PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Double Reflections from corrugated surfaces
Jorge PETROSINO1; Nicolás CASAIS DASSIE2; Damián Andrés FERNANDEZ2; Georgina LIZASO2
1,2

Universidad Nacional de Lanús, Argentina

ABSTRACT
Reflections of acoustic waves from corrugated surfaces are discussed. A proposal made by Yi-Fan Zhu for ametasurface model considers that each groove on a corrugated surface is analogous to each individual source
in a line array, where the phase difference between sources is determined by the depth of each groove. Zhu’soriginal proposal suggests using the phase gradient of the arrangement as a design parameter and derives
conditions for reflections without dispersion. In this paper, we expose the advantages of considering thetime-delay gradient instead of the phase gradient. Also, designs that combine two corrugated surfaces
interspersed with different gradients to produce double reflection patterns are presented. The angles anddelays of both reflections can be selected independently.
Keywords: Dispersionless reflection; Acoustic metasurfaces; Reflection control.

1. INTRODUCTIONIn the last few years, there is raising interest in the study of artificial structures with subwavelengthcharacteristics which could enable wave manipulation features that could hardly be obtained withconventional materials. These structures are known as metamaterials and the possibility of obtaining
practical solutions to acoustical problems applying these designs continues to increase . A
metamaterial is a structure obtained by adding units with a particular design related to its geometry,orientation, and internal organization, which enables acoustical properties that are not present inmaterials found in nature (1). The term ‘metasurface’ is used to refer to a bi-dimensional metamaterialstructure.
In this article, an analysis of a particular kind of structure is presented as a possible link between
classic theory and recently proposed designs. The work presented by Zhu et al. (2,3) is used as areference, where the characteristics of a corrugated surface that enables dispersionless wavefront
manipulation are analyzed. The original process requires taking the phase gradient as a parameter ofdesign. This forces to choose a reference frequency for the calculations, even when the design isintended to achieve has no dispersion, so that it is independent of the frequency. The proposal
presented here suggests that the design becomes even easier if you take the time delay gradient delays
as a design parameter, instead of the phase gradient. On the other hand, it will be shown that the timedelay gradient is not altered if we add a constant delay to the whole design, which allows selecting
independently a specific reflection profile and an arbitrary delay of the reflected waves. In addition, itis proposed to combine two different profiles of reflection in the same design. This is accomplished byperforming the calculations for each profile independently and alternating then the values of depth ofgrooves of each one of them. A proof-of-concept of this proposal is simulated using COMSOLMultiphysics.
1.1 Wavefronts Emitted by a Linear Array.
The proposal presented by Zhu begins with the well-known theory that describes the behavior of alinear array of n elements separated by a distance Δx. A cylindrical wavefront will be formed in thenear field by a linear array with n number of sources radiating in phase, within a certain bandwidth,and in a direction that will be perpendicular to the array (this is, with a radiating angle of 0°). The
lower cutoff frequency of the band is related to the total length of the array L=n.Δx, while the upper
cutoff frequency depends on the distance Δx between sources.
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jorgepetrosino@gmail.com

Constant phase delays between adjacent sources will result in a wavefront with radiating angle θ(Figure 1), which is related to the distance Δr among the source B and a point A' with the same p haseas the adjacent source A, as shown in Eq. (1).

T§ 'r ·¸
© 'x ¹

arcsin ¨(1)Figure 1 – A linear array of sources with constant phase difference will radiate a wavefront with angle θ.
Points labeled with A’ and A in Figure 1 are in phase if B is ahead of A by a value of Δɸ. For a given
wavelength, the relationship between the parameters is expressed by Eq. (2).

T

§ O 'I · ¸© 2S 'x ¹

arcsin ¨(2)Considering a continuous arrangement of sources, Δɸ /Δx turns into dɸ /dx, this is the phasegradient of the array. Eq. (2) allows analyzing the necessary conditions for the angle of emission θ tobe non-dispersive (that is, to be independent of the frequency). As the argument of the arcsine depends
explicitly on the wavelength it is necessary that the gradient of phase dɸ/dx be inversely proportionalto λ.
1.2 Wavefront reflected by a corrugated surface
Figure 2 shows a corrugated surface formed by a set of rectangular grooves of different depths. Aplane wave incident will face each open inlet of the grooves; propagate along its length and return to beemitted by the open outlet with a change on its phase. The phase shift derives from the difference in thewave path which will result in a phase-gradient inversely proportional to λ, satisfying the conditionindicated by Zhu to ensure dispersion-less reflection.

Figure 2 – Corrugated surface. (a) section drawing, (b) samples of 3D printed surfaces

The lower cutoff frequency of the bandwidth will be related to the total length of the array (in thex-direction of Figure 2b), and its upper frequency with the gap between grooves (Δx).
Given a groove depth Δh, the necessary time to travel through it with propagation velocity c isshown in Eq. (3), and the phase change in Eq. (4), where c stands for the propagation velocity.

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't2  'hc(3)'I2S 2  'hO(4)Replacing equation (4) in Eq. (2) cancels the factor λ/2π. Eq. (5) relates the depth gradient of thegrooves Δh/Δx with the reflection angle without dispersion.

T§ 'h ·¸© 'x ¹arcsin ¨ 2 (5)Figure 3 – Reflection angle comparison on a corrugated surface and a plane surface
Figure 3a shows the reflection angle θ produced by a corrugated surface. The reflected angle isequal to arcsin(2.tan(α)), because α = tan(Δh/Δx). On the plane surface (Figure 3b), reflected angle isequal to twice the angle α. Figure 4 shows a simulation of surfaces reflecting a plane wave, both withthe same value of α.

Figure 4 – Wave reflection simulation on a corrugated surface and a plane surface

2. DESIGN OF A SPECIFIC PROFILE
In the previous descriptions, it has been assumed a constant value Δh/Δx between adjacentgrooves. In general, the non-dispersive condition can be satisfied by deriving the Eq. (4) for acontinuous case in which the groove depth profile h depends on x (Eq. 6).
(6)
dI
4S dh

dx
O dx
By integration, Eq. (7) is obtained which allows the design of a specific profile h(x) from thedesired phase gradient.

2002h (x)´O µ dIdx
4S µ dx¶(7)

The Eq. (7) expresses in general terms the proposal of Zhu. This process starts by selecting areference frequency f 0 to determine λ, and hence the depth profile h(x).
There is an alternative that does not require using a reference frequency f0 for the design expressing
the phase gradient dh/dx in terms of time-delay gradient dτ/dx (Eq. 8), to obtain Eq. (9)

dIdx

2S dW
c dxO(8)(9)
Eq. (9) derives the groove-depth profile h(x) from the time-delay of the array τ(x). In practice, theconstant of integration C corresponds to a parallel displacement of the entire depth profile h(x) and it
results in a constant time delay for all elements of the array. This delay can be controlled independently
from the previously selected reflection angle. Figure 5 shows simulations of profiles with the sameangle of reflection and different delays.

Figure 5 – Different delays in corrugated surfaces

3. DOUBLE REFLECTION PROFILES
A surface with double reflection can be obtained with two depth profiles h1(x) and h2(x), withalternating values for the depth of each slit, by distributing, for example, to the odds slits the profile
h1(x) and to the evens h2(x). On the basis of what has been discussed in the previous section, different
delays for each reflection can be selected. Each of the reflections will be generated by sets of slots (oddor even) that are spaced further apart, so the maximum frequency limit, which depends on theseparation between the slits, will be lower. Figure 6 shows a simulation of a profile designed to have adouble reflection (one matches to θ1 = 45° and the other to θ2 = 0°).

Figure 6 – First two reflections of plane wave on a corrugated surface2003

A double-reflection design generates, in fact, multiple sets of reflections in sequence due to theinteraction between adjacent slots. Each wavefront reflected by the set of slots of orde r even(corresponding to θ2 = 0° in the example of Figure 6) will cause new pressure variations in the inputsof the adjacent slots odd-numbered. This will cause extra waves crossing odd slots, and drive to afurther reflection with a new time delay, as shown in Figure 7.

Figure 7 – Later reflections (reverberant tail) generated by a corrugated surface

4. CONCLUSIONSThe proposal of Zhu et al. allows for the design of corrugated surfaces to achieve reflection without
dispersion in a controlled manner within a range of frequencies from knowing the gradient of the phasedesired. The proposal presented here replaces the phase gradient for a time-delay gradient, which
avoids the selection of a reference frequency. On the other hand, it is possible to select a time delay of
the reflected pattern in an independent way of the angle of reflection, simply by shifting horizontallythe depth profile of the grooves. The preliminary simulations carried out test the behavior of surfaces
with a double reflection pattern, which in turn causes that the reflections do not become extinct
immediately, but have an exponential decay. Such profiles with double reflection and with exponential
decay of reflections may have interesting applications in room acoustics. Currently, the authors aredeveloping physical models of such profiles in order to complement the study of these profiles withexperimental measurements.

ACKNOWLEDGEMENTS
Authors are grateful for the willingness and the commitment on the part of Andres Ruscitti,
Guillermo Andrade, and students Lucas Bujan and Sabrina Accotto that are dealing with modeling andprint the first prototypes for future measurements. The present work is part of a research p roject on thepropagation of acoustic waves carried out at the Universidad Nacional de Lanús in conjunction withAndres Bonino Reta, Ianina Canalis, Ian Kuri, and Lucas Landini.

REFERENCES
1. Ma G, Sheng P. Acoustic metamaterials: From local resonances to broad horizons. Sci adv. 2016;
2(2):1501595.
2. Zhu YF, Zou XY, Li RQ, Jiang X, Tu J, Liang B, Cheng JC. Dispersionless manipulation of reflected
acoustic wavefront by subwavelength corrugated surface. Sci rep. 2015; 5:10966.
3. Zhu X, Zou XY, Liang B, Cheng J. Acoustic one-way open tunnel by using metasurface. Appl Phys Lett.
2015; 107:113501.

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