Page 16 - Fall 2008

(cannot exist), and are pushed down to the equilibrium mode “a.” However, some of the modes can exist and survive unchanged, because when the infinite masses are placed on the string, they wind up at the nodal points of the mode. Because the nodal points never move anyway, placing infinite masses there has no effect. Such a situation occurs for modes “i” and “q” in Fig. 5a and are shown in Fig. 5b. These modes have nodal points that divide the length of string shown into four parts, exactly where the infinite masses are placed. For modes “j” through “p,” the effect of the infinite masses is to “pull” the nodal points so that the transverse displacement looks just like mode “i,” and they become equivalent to that mode. Mode “q” also picks up duplicate modes, but these modes (from above “q”) were not included in Fig. 5a. The result of placing scatterers with infinite mass, dividing the length l into four parts, is illustrated in Fig. 5b. Apart from the equilibrium mode “a,” only two distinct modes survive from Fig. 5a. However, it is important to note that there are still an infinite number of modes above mode “q” that corre- spond to fitting an integer number of half-wavelengths inside the lattice constant, a.
A bit of an explanation is required here. We have used the expressions “suppressed” and “pulled” to describe what is happening to the modes when the masses are abruptly changed from zero to infinity. This is not whimsy. It clearly describes the effect on the modes when the masses are slow- ly increased from zero to infinity, as will be described next.
Finite masses
If we mathematically derive and plot the frequencies of the seventeen modes as a function of increasing mass from zero to infinity, we obtain the graph illustrated in Fig. 6. Because m=∞ cannot fit on a finite size figure, we can use an artifact to make the plot fit. We will let the variation of the mass be reflected in a parameter given by arctan(m/ρLa). This parameter varies from zero to π/2 as the mass varies from zero to infinity. The dimensionless combination m/ρLa gives the mass of the scatterer relative to the mass of the string between the scatterers because ρL is the mass per unit length and a is the length of a unit cell. A plot indicating how the frequencies of the modes (the vertical positions of the modes in Fig. 5a and Fig. 5b) change as the mass goes from zero to infinity is shown in Fig. 6.
On the far left of Fig. 6, where m=0, the harmonic sequence of the seventeen frequencies of the string without any masses (Fig. 5a) is evident. The letters on the left indicate the modes that have these frequencies (the modes are two- fold degenerate except for “a”). Note that the harmonic sequence of frequencies continues upward to infinity, with the sequence generated by fitting an integral number of half- wavelengths into the length l. The far right of Fig. 6 shows some frequencies when the mass of the scatterers is infinite as is seen in Fig. 5b. These are the first three frequencies of a sequence that also continues upward to infinity, but in this case the sequence is generated by fitting an integral number of half-wavelengths inside the length a, rather than inside the length l. The letters on the left of Fig. 5b indicate the modes associated with the frequencies on the far right of Fig. 6. The
Fig. 6. A plot indicating how the frequencies of modes vary as the mass of the scat- terer in a periodic array goes from zero to infinity. The letters refer to the labels of Figs. 5 (a) and (b). The frequency levels on the left are the equally spaced harmon- ics of the string without scatterers, and the frequency levels on the right are the nat- ural frequencies of the “local oscillators” formed between the infinite masses.
continuous lines connecting the frequencies on the far left with the frequencies on the far right indicate how sets of fre- quencies evolve as the mass of the scatterers is continuously varied from zero to infinity. The connecting lines consist of sets of four curved lines lying in the regions located between pairs of horizontal straight lines. One can see that, for exam- ple, lines “b,” “c,” “d,” “e,” “f,” “g,” and “h” in Fig. 6 are lower in frequency when the masses are greater than zero and indeed, are “pulled” toward line “a” as described.
The vertical dashed line in Fig. 6 indicates an example when the mass of the scatterers lies somewhere between zero and infinity; the frequencies are given by the symbols (circles, squares, and triangles) at the intersections of the dashed line with the curved lines. The frequencies of each mode with a non-zero mass is lower than the same mode with a zero- mass. The collection of frequencies given by the symbols on one horizontal line and the curved lines just above it (and below the next horizontal line) is referred to as a “pass band,” or simply “band.” In Fig. 6, the circles constitute one band, and the squares constitute a second band; this continues upward to infinity. The empty space between the uppermost symbol on a curved line in a band and the lower symbol on the horizontal line in the next band is referred to as a “stop band” or “gap.” The pattern of frequencies, with bands and gaps, along the dashed line, continued up to infinity, is referred to as “band structure;” note that the pattern of fre- quencies is not harmonic. It should be kept in mind that Fig. 6 is for the particular case where l = 4a. For the general case where l = Na, with N an integer, the number of frequencies in one band would be N+1. If the lattice constant a is fixed, then as l becomes large (and N becomes large), then the frequen- cies in a band get close together, and the band approaches a continuous distribution of frequencies.
Another way of describing the evolution of the frequen- cies as a function of the scatterer mass is to begin with the case where the mass is infinite, and then consider what hap- pens when the mass is decreased. This corresponds to start- ing at the far right side of Fig. 6 and moving to the left. This description will also give rise to another, far-reaching appli-
Wave Propagation in Scatterers 15