Page 19 - Fall 2008

Fig. 10. Apparatus for demonstrating wave propagation in periodic or disordered arrays of scatterers. The wave medium is a steel wire (guitar string), with tension maintained by a weight, and with scatterers consisting of small masses (split shot for fishing line) positioned along the wire.
used in a research program to study the additional effects of time-varying mass positions11 and finite amplitude (nonlin-
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with struts angled at 45 degrees at each end. A steel wire (a gui- tar string) is attached between the extreme ends of the struts. Starting at the left end in the figure, the wire travels at 45 degrees until it passes through an eyelet and proceeds hori- zontally; an electromechanical vibrator is attached at this point. The wire continues horizontally to the right end where a large weight is attached, after which the wire continues at 45 degrees to the attachment point on the right strut. The large weight is supported by the wire only, and given the configura- tion of the wire as shown in the figure, the weight maintains constant tension in the wire. Positioned along the horizontal section of the wire are small masses (“split shot” for fishing line) whose positions are carefully set using precision calipers.
The base of the apparatus acts as a track for a trolley that carries a wave amplitude detector and meter, fashioned from an electric guitar pick-up and a modified sound-level meter. As the trolley is moved along the wire, the meter indicates the transverse displacement of the wire as a function of the hor- izontal position, i.e., ψ(x). A photograph of the detector, with the wire and some masses just visible, is presented in Fig. 11.
A model of a second apparatus that could be used to study two-dimensional phenomena is shown in Fig. 12. The local oscillators in this apparatus are musical tuning forks
ear) transverse waves.
As shown in Fig. 10, the experimental apparatus has a base
Fig. 11. Photograph of part of the apparatus shown schematically in Fig. 10.
tant systems of scatterers where the positions are time- dependent. An important example occurs when the scatter- ers are the atomic ions in a metallic crystal and the waves are the quantum mechanical Schrödinger waves of the electrons. In this case the positions of the scatterers are changing because of the thermal motion of the ions. Another example would be when the scatterers are small particles suspended in a fluid. Here, the scatterers change positions because of another thermal effect—Brownian motion.
(4) Although not explicitly stated, all of the results dis- cussed so far have been obtained assuming that the wave propagation in the system is linear; that is, none of the equa- tions governing the system contain terms that are not linear- ly proportional to the wavefunction. However, if a wave sys- tem is driven to sufficiently high amplitude, then classical force laws will result in the appearance of nonlinear terms. For example, the wave equation for a string is based on the assumption that the tension in the string is constant. However, if the string has a finite transverse displacement, then the arc length of the string is increased, and if the string is rigidly clamped at two ends, then the presence of the finite amplitude wave will increase the tension. This will cause a nonlinear term to appear in the wave equation.
Each of the deviations discussed above have individually been the subject of significant research. However, more possi- bilities arise when one studies systems that involve combina- tions of these deviations, and for these systems theoretical treatment, and even numerical treatment, is very difficult. In these cases, there is significant opportunity for very interesting experimental research. An important consideration is that if such experimental research is to be undertaken, then the experiment must be able to pass a crucial test—the experiment should be first configured with a static and periodic (as accu- rately as possible) array of scatterers and a linear wave medi- um, and under these conditions the experiment must be able to verify Floquet’s theorem. Only when this test is passed may the experiment be considered valid for the study of deviations. Thus, the understanding of wave propagation in periodic sys- tems maintains its importance even in novel experiments.
An acoustical experiment
An experimental apparatus that is used to demonstrate phenomena in both periodic and disordered arrays of scat- terers is illustrated in Fig. 10; it is a realization of the model discussed in this article, involving a string with masses as scatterers. A larger version, with more than 50 masses, was
Fig. 12. An apparatus used to demonstrate phenomena for wave propagation in two-dimensional arrays of scatterers. The model is based on musical tuning forks as local oscillators, which are coupled together with arcs of steel wires attached to the tines of the tuning forks.
18 Acoustics Today, October 2008