- wavelengths in the length l (so that λ / 2l = integer) and (b) when scatterers are added, they will be placed with a lattice constant a such that an integer number of lattice constants will fit in the length l (l / a = integer). These last relationships constitute forming “periodic boundary conditions” applied to the system at x=0 and x=2l. It should be noted that with these conditions, the frequencies of the modes (for the case
standing waves.
of electricity is due to two features: (a) because of quantum mechanics, the electron behaves as if it were a wave, and (b) in a crystal the ions are arranged periodically.
A one-dimensional acoustic array
The properties of soundwaves (or electrons) in periodic arrays of scatterers may be thought of as resulting from some special coherence in the multiple scattering within the peri- odic system of scatterers. These properties have a rigorous mathematical foundation in Floquet's theorem5 that is gener- alized in physics and called Bloch's theorem.6 Rather than use a highly mathematical approach, we will explain the behavior of wave propagation in a periodic array of scatterers with a simpler, graphical presentation.
To understand how sound propagates in a periodic sys- tem, we will study a one-dimensional periodic system because the one-dimensional approach may be readily applied to three orthogonal dimensions. First it will be nec- essary to establish a model and some notation. The one-
Fig. 3. An illustration of the importance of understanding the vibration of a system with a periodic array of scatterers. (a) Waves propagating down a plate may make noise at a far end. (b) A rib would reflect the wave, reducing the noise. (c) Remarkably, more ribs, if arranged periodically, would not further reduce the noise.
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