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Description
Tapered edges in beams have been attracting attention in engineering science for their passive vibration damping properties [1]. Flexural waves propagating into a power-law-shaped beam edge slow down with decreasing beam thickness until the phase and group velocities are zero at the tip. Standing waves, often undesired in engineering, cannot form in a beam with a perfectly tapered edge due to the inhibited wave propagation and reflection at the tip. Due to their similarities with astronomical black holes, structures like those described are called ‘acoustic black holes’ (ABH).
Theoretical and experimental research of ABH so far has focused only on elastic waves in solids that are substantially thinner than the acoustic wavelengths of interest [1]. Already in the first literature [2], equations approximated the problem in the long wavelength regime. Regarding thicker plates, however, formulas for the phase and group velocities reached unphysically large values exceeding even the longitudinal wave velocity.
According to the Rayleigh-Lamb theory, in plates that are thicker than the wavelengths, flexural waves turn into surface acoustic waves (SAW) [3]. Reflection reduction of SAW is of interest in microelectromechanical systems like SAW sensors or microfluidic actuators [4]. In our study, we investigate the applicability of the Rayleigh-Lamb theory to solve ABH for thicker plates and unlock the concept of ABH for SAW. We present a procedure to find analytical solutions of the Ralyeigh-Lamb equation in an ABH via MATLAB® and compare the results with numerical simulation.
Our results show that the Rayleigh-Lamb theory is useful for understanding the interplay of surface and flexural waves along an ABH and can help optimize the design of ABH when SAW reflections must be suppressed.
References
[1] A. Pelat, F. Gautier, S. C. Conlon and F. Semperlotti, "The acoustic black hole: A review of theory and applications," Journal of Sound and Vibration, p. 115316, 2020.
[2] M. A. Mironov, “Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval,” Akust. Zh. , pp. 546-547, 1988.
[3] H. Lamb, “On waves in an elastic plate,” Proceedings of the Royal Society of London, pp. 114-128, 1917.
[4] L. Y. Yeo and J. R. Friend, “Surface Acoustic Wave Microfluidics,” Annual Review of Fluid Mechanics, pp. 379-406, 2014.
