Dragon curve ultra fractal11/11/2022 Additionally, the transmissions of acoustic waves in solids 13, 14 and water 15 were also manipulated to change the relative refractive indices and acoustic velocities, in which distinct acoustic lens were created. Furthermore, a simple and compact structure of one string of side pipes arranged along a waveguide was present to obtain diverse group velocities, as slow acoustic waves, fast acoustic wave and even negative group velocity 12. When the resonant frequencies of the resonator pairs are close to each other, slow sound waves were achieved at a frequency between both resonant frequencies 9, 10, which resembles the phenomenon of electromagnetically induced transparency in optics 11. DRAGON CURVE ULTRA FRACTAL SERIESFurthermore, on the analogy of dual coupled resonators used to produce slow light 4, an acoustic metamaterial was created based on a series of detuned Helmholtz resonator pairs arranged along a tube. By introducing a line defect in a triangular array of aluminum cylinders, slow waves were obtained at audio frequencies 8. Defected phononic crystals were widely used to produce slow acoustic waves. An acoustic metamaterial composed of an array of grooves perforated on a rigid bar was presented to trap acoustic waves at separate locations according to frequencies, which was called to be a phenomena of rainbow trapping 7. On the other hand, studies were performed for the realizations of slow waves and energy trapping. Then, extremely large and even negative group velocities were also realized in a suspension of elastic microspheres 6. First, faster-than-light acoustic waves were created in a Herschel–Quincke tube, in which the envelope of the output acoustic pulse preceded that of the input pulse 5. Encouraged by the achievements in the control of electromagnetic waves and light, variant methods were adopted to manipulate the transmissions of acoustic waves. On the contrary, slow light was produced in a dendritic cell cluster metasurface waveguide and dual coupled-resonator system 3, 4. Superluminal transmissions and negative group velocities of light were obtained in highly nonlinear fibers and gaseous nanolayers 1, 2. The manipulation of wave transmission has attracted considerable attention in the fields of electromagnetics, optics and acoustics, in which numerous endeavors were provided to achieve extremely fast or slow waves.
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