Page 13 - Fall 2008

WAVE PROPAGATION IN ARRAYS OF SCATTERERS TUTORIAL: PART 1
Julian D. Maynard
Department of Physics, The Pennsylvania State University University Park, Pennsylvania 16802
When sound waves, propagating in
a uniform medium, encounter a
foreign object, the sound is scat-
tered in all directions. When the size of
the object is about the same size as the
wavelength of the sound, the scattering
may be called “strong” and its pattern may
be complicated. If the geometry of the
object is relatively simple, it would be pos-
sible to calculate the scattered sound
field—such a situation is represented by
Fig. 1a. However, if there is a second object, located a few wavelengths away from the first object, then the scattered sound may reflect back and forth between the objects, undergoing “multiple scattering,” as shown in Fig. 1b. The multiple scattering makes any calculation far more difficult, and the problem would rapidly grow in difficulty if several more scattering objects were added. What then if a medium were filled with such scatterers, as illustrated in Fig. 2? One might imagine that such a situation would be hopeless. However, if the scatterers are arranged periodically, like the atoms in a crystal, then a sound field may be readily calcu-
1
lated to high accuracy. Furthermore, if the scatterers were
in a disordered configuration, one could make significant
qualitative and quantitative predictions as to how sound
2
would behave if it encountered such a system. There is also
a possible configuration of scatterers intermediate between periodic and disordered, referred to as quasicrystalline,3 for which an accurate understanding of the behavior of the sound is again possible. This article is an introduction to understanding sound propagation in periodic arrays of scatterers. Later articles will treat the cases of disordered and quasicrystalline arrays of scatterers.
“If scatterers are arranged periodically like atoms in a crystal, then a sound field may be readily calculated to high accuracy.”
Periodic array of scatterers
The importance of understanding wave propagation in a periodic array of scatterers is illustrated in Fig. 3. Consider a plate (e.g., a floorboard) with a source of vibration at one end and some listeners at the far end. The source generates trans- verse (flexural) waves, and these waves propagate to the far end of the plate, radi- ate sound and create an annoyance for the listeners (Fig. 3a). Usually, for structural
reasons, a plate will have a rib on it, and that rib reflects the vibration (as shown in Fig. 3b) so there is less vibration trans- mitted, and less noise at the other end of the plate. If one rib reflects the vibration and reduces the annoyance, why not a series of many ribs? For ease of manufacture, the ribs may be identical and arranged periodically, as illustrated in Fig. 3c. One might expect that the array of ribs would greatly decrease the vibration, with transmission proportional to some large power of the single-rib transmission coefficient. However, this is not at all what happens. The remarkable result for the periodic array is that the first rib reflects the vibration, but all the rest transmit without any further reduc- tion, at least within ranges of frequencies, called “pass bands.”
This remarkable property of periodic arrays of scatterers is well known in solid state physics,4 where a typical applica- tion would be the understanding of the electrical conductiv- ity of a metallic crystal. In a crystalline wire consisting of moving electrons and fixed positive ions, as illustrated in Fig. 4, a classical electron would be very strongly scattered by the ions as it tried to move down the wire, and one would have to conclude that a metal should be a very poor conductor of electricity. The fact that metals are actually good conductors
Fig. 1. Illustrations of sound scattering from relatively simple objects. (a) Single object. (b) Multiple scattering between two objects greatly complicates the scattering problem.
12 Acoustics Today, October 2008