Fig. 1

A brief illustration of the phenomenon of localization and amplification of elastic wave energy via defective PnCs and their potential applications to bandpass filters, energy harvesters, ultrasonic receivers, and ultrasonic actuators when the defective PnCs are combined with piezoelectric materials (color online)"

Fig. 2

An overview of previous research on the use of defective PnCs integrated with piezoelectric materials for excitation of either longitudinal or bending waves, with a particular focus on the amplification of velocity amplitudes. The research trend begins with single defect scenarios under longitudinal waves and progresses to double defect scenarios under bending waves (color online)"

Fig. 3

The front view of the targeted concurrent wave-generation system that consists of a one-dimensional defective PnC with bimorph piezoelectric elements connected to two independent voltage sources vPT and vPB. The defective PnC is employed in band-structure analysis, whereas the defective PnC attached to semi-infinite structures is utilized in wave-generation analysis (color online)"

Fig. 4

Intuitive illustration of band-structure analysis at the unit cell and defective PnC levels: (a) the transfer matrix method for band-gap analysis at the unit-cell level, (b) the transfer matrix method for defect-band analysis at the defective PnC level, and (c) the sequential results expected, including band-gap frequencies, defect-band frequencies, and defect-mode shapes (color online)"

Fig. 5

Intuitive illustration of wave-generation analysis at the system level: (a) the S-parameter method, which considers two input voltage sources, to predict the velocity amplitude of outgoing waves and (b) the sequential results expected, including the velocity-amplitude frequency response curves and the effects of temporal phase difference and voltage magnitude difference on peak velocity amplitudes (color online)"

Fig. 6

Results of the band-structure analysis. The x-axes in (a) and (b) correspond to the real and imaginary parts of the normalized Bloch wavenumber, respectively. The solid red and blue lines indicate the results obtained from the proposed analytical model for longitudinal and bending waves, respectively. In contrast, the results from the COMSOL simulation are depicted using the black dashed lines. The phononic band gaps identified at the unit-cell level are highlighted with boxes, and the lines within these boxes indicate the defect bands identified at the defective PnC level (color online)"

Fig. 7

Results of the defect-mode-shape analysis: (a) the axial displacement field under longitudinal waves, (b) the transverse displacement field under bending waves, (c) the axial strain field under longitudinal waves, and (d) the axial strain field under bending waves. In (c), the strain field is determined at zPnC=0. In (d), the strain field is determined at zPnC=hPnC/2 (color online)"

Fig. 8

Results of the wave-generation analysis: the frequency response curves for (a) the axial velocity amplitudes when (vPT, vPB)=(1 V, 1 V), (b) the transverse velocity amplitudes when (vPT, vPB)=(1 V, −1 V), (c) the axial velocity amplitudes and (d) the transverse velocity amplitudes when (vPT, vPB)=(1 V, 0 V) (color online)"

Fig. 9

Results of the parametric study when the magnitude and temporal phase differences are examined separately: the effects of (a) the magnitude difference and (b) the temporal phase difference on the axial (45.68 kHz) and transverse (25.51 kHz) velocity amplitudes. In (a), a temporal phase difference of 0 is present. Conversely, in (b), a magnitude difference of 1 V is present (color online)"

Fig. 10

Results of the parametric study when the magnitude and temporal phase differences are coupled: the effects of (a) the magnitude difference and (b) the temporal phase difference on the axial (45.68 kHz) and transverse (25.51 kHz) velocity amplitudes. In (a), a temporal phase difference of π/4 is present. Conversely, in (b), a magnitude difference of 2V is present (color online)"