Ting-Ting Wang

Yan-Feng Wang

Zi-Chen Deng

Vincent Laude

Yue-Sheng Wang

Reconfigurable coupled-resonator acoustoelastic waveguides in fluid-filled phononic metaplates

Keywords: Phononic crystal, Reconfigurable waveguide, Coupled-resonator acoustoelastic waveguides

We study numerically and experimentally acoustoelastic wave propagation in a two-dimensional phononic metaplate consisting of a periodic array of cups sitting on a thin epoxy plate that is perforated with cross holes. When all cups are filled with water, the metaplate possesses a complete band gap. Reconfigurable coupled-resonator acoustoelastic waveguides (CRAEWs) are created by locally emptying certain cups, thus introducing local resonances that are evanescently coupled. Straight and 90 • bent periodic waveguides are considered, together with an aperiodic chain of 11 coupled resonators. The aperiodic chain has no definite spatial periodicity but supports collective resonances resulting from the coupling of nearest resonators. Lamb waves are experimentally excited by a piezoelectric patch and received by a scanning optical vibrometer. Experimental results for acoustoelastic wave propagation along both periodic and aperiodic CRAEWs are compared to a three-dimensional finite element model taking fluid-structure interaction into account. The propagation of confined acoustoelastic waves in the 90 • bent waveguides and the collective resonances of the aperiodic chain of defected resonators are observed experimentally. Reconfigurability are realized based on the coupling of acoustoelastic waves in a phononic metaplate. Our results show plenty of potential possibilities for the practical design of reconfigurable and programmable elastic wave devices.

Introduction

Phononic crystal (PC) is a new type of periodic functional composite material [START_REF] Yablonovitch | Inhibited spontaneous emission in solid-state physics and electronics[END_REF][START_REF] Kushwaha | Acoustic band structure of periodic elastic composites[END_REF] possessing frequency band gaps within which the propagation of acoustic/elastic waves is completely forbidden [START_REF] Sigalas | Band structure of elastic waves in two dimensional systems[END_REF]. Stated otherwise, Bloch waves become evanescent inside band gaps [START_REF] Laude | Evanescent Bloch waves and the complex band structure of phononic crystals[END_REF]. A coupling mechanism for the vibrations of chains of masses connected by springs [START_REF] Jensen | Phononic band gaps and vibrations in one-and twodimensional mass-spring structures[END_REF][START_REF] An | Elastic wave and vibration bandgaps in two-dimensional acoustic metamaterials with resonators and disorders[END_REF] or of coupled-resonator waveguides [START_REF] Wang | Evanescentwave tuning of a locally resonant sonic crystal[END_REF] is provided by those evanescent waves. Furthermore, strong localization of waves in the phononic band gap can be achieved. [START_REF] Torres | Sonic band gaps in finite elastic media: surface states and localization phenomena in linear and point defects[END_REF][START_REF] Pennec | Two-dimensional phononic crystals: Examples and applications[END_REF][START_REF] Akahane | Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab[END_REF]. When the periodicity of a perfect phononic crystal is destroyed locally, defect states appear. Waves within the band gap are confined into the defects [START_REF] Anufriev | Heat conduction engineering in pillarbased phononic crystals[END_REF] and decay rapidly far away from them. The guidance of acoustic or elastic waves can thus be realized by designing defects in a perfect phononic crystal. The introduction of defects provides new ideas to manipulate waves, and to design and manufacture novel acoustic devices with phononic crystals, and has attracted widespread attention. Linear lines of defects [START_REF] Baboly | Demonstration of acoustic waveguiding and tight bending in phononic crystals[END_REF][START_REF] Otsuka | Broadband evolution of phononic-crystal-waveguide eigenstates in real-and k-spaces[END_REF][START_REF] Baboly | Acoustic waveguiding in a silicon carbide phononic crystals at microwave frequencies[END_REF], as the most commonly used guidance mechanism, are thus formed to channel waves at selected frequencies in the band gap with strong confinement [START_REF] Pennec | Tunable filtering and demultiplexing in phononic crystals with hollow cylinders[END_REF][START_REF] Jin | Phononic crystal plate with hollow pillars actively controlled by fluid filling[END_REF], promising a variety of potential applications [START_REF] Singh | Phonon conductivity of nanoparticles embedded in dielectric material[END_REF][START_REF] Aly | High-performance phoxonic cavity designs for enhanced acousto-optical interaction[END_REF] such as sensing [START_REF] Rostami | Acoustic 1× 2 demultiplexer based on fluid-fluid phononic crystal ring resonators[END_REF][START_REF] Nagaty | Acoustic wave sensor based on piezomagnetic phononic crystal[END_REF][START_REF] Shaban | Determination of 1propanol, ethanol, and methanol concentrations in water based on a one-dimensional phoxonic crystal sensor[END_REF][START_REF] Aly | One-dimensional phononic crystals that incorporate a defective piezoelectric/piezomagnetic as a new sensor[END_REF][START_REF] Zaki | Highperformance liquid sensor based one-dimensional phononic crystal with demultiplexing capability[END_REF][START_REF] Zaki | Fano resonance based defected 1D phononic crystal for highly sensitive gas sensing applications[END_REF], filtering [START_REF] Zhu | Metamaterial based embedded acoustic filters for structural applications[END_REF], or waveguiding [START_REF] Baboly | Acoustic waveguiding in a silicon carbide phononic crystals at microwave frequencies[END_REF][START_REF] Wang | Guiding and splitting Lamb waves in coupled-resonator elastic waveguides[END_REF][START_REF] Deymier | Acoustic metamaterials and phononic crystals[END_REF].

The concept of the coupled-resonator optical waveguide (CROW) defined in a photonic crystal [START_REF] Yariv | Coupled-resonator optical waveguide:? a proposal and analysis[END_REF][START_REF] Notomi | Large-scale arrays of ultrahigh-Q coupled nanocavities[END_REF] has been extended in recent years to the field of phononic crystals. Overall, the propagation of waves in evanescently coupled waveguides stands as simple and efficient among the various physical mechanisms for guiding waves in artificial crystals. In contrast to lineardefect waveguides, the coupled-resonator waveguide is based on the evanescent coupling of defect cavities or resonators [START_REF] Jensen | Phononic band gaps and vibrations in one-and twodimensional mass-spring structures[END_REF][START_REF] An | Elastic wave and vibration bandgaps in two-dimensional acoustic metamaterials with resonators and disorders[END_REF], permitting the design of rather arbitrary acoustic circuits [START_REF] Sainidou | Linear chain of weakly coupled defects in a three-dimensional phononic crystal: A model acoustic waveguide[END_REF][START_REF] Escalante | Dispersion relation of coupled-resonator acoustic waveguides formed by defect cavities in a phononic crystal[END_REF]. The coupling between neighboring defect cavities or resonators creates a propagating dispersion band with a small slope (slow sound) appearing around the flat band generated by the isolated point defect. If the frequency of the defect state is inside a complete band gap of the perfect structure, the defect state is completely localized. The field beyond the defect is evanescent, decaying exponentially away from the defect. Waveguides based on linear chains of coupled cavities have been shown theoretically to allow simultaneously for very strong wave confinement [START_REF] Torres | Sonic band gaps in finite elastic media: surface states and localization phenomena in linear and point defects[END_REF][START_REF] Pennec | Two-dimensional phononic crystals: Examples and applications[END_REF][START_REF] Akahane | Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two-dimensional photonic crystal slab[END_REF] and for low group velocity transmission [START_REF] Escalante | Dispersion relation of coupled-resonator acoustic waveguides formed by defect cavities in a phononic crystal[END_REF]. It is conducive to design new and efficient acoustic devices. In fact, as long as the distance between resonators remains limited and the resonance frequencies are the same, any defect chain of defect cavities or resonators can form a waveguide [START_REF] Khelif | Trapping and guiding of acoustic waves by defect modes in a full-band-gap ultrasonic crystal[END_REF]. This is the basic idea of a coupled-resonator waveguide. Coupled-resonator waveguides do not only manipulate wave propagation by changing the length of the waveguides [START_REF] Wang | Phononic coupled-resonator waveguide microcavities[END_REF], they also regulate the dispersion relation by changing the distance between adjacent resonators [START_REF] Wang | Reconfigurable phononic-crystal circuits formed by coupled acoustoelastic resonators[END_REF], the number of resonators along the circuit, or the coupling coefficient between resonators.

To date, some works have focused experimentally on elastic or acoustic wave guiding along coupledresonator waveguides. Mohammadi et al. realized an effective band-pass filter for wireless communication based on a coupled-resonator acoustic waveguide (CRAW) designed in a phononic crystal plate [START_REF] Mohammadi | VHF phononic band gap band pass filters using coupled resonator acoustic waveguides (craw)[END_REF]. Wang et al. investigated coupled resonator elastic waveguides designed in a PC slab with cross holes [START_REF] Wang | Guiding and splitting Lamb waves in coupled-resonator elastic waveguides[END_REF]. The transmission of strongly confined Lamb waves along a straight waveguide and in a wave splitter circuit with 90 • bends were observed experimentally. However, the conventionally designed and fabricated structures can hardly have tunable (or reconfigurable) topologies or material parameters, limiting the manipulation of waves.

Many researchers have devoted a lot of efforts to the design, development, and demonstration of tunable PCs and metamaterials resulting in an emerging revolution for tunable, active, or even smart control of acoustic or elastic waves. Tunable or active ways of manipulating waves either based on multifield coupling effects [START_REF] Hou | Phononic crystals containing piezoelectric material[END_REF][START_REF] Wu | Band gaps and the electromechanical coupling coefficient of a surface acoustic wave in a two-dimensional piezoelectric phononic crystal[END_REF][START_REF] Laude | Full band gap for surface acoustic waves in a piezoelectric phononic crystal[END_REF] or by mechanical means [START_REF] Wang | Guiding and splitting Lamb waves in coupled-resonator elastic waveguides[END_REF][START_REF] Trainiti | Non-reciprocal elastic wave propagation in spatiotemporal periodic structures[END_REF] are investigated. Li et al. tuned the propagation direction of the flexural wave by active control system behaved as the piezoelectric patches on a plate with T-shaped waveguide [START_REF] Li | Active control on switchable waveguide of elastic wave metamaterials with the 3d printing technology[END_REF]. Pennec et al. confined and guided sound and light waves with certain frequencies in the as-sociated band gaps by engineering the point and linear defects [START_REF] Pennec | Modeling light-sound interaction in nanoscale cavities and waveguides[END_REF]. Hu et al. investigated the temperature effects on the defect states by changing the temperature of the central rod of the two-dimensional ferroelectric ceramic plate and realize the manipulation of elastic waves in the band gaps [START_REF] Hu | Temperature effects on the defect states in two-dimensional phononic crystals[END_REF]. Mazzotti et al. investigated the effect of a generic state of prestress on the passbands and bandgaps of a phononic crystal plate [START_REF] Mazzotti | Modeling bloch waves in prestressed phononic crystal plates[END_REF].

Among the means for tunable or active manipulation of waves, fluid-solid coupling is a rather easy way to realize reconfigurability for a phononic crystal [START_REF] Wang | Tunable fluidfilled phononic metastrip[END_REF][START_REF] Wang | Reconfigurable waveguides defined by selective fluid filling in two-dimensional phononic metaplates[END_REF]. Fluid-solid coupling is also suitable to achieve active and smart control of acoustic/elastic waves. How to realize reconfigurability of coupled resonator waveguides is still an urgent problem to be solved. In general, almost all works tackling this problem are limited to numerical simulations and a small amount of theoretical analysis, and lack experimental verification. In previous researches, some unnecessary limitations to the periodic array of resonator chains were set on the design of coupled resonator waveguides [START_REF] Sainidou | Linear chain of weakly coupled defects in a three-dimensional phononic crystal: A model acoustic waveguide[END_REF][START_REF] Escalante | Dispersion relation of coupled-resonator acoustic waveguides formed by defect cavities in a phononic crystal[END_REF]. However, evanescent coupling of waves across a band gap is omnidirectional, decreasing exponentially away from a resonator. Therefore, evanescent waves suitably couple adjacent resonators placed along a rather arbitrary path, forming an aperiodic coupled-resonator chain [START_REF] Wang | Collective resonances of a chain of coupled phononic microresonators[END_REF].

In this paper, we aim at investigating twodimensional phononic metaplates consisting of a periodic array of cups sitting on a thin epoxy plate that is perforated with periodic cross holes [START_REF] Wang | Multiple wide complete bandgaps of two-dimensional phononic crystal slabs with cross-like holes[END_REF][START_REF] Miniaci | Experimental observation of a large low-frequency band gap in a polymer waveguide[END_REF]. When all cups are filled with water, the metaplate possesses a complete band gap. Reconfigurable coupled-resonator acoustoelastic waveguides (CRAEWs) are created by locally emptying certain cups, thus introducing local resonances that are evanescently coupled. We first discuss the dispersion relation of the bare phononic crystal. Then, different waveguides are designed by emptying selected cups. An aperiodic negative chain with 11 coupled resonators is formed respecting an equal-coupling scheme. 3D finite element computations accounting for fluid-structure interaction are compared with experimental measurements. In general, Numerical and experimental results are found to be in good agreement with a slight resonance frequency shifts. Strong acoustoelastic waves confinement is effectively observed in all cases.

Methods

In this work, the unit cell of the metaplate consists of a single cup grafted onto a plate with periodic cross holes, as shown in Figure 1(a). Following Ref. [START_REF] Wang | Reconfigurable waveguides defined by selective fluid filling in two-dimensional phononic metaplates[END_REF], geometrical parameters of the unit cell are a=5 cm, h=0.8a, b=0.2a, c 1 =0.1a, r 1 =0.38a, r 2 =0.33a, and c 2 =0.11a. With this set of parameters, certainly wide band gap is obtained. The epoxy metaplate sample shown in Fig. 1(b) is processed by 3D printing technique.

Wave propagation is measured at the bottom surface of the epoxy metaplate by Polytec PSV-500 scanning vibrometer, associating with the periodic chirp as the source waveform. Then the vertical displacement vibration is formed by a vertically polarized piezoelectric patch bonded to the sample. Experimental transmissions and displacement distributions are finally integrated by detecting and averaging the vertical displacements around the scan points at the bottom side of the sample.

To evaluate numerically the transmission properties of the considered systems, the 3D finite element method is used. Phononic band structures are obtained by solving an eigenvalue problem [START_REF] Escalante | Dispersion relation of coupled-resonator acoustic waveguides formed by defect cavities in a phononic crystal[END_REF]. Based on Bloch's theorem, two-dimensional Bloch-Floquet periodic boundary conditions are applied on pairs of opposite boundaries of the unit cell depicted in Fig. 1(a) with free bottom and top surfaces. After the finite mesh of the unit cell is created adaptively, it is divided into finite elements connected by nodes. At the interface between fluid and solid, a fluid-solid boundary condition relating the pressure in the fluid to the normal displacement of the solid boundary is imposed. For the filled cup, a sound-soft boundary condition is applied on the top surface of the liquid column. Considering fluid-solid interaction, the discrete form of the acousto-elastic equations is [START_REF] Wang | Hybridization of resonant modes and Bloch waves in acoustoelastic phononic crystals[END_REF]

K s S T f s 0 K f u p -ω 2 M s 0 -S f s M f u p = F 0 , (1)
where u and p represent the displacements and the pressure at the nodes of the solid and fluid field mesh, respectively. F are nodal forces. K s and K f are the stiffness matrices of the solid and fluid; M s and M f are the mass matrices of the solid and fluid; S f s represents the fluid-solid coupling matrix and S T f s is its transpose. Bloch's theorem is applied on the boundaries of the unit cell in the direction where periodicity applies, yielding the following relation between displacements at the nodes on the boundary of the unit cell:

u(r + a) = e ik•a u(r), ( 2
)
where r is located at the boundary nodes and a is the lattice constant vector. We solve directly the eigenvalue problem Eq. ( 1) given the wavevector k under the complex boundary condition Eq. ( 2). We thus get the whole band structure when the wavevector k sweep the irreducible Brillouin zone. According to the dynamic equilibrium Eq. ( 1), we obtain both the pressure field in the fluid and the displacements field in the solid.

Then the frequency response is estimated as follows. Since a finite phononic crystal and an external source of waves are considered, the finite computational domain has to be terminated with radiation boundary conditions to minimized unwanted reflections. For simulation, a time harmonic and spatially random wave source of vertical polarization is used at the source region where a cup is removed to allow a direct comparison with experiments [START_REF] Laude | Stochastic excitation method for calculating the resolvent band structure of periodic media and waveguides[END_REF]. By sweeping the excitation frequency f , we evaluate the frequency response function (FRF) R( f ) in decibels units by considering the ratio of the z-component of the displacements integrated over the source and the receiver as

R( f ) = 20 log 10                 S r u z ds S s u z ds                 . ( 3
)
where u z is the vertical displacement along S r , the area of the receiver and S s , the area of the source. In this work, the solid material parameters for epoxy are mass density ρ s = 1175 kg/m 3 , Poisson's ratio υ = 0.41, and Young's modulus E = 3.2 GPa. The fluid material is water with mass density ρ f = 1000 kg/m 3 and sound velocity c = 1490 m/s. Here, the influence on transmission of the viscosity of water and the viscoelasticity of epoxy which are the main sources of damping is neglected for large lattice constants and low frequencies [START_REF] Wang | Reconfigurable phononic-crystal circuits formed by coupled acoustoelastic resonators[END_REF].

We first consider the perfect PC metaplate for later Experimentally, partial transmission over an even wider bandwidth is observed. As a result, negative contrast can be used to define resonating defects in the phononic metaplate, i.e. by emptying given cups in the array of initially filled cups [START_REF] Wang | Reconfigurable waveguides defined by selective fluid filling in two-dimensional phononic metaplates[END_REF].

The numerical and experimental FRFs are fairly consistent without any parametric adjustment, as a comparison of Fig. 2(e) and Fig. 2(f) shows. In detail, the experimental FRFs appear to be shifted in frequency and have a wider frequency extension compared to the numerical results. The slight frequency shifts may be attributed to the difference of machining and material properties or the underestimated acoustoelastic coupling in the numerical simulation. Generally, the complete band gap for filled cups is suitable for the design of highly confined coupled resonator waveguides, as we discuss next. In the following, we investigate wave propagation in either periodic or aperiodic coupled-resonator acoustoelastic waveguides (CRAEWs) defined by negative contrast (defects are empty cups). It is expected that elastic waves can be spatially localized around the defects at the frequencies inside the complete band gaps, decaying exponentially away from the defect center [START_REF] Torres | Sonic band gaps in finite elastic media: surface states and localization phenomena in linear and point defects[END_REF][START_REF] Psarobas | Phononic crystals with planar defects[END_REF][START_REF] Khelif | Trapping and guiding of acoustic waves by defect modes in a full-band-gap ultrasonic crystal[END_REF].

Results and discussion

We now consider CRAEWs formed in the perfect PC metaplate by locally emptying certain cups. We first consider a straight line of defects, with adjacent cavities separated by two lattice constants, as shown in Fig. 3(a). Figs. 3(b-d) present the phononic band structure of the CRAEW, as well as the numerical and experimental frequency response functions for the finite sample. The CRAEW supercell is shown in Fig. 3(e) together with a representative modal shape for the guided wave at point N L . Looking closer at the band structure in Fig. 3(b), there are additional guiding bands that appear throughout the frequency range between 8.09 kHz to 8.27 kHz which is extremely sensitive to local changes in the resonators. In other words, its dispersion relationship is ultimately determined by the coupling strength between the resonators. The simulated and experimental frequency response functions in Figs. 3(c,d) indicate that transmission through the waveguide is indeed obtained. As in the case of the perfect crystals, the FRFs agree fairly well. It is noticed that there are 10 dB differences between the numerical simulations and the experiments in the FRF. This can possibly be attributed to the slightly inaccurate modeling of the sample, inaccurate material properties, evaporation of water during experiments, and the neglection of certain aspects of acoustoelastic coupling. On the other hand, the excitation sources of the simulations and experiments are not exactly the same. The vibrations for the eigenmode in Fig. 3(e) are mainly concentrated on the defect. There is some energy leakage to the adjacent unit cells ensuring coupling between subsequent defects. The eigenmode is also symmetric with respect to the direction of wave propagation. The out-of-plane displacement field for the finite sample is shown (f) for numerical simulation at 8.27 kHz and (g) for experiment at 8.16 kHz. Movies of the propagation of the guided waves are further shown in the Supplementary Material [52]. Propagation along the straight CRAEW is observed neatly. After the first row of the crystal, elastic energy is well confined in the defects along the waveguide.

Wave confinement along straight linear waveguides was discussed quantitatively before [START_REF] Wang | Reconfigurable waveguides defined by selective fluid filling in two-dimensional phononic metaplates[END_REF] and is here extended to the CRAEW case. A confinement degree [START_REF] Wang | Bandgaps and directional propagation of elastic waves in 2D square zigzag lattice structures[END_REF] was proposed as follows:

C x = 1 l y 1 l x |u z | |u z | max |x| 2 dxdy -1 (4)
where l x and l y are the lengths for one row of the finite structure in the x and the y directions. |x| is the distance to the excitation source. The confinement degree calculated for the straight CRAEW is C x = 32.64 m -2 , which implies a stronger confinement than inside the linear straight waveguide in the same phononic metaplate [START_REF] Wang | Reconfigurable waveguides defined by selective fluid filling in two-dimensional phononic metaplates[END_REF].

Next, we consider a bent CRAEW including a sharp corner with a 90 • bend. Fig. 4(a) shows a schematic representation of the negative contrast bent CRAEW. In this case, the phononic band structure cannot be obtained. Instead, we can still obtain the frequency response function and compare it to the experimental result. The numerical (b) and experimental (c) FRFs are not significantly different from the straight CRAEW case. In particular, the transmission bandwidth are similar. The out-of-plane displacement field for the finite sample is shown (d) for numerical simulation at 8.12 kHz and (e) for experiment at 8.16 kHz. Movies of the propagation of the guided waves are further shown in the Supplementary Material [52]. Vibrations are again mainly confined at the defect sites where the cups are empty along the circuits. When looking closer to the displacement distributions, vibrations are similarly symmetric with respect to the direction of wave propagation before the bend, as for the straight CRAEW. Then symmetry with respect to the direction of propagation is broken. Vibrations are found to be a superposition of two orthogonal dipolar components, oriented along the x and the y axes, effectively inducing an elliptical vibration [START_REF] Wang | Reconfigurable waveguides defined by selective fluid filling in two-dimensional phononic metaplates[END_REF]. This effect can especially be observed in the animations shown in Supplementary Material [52]. A good consistency is perceived between numerical and experimental results. In addition, different from topological waveguides [START_REF] Miniaci | Experimental observation of a large low-frequency band gap in a polymer waveguide[END_REF], the energy in bent CRAEW will be either reflected by the corner or confined by the periodicity. Obtaining a frequencydependent number, such as a coefficient of transmission, indeed appears difficult.

As we know, evanescent coupling of waves across a band gap is omnidirectional, decreasing exponentially away from a resonator, that is why the coupling between adjacent resonators is not limited to the privileged crystallographic directions [START_REF] Wang | Collective resonances of a chain of coupled phononic microresonators[END_REF]. Therefore, evanescent waves suitably couple adjacent resonators placed along a rather arbitrary path, forming an aperiodic coupledresonator chain [START_REF] Benchabane | Surface-wave coupling to single phononic subwavelength resonators[END_REF][START_REF] Raguin | Dipole states and coherent interaction in surface-acoustic-wave coupled phononic resonators[END_REF]. The collective resonances of a chain of coupled phononic microresonators have been achieved in the pure silica phononic structure in our previous work [START_REF] Wang | Collective resonances of a chain of coupled phononic microresonators[END_REF] where the defects are designed by omitting the etching of selected holes in a solid plate. Here, a similar chain of coupled acoustoelastic resonators, or an aperiodic CRAEW, is designed by locally emptying certain cups separated by (±2, ±1) or (±1, ±2) lattice shifts, as shown in Fig. 5(a).

The numerical (b) and experimental (c) FRFs are presented in Fig. 5. A series of sharp resonances are clearly observed inside the initial complete band gap. As argued in Ref. [START_REF] Wang | Collective resonances of a chain of coupled phononic microresonators[END_REF], it can be explained from the discrete sequence of eigenfrequencies of the chain modeled as a phononic polymer. The maximum amplitude of the out-of-plane displacement for each peak varies notably, indicating that vibration modes are variously matched to the excitation source. The out-of-plane displacement field for the finite sample is shown in Fig. 5(d) for numerical simulation at 8.12 kHz and in Fig. 5(e) for experiment at 8.16 kHz. Movies of the collective vibrations are further shown in the Supplementary Material [52]. Clearly, the full chain of defects vibrates coherently. It can be noticed that the vibrations of the defect resonators along the chain are elliptical, in correspondence with the vibrations in the bent CRAEW after the bend. This effect results from the lack of periodic-ity of the chain and of the absence of symmetry with respect to the direction of propagation. Overall, guided waves are rather well confined inside the defect chain.

In the following, a simplified periodic version of the aperiodic CRAEW is considered to compare to some extent with the discrete structure. The supercell is depicted in Fig. 6(a). The sequence of lattice translations is (2, -1)a then (2, +1)a so that the spatial period along axis x is 4a. The periodic CRAEW thus defined can be obtained from a continuous deformation of the chain of resonators. The phononic band structure for the supercell is shown in Fig. 6(b). A number of additional bands appear inside the complete band gap, as in the case of Fig. 3(b). These bands have different polarization contents and couple differently with the source of vibrations. As a remark, since two periods of the chain are actually included in the supercell compared to the periodic straight CRAEW as shown in Fig. 3, the 4a period causes spurious foldings at the Brillouin zone edges. As a result, the additional bands around 8.18 kHz extend almost asymmetrically and continuously toward the Γ and X points, in contrast to the cosine shape of the bands of the straight CRAEW in Fig. 3(b). Therefore, coupling coefficients are almost independent of the direction of coupling. Figs. 6(c,d) also illustrate the eigenmodes at the high-symmetry points of the Brillouin zone around 8.18 kHz corresponding to points A and B. The elliptical vibration directions of both defect resonators are reversed at the Γ and X points. More significantly, the eigenmode shapes are clearly very similar to those observed in the collective vibrations of Fig. 6.

Conclusions

In this paper, wave propagation in coupled-resonator acoustoelastic waveguides formed by evanescent coupling of chains of defect cavities has been studied numerically and experimentally. Straight and bent periodic waveguides, and aperiodic circuits which are formed by locally emptying certain cups have been investigated. Localized defect modes existing inside the complete band gap are the basis of the wave guidance in CRAEWs and lead to strong wave confine-ment in the defects. Experimental results are found to be in fair agreement with numerical results in all cases. This work provides the first numerical and experimental realization of two-dimensional reconfigurable coupled-resonator acoustoelastic waveguides. Furthermore, to our knowledge it is the first time that acoustoelastic wave propagation along aperiodic CRAEWs is achieved, resulting in a larger choice for the definition of phononic circuits. This work provides prospects for the reconfigurable manipulation of acoustoelastic wave transmission in coupled-resonator waveguides. Different chains of CRAEWs with a rather arbitrary shape can indeed be straightforwardly realized without changing the solid metaplate. Active or even smart manipulation of localized resonators is thus expected. Tianjin (20JCQNJC01030). V.L. acknowledges financial support by the EIPHI Graduate School (ANR-17-EURE-0002).

Figure 1 :

 1 Figure 1: (a) Schematic of the PC unit cell and definition of geometrical dimensions. (b) A finite 3D-printed epoxy metaplate sample consisting of 13 × 12 periodic unit cells. One cup is removed inside the phononic structure to bond a piezoelectric patch for excitation of Lamb waves. Each cup can be individually filled with water. Coupled-resonator acoustoelastic waveguides are thus defined in a reconfigurable and re-usable manner. Imaging of wave propagation is performed on the bottom side of the metaplate.

Figure 2 :

 2 Figure2: Schematic of the perfect epoxy metaplate (a) with all cups empty and (b) with all cups filled with water. The green and gray parts mark water and epoxy materials, respectively. The phononic band structure of the perfect epoxy metaplate without (c) and with (d) water filling the cups for a selected frequency range. The color scale measures the polarization amount of the out-of-plane component of displacement. Frequency response functions (FRFs) for the perfect phononic metaplate with all cups empty (red line) and filled with water (black line) obtained from simulation (e) and experiment (f) for a selected frequency range.

Figure 3 :

 3 Figure 3: Schematic of the epoxy metaplate with a negative contrast straight CRAEW formed by locally emptying a line of cups (a). The green and gray parts mark water and epoxy materials, respectively. The phononic band structure (b) of perfect phononic metaplate with all cups filled with water (solid lines) the straight CRAEW (dash lines) for a selected frequency range. The frequency response functions (FRFs) of the perfect phononic metaplate with all cups filled with water (black line) and for the straight CRAEW (blue line) obtained from simulation (c) and experiment (d) for a selected frequency range. The CRAEW supercell and eigenmode N L at the marked dispersion point are depicted in (e). The numerical (f) and experimental (g) out-of-plane displacement distributions are shown at 8.27 kHz and 8.16 kHz, respectively. The propagating circuits are surrounded by the pink lines. The black (red) disks indicate the wave source (receiver) positions. The color scales represent the amplitude of the out-of-plane displacement field from 0 (blue) to maximum (red).

Figure 4 :

 4 Figure 4: Schematic of the epoxy metaplate with a negative contrast bent CRAEW formed by locally emptying a bent line of cups (a). The green and gray parts mark water and epoxy materials, respectively. The frequency response functions (FRFs) of the perfect phononic metaplate with all cups filled with water (black line) and for the bent CRAEW (purple line) obtained from simulation (b) and experiment (c) for a selected frequency range. The out-of-plane displacement distributions at 8.12 kHz in the simulation (d) and 8.16 kHz in the experiment (e). The propagating circuits are surrounded by the pink lines. The black (red) disks indicate the wave source (receiver) positions. The color scale represents the amplitude of the out-of-plane displacement field from 0 (blue) to maximum (red).

Figure 5 :

 5 Figure 5: Schematic of the epoxy metaplate with a negative contrast aperiodic CRAEW of 11 coupled resonators formed by locally emptying a chain of cups (a). The absolute vertical displacements obtained from simulation (b) and experiment (c) at the position of the last resonator in the chain around selected frequencies. Vertical-displacement maps at frequencies of 8.18 kHz in the numerical simulation (d) and 8.19 kHz in the experiment (e). All the defect resonators along the chain are figured by the pink square. The color scale represents the amplitude of the out-of-plane displacement field from 0 (blue) to maximum (red).

corresponding author. E-mail address: wangyanfeng@tju.edu.cn

corresponding author. E-mail address: vincent.laude@femtost.fr

Acknowledgments

The authors acknowledge financial support by the National Natural Science Foundation of China (12102350, 12122207, and 12021002). YW acknowledges support by the Natural Science Foundation of


Izabela Jelovac
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Catalina Bordoy

Pedro P Barros

Stefano Comino

Miguel Gouveia

Inés Macho-Stadler

Xavier Martínez Giralt

Pau Olivella

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Pricing and Welfare Implications

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Introduction

With this paper, we participate to the ongoing debate over the benefits and drawbacks from permitting parallel imports among countries. In particular, we investigate the pricing and welfare implications of parallel trade of pharmaceuticals between two countries. Parallel imports are goods produced genuinely under intellectual property right (IPR) protection, placed into circulation in one market, and then imported into a second market without the authorization of the IPR owner. They are identical to the legitimate products, except that they may be packaged differently and may not carry the original manufacturer's warranty [START_REF] Maskus | Parallel imports[END_REF].

One important reason why parallel imports might arise, if they are permitted, is to arbitrage away international price discrimination, which is widely observed for pharmaceutical products (see [START_REF] Maskus | Parallel imports[END_REF]. One expected effect of permitting parallel imports is a convergence in prices between countries. [START_REF] Ganslandt | Parallel imports of pharmaceutical products in the European Union[END_REF] provide evidence on price convergence resulting from parallel trade of pharmaceuticals in EU countries.

The expected effect of parallel trade in terms of social welfare is not so clear-cut.

Welfare is shown to either increase or decrease with parallel imports, depending on whether authors consider any of the following aspects: different drug prices regulations across countries [START_REF] Pecorino | Should the US allow prescription drug reimports from Canada?[END_REF]; efforts of IPR owners to exert vertical price control [START_REF] Maskus | Parallel imports in a model of vertical distribution: Theory, evidence and policy[END_REF]; the level of demand dispersion across markets [START_REF] Malueg | Parallel imports, demand dispersion, and international price discrimination[END_REF]; and the need for manufacturers to recoup their global research and development costs [START_REF] Danzon | The economics of parallel trade[END_REF].

The main contribution of our paper is that it stresses the importance of identifying the main determinants of international price discrimination to understand the welfare effects associated with parallel trade. We use a model that accounts for the differences between countries in terms of health insurance reimbursement policies and in terms of drug needs reflected in the patients' valuation for a drug. We neglect the effects associated with different income levels across countries, even though this difference is likely to be an important determinant of international price discrimination. When we consider differences in income only, the parallel imports are expected to flow from low-income countries towards high-income countries. Since parallel imports generate price convergence between countries, richer countries might benefit from parallel imports while poorer countries might be worse off (see [START_REF] Danzon | The economics of parallel trade[END_REF]. However, international price discrimination is likely to be caused not only by differences in income across countries, but also differences in other relevant characteristics of the demand. Otherwise, how could we answer the question raised by [START_REF] Maskus | Parallel imports in pharmaceuticals: implications for competition and prices in developing countries[END_REF]: Why might prices be higher in poor countries? 1 Characteristics of the demand that are especially relevant for pharmaceuticals rely both on insurance and on drug needs. Both can be specific to countries. On the one hand, the huge variations among national health systems can influence the pricing strategies of pharmaceutical firms. In particular, the level of insurance reimbursement influences the pricing of drugs, since it directly affects the price elasticity of the demand for drugs. If 1 [START_REF] Maskus | Parallel imports in pharmaceuticals: implications for competition and prices in developing countries[END_REF] reports the finding that prices are elevated in such countries as South Africa, Mexico and Brazil relative to those in Canada, Spain and Italy.

there is no other regulation on drug prices as it is the case in Germany and in Denmark among other countries, the pharmaceutical manufacturers would charge higher prices in countries where insurance is more generous, taking advantage of a lower price elasticity of demand. 2 Pavcnik (2002) provides evidence on this relationship between the patients' copayment for buying drugs and drug prices in Germany. On the other hand, pharmaceutical firms might also take advantage of differences in needs for a given drug among countries, charging higher prices where some endemic illness makes the need for the appropriate drug higher than in countries where this illness is not active, for example.

We tackle these issues using a model with the following timing. In the first stage, a multinational monopoly producer sets the price of a patented drug to be sold in two countries. If the prices are different between the two countries, parallel traders can buy, in the second stage of the game, drugs in the low-price country and re-sell them in the highprice country at a price depending on whether the market for parallel imports is monopolistic or competitive. In the third stage of the game, the individuals in both countries choose to consume either one unit of the drug supplied by the monopolist, or one unit of the parallel imported drug, or nothing, so as to maximize their utility.

We first confirm a result already reported in the literature: Parallel trade makes the prices converge between countries. As a reaction to the possible entry of parallel traders in the market, the pharmaceutical monopoly producer trades-off the benefits from price discrimination with the losses of facing competition from parallel imports in the high-price country. Therefore, the monopolist increases the price in the low-price country, and decreases the price in the high-price country so as to deter some amount of parallel imports. This does not mean that permitting parallel trade results in global uniform pricing. Contrary to other papers [START_REF] Malueg | Parallel imports, demand dispersion, and international price discrimination[END_REF][START_REF] Malueg | Parallel imports, demand dispersion, and international price discrimination[END_REF][START_REF] Richardson | An elementary proposition concerning parallel imports[END_REF] in which parallel imports are assumed to imply de facto global uniform pricing, we obtain global uniform pricing only if consumers value the original drug and the parallel imported drug equally.

However, as noted by [START_REF] Maskus | Parallel imports in pharmaceuticals: implications for competition and prices in developing countries[END_REF], goods that are parallel imported may not be perceived to be of the same quality between markets, even if the manufacturer placed them on the market originally, because of differences in packaging or guarantees. This difference in perception leads in our model to the persistence of some level of price discrimination between countries, even when parallel imports are permitted.

Furthermore, we show that the effect of parallel imports on the total welfare is ambiguous. We identify three cases in which the effect of parallel trade in terms of total welfare can be stated unambiguously. We show that parallel trade increases the total welfare when it takes place between countries differing in their drug needs only. The rationale behind this positive effect relies on the re-allocation of consumption from individuals with relatively lower needs in the exporting country, towards individuals with relatively higher needs. The opposite re-allocation of consumption is the result of parallel trade when countries differ only in their health insurance reimbursement policies. In that case, the total welfare decreases with parallel trade. Allowing parallel imports would also decrease the total welfare if it induces the monopolist to stop selling drugs in the originally low-price country. The rationale for this case follows a result of [START_REF] Hausman | Price discrimination and patent policy[END_REF]: If one market is not served under uniform pricing, then price discrimination yields a Pareto improvement.

One specific feature of our model is worth mentioning in order to contrast our results with an existing general result over the welfare effect of third-degree price discrimination. In our model, the use of linear demand functions results in the same total quantity of drugs purchased, no matter whether parallel imports are permitted or not. This is analogous to the result of [START_REF] Robinson | Economics of Imperfect Competition[END_REF]: If a single price monopoly selling in two markets under constant costs is allowed to discriminate between them, total output is unchanged if both markets have linear demand curves and both markets are served under both regimes (either uniform pricing or third-degree price discrimination). Generalizing and extending Robinson's result, [START_REF] Schmalensee | Output and welfare implications of monopolistic third-degree price discrimination[END_REF] shows that price discrimination reduces welfare if it does not increase total output. Even though our aim is not to compare uniform pricing with price discrimination, our result departs from this classical result: The welfare effect of lowering price discrimination by permitting parallel imports is not necessarily positive, even though the total output remains constant. It is the presence in our model of differentiated co-payments for buying pharmaceuticals that explain this discrepancy: The degree of discrimination in prices does not need to coincide with the degree of discrimination in consumers' prices. However, as shown in [START_REF] Jelovac | Monopolistic third-degree price discrimination of subsidized goods[END_REF], consumers' prices rather than full prices matter to evaluate the welfare effect of thirddegree price discrimination.

In the next section we describe the model. Section 3 discusses the equilibrium of the game. In Section 4 we analyze the welfare implications of allowing parallel trade. Finally, we conclude in Section 5.

The Model

We consider a multinational firm producing a patented drug. We assume that the variable cost of producing the drug is zero. This assumption reflects the empirical observation that variable production costs are very low for pharmaceuticals. The producer acts as a monopolist given the patent on his product. He sells the drug in two countries, A and B, at prices p A and p B , respectively. If parallel imports are permitted, one or more wholesalers can buy the drug in country i, i = A, B, at price p i , and re-sell it in the other country at price p w , and at no cost except the price paid for the drug in the first country. 3Each country has a population whose size is normalized to one. For simplicity, individuals in both countries are assumed to have a utility additively separable in the consumption of a numeraire composite good and the consumption x of the drug, with . They have an income I at their disposal to buy the composite good, and one or zero unit of the drug. In each country, individuals differ in their valuation of the drug, θ,

which is uniformly distributed on the support [ { } 1 , 0 = x i θ , i θ ] in country i, i = A, B: θ ~ U [ i θ , i θ ].
We assume, for simplicity, that 1 = i i θ θ . To guarantee that the equilibrium demand solutions are interior in the benchmark, we also assume that

2 0 ≤ ≤ i θ , i = A, B.
Moreover, we assume that individuals prefer to consume the drug supplied by the monopolist to the one supplied by the parallel importer. Therefore, their valuation of the parallel imported drug is weighted by ρ < 1. This assumption reflects the fact that, according to [START_REF] Maskus | Parallel imports in pharmaceuticals: implications for competition and prices in developing countries[END_REF] among others, "goods that are parallel imported may not be perceiv rer in both countries, so that the individuals only pay a share α i of it in country i. Therefore, the indirect utility function of an individual with valuation θ in country i can be written as:

ed to be of the same quality between markets, even if they were placed on the market originally by the manufacturer, because of differences in packaging or guarantees".

We assume that the expenses for drug consumption of an individual, p i x i , are partially reimbursed by some public health insu { } 0 ; ;

w i i i i p p Max I U α ρθ α θ - - + = ,
if there are parallel imports available in country i. Otherwise, the utility function reduces to:

{ } 0 ; i i i p Max I U α θ - + = .
We assume throughout the paper the following inequality:

B B A A α θ α θ <
, to acco i that parallel trade takes place from country A towards country B, unt for the differences between the countries, and to guarantee that if parallel trade takes place, it does so from country A towards country B.

The timing of our game is assumed to be as follows. If parallel trade is permitted, and assuming a prior then the monopolist sets the prices p A and p B in the first stage of the game so as to maximize his profits:

B B w A A m D p D D p + + = Π ) ( ,
where D i , i = A, B, stand for the demand for the drug directly supplied by the monopolist in country i, and D w stands for the demand faced by the parallel importer in the importing country, B. Then, in the second stage of the game, the parallel importer sets the price p w , as a Stackelberg follower. If the parallel importer is unique, he sets p w so as to maximize his profit:

. ) (

w A w w D p p - = Π
If there are many wholesalers competing with each other in the parallel imports market, then they set a price equal to their marginal cost: p w = p A . In the third stage of the game, the individuals in both countries choose to consume either one unit of the drug supplied by the monopolist, or one unit of the parallel import if it is available, or nothing, so as to maxim ly described vanishes, and D w = 0. We first present, as a benchmark case, the equilibrium of the game when parallel imports are legally forbidden. In the last stage of the game, individuals choose to consume either one unit of the good supplied directly by the monopolist in their country, or nothing. Given the utility:

{ } 0 ; i i i p Max I U α θ - + = ,
only the individuals in country i with a valuation for the drug

i i p α θ ≥
are going to buy one unit of the good. Therefore, the demand faced by the monopoly in country i is:

i i i i p D α θ - = , i = A, B.
Given these demands, the monopolist sets the prices p A and p B so as to maximize his profit:

) ( ) ( B B B B A A A A B B A A m p p p p D p D p α θ α θ - + - = + = Π .
The equilibrium prices that maximize this profit (p ) for paying a share 1 -α i , i = A, B, of the expenses associated with drug consumption, are also presented in Table 1.

Table 1

Country

A Country B Price A A A p α θ 2 * = B B B p α θ 2 * = Demand 2 * A A D θ = 2 * B B D θ = Cons. surplus 8 2 * A A I CS θ + = 8 2 * B B I CS θ + = Profit ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = Π B B A A m α θ α θ 2 2 * 4 1 Public expense s 2 * 2 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = A A A A PE θ α α 2 * 2 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = B B B B PE θ α α
In both countries, the equilibrium monopoly prices increase with the ratio representing the maximum effective willingness to pay for the drug:

4 i i α θ , i = A, B.
In particular, the monopoly manufacturer can charge higher prices when the co-payment rate is lower, taking advantage of a lower price elasticity of demand when the co-payment is lower. The equilibrium demands do not depend on the level of the patients' co-payment for buying the drug, since the price faced by the individuals in both countries, α i p i , only depends on the maximum valuation for the drug in their country. Furthermore, the price is lower in country A, which results from the assumption that the maximum effective willingness to pay is lower in country A. In this benchmark case, the monopoly producer discriminates as much as possible the prices between the two countries.

Parallel imports are legally permitted

When parallel imports are legally permitted, the demands for both the parallel import and the drug supplied by the producer are realized in the third stage of the game.

Parallel trade, if it takes place, does so from country A towards country B by assumption.

Then, in country A where the drug is not available as a parallel import, the individuals with a valuation

A A p α θ ≥
buy one unit of the drug supplied by the monopoly producer in this country. Therefore,

A A A A p D α θ - = .
In country B where parallel imports are available, only the individuals with a valuation:

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - - ∈ ρ α ρ α θ 1 ) ( ; w B B w B p p p ,
maximize their utility buying one unit of the parallel import:

{ } 0 ; B B w B p Max p α θ α ρθ - ≥ - .
Therefore,

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ - - - = 0 ; 1 ) ( ρ α ρ α w B w B B w p p p Max D ,
which is equivalent to:

⎪ ⎩ ⎪ ⎨ ⎧ ≤ - - ≥ = . ) 1 ( ) ( , 0 B w w B B B w w p p if p p p p if D ρ ρ ρ ρ α ρ
For parallel trade to be attractive to the individuals in country B, we need a price p w not only lower than p B , but lower than ρp B , to account for the fact that, ceteris paribus, individuals prefer the drug supplied by the monopolist to the parallel import.

Individuals in country B with a valuation:

ρ α θ - - ≥ 1
) ( Therefore, the demand for the drug supplied directly by the monopolist in country B is:

⎪ ⎩ ⎪ ⎨ ⎧ ≤ - - - ≥ - = . 1 ) ( , B w w B B B B w B B B B p p if p p p p if p D ρ ρ α θ ρ α θ
In the second stage of the game, the parallel importer(s) can buy drugs in country A, and decide upon the price p w , anticipating the demands D A , D B and D w previously derived.

If the parallel imports market is a competitive one, then the equilibrium parallel import price is:

p w = p A .
This price attracts consumers in country B only if

B A w p p p ρ ≤ =
. That is, only if the difference in the prices charged by the monopolist in both countries is big enough.

Otherwise, i.e. if prices p A and p B are too similar, there would be no room for the parallel importers to enter the market in country B.

If there is only one monopolistic parallel importer, then the equilibrium price is the one that maximizes his profit:

⎪ ⎩ ⎪ ⎨ ⎧ ≤ - - - ≥ = - = Π . ) 1 ( ) ( ) ( , 0 ) ( B w w B B A w B w w A w w p p if p p p p p p if D p p ρ ρ ρ ρ α ρ
If the difference between p A and p B is high enough:

B A p p ρ ≤
, then the parallel importer enters the market with the following equilibrium price :

2 B A w p p p ρ + = .
Otherwise, i.e. if

B A p p ρ >
, there would be no parallel imports available in country B.

In stage 1, the monopoly producer sets the prices p A and p B to maximize his profit, anticipating the parallel import price and the demands D A , D B and D w . The demand for the drug supplied by the monopoly producer in country A is unaffected by the decision of the parallel importer in stage 2. Therefore, the demand D A that is anticipated in stage 1 is:

A A A A p D α θ - = .
Given the sub-game perfect equilibrium in stage 2, the demand for parallel imports in country B that is anticipated in stage 1 is:

⎪ ⎩ ⎪ ⎨ ⎧ < - - ≥ = , ) 1 ( ) ( , 0 B A A B B B A w p p if p p p p if D ρ ρ ρ ρ α ρ
whenever the market for parallel imports is competitive or monopolistic.

The demand D B that is anticipated in stage 1 depends on the market for parallel imports. If it is a competitive market, then:

⎪ ⎩ ⎪ ⎨ ⎧ < - - - ≥ - = . 1 ) ( , B A A B B B B A B B B B p p if p p p p if p D ρ ρ α θ ρ α θ
If it is a monopolistic market, then:

⎪ ⎩ ⎪ ⎨ ⎧ < - - - - ≥ - = . ) 1 ( 2 ) ) 2 (( , B A A B B B B A B B B B p p if p p p p if p D ρ ρ ρ α θ ρ α θ
Given these demands, the equilibrium prices p A and p B that maximize the producer's profit:

B B w A A m D p D D p + + = Π ) ( ,
are presented in Table 2 for an interior solution. The corresponding equilibrium demands (D A , D B and D w ), profits (Π A and Π B ), consumers' surpluses (CS A and CS B ), and public expenses (PE A and PE B ), are also presented in Table 2.

For the sake of clarity, we use the following notation:

{ } c m ∆ ∆ ∆ ∈ ∆ , , 0 , with: - , if 0 0 = ∆ = ∆ A B B A θ α θ ρα ≤
, and/ or if parallel imports are legally forbidden.

-

( ) B A A B B A m α α ρ ρ θ α θ ρα + - - = ∆ ) 2 ( 2 , if A B B A θ α θ ρα >
, parallel imports are permitted, and their market is monopolistic.

-

( ) B A A B B A c α ρα θ α θ ρα + - = ∆ 2 , if A B B A θ α θ ρα >
, parallel imports are permitted, and their market is competitive.

The term ∆ allows us to present the equilibrium solution in Table 2 in an uniform way, independently on the situation considered: either no market for parallel imports, or monopolistic parallel imports market, or competitive parallel imports market. Thus, in order to compare these three situations, it is enough to focus on ∆, with .

c m ∆ < ∆ < ∆ 0 Table 2 Country A Country B Parallel importer Price ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ + = 2 1 A A A p θ α ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ - = ρ θ α 2 1 B B B p ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ - = 2 B B w p θ α ρ Demand ∆ - = 2 A A D θ 2 B B D θ = ∆ = w D Cons. surplus ( ) ∆ - ∆ - = A A A CS CS θ 2 * ( ) ∆ + ∆ + = B B B CS CS θ ρ 2 * Profit ( ) A B B A B A m m θ α θ ρα α α - ∆ - Π = Π 2 * ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ + - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ - ∆ = Π 2 1 2 A A B B w θ α θ α ρ Public expenses ( ) ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ∆ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = 2 2 2 1 A A A A PE θ α α ( ) ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ∆ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = 2 2 2 1 ρ θ α α B B B B PE 16 If A B B A θ α θ ρα ≤
, then allowing parallel imports or not does not make any difference, since . In that case, the market conditions in both countries are very similar. That can be seen adding our assumption on the market asymmetry:

0 0 = ∆ = ∆ B B A A α θ α θ <
, to the condition characterizing the situation discussed now:

B A A B B A θ α θ α θ ρα < ≤ .
With such a similarity between the market conditions of both countries, the room for the monopoly producer to price discriminate is very limited, no matter whether parallel imports are tolerated or not. Therefore, no parallel importer could take advantage of this price difference to attract clients in country B.

If A B B A θ α θ ρα >
, market conditions in both countries are different enough, so that parallel trade occurs if it is allowed. We can discuss the effects of allowing parallel imports, independently on whether the parallel imports market is competitive or monopolistic. In both cases, ∆ > 0, which can be compared to the benchmark situation where parallel imports are forbidden and ∆ = 0.

In Table 2, we see that allowing parallel imports makes the prices in both countries converge : p A increases and p B decreases. This is an intuitive result, and it is explained by the following trade-off faced by the monopoly producer. The latter would like to enjoy the benefits associated with the price discrimination, and limit the competition associated with the parallel trade in country B. The trade-off arises because the stronger the price discrimination, the bigger the room for parallel imports.

The main difference between a competitive parallel imports market and a monopolistic one can be understood when realizing that

c m ∆ < ∆
. This implies that the price convergence due to parallel imports is stronger when the parallel import market is competitive. This happens because the afore-mentioned trade-off and its resulting price effect are stronger when the threat of competition from parallel importers is stronger, thus when the parallel imports market is competitive. Consequently, all the remaining effects associated with parallel imports are stronger when the parallel import market is competitive.

The price set by the parallel importer is naturally higher or equal than the price paid in country A, and it is lower than the price of the competing drug supplied by the producer in country B. Given the convergence in price, we have that:

* * B B w A A p p p p p < < ≤ < .
However, parallel imports do not necessarily lead to a convergence in consumers' prices. Analyzing the demands in Table 2, we depict a re-allocation of the drug consumption from country A to country B, the total output remaining unchanged. This can be seen graphically in Figure 1:

Indeed, neither

θ Α A θ θ B B θ ∆ ∆ ∆ - 2 A θ 2 B θ Country A : Country B : Figure 1
Clearly, the consumers in country B are better off, enjoying more consumption at lower prices. The opposite happens in country A. This explains why CS A is lower and CS B is higher when ∆ > 0 than when ∆ = 0.

The fact that total output is the same, no matter whether parallel imports are permitted or not, is in line with the result of [START_REF] Robinson | Economics of Imperfect Competition[END_REF]: When two markets have linear demands and both are served whenever the monopolist price discriminates or not, then a monopolist's equilibrium supply does not depend on whether price discrimination is feasible or not. In our model, the demand for drugs is linear because the valuation of the drug, θ, is assumed to be uniformly distributed and consumers' utility is assumed to be additively separable in the consumption of a numeraire and the consumption of the drug.

The monopoly producer profit is reduced due to both the competition from parallel imports in country B, and the lower price discrimination. The profit of the parallel importer(s) is obviously at least as high as when they do not operate on the market.

Last, the public expenses are lower in both countries when parallel imports are permitted.

So far, we have analyzed the results reported in Table 2, which strictly focuses on an interior solution. However, our model does not prevent some corner solution for the case where parallel imports are permitted. In particular, when the market conditions are too different between the countries:

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + > B A A B B A α ρα θ α θ ρα 1 1 ,
the monopolist is better off setting a price in country A so high that neither the consumers in that country nor the competing parallel traders buy the drug:

* B A p p ρ = and 0 = = w A D D .
Consequently, the price in country B is the same as if parallel imports were forbidden, and so is the resulting demand:

The welfare analysis

We now analyze how the changes induced by the parallel imports affect the total welfare. We define the total welfare as the sum of consumers' surpluses net of the public expenses in both countries, and profits of both the monopoly producer and the parallel importer(s):

.

w m B A B A PE PE CS CS TW Π + Π + - - + = 20
We start our welfare analysis focusing on the interior solution as reported in Table 2. We already know that, on the one hand, parallel imports, when they take place, cause a positive effect on the total welfare through CS B , PE A , PE B , and eventually Π w (if the parallel importer is a monopoly; otherwise, Π w = 0). On the other hand, they have a negative effect on the total welfare through CS A and Π m . In order to determine the circumstances under which the positive effect out-weights the negative one, it is useful to compare the total welfare when parallel imports are allowed, TW, with the one characterizing the benchmark case, TW * :

( ) ( ) ∆ + - - ∆ + = ρ θ θ ρ 1 2 * A B TW TW ,
where:

* * * * * * m B A B A PE PE CS CS TW Π + - - + = .
Therefore, the necessary and sufficient condition for parallel imports to increase total welfare is:

∆ + > - ) 1 ( ρ θ θ ρ A B
.

[NSC]

Given our assumptions, and the condition for the parallel imports to take place

( A B B A θ α θ ρα >
), parallel trade can result either in an increase or in a decrease of the total welfare. The sum of the profits net of the public expenses always decreases as a result of parallel trade:

2 , * , * , , ) 1 ( ∆ + - - Π = - Π ∑ ∑ ∑ ∑ = = = = ρ B A i i w m j j B A i i w m j j PE PE .
Therefore, the total welfare can increase with parallel trade only when the gain for the consumers in country B is sufficiently large to compensate the loss for the consumers in country A. A necessary but not sufficient condition for is thus:

* TW TW > ( ) ∑ ∑ ∑ = = = > ∆ + + - ∆ + = B A i i A B B A i i B A i i CS CS CS , * , * , ) 1 ( 2 ρ θ θ ρ , which is equivalent to: 0 ) 1 ( > ∆ + + - ρ θ θ ρ A B
.

[NC]

Both conditions [NC] and [NSC] hold when countries only differ in the distribution of valuations for the drug, reflected in i θ , i = A, B. This happens when we consider countries with a similar health system, but with different valuations for the drug due to differences in the endemic illnesses suffered by their populations, for example. In that case, the condition under which parallel imports take place reduces to A B θ θ ρ > . Therefore, in this situation, the increase in the consumers' surplus in country B more than compensates the decrease in the one of country A. One explanation for that relies on the re-allocation of the drug consumption from country A towards country B. The parallel imports would make the individuals in country A with a valuation:

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∆ + ∈ 2
start consuming the drug thanks to the parallel trade. Therefore, we have a re-allocation from individuals valuing the drug less towards individuals valuing the drug more, since:

∆ - < ∆ + 2
model, parallel imports decrease rather than prohibit third-degree discrimination. In that sense, we compare different degrees of third-degree discrimination while Schmalensee (1981) compares unrestricted third-degree discrimination with uniform pricing. Therefore, the result in Proposition 1 can be seen to a certain extent as a generalization of Schmalensee's result.

Another interesting case considers two countries differing only in their health care system, reflected in the co-payment for buying the drug. We can think of countries with similar health needs and different social security systems. Some countries in the European Union satisfy these characteristics. In this case,

θ θ θ = = B A
, and the condition for parallel trade to take place is B A α ρα > . The total welfare can be rewritten as:

( ) ( ) ( ) ∆ + + - ∆ - = ρ θ ρ 1 1 2 * TW TW .
Therefore, parallel imports decrease the total welfare in this case, even when the sum of the consumers' surpluses is positive, which occurs only when:

∆ + ∆ - > θ θ ρ .
We have now a re-allocation of the drug consumption from individuals in country A with a higher valuation:

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∆ + ∈ TW TW ≤ .
At first sight, the result of Proposition 2 seems to contradict Schmalensee's result.

However, as shown in [START_REF] Jelovac | Monopolistic third-degree price discrimination of subsidized goods[END_REF], we must consider discrimination (or convergence) in consumers' prices to interpret Schmalensee's result in the presence of differentiated copayments. In the case considered in Proposition 2, parallel imports actually imply a divergence in consumers' prices together with a convergence in full prices:

. 2 2 2 2 * * ∆ + = < = = < ∆ - = < ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ - = θ α θ α α ρ θ α θ ρ α A A B B A A B B w B p p p p p
This divergence in consumers' prices explains why parallel imports decrease total welfare. Therefore, our result in Proposition 2 confirms rather than contradicts Schmalensee's. 5 Last, we discuss the case where market conditions of country A and country B are so different that allowing parallel trade induces the following corner solution:

Conclusions

With this paper, we participate to the ongoing debate over the benefits and drawbacks from allowing parallel trade among countries. We use a model that accounts for the differences between countries in terms of health system (reflected in the level of patients co-payments), and in terms of drug needs (reflected in the patients valuation for the drug). Our main findings are the following.

First, we confirm some results already discussed in the ongoing debate: Parallel trade makes the prices converge between countries, it makes the individuals of the 5 See [START_REF] Jelovac | Monopolistic third-degree price discrimination of subsidized goods[END_REF] for a more detailed analysis of the role of differentiated subsidies on the welfare effects of third-degree price discrimination versus uniform pricing. importing country better off, while making the ones of the exporting country worse off, and they decrease the profit of the monopoly producer. Moreover, we show that the public expenses in both the importing and the exporting countries are reduced with parallel trade.

Second, we show that the effect of parallel imports on the total welfare is ambiguous, even though the total output remains unchanged. This certainly contrasts with the classical result over the negative effect associated with price discrimination when total output is unchanged. This discrepancy relies on the presence of differentiated co-payments in our model.

We identify three cases where the effect of allowing parallel trade on the total welfare can be stated unambiguously. On the one hand, we show that parallel trade increases the total welfare when it takes place between two countries differing in their health needs only. The rationale behind this positive effect relies on the re-allocation of the drug consumption from individuals with relatively less drug needs in the exporting country, towards individuals with relatively higher drug needs.

On the other hand, we show that parallel trade decreases the total welfare when it takes place between countries differing in their health system only. In that case, the drug consumption is re-allocated from individuals with relatively more drug needs to individuals with relatively less drug needs. Last, parallel trade also decreases the total welfare when the two countries are too different. In that case, the weak market is not served when parallel trade is permitted.

Our analysis is made maintaining the level of income equal between the countries. Therefore, our results are applicable to trade taking place between countries of similar income levels. A direct interpretation of our results would be the following: On the one hand, parallel trade would increase the total welfare when it takes place between two developing countries with the same level of income and patients co-payments, and different drug needs, to account for the higher needs for malaria or AIDS treatment in some developing countries than in other ones. On the other hand, parallel trade between industrialized countries, characterized by similar levels of (high) income and similar epidemiological conditions, and different drug reimbursement levels, would decrease the total welfare.

When we consider parallel trade between countries with different income levels, such as the trade between developing countries and developed ones, we should carefully add the well known effects of parallel trade between a poor country and a rich country (reallocation of the consumption from the poor country towards the rich one) to the effects identified in the present paper.

Given the results obtained in this paper, it would be interesting to explore the optimal decision of governments about whether to permit parallel imports or not, together with an endogenous decision about the optimal health insurance reimbursement policy. This additional step is left for further research. In that sense, it is worth acknowledging the contribution of [START_REF] Richardson | An elementary proposition concerning parallel imports[END_REF] who demonstrates that, when countries individually choose whether or not to prohibit parallel imports, a global Nash equilibrium involves the permitting of parallel importing into all relevant foreign markets.

  We solve the game by backwards induction to derive the Nash sub-game perfect equilibrium. ize their utility. If parallel trade is legally forbidden, then the second stage of the game previous 3. The equilibrium of the game 3.1. Benchmark: Parallel imports are forbidden legally

  buying one unit of the good supplied by the monopolist, if the parallel import attracts some individuals in country B, i.e. if of the good supplied by the monopolist in country B.

  In this case, the market is not served in country A while country B is in the same situation as if parallel imports were legally forbidden. Therefore, forbidding parallel trade would yield a Pareto improvement since it would open the market in country A without damaging the market in country B (see Hausman and MacKie-Mason for a similar argument).

In the presence of additional price regulations, which is not the focus of our paper, this relationship may not hold anymore. See[START_REF] Jelovac | On the relationship between the negotiated prices of pharmaceuticals and the patients' co-payment[END_REF] for the effect of item-by-item negotiation on this relationship.

For the role of transportation costs, see[START_REF] Ganslandt | Parallel imports of pharmaceutical products in the European Union[END_REF].

In this model, the maximum valuation for the drug is corrected by the co-payment rate to obtain the maximum effective willingness to pay for the drug.