Extraordinary acoustic reflection enhancement by
acoustically-transparent thin plates
- AUXILIARY MATERIAL -
Zhaojian He, Shasha Peng, Rui Hao, Chunyin Qiu, Manzhu Ke,
Jun Mei, and Zhengyou Liu

Corresponding authors. cyqiu@whu.edu.cn; zyliu@whu.edu.cn
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In this supplementary material, we will discuss the physics origin of the weak
transmission kinks that are ignored in the text of the paper. As an example, here we
only present the details for the system consisting of a periodic array of epoxy strips,
with the strip width w  1.8mm and thickness t  0.4mm , and the structure period
a  2.3mm . The unified picture is further verified by the systems with different
geometries.
1.0
(a)
Transmission
0.5
0.0
1.0


A
(b)

0.5

D
B
0.0
0.9
(c)
0.8


0.7
Single strip
C
0.6
1.5
2.0
2.5
3.0
3.5
4.0
wavelength (mm)
FIG. S1 (Color Online). (a) The numerical transmission spectrum for a plane wave incident
normally onto the sample consisting of a periodical array of epoxy strips immersed in water, with
the strip width w  1.8mm and thickness t  0.4mm , and the structure period a  2.3mm . (b)
The same as (a), but at an incident angle   10o . (c) The transmission spectrum for a Gaussian
beam incident normally onto a single strip. Here the horizontal-axis gives the wavelength in water
and the arrows mark the transmission kinks discussed in this supplementary material.
As shown in Figs. S1(a) and S1(b), both transmission spectra at incident angles
  0o and   10o display a fine kink around the wavelength 1.6mm. Again, the
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weak angular dependence of the kink wavelength indicates a character of the local
mode. This is further verified by the position-invariant kink appearing in Fig.S1(c),
the normal transmission spectrum for a single strip case. In order to visually illustrate
the physics origin of the local mode, in Figs. S2(a), S2(b) and S2(c) we present the
corresponding vibration fields around the strip for the above three situations,
respectively. Featured by four distorted vortices inside the strip, all of the three figures
display strong flexural motions. This suggests that the local resonance may stem from
the resonant excitation of the standing wave mode of the antisymmetric Lamb wave.
To further verify this physics picture, from the dispersion curve in Fig. S3 we derive
quantitatively the wavelength of the antisymmetric Lamb wave S  1.0mm . This
wavelength corresponds to the fourth order standing wave mode of the antisymmetric
Lamb wave supported in the single strip, since n  2( w  t ) / S  4 , considering a
correction t  0.4 for the strip width.
It is of interest that in Fig. S1(b) the transmission spectrum at oblique incidence
shows another kink around the wavelength 2.7mm. This wavelength corresponds to
S  1.4mm for the antisymmetric Lamb wave, as shown in Fig. S3. From the
formula n  2( w  t ) / S , we obtain n  3 . The third order standing wave mode is
also characterized by the three distorted vortices displayed in Fig. S2(d). This
transmission kink is missing in the case of normal incidence, see Figs. S1(a) and S1(c).
It is reasonable since the vibration field of the normally incident wave is symmetric
about the mirror plane x  0 , whereas the third order standing wave mode is
antisymmetric about the mirror plane. Once the incidence is tilted, the
perfect-mismatch of the symmetry is broken and thus the third order standing wave
mode can be excited.
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FIG. S2 (Color Online). Cross section views of the field distributions inside and around the epoxy
strip, where (a), (b), (c) and (d) respectively correspond to the transmission kinks A, B, C and D
marked in FIG. S1. Here the color scale displays the field amplitude (normalized respectively by
the strongest field in each figure), and arrows indicate the field direction.
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1.8
s(mm)
1.6
1.4
1.2
1.0
0.8
1.5
2.0
2.5
3.0
3.5
4.0
wavelength (mm)
FIG. S3 (Color Online). Dispersion curve for the antisymmetric Lamb wave in a flat plate of
thickness 0.4mm , where the horizontal-axis and vertical-axis give the wavelengths for water and
antisymmetric Lamb wave, respectively. Here the blue circles denote the positions of the
transmission kinks in FIG. S1.
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