Tailored elastic surface to body wave Umklapp conversion

Download PDF Article Open access Published: 29 June 2020 Tailored elastic surface to body wave Umklapp conversion Gregory J. Chaplain  ORCID: orcid.org/0000-0001-9868-84531, Jacopo M. De

Ponti2,3, Andrea Colombi  ORCID: orcid.org/0000-0003-2480-978X4, Rafael Fuentes-Dominguez5, Paul Dryburg  ORCID: orcid.org/0000-0001-5389-67695, Don Pieris5, Richard J. Smith  ORCID:

orcid.org/0000-0003-2502-50985, Adam Clare6, Matt Clark  ORCID: orcid.org/0000-0002-1284-31825 & …Richard V. Craster  ORCID: orcid.org/0000-0001-9799-96391,7,8 Show authors Nature

Communications volume 11, Article number: 3267 (2020) Cite this article

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Elastic waves guided along surfaces dominate applications in geophysics, ultrasonic inspection, mechanical vibration, and surface acoustic wave devices; precise manipulation of surface

Rayleigh waves and their coupling with polarised body waves presents a challenge that offers to unlock the flexibility in wave transport required for efficient energy harvesting and

vibration mitigation devices. We design elastic metasurfaces, consisting of a graded array of rod resonators attached to an elastic substrate that, together with critical insight from

Umklapp scattering in phonon-electron systems, allow us to leverage the transfer of crystal momentum; we mode-convert Rayleigh surface waves into bulk waves that form tunable beams.

Experiments, theory and simulation verify that these tailored Umklapp mechanisms play a key role in coupling surface Rayleigh waves to reversed bulk shear and compressional waves

independently, thereby creating passive self-phased arrays allowing for tunable redirection and wave focusing within the bulk medium.

of ultrathin waveguides by elastic metagratings Article Open access 17 March 2022 Sub- and supersonic elastic waves in an annular hole phononic metamaterial Article Open access 03 June 2021

Generalized acoustic impedance metasurface Article Open access 15 January 2024 Introduction

The Umklapp, or flip-over process first hypothesised by Peierls1 is conventionally concerned with describing phonon–phonon scattering to explain thermal conductivity at high temperatures,

and has a rich history in the quantum theory of thermal transport2,3. Concepts based around the Umklapp process are not traditionally incorporated in areas of wave physics concerning designs

of elastic metasurfaces; deep elastic substrates support surface Rayleigh waves that propagate along the surface, often over large distances, which are an essential component of, for

instance, surface acoustic wave microfluidic devices4, acoustic microscopy5, at small-scales and of seismic wave and groundborne vibration propagation at the geophysical scale6. An

isotropic, and homogeneous, elastic medium supports two types of bulk waves: compressional, P, and shear, SV and SH, waves polarised vertically and horizontally that propagate with different

wavespeeds cp and cs with cp > cs7 with the Rayleigh wavespeed cr slower than both.

Recently emerging ideas in graded metamaterial arrays and so-called rainbow-trapping devices have presented novel ways to manipulate wave propagation. Taking advantage of ideas that emerged

in optics around slow-light devices and optical “rainbow” trapping8 have in turn motivated tailored designs of graded Helmholtz resonator arrays in acoustics9, and their analogues in water

waves10, to slow the array-guided waves and trap the waves at different spatial positions, with application to broadband sound absorbers11,12. In almost all wave regimes where these graded

systems are designed, Umklapp effects have been neglected.

Neither acoustic nor electromagnetic waves have the additional complications of elasticity, that is, having both shear and compressional wavespeeds, mode coupling at interfaces and Rayleigh

surface waves; this presents an opportunity as the additional degree of freedom in elasticity can be exploited. Elastic graded resonator arrays use rods or beams, whose length determines the

resonance frequency, and grading creates a metawedge13. Trapping and slow-wave phenomena occur but now with the additional physics of mode conversion from the Rayleigh wave into a

forward-propagating shear wave in the bulk as confirmed experimentally14. The trapping phenomena has potential for energy harvesting15; the metawedge provides spatial segregation by

frequency thereby amplifying elastic energy at specific resonators that can be coupled to piezoelectric patches16,17.

The physics of a graded structured array can also be interpreted in terms of phase changes, induced by changes in the array elements, as the wave transits the array. In this vision, a graded

array acts as a self-phased array and one can induce backward-directed radiation from an array (Supplementary Note 5, Supplementary Fig. 4). In the context of a model for flexural waves on

thin elastic plates, a simple scalar model, a graded line array has been shown18 to create focusing and flat-lensing effects, that emulate negative refraction on a line and Pendry–Veselago

flat lenses. Although motivational, the thin plate model also lacks the multiple wavespeeds of the deep elastic substrate, and the array-guided wave only exists below the sound-line, i.e. in

the first Brillouin zone.

In this article, by taking an elastic half-space, patterned by a graded resonant array of rods (Fig. 1), we show that, one can obtain mode conversion from surface waves to compressional, P

waves, and not just to shear, S waves in the bulk. Furthermore, this can be directed backward, and used to create focusing, and these features are due to Umklapp scattering from outside the

first Brillouin zone. These striking results are confirmed experimentally from the predictions made by the theory and by simulation. The elastic wave system, having distinct wave types with

different wavespeeds, is imbued with richer physics than acoustic/electromagnetic systems and this yields a greater degree of flexibility and the opportunity for novel effects; in elasticity

we have two distinct sound-lines, also as the Rayleigh surface wave is, unlike spoof surface plasmons, not induced by periodic geometrical changes it is not confined to lie below the

sound-line nor is it confined to be within the first Brillouin zone.

Snapshots in time of temporal–spatially filtered scans along top and bottom surfaces observing S conversion (a–c) and P conversion (d–f), filtered between 1.1–1.2 and 1.45–1.55 MHz,

respectively, normalised to the maximum of displacement of the top surface (uz). Solid red lines show where graded array begins, with increasing rod height in the direction of wave

propagation on the top surface. The reversed conversion is clear; the forward propagating surface wave on the top surface is reverse converted into the bulk and is seen to excite reversed

propagating surface waves on the bottom surface. The measured angles of reversed conversion,  −131.8∘ and  −106.9∘ for S and P, respectively, match the predicted angles from Fig. 3, with the

separate polarisations evident from the difference in excitation wavelength on the bottom surface, marked λS, λP. The experimental setup is shown g, as is a schematic of array geometry h

detailing the rod diameter, t, periodicity a and grading through the changes in height of the nth rod, hn, as hn = h0 + nΔh. Fabrication details are given in the “Methods” section, with the

array parameters given in Supplementary Table 1.

When operating outside the first Brillouin zone, concepts of crystal momentum become important19. The conventional definition of an Umklapp process, or U-process, is elucidated in Fig. 2a,

b, whereby the resultant wavevector of a scattering process within a periodic crystal lies outside the first Brillouin zone; promotion of a backward-propagating phonon results through

crystal momentum transfer, since the wave vector, q is defined modulo G, i.e. up to a reciprocal lattice vector. The textbook distinction between normal (N-processes) and U-processes is then

given by

{\bf{G}}&{\rm{U}}{\hbox{-}}{\rm{process}}.\end{array}\right.$$ (1)

where the qi are the wave vectors in Fig. 2a, b. We stress that these mechanisms do not violate momentum conservation1; unlike in conventional graded structures we are not considering only

the true momenta of interfering waves (phonons) within a crystal, but taking advantage of the momentum of the system as a whole. Nuances of this simplistic definition arise due to the

apparent interchangeability of N-process and U-process through altering the primitive cell, and the associated (quasi)momentum conservation19,20,21. The distinction is achieved throughout by

analysing the dispersion curves of the locally periodic elements, along with simplified isofrequency contours (Supplementary Figs. 1 and 2). Recently the importance of the Umklapp process

has been solidified for electron–electron scattering through experimental verification, highlighting the fundamental role it plays in electrical resistance in pure metals22, along with the

utility to probe electronic structures23. Further to this, new breeds of entirely flat lens devices have been theorised which incorporate Umklapp-scattering processes for surface waves on

dielectric substrates24 by virtue of abrupt, not adiabatic, gradings.

a Wave vector scattering in the first Brillouin zone showing conventional definitions of an N-process and a U-process b; the ‘flip-over' description is only strictly true for collinear

scattering19. c Longitudinal dispersion curves within second Brillouin zone; X marks the edge of the first Brillouin zone, whilst \(X^{\prime}\) marks the edge of the second Brillouin zone.

Each dispersion curve represents the second longitudinal mode for a perfectly periodic array of rods, with the fixed heights with R(S) sound lines shown in red(green). d Isofrequency

contours of rod where last longitudinal mode supported, after which an effective bandgap is reached, when exciting at a frequency of 1.340 MHz. At this position U-processes are preferential

to a change in rod motion and, resulting in an effective reversed wavenumber κ − G, capable of mode converting into a P wave (black isofrequency contour, shown in full dispersion curves in

Supplementary Fig. 1). e Graded array schematic colours corresponding to the curves in c.

Undeterred by its prevalence in quantum, discrete systems we demonstrate the efficacy of U-processes which can preside over purely passive, classical, continuous elastic devices. The extent

of this mechanism is often not considered in the wide range of structures based on (locally) periodic material systems, and indeed neglected in elasticity theories25. Incorporating this

effect is achieved by utilising the existence of surface waves outwith the periodic structure, and marrying the transition of such a wave to the excited wave within a graded structuring. By

doing so, striking reversed conversion into S and P waves can be achieved, and controlled, as predicted in Fig. 3 and experimentally verified in Fig. 1. We present here the design

methodology for structures capable of the conversion of Rayleigh waves directly into both S and P waves independently, by utilising the dispersion curves and isofrequency contours of

perfectly periodic arrays of rods. Due to the adiabatic changes of the array parameters the global spatial properties of the array are determined by the locally periodic structures; the

dispersive properties of the array at a given position are inferred from the periodic dispersion curves of the constituent element at that position11,13.

conversion by Umklapp scattering.

a S conversion for excitation at f = 1.2 MHz, predicted using isofrequency contours shown in b. c P conversion for excitation at f = 1.45 MHz, predicted using isofrequency contours shown in

d. In each case at the position of conversion is determined by the change from longitudinal to flexural motion of neighbouring rods. The ratio of the relative magnitude of longitudinal to

flexural motion is shown by the ellipses above the rods. The angle of conversion, computed directly from the simulated field transformed in the wavenumber space, matches the predictions from

the isofrequency curves of the last rods supporting the longitudinal mode, similarly to Fig. 2c. The difference between the converted wave types (S and P) can be seen clearly from the

streamline analysis. Wavenumber analysis on the simulated data also confirmed the two different velocities ~3000 m s−1 for S and 6000 m s−1 for P. Array parameters are detailed in

Supplementary Table 1.

Shown in Fig. 2c, are the dispersion curves for the longitudinal motion of rods with fixed heights within the second Brillouin zone. These curves are obtained through an averaging process of

the fully polarised dispersion curves (Supplementary Note 2, Supplementary Fig. 1), since incident R-waves excite a superposition of longitudinal and flexural rod motion26. Utilising modes

in this region of reciprocal space, i.e. outwith the first Brillouin Zone, is somewhat counter-intuitive, in that the waves which appear above the light-line in an irreducible representation

of the Brillouin Zone are traditionally ignored when considering spoof surface plasmons27, which are the closest electromagnetic analogue to the Rayleigh wave. The behaviour of the

adiabatically graded array, as shown in Fig. 2e is predicted from these locally periodic curves. For a given frequency, longitudinal motion is preferentially supported by a number of rods;

at some rod height however, there is a transition in preferred rod motion from longitudinal to flexural (or vice versa). This is seen as an effective band gap in Fig. 2c for the fourth rod,

where U-processes dominate; the transfer of crystal momentum results in an effective reversed wavenumber, κ − G. Depending on the operational frequency, this can lie within the isocircle of

the free S or P body waves; reversed conversion is achieved by conservation of the tangential component of the wavevector. Figure 2d shows the prediction of this angle by inspection of the

last supported longitudinal mode (corresponding to the third resonator). The projected dispersion curves show the resultant wave vector inside the first Brillouin zone, as a result of

transfer of crystal momentum, by the white star corresponding to that in Fig. 2c. The wave can hybridise with a reversed P wave by the phase matching with the isofrequency circle of the P

wave, shown in black. A similar analysis can be carried out for conversion into S waves, by operating at a lower frequency. In this way, either longitudinal or flexural rod motion can excite

either S or P reversed waves by the inspection of the transition from one dominant rod motion to another.

When operating at higher frequencies, corresponding to wavevectors outside the first Brillouin zone, U-processes occur regardless of any grading; energy is continually shed along the array

(Supplementary Note 3, Supplementary Fig. 1c), resembling an elastic leaky wave antenna28. This can be manufactured to take place at a desired position by either adiabatic or abrupt

gradings24. A similar effect is observed when exciting along the opposite array direction; the grading experienced is then from high to low rods. In this case, since the effective bandgap is

not a true bandgap, propagation occurs through the array with Umklapp processes taking place all along the array (Supplementary Note 6, Supplementary Figs. 5 and 6).

Within this simple metawedge design paradigm lies many degrees of freedom: the position, angle and wave-type of reversed conversion, all allow a tailored conversion which simply relies upon

a change in primary mode behaviour between neighbouring rods, within a higher Brillouin Zone. Demonstrated in Fig. 3, modelled using SPECFEM29, is the reversed conversion into S and P modes

via the transition from longitudinal to flexural rod motion for an array with parameters detailed in Supplementary Table 1. The ellipses above the rods convey the ratio of the two dominant

rod behaviours via their semi-minor and major axis. The angle of conversion comes from the conserving the tangential component of the wavevector, determined via the isofrequency contours of

the rod prior to the change in rod motion (Supplementary Fig. 2), showing S conversion for 1.2 MHz and P conversion for 1.45 MHz.

Simulation results are confirmed experimentally in a 1.8 cm-thick slab of aluminium patterned using 3D printing, with a graded array of aluminium microrods on the surface (Fig. 1,

Supplementary Fig. 6). A laser adaptive photorefractive interferometer scans the surface of the aluminium sample providing a reading of the displacement field in the out of plane direction

uz. The block is attached to a moving platform. Pure Rayleigh waves are generated by an ultrasonic transducer attached at the surface of the aluminium sample (see “Methods” section for

details on the experimental set-up). To enhance the visualisation of the conversion spatial and frequency filters have been applied. Time series have been bandpassed between 1.1–1.2  and

1.45–1.55 MHz for S and P conversion, respectively. The wavefield scans have been filtered selecting wavevectors pointing towards (away) from the resonators for the top (bottom) surface.

This procedure mainly remove echoes from the boundaries as well as leakage from the transducer. The wavelength difference between input Rayleigh and converted S-waves suggests that the

wavefront is mainly reflected upward in the bulk according to Snell’s law only partly converting into backward travelling Rayleigh waves along the bottom surface. In the P conversion case,

the effect is exacerbated and differences in wavelength, velocity and propagation direction between top and bottom signal are clear.

To conclude, we have shown that crystal momentum transfer via Umklapp scattering is of paramount importance to furthering the modalities of many metamaterial devices. Leveraging this decades

old phenomenon with modern, advanced structured materials permits these remarkable devices to harness further powers of wave manipulation. For elasticity we have experimentally verified a

tailored surface wave to body wave reversed conversion effect, which allows the distinct bulk wave-types to be separated at will (Supplementary Note 4, Supplementary Fig. 3). The

incorporation of these mechanisms motivates hybrid effects with self-phased systems, with potential to spark new paradigms of control across all wave regimes.

The aluminium resonators were printed by selective laser melting (SLM). The Rayleigh wave was generated by a Panametrics Videoscan V414 0.5 MHz plane wave transducer and coupled into the

sample by a 65° polymer wedge. Phenyl salicylate was used to glue the transducer and wedge to the sample providing good coupling and long-term stability. A Ritec RPR-4000 programmable pulser

drove the transducer using a three-cycle sinusoidal burst at 1 and 1.5 MHz for S and P-conversion, respectively, with an amplitude of 300 V peak-to-peak and repetition rate of 500 Hz. At

this repetition rate, there were no echoes from previous pulses. The sample was mounted on scanning stages and measured with a rough-surface capable optical detector (Bossanova Tempo-10HF)

over an area of 100 × 30 mm and we used a 0.25 mm step-size. An Agilent digital oscilloscope was used to captured the signal with 125 MSa s−1 and 512 averages before transfer to a desktop

computer.

The 2D wavefield in the halfspace characterised by Rayleigh, P and S waves is simulated using SPECFEM2D, a code that solves the elastic wave equation using a second-order Newmark integration

scheme in time and the spectral element method in space. The code is parallelised using domain decomposition and the MPI instruction set. The quadrilateral mesh is created using the

commercial software Trelis. Stress-free conditions are applied to the rod’s boundary and the top surface, while the lateral and bottom borders are supplied with absorbing boundaries to avoid

undesired reflections. The code runs on a parallel cluster on 32 CPUs. A typical simulation for this work runs in <5 min. 2D and 3D figures have been created using Python library

Matplotlib, Matlab and Blender. Independent corroborations using Abaqus opposed to SPECFEM2D are presented (Supplementary Note 1).

The data that supports the findings of this study are available from the corresponding author upon reasonable request.

The software SPECFEM2D is available as an open source code at: https://geodynamics.org/cig/software/specfem2d/ Meshing and configuration files for the simulations presented here can be

obtained upon reasonable request from Dr. Colombi ([email protected]).

References Peierls, R. Zur kinetischen theorie der wärmeleitung in kristallen. Ann. Phys. 395, 1055–1101 (1929).

Article  Google Scholar 

Hoddeson, L., Baym, G. & Eckert, M. The development of the quantum-mechanical electron theory of metals: 1928–1933. Rev. Mod. Phys. 59, 287–327 (1987).

Article  ADS  CAS  Google Scholar 

Taylor, P. L., Taylor, P. L. & Heinonen, O. A Quantum Approach to Condensed Matter Physics (Cambridge University Press, 2002).

Yeo, L. Y. & Friend, J. R. Surface acoustic wave microfluidics. Annu. Rev. Fluid Mech. 46, 379–406 (2014).

Article  ADS  MathSciNet  Google Scholar 

Briggs, A. Acoustic Microscopy (Clarendon Press, 1992).

Dahlen, F. A. & Tromp, J. Theoretical Global Seismology (Princeton University Press, Princeton, NJ, 1998).

Google Scholar 

Graff, K. F. Wave Motion in Elastic Solids (Oxford University Press, 1975).

Tsakmakidis, K. L., Boardman, A. D. & Hess, O. Trapped rainbow storage of light in metamaterials. Nature 450, 397–401 (2007).

Article  ADS  CAS  Google Scholar 

Zhu, J. et al. Acoustic rainbow trapping. Sci. Rep. 3, 1728 EP (2013).

Article  Google Scholar 

Bennetts, L., Peter, M. & Craster, R. V. Graded resonator arrays for spatial frequency separation and amplification of water waves. J. Fluid Mech. 854, R4 (2018).

Article  ADS  MathSciNet  Google Scholar 

Romero-García, V., Picó, R., Cebrecos, A., Sánchez-Morcillo, V. J. & Staliunas, K. Enhancement of sound in chirped sonic crystals. Appl. Phys. Lett. 102, 091906 (2013).

Article  ADS  Google Scholar 

Jimenez, N., Romero-Garcia, V., Pagneux, V. & Groby, J.-P. Rainbow-trapping absorbers: broadband, perfect and asymmetric sound absorption by subwavelength panels for transmission problems.

Sci. Rep. 7, 13595 (2017).

Article  ADS  Google Scholar 

Colombi, A., Colquitt, D., Roux, P., Guenneau, S. & Craster, R. V. A seismic metamaterial: the resonant metawedge. Sci. Rep. 6, 27717 (2016).

Article  ADS  CAS  Google Scholar 

Colombi, A. et al. Enhanced sensing and conversion of ultrasonic Rayleigh waves by elastic metasurfaces. Sci. Rep. 7, 6750 (2017).

Article  ADS  Google Scholar 

Cardella, D., Celli, P. & Gonella, S. Manipulating waves by distilling frequencies: a tunable shunt-enabled rainbow trap. Smart Mater. Struct. 25, 085017 (2016).

Article  ADS  Google Scholar 

De Ponti, J. M. et al. Graded elastic metasurface for enhanced energy harvesting. New J. Phys. 22, 013013 (2019).

Article  Google Scholar 

Alan, S., Allam, A. & Erturk, A. Programmable mode conversion and bandgap formation for surface acoustic waves using piezoelectric metamaterials. Appl. Phys. Lett. 115, 093502 (2019).

Article  ADS  Google Scholar 

Chaplain, G. J. & Craster, R. V. Flat lensing by graded line meta-arrays. Phys. Rev. B(R.) 99, 220102 (2019).

Article  ADS  CAS  Google Scholar 

Maznev, A. A. & Wright, O. B. Demystifying umklapp vs normal scattering in lattice thermal conductivity. Am. J. Phys. 82, 1062 (2014).

Article  ADS  Google Scholar 

Peierls, R. E.Quantum Theory of Solids (Clarendon Press, 1996).

Ding, Z. et al. Umklapp scattering is not necessarily resistive. Phys. Rev. B 98, 180302 (2018).

Article  ADS  CAS  Google Scholar 

Wallbank, J. R. et al. Excess resistivity in graphene superlattices caused by umklapp electron–electron scattering. Nat. Phys. 15, 32–36 (2019).

Article  CAS  Google Scholar 

Rader, O. & Shikin, A. An elastic sieve to probe momentum space: Gd chains on w (110). Phys. Rev. Lett. 93, 256802 (2004).

Article  ADS  CAS  Google Scholar 

Chaplain, G. J. & Craster, R. V. Ultrathin entirely flat umklapp lenses. Phys. Rev. B 101, 155430 (2020).

Article  ADS  CAS  Google Scholar 

Gurevich, V. & Thellung, A. Quasimomentum in the theory of elasticity and its conservation. Phys. Rev. B 42, 7345 (1990).

Article  ADS  CAS  Google Scholar 

Colquitt, D., Colombi, A., Craster, R., Roux, P. & Guenneau, S. Seismic metasurfaces: sub-wavelength resonators and Rayleigh wave interaction. J. Mech. Phys. Solids 99, 379–393 (2017).

Article  ADS  MathSciNet  CAS  Google Scholar 

Pendry, J. B., Martín-Moreno, L. & Garcia-Vidal, F. J. Mimicking surface plasmons with structured surfaces. Science (New York, NY) 305, 847–8 (2004).

Article  ADS  CAS  Google Scholar 

Balanis, C. A. Modern Antenna Handbook (Wiley, 2008).

Peter, D. et al. Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes. Geophys. J. Int. 186, 721–739 (2011).

Article  ADS  Google Scholar 

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The authors would like to thank Richard Sélos for printing the sample. A.C. was supported by the Ambizione Fellowship PZ00P2-174009. The support of the UK EPSRC through grants EP/K021877/1

and EP/T002654/1 is acknowledged as is that of the ERC H2020 FETOpen project BOHEME under grant agreement No. 863179.

Author informationAuthors and Affiliations Department of Mathematics, Imperial College London, London, SW7 2AZ, UK

Gregory J. Chaplain & Richard V. Craster

Department of Civil and Environmental Engineering, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133, Milano, Italy

Jacopo M. De Ponti

Department of Mechanical Engineering, Politecnico di Milano, Via Giuseppe La Masa, 1, 20156, Milano, Italy

Jacopo M. De Ponti

Department of Civil, Environmental and Geomatic Engineering, ETH, Stefano-Franscini-Platz 5, 8093, Zürich, Switzerland

Andrea Colombi

Optics and Photonics, Faculty of Engineering, University of Nottingham, Nottingham, NG7 2RD, UK

Rafael Fuentes-Dominguez, Paul Dryburg, Don Pieris, Richard J. Smith & Matt Clark

Advanced Component Engineering Laboratory (ACEL), Faculty of Engineering, University of Nottingham, NG7 2RD, Nottingham, UK

Adam Clare

Department of Mechanical Engineering, Imperial College London, London, SW7 2AZ, UK

Richard V. Craster

UMI 2004 Abraham de Moivre-CNRS, Imperial College London, London, SW7 2AZ, UK

Richard V. Craster

AuthorsGregory J. ChaplainView author publications You can also search for this author inPubMed Google Scholar

Jacopo M. De PontiView author publications You can also search for this author inPubMed Google Scholar

Andrea ColombiView author publications You can also search for this author inPubMed Google Scholar

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Don PierisView author publications You can also search for this author inPubMed Google Scholar

Richard J. SmithView author publications You can also search for this author inPubMed Google Scholar

Adam ClareView author publications You can also search for this author inPubMed Google Scholar

Matt ClarkView author publications You can also search for this author inPubMed Google Scholar

Richard V. CrasterView author publications You can also search for this author inPubMed Google Scholar

G.J.C. and R.V.C. applied the theory of Umklapp scattering to elastic devices. J.M.D.P. and A.Co. simulated the structures. R.F.-D., P.D., D.P., R.J.S., A.Cl. and M.C. performed the

experiments. G.J.C. and A.Co. carried out post processing. G.J.C., R.V.C., J.M.D.P., A.Co., R.F.-D. and M.C. contributed to the writing of the manuscript.

Corresponding author Correspondence to Gregory J. Chaplain.

The authors declare no competing interests.

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About this articleCite this article Chaplain, G.J., De Ponti, J.M., Colombi, A. et al. Tailored elastic surface to body wave Umklapp conversion. Nat Commun 11, 3267 (2020).

https://doi.org/10.1038/s41467-020-17021-x

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Received: 12 January 2020

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Published: 29 June 2020

DOI: https://doi.org/10.1038/s41467-020-17021-x

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