Acoustic Bandgaps of Phononic Penrose and Quasicrystals
AA COUSTIC B ANDGAPS OF P HONONIC P ENROSE AND Q UASICRYSTALS
Bryan Eastwood
Department of Computer ScienceUniversity of Massachusetts Amherst [email protected]
Edward A. Rietman
Department of Computer ScienceUniversity of Massachusetts Amherst [email protected]
July 8, 2020
Using 3D printing we manufactured two rodlike phononic crystals. Viewed from thetop, one is a Penrose tile and the second is a quasicrystal. We explored the acousticproperties and band gaps for both in the frequency range of 5kHz to 25kHz. K eywords Penrose · quasi crystal · phononic · band gap In our experiment, we measured the acoustic effect on different constructions ofsolids to the transmission of sound waves. The purpose of this experiment was tocorroborate the findings of other researchers on ordered but aperiodic solids on thetransmission spectra of waves. Many other experiments have demonstrated a causalrelationship between these sort of patterns and the appearance of band-gaps in thespectra, both in the phonoic and photonic domains. Our experiment was designedspecifically to determine if this relationship held for our two phononic crystals, thefirst being structurally based on the Penrose tiling and the second being a quasicrystal.We measured at incidence angles comprising the entire circumference of the crystalto capture the spectra associated with waves bent by its interior constitution.
The design of the two crystals is shown in figure 1. The first is a quasicrystal, whichis ordered but not periodic. The second is based off of the Penrose tiling, whichis aperiodic. Both crystals have a rod diameter of 6.19mm and a rod spacing of a r X i v : . [ phy s i c s . c l a ss - ph ] J u l .000cm. (3.81mm gap between rods) Both are composed of 3D printed plastic. Theexperiments were done in air. Figure 1: construction of the crystals
The physical dimensions of the apparatus are shown in figure 2. Each of the micro-phones is attached to a 16-channel multiplexer, the output of which was connectedto an oscilloscope. For each frequency in the range we investigated, a tone of thefrequency is played through a speaker. The multiplexer cycles through each of thenine microphones and records the intensity of the frequency using the FFT functionof the oscilloscope. Specifically, we record the value of the peak in the frequencyspectrum corresponding to the frequency being tested. The frequency spectra shownin this paper are the result of subtracting the results of a trial with each solid placedin the apparatus by the results of a trial with nothing in the apparatus (i.e. the absenceof the crystal). This was done because the microphones do not have a flat frequencyresponse.
Figure 4 shows the results of the experiment for both solids. Each plot in the 3x3grid has an abscissa representing our frequency range and an ordinate representingthe intensity of the frequency after being picked up by the microphone. We canclearly see that in the central microphone, the band-gap due to the Penrose tiling is wider than that of the quasicrystal, although they are both significant. The spectrafrom bent waves on either side of the crystal appear to show some kind of periodicbehavior, but no band gaps can be discerned.
Several other papers have described experiments on the band gaps inherent to bothphotonic and phononic Penrose[1][2] and Quasicrystals.[3] The main purpose ofthis experiment was to investigate and attempt to replicate both the theories andexperiments of other researchers in this field, and to document our procedure withgreater detail. As shown above, we have clearly replicated the presence of band gapsin the transmission spectra of both structures. The isotropy discussed in [3] was alsodemonstrated, with a 45 ◦ rotation of the two structures yielding essentially the sameresult, with the slightly more defined band gap in the 45 ◦ case (figure 5) possiblyresulting from the greater amount of crystal between the microphone (due to theirorientation and the square shape of the crystals). This experiment could be expanded upon by testing the transmission spectra for much higher frequencies in the order of100 kHz, in order to replicate the results of the simulations discussed in [4]. The source code used on the Arduino boards and main computer can be found at https://github.com/bryan-eastwood/phononic-code . References [1] I. B. M. Bayindir, E. Cubukcu and E. Ozbay, “Photonic band gaps and localiza-tion in two-dimensional metallic quasicrystals,”
Europhys. Lett. , vol. 56, pp. 41–46, 2001.[2] I. B. M. Bayindir, E. Cubukcu and E. Ozbay, “Photonic band-gap effect, localiza-tion, and waveguiding in the two-dimensional penrose lattice,”
Physical ReviewB , vol. 63, 2001.[3] D. Sutter-Widmer and W. Steurer, “Prediction of band gaps in phononic qua-sicrystals based on single-rod resonances,”
Physical Review B , vol. 75, 2007.[4] E. A. Rietman and J. M. Glynn, “Band-gap engineering of phononic crystals: Acomputational survey of two-dimensional systems,” 2007. ◦◦