Band-gap Structure of Phononic Crystals

The emergence of bandgaps within phononic crystals can be elucidated through three principal mechanisms: Bragg-scattering, local resonance, and hybridization. Bragg-scattering bandgaps arise due to the interference between waves originating from periodic inclusions, heavily reliant on the periodicity order and structural dimensions [1]. Local resonance exploits the negative mass effect, occurring when the excitation frequency aligns with the natural frequency of the resonating structure. The frequency span of local resonance is contingent upon the oscillator's frequency [2-6]. To provide further details, Structures which can have a negative momentum for a positive momentum excitation are characterized by negative mass effect, this effect is explained by Milton Willis using newton’ equation of motion [7]. Hybridization, the third mechanism, involves the coupling influence of local resonance and Bragg-scattering. This interaction can engender multiple bandgaps of diverse widths and shapes, with the resultant bandgap properties shaped by the interplay between local resonance and Bragg-scattering mechanisms [8].

Bragg Mechanism

One way to control waves is to introduce a secondary wave as a disturbance that changes the behavior of the primary wave as desired. This method of controlling wave propagation is known as the "Bragg mechanism." To model the behavior of a wave using this method, we can use the following equation:

1

The formula introduced is the wave propagation formula in a one-dimensional medium, where K is the wave number, ω is the angular frequency, and ϕ is the phase angle, we will use also following relations:

2

Where T is the period, f is the frequency, and C is the wave speed. It should be noted that the introduced wave speed is only valid and constant in media without scattering.

Consider Figure 1. In this image, it is assumed that a wave has passed through a one-dimensional discrete medium whose components are separated from each other by a distance of Δx. As it is evident, the propagated wave from the component located at a distance of Δx differs by KΔx from the propagated wave of the previous component (due to the nature of wave propagation). Now, if this parameter is adjusted in such a way that its value is equal to 180 degree, then the interaction of these two waves will be destructive, and their resultant will be zero, and therefore, the propagated wave in the system will be neutralized. Therefore, we have.

3

Therefore, if we want to manage the passage of a wave in a medium where the wave is propagated at the speed of sound, we need to be able to control a distance equal to half the wavelength or C/2f. So, if we need to control a seismic wave with a speed of 3000 kilometers at a frequency of 100 Hz, we need to dig holes with a distance of 5 kilometers! Therefore, this method is heavily dependent on distance, which is the same problem we face in wave control.

Local Resonance

Another way of wave propagation is to couple the propagation medium to another medium. This coupling allows us to create changes and adjust the frequency of wave propagation. For example, consider a system consisting of sub waves. These two systems are coupled to each other by a spring, so through this parameter, the degree of coupling and consequently the wave propagation in the system can be controlled

Considering the system consists of two waves that are coupled to each other by a spring (above figure), we can write the following governing equations.

4
Or :
5

Finally

6

As per Relationship 6, one of the frequencies of the system is defined by the coupling, and consequently, the wave propagation can be control by the coupling frequency . However, the mentioned problem regarding the dependence on the distance and geometry of the problem is still implicitly hidden in this method, and we have not yet been able to free ourselves from the constraint of distance or geometry in controlling the waves. Since we recognize this dilemma in wave propagation control, the only solution ahead is to resort to extraordinary materials to eliminate the need for wave control.

References:

[1] J. V. Sánchez-Pérez et al., "Sound Attenuation by a Two-Dimensional Array of Rigid Cylinders," Physical Review Letters, vol. 80, no. 24, pp. 5325-5328, 06/15/ 1998.

[2] Z. Liu et al., "Locally resonant sonic materials," science, vol. 289, no. 5485, pp. 1734-1736, 2000.

[3] J. Li and C. T. Chan, "Double-negative acoustic metamaterial," Physical Review E, vol. 70, no. 5, p. 055602, 2004.

[4] G. Wang, D. Yu, J. Wen, Y. Liu, and X. Wen, "One-dimensional phononic crystals with locally resonant structures," Physics Letters A, vol. 327, no. 5-6, pp. 512-521, 2004.

[5] Y. Ding, Z. Liu, C. Qiu, and J. Shi, "Metamaterial with simultaneously negative bulk modulus and mass density," Physical review letters, vol. 99, no. 9, p. 093904, 2007.

[6] Y. Achaoui, A. Khelif, S. Benchabane, L. Robert, and V. Laude, "Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars," Physical Review B, vol. 83, no. 10, p. 104201, 2011.

[7] G. W. Milton and J. R. Willis, "On modifications of Newton's second law and linear continuum elastodynamics," Proceedings of the Royal Society A: Mathematical, Physical

Engineering Sciences, vol. 463, no. 2079, pp. 855-880, 2007.

[8] Y. Chen and L. Wang, "Periodic co-continuous acoustic metamaterials with overlapping locally resonant and Bragg band gaps," Applied Physics Letters, vol. 105, no. 19, 2014.