Pierric Mora
email: pierric.mora@univ-eiffel.fr

Mathieu Chekroun

Samuel Raetz

Vincent Tournat

Nonlinear generation of a zero group velocity mode in an elastic plate by non-collinear mixing

Keywords: Lamb waves, Non-collinear interaction, Quadratic non-linearity, Harmonic generation

Zero Group Velocity (ZGV) modes are peculiar guided waves that can exist in elastic plates or cylinders, and have proved to be very sensitive tools in characterizing materials or thickness variations with sub-percent accuracy at space resolutions of about the plate thickness. In this article we show theoretically and experimentally how such a mode can be generated as the sum-frequency interaction of two high amplitude primary waves, and then serve as a local probe of material non-linearity. The solutions to the phase matching condition, i.e. condition for a constructive non-linear effect, are obtained numerically in the mark of classical, quadratic non-linearity. The coupling coefficients that measure the transfer rate of energy as a function of time from primary to secondary modes are derived. Experiments are conducted on an aluminum plate using piezo-electric transducers and a laser interferometer, and explore the interaction for incident symmetric and anti-symmetric fundamental Lamb modes. In an experiment operated without voltage amplifier we demonstrate that the resonant nature of these ZGV modes can be leveraged to accumulate energy from long excitations and produce detectable effects at extraordinarily low input power even in such weakly non-linear material.

Introduction

Nonlinear ultrasonics in non-destructive testing has received considerable interest in the last decades owing to its promise to detect material degradation at much earlier stages than linear techniques do [START_REF] Nagy | Fatigue damage assessment by nonlinear ultrasonic materi-710 als characterization[END_REF].

As one among other non-linear effects that have shown great sensitivity to micro-structural changes, harmonic generation is often the ground of a strategy to interrogate a medium [START_REF] Matlack | Review of second harmonic generation measurement techniques for material state determination in metals[END_REF]. In this context, appealing guided wave based techniques have emerged (3; 4; 5; 6), although their development has been hindered by the complexity of wave propagation in these structures which adds to the other difficulties that any non-linear based method must cope with. Indeed, due to dispersion, the phase matching condition without which no significant magnitudes are attained is in general not fulfilled for collinear interaction, except at some very specific combinations of modes and frequencies that have to be carefully identified (7; 8), and among which only few are at reach in practice given the state of the art in transduction.

In an attempt to broaden the design space of this class of methods, recent works have begun to explore the new possibilities offered by non-collinear mixing (9; 10; 11; 12; 13). Noncollinear mixing is the name given to the phenomenon of harmonic generation when the two primary waves come from different directions. Although well known (14; 15; 16; 17) and with identified advantages applied in non-destructive evaluation in the case of bulk waves (see e.g. (18; 19)), and despite pioneer experimental demonstrations in plates in the 70s (20; 21), this effect has remained out of focus for Lamb waves until the articles cited above. The benefits of non-collinear mixing are in 30 general that it allows an easier separation between system and sample non-linearity. In the case of guided waves, the other benefit is that the solutions to the phase matching condition are tremendously more numerous, allowing most modes to be generated by a well chosen pair of primary waves. We show in this 35 article how a zero group velocity mode can be excited this way. Waves having zero group velocity (ZGV) and non-zero wavenumber can exist in elastic waveguides. They behave like thickness resonances although, due to their finite wavelength (about the plate thickness), they remain strongly localized in 40 space and can serve as local probes of sample properties (e.g. defects [START_REF] Holland | Air-coupled acoustic imaging with zerogroup-velocity lamb modes[END_REF], thickness changes (23; 24; 25; 26), mechanical constants (27; 28; 29; 30), bonding (31; 32; 33)) with a single point measurement. Owing to their high quality factor, these modes are usually relatively easy to generate and can be de-45 tected at great precision with non-contact setups. For instance, Q ≈ 15000 was reported [START_REF] Clorennec | Laser impulse generation and interferometer detection of zero group velocity lamb mode resonance[END_REF] in aluminum alloys, which enabled detecting and mapping (23; 24) relative thickness variations as small as 2 × 10 -4 . Successful applications in nondestructive testing were reported with air-coupled transducers tection that is usually required is found in civil engineering (the "impact-echo" technique [START_REF] Gibson | Lamb wave basis for impact-echo method analysis[END_REF]) where, due to the small size of the sensors regarding the thickness of the walls being probed, contact detection can be meaningful without significantly affecting the modes. Lastly, let us also mention that ZGV modes have led to the design of high-Q micro resonators, either optical [START_REF] Ibanescu | Microcavity confinement based on an anomalous zero group-velocity waveguide mode[END_REF] or mechanical [START_REF] Yantchev | Thin-film zero-groupvelocity lamb wave resonator[END_REF] for electroacoustic components. Insights on the ranges of existence of ZGV modes as a function of Poisson's ratio can be found in Ref. [START_REF] Prada | Local vibration of an elastic plate and zero-group velocity lamb modes[END_REF]. Other aspects have been discussed such as the influence of boundary conditions and layering [START_REF] Kausel | Number and location of zero-group-velocity modes[END_REF], anisotropy [START_REF] Prada | Influence of the anisotropy on zero-group velocity lamb modes[END_REF], or their time decay behavior due to dispersion and material damping [START_REF] Prada | Power law decay of zero group velocity lamb modes[END_REF].

Altogether, these qualities make ZGV modes attractive candidates to achieve new performances in evaluating the nonlinear properties of a medium or a bonded assembly (41; 42; 43). However -except from Ref. [START_REF] Gusev | Self-action of propagating and standing lamb waves in the plates exhibiting hysteretic nonlinearity: Nonlinear zero-group velocity modes[END_REF] which explores theoretically self-action effects in the mark of hysteretic non-linearity -reported works cover only linear properties. One obstacle regarding harmonic generation is that it is in general not possible to find a primary mode that collinearly satisfies the phase matching condition with a ZGV mode as target. The natural way to overcome this difficulty is to resort to non-collinear mixing. This is the starting point of the present article.

Let us give a brief illustration before going into details. Figure 1 shows a setup in which two narrow-banded, high amplitude wave trains are launched in an aluminum plate in such a way that their wavevectors and frequencies sum to a ZGV point of the plate. After an interaction time of half a hundred cycles, the remaining field has a clear component at the expected ZGV frequency.

This article is organized as follows. We first obtain the solutions to the phase matching condition considering the "sum frequency" interaction and the so-called S 1 S 2 ZGV mode as target. This mode is the lowest frequency and most widely known and exploited ZGV mode that is supported by plates in a very wide range of Poisson's ratio, covering most engineering isotropic materials (metals, concrete, plastics, etc). We also derive and discuss the coefficients that quantify the interaction between selected modes. We then describe experimental realizations, the first two of which giving evidence of the phenomenon for different choices in the primary modes. Finally, a third experiment emphasizes one of the unique benefits that a ZGV mode can provide by demonstrating detectable effects obtained at low driving amplitudes.

Theoretical background

General considerations

We recall here a few background results of non-collinear harmonic generation in hyperelastic materials showing classical, quadratic non-linearity. These results can be obtained by considering the equations of motion for finite amplitudes in the perturbative approximation. They were first derived for bulk waves (14; 15; 16; 17) and are qualitatively similar for guided waves (9; 10) as the methodology to establish them extends naturally.

Non-collinear mixing is also possible for other types of nonlinearities, although it involves other angles, admissible symmetry pairings and harmonic orders -see Ref. [START_REF] Gusev | Theory of non-collinear interactions of acoustic waves in an isotropic material with hysteretic quadratic nonlinearity[END_REF] for bulk 110 waves and hysteretic quadratic non-linearity. Contact acoustics non-linearity, another family of non-classical non-linearities, could also be relevant in the future in the case of the non-linear and non-collinear interactions at a crack (13; 46).

Let us consider two primary displacement fields u 1 and u 2

115

propagating in a plate made of such a material. u 1 and u 2 are assumed to be monochromatic and composed of a single Lamb mode each:

u 1 = Re A 1 U 1 (z)e i(ω 1 t-k T 1 x) , (1a)
u 2 = Re A 2 U 2 (z)e i(ω 2 t-k T 2 x) , (1b)
where the coordinate system x = (x, y, z) T is such that the free surfaces of the plate are parallel to the (x, y) plane, and with the 120 convention ω 1 ≥ ω 2 > 0. The wave vectors k i are contained in the (x, y) plane. Both pairs (ω i , k i ) satisfy the dispersion relations of the plate and are associated with mode shapes U i . From the non-linear interaction between u 1 and u 2 in the region where they intersect, small secondary waves are produced whose am-125 plitudes are proportional to the product A 1 A 2 . If furthermore (ω 1 , k 1 ) and (ω 2 , k 2 ) satisfy the phase-matching condition, that is, if their sum or difference

ω 3 = ω 1 ± ω 2 , (2a)
k 3 = k 1 ± k 2 , (2b)
is such that (ω 3 , k 3 ) satisfies the dispersion relations of the plate, then one mode1 , called u 3 , emerges from the secondary 130 field. Its amplitude grows in proportion with the interaction time t:

u 3 = Re A 3 (t)U 3 (z)e i(ω 3 t-k T 3 x) , (3a)
A 3 (t) = A 1 A 2 γ (t) 1,2,3 t, (3b)
with U 3 the mode shape related to (ω 3 , k 3 ), and γ (t) 1,2,3 a coupling constant which depends on the triplet of modes and on the elastic constants of the material -a derivation is given in 135 Appendix A. In Eqs. (2), "+" and "-" signs correspond respectively to the "sum" and "difference" interactions (A 2 must be replaced by its conjugate A * 2 for the "-" case). Equation (3b) is often written as a function of the propagation distance. Here the choice of the time variable is made necessary by the ZGV 140 modes in which we are ultimately interested, which, because of their zero power flux, would otherwise yield infinite terms.

The coupling constants γ (t) 123 are zero when triplets of inadequate symmetries are involved, prohibiting several families of interactions. A complete discussion can be found in (9; 10).

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For our specific case of a target of P-SV symmetric type, the possible combinations are those of two antisymmetric or two symmetric modes, regardless their belonging to the P-SV or SH families.

System under interest

From here on in this theoretical section we consider a plate made of a homogeneous and isotropic material, of thickness h (see Fig. 2-d). The material is modeled using elastic constants characteristic of aluminum (see Tab. 1). For such Poisson's ratio (ν = 0.35), the S 1 S 2 ZGV mode (or merely "ZGV mode" for short) is around (ω ZGV , k ZGV ) ≈ (0.93c s , 0.25) × 2π/h, with c s the shear wave velocity.

Phase matching condition

Here we seek the solutions to Eqs. (2) for the sum interaction when the target is the ZGV mode, i.e. (ω 3 , ||k 3 ||) = (ω ZGV , k ZGV ), and given a phase direction that is arbitrarily set along the y axis, i.e. k 3 /||k 3 || = (0, 1, 0) T .

Methodology. We first numerically obtain (see e.g. [START_REF] Achenbach | Wave Propagation in Elastic Solids[END_REF]) the dispersion relations of the plate ω n (k) and identify the modal branches that we want to be considered as primary fields: e.g. the S 0 branch for the S 0 -S 0 → ZGV case. We vary k 1 on a square grid [-k max , k max ] × [-k max , k max ] whose bounds k max are empirically set large enough to obtain all solutions. At each test point of the grid, k 2 = k 3 -k 1 is deduced from Eq. (2b), and the sum angular frequency

ω + = ω 1 (||k 1 ||) + ω 2 (||k 2 (k 1 )||) is stored.
Finally, a contour finder (49) (the Marching squares algorithm) is called to find the iso-values ω + (k 1 ) -ω 3 = 0 and returns the

Primary modes

Frequency range (in units of 1 2 f ZGV )

Min. angle

S 0 -S 0 63% -137% 127 • A 0 -A 0 73% -127% 155 • SH 0 -SH 0 73% -127% 149 • S 0 -SH 0 S 0 : 92% -150% SH 0 : 50% -108% 139 • A 0 -A 1 A 0 : 16% -70% A 1 : 130% -184% 132 • A 0 -SH 1 A 0 : 41% -81% SH 1 : 119% -159% 144 •
Table 2: Frequency ranges and minimum interaction angles for mode pairs of admissible types giving solutions to the phase-matching condition for the generation of the S 1 S 2 ZGV mode as a sum interaction.

admissible k 1 vectors, from which the corresponding admissible k 2 , angular frequencies ω 1 and ω 2 , and interaction angles arccos (k

T 1 k 2 /||k 1 || ||k 2 ||) can be post-treated. 175
Results. The methodology is applied to all possible primary pairings. The results for the three most canonical cases A 0 -A 0 → ZGV, S 0 -S 0 → ZGV and SH 0 -SH 0 → ZGV are represented in Fig. 2. Solutions are also found to the more exotic cases S 0 -SH 0 → ZGV, A 0 -A 1 → ZGV and A 0 -SH 1 → 180 ZGV and are summarized in Tab. 2. Other combinations are either of wrong symmetry parity (e.g. A 0 -S 0 cannot generate a symmetric mode) or start at frequencies above the target. Figure 2-a represents (thick lines) the portions of the fundamental modal branches where the phase-matching condition 185 can be satisfied by a non-collinear arrangement. Triangles mark the lowest and highest possible frequencies and correspond to a 180 • interaction angle, while circles mark the case f 1 = f 2 where the interaction angle is minimal. Note that the interaction angle in this symmetric case can be easily obtained by geomet-standing, the wavevectors corresponding to the points marked in Fig. 2-a are drawn for the S 0 -S 0 → ZGV case. Finally, Fig. 2-c represents the angle that must be set between k 1 and k 2 , as a function of the driving frequency f 1 .

Closed-form approximate solution. Note that, because the dispersion of the A 0 and S 0 modes is small in the allowed frequency range, relatively good approximate results can be obtained by using the closed-form solution to non-collinear interaction of bulk waves of speed c 1 and c 3 with c 1 ≤ c 3 -this solution is exact for the case of SH 0 -SH 0 → ZGV. The solu-205 tion space of wavevectors is an ellipse with foci 0 and k 3 , with extreme values

k 1 = 1 2 k 3 (1 ± c 3 c 1 ) at ω 1 = 1 2 ω 3 (1 ± c 1 c 3 )
. Using this formula with the phase velocity of A 0 or S 0 taken at half the ZGV frequency predicts bounds of 1 2 f ZGV ± 22% (exact: ±27%) for the A 0 -A 0 → ZGV interaction, and of 1 2 f ZGV ±44% (exact: 210 ±37%) for the S 0 -S 0 → ZGV interaction.

Interaction coefficients

Here we give numerical values of the interaction coefficients γ (t) 1,2,3 appearing in Eqs. (3b) and (A.12).

Normalization of the mode shapes. The coefficients γ (t) 1,2,3 are defined up to a factor that depends on the normalization of the mode shapes U i . We choose to adopt a definition that allows a comparison with the well-known, scalar case of harmonic generation caused by two collinear longitudinal bulk waves. A detailed discussion of this choice is given in Appendix B. In a nutshell, we require the Lamb mode Re A i U i (z)e i(ωt-k T x) to carry the same thickness-average mechanical energy as a plane bulk wave A i cos ω(tx/c l ), leading to 1 h h 0 ||U i ||2 dz = 1 for a homogeneous plate. We set the phase by requiring in-plane components to be pure real, with a non-negative value at z = 0 along the radial and azimuthal directions:

U i,x | z=0 ≥ 0 (P-SV) and U i,y | z=0 > 0 (SH) for k/||k|| = (1, 0, 0) T .
Generalized acoustic non-linear parameter. We introduce the following dimensionless coefficients to characterize the nonlinear coupling within a given mode triplet:

β 1,2,3 = 2A 3 c l A 1 A 2 ω 1 ω 2 t , (4)
with c l the longitudinal wave velocity. Equation ( 4) is meant to be applied to measure β 1,2,3 from a time signal, under a plane wave assumption. When applied to bulk longitudinal waves with ω 1 ω 2 , β 1,2,3 coincides with the acoustic parameter of quadratic non-linearity β defined as a strain-hardening 2 factor σ = ρc 2 l (ε + βε 2 ), where σ is the stress, ε the strain, and β = 3 2 + l+2m λ+2µ . Equation ( 4) can be related to material parameters using Eq. (3b), giving β 1,2,3 = 2γ (t) 1,2,3 c l /ω 1 ω 2 . Furthermore, as γ (t) 1,2,3 depends linearly on the Murnaghan's constants l, m, n, we can write:

β 1,2,3 = β (0) 1,2,3 + ∂β 1,2,3 ∂l l + ∂β 1,2,3 ∂m m + ∂β 1,2,3 ∂n n. (5)
The four coefficients that appear in the right-hand side of Eq. ( 5) depend only on the linear material constants (ρ, λ, µ) and on the triplet of modes. Comparing again to bulk longitudinal 230 waves, one obtains

β (0) = 3/2, ∂ l β = 1/(λ + 2µ), ∂ m β = 2/(λ + 2µ) and ∂ n β = 0.
Conversion factor for measurements of the out-of-plane component. In this work the observable is the surface out-of-plane displacement u z | z=0 . It is therefore convenient to use the alternative normalization (superscript (z) )

u = Re A (z) i U (z) i (z)e i(ωt-k T x) such that U (z) i,z | z=0 = 1 and A (z)
i is the measured amplitude. We call

α 1,2,3 = U 1,z U 2,z U 3,z z=0 (6)
the conversion factor to the former normalization such that Eq. ( 4) expresses as:

β 1,2,3 = α 1,2,3 2A (z) 3 c l A (z) 1 A (z) 2 ω 1 ω 2 t . (7)
Results. The coefficients appearing in Eq. ( 5) are represented in Fig. 3 for the interactions S 0 -S 0 → ZGV, A 0 -A 0 → ZGV and SH 0 -SH 0 → ZGV, in the allowed frequency ranges. Se-

235
lected numerical values are given in Tab. 3, along with the conversion factor (note that α 1,2,3 is pure imaginary). The derivatives with respect to l, m and n are normalized by the P-wave modulus λ + 2µ to facilitate a comparison with β (β = -5.1 to -9.1 with our set of constants).

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Let us explain on an example how to read Fig. 3. Suppose one wants to calculate the theoretical value of β 1,2,3 for the S 0 -S 0 → ZGV interaction, with equal primary frequencies

f 1 = f 2 = 1 2 f ZGV .
Then, one has to read on Fig. 3-a at abscissa 1 the values of the solid (-1.44) and dashed curves (-2.40, -1.61, 245 0.31) -also reported in Tab. 3. These three latter values must then be multiplied respectively by l, m and n, and divided by λ + 2µ. Finally, β 1,2,3 is obtained by summing these four values according to Eq. ( 5). Now, if one wants to calculate β 1,2,3 for another combination of primary frequencies, the procedure is 250 similar but the curves must be read at another abscissa by taking

either f = f 1 or f = f 2 .
As expected, these three interactions exhibit complementary sensitivities to l, m, n. As Murnaghan's constants are usually of the same sign, the SH 0 -SH 0 → ZGV interaction is the one them all from experiments repeated at different driving frequencies and interaction angles. Another noteworthy characteristic of this interaction is that the geometric non-linear contribution β (0) 1,2,3 is quite small at the extreme limits of the frequency range, i.e. for a 180 • interaction angle: this point is the most sensitive choice to weak material non-linearities.

c. b. a. ZGV mode SH 0 -SH 0 S 0 -S 0 A 0 -A 0 x y z k 1 k 2 k ZGV ± 37% ± 27% ± 27% f 1,2 -f ZGV f ZGV 1 2 1 2 d.

Experiments

Three experiments performed on a 1.5 mm thick aluminum plate (alloy Al 5754-H111, lateral dimensions 20 × 30 cm 2 ) are reported in the following sections. Prior to these experiments, the exact value of the S 1 S 2 ZGV resonance in the plate was determined using a classical experimental setup -we measured f ZGV = 1.98 MHz.

Setup. Figure 4 represents the experimental setup that is common to Exps. #1 and #2, and of which Exp. #3 only slightly differs. Two independent tonebursts at frequencies f 1 and f 2 close to 0.99 MHz are sent by a signal generator (Tektronix 280 AWG-3022C) into a high power gated amplifier (RITEC SNAP 5000), and then fed into piezoelectric disks (25 mm diameter, 2 mm thickness) glued on heads. The heads (wedges for Exps. #1 and #3, metallic footprints for Exp. #2, see Figs. [START_REF] Lissenden | Nonlinear ultrasonic guided waves-principles for nondestructive evaluation[END_REF]-a and 7-a) are designed to select a desired Lamb mode and are them-285 selves glued onto the aluminum plate. The plate is mounted on a 2D translation stage (Newport UTS50CC and SMC100CC) that can span 5 cm on each axis at micrometric precision. Waves are detected with a laser interferometer (Tempo 1D, Bossa Nova Technologies; bandwidth of 20 kHz -1 GHz) that measures the 290 absolute out-of-plane displacement on a focused spot that is pointwise regarding the millimetric wavelenghts involved. This device is chosen for its very good sensitivity as well as for allowing non-contact detection, which is essential to detect a ZGV mode without dramatically affecting it. It has however a 295 drawback in that it saturates for peak amplitudes above 5-6 nm, levels that are easily reached with our transducers when driven at amplified voltages. Digitization (PicoScope 5444D) is done on [START_REF] Rollins | Interaction of ultrasonic waves in solid media[END_REF]

β (0) 1,2,3 (λ + 2µ)× ∂β 1,2,3 ∂l ∂β 1,2,3 ∂m ∂β 1,2,3 ∂n α 1,2,3 Typical β 1,2,3 in aluminum S 0 -S 0 f 1 = 1 2 f ZGV k 1 , k 2 = 127 • -1.44 -2
.40 -1.61 0.31 -0.39 i 6.4 to 11.9

f 1 = 1 2 f ZGV ± 37% k 1 , k 2 = 180 • -2.64 -2.80 -3.01 0 -0.36 i 10.0 to 17.8 A 0 -A 0 f 1 = 1 2 f ZGV k 1 , k 2 = 155 • -1.15
0.66 0.10 -1.23 -0.99 i -0.2 to 1.4

f 1 = 1 2 f ZGV ± 27% k 1 , k 2 = 180 • -0.19 0.68 0.73 0 -1.0 i -3.3 to -5.2 SH 0 -SH 0 f 1 = 1 2 f ZGV k 1 , k 2 = 149 •
-0.28 0 -0.43 0.28 0.0 to 0.3

f 1 = 1 2 f ZGV ± 27% k 1 , k 2 = 180 •
-0.36 0 -0.90 0.60 0.1 to 0.9

Table 3: Selected theoretical values of the coefficients from which the non-linear acoustic parameter β 1,2,3 can be calculated, for the three canonical combinations, and of the conversion factor α 1,2,3 for measurements of the out-of-plane displacement (see Eqs. ( 5) and ( 6)). Ranges for β 1,2,3 are min-max values from the sets of third order moduli given in Tab. 1 which represent different alloys.

Generator Amplifier The primary field is generated by two contact transducers fed with high voltage.

Computer

The surface out-of-plane displacement is recorded as a point detection over a 2D grid.

(2.7 mm/µs for the longitudinal wave in Nylon, and 5.0 mm/µs phase velocity for the S 0 Lamb mode at 1 MHz) predicts an optimal angle of incidence of 33 • . According to the predictions of Section 2.3, the wedges are glued on the plate to form an angle of 127 • . The driving frequencies are set with a small difference to facilitate the distinction between input and sample non-linearity: f 1 = 1.09 MHz and f 2 = 0.89 MHz (i.e. ±10%). This small | f 1f 2 |/2 does not affect significantly the interaction angle which is almost constant around | f 1f 2 |/2 = 0 (see Fig.

2-c).

Results and discussion. Results are shown in Figs. 5 and6.

A typical signal recorded at the center of the region where beams #1 and #2 cross is represented in Fig. 5-a. It consists of two parts. During the first tens of µs the interferometer detects the intense, primary S 0 field, and saturates -the inset shows in thinner line the actual amplitude reconstructed from an acquisition at lower driving voltage. For harmonics analysis, this first part is blind. Once the 50 cycles trains are gone, the interferometer detects a much lower field, most of 330 which is linear and composed of unwanted, direct A 0 modes, and of a diffuse A 0 + S 0 field due to edges reflections. Figure 5-b shows a time-frequency analysis of the signal. Different spectral contributions appear after the saturated part. Of those, 2 f 1 , 2 f 2 , 3 f 1 and 3 f 2 are thought to be mainly due to 335 the wedge/plate and wedge/piezo adhesive bonds because they can vary from absent to strong when unmounting/remounting the wedges and repeating the measurement. The component at f 1 + f 2 = 1.98 MHz = f ZGV distinguishes from all others in that, besides being detectable only when both piezo are acti-340 vated, it is much more monochromatic and of almost constant amplitude over long times -remember that 1000 µs represent nearly 2000 periods. This clearly indicates that we are observing the expected phenomenon. Figure 5-c shows a spectrum of the non-saturated part of the signal (the time window is depicted 345 in black in Fig. 5-a) together with spectra of #1 and #2 alone, allowing the previous remark to be appreciated more quantitatively. Then, Figs. 5-d to -g show how the peak amplitude at f 1 + f 2 varies along with several control parameters. One indeed observes a quadratic trend with the driving voltage (Fig. 5-d)

350
and a linear trend with the number of cycles (Fig. 5-e). Figure 5-f gives an idea of how sharp the effect is in matching f 1 + f 2 with f ZGV , although strictly speaking the peak width depends also critically on the number of input cycles. The fact that no harmonics is generated for f 1 + f 2 < f ZGV is not surprising be-355 cause the S 1 -S 2 branch is not propagative -and as such cannot accumulate energy over long excitations, and other interactions S 0 -S 0 → S 0 , SH 0 , A 0 , SH 1 , A 1 are not possible due to wrong symmetry pairings or absence of solution to the phase matching condition. However, as emphasized by the asymmetry of 360 the peak, the side f 1 + f 2 > f ZGV requires another explanation. We think that several adverse features combine to this absence of detection, namely a window effect and two physical reasons to why higher frequencies should be generated with much less efficiency:

• as we are blind during forcing, we only observe whatever remains without traveling (the ZGV mode) or comes back as a smaller, diffuse field,

• the group velocity becomes significant, so the secondary wave soon leaves the interaction area without accumulating much energy,

• there is a strong mismatch in the phase matching condition: because the dispersion relation f (k) is flat at k ZGV , increasing slightly f 1 + f 2 means a large change on ||k 3 ||, and hence on the interaction angle.

Figure 5-g shows a study on the influence of the difference

| f 1 - f 2 |/2.
It can be seen that despite the relatively high level of input non-linearity observed at

| f 1 -f 2 |/2 = 0 (for which 2 f 1 = 2 f 2 = f 1 + f 2 ,
thus comprising second harmonic generation at 2 f 1 and 2 f 2 from each excited beam, and the non-collinear wave 380 mixing effect f 1 + f 2 exciting the ZGV mode), the subtraction step (#1&#2)-#1-#2 (red squares) is effective in recovering at good precision the true amplitude resulting only from the nonlinear and non-collinear wave-mixing effect at f 1 + f 2 and allows to isolate the sample non linearity. We rely on this conclusion for Exp. #2 described in the following section. Finally, (x, y) scans (4 × 4 cm 2 sampled in 64 × 64 points) are presented in Fig. 6 as space and wavenumber analyses. Figure 6-b shows a snapshot of the primary field at early times, when arriving in the interaction region. Figure 6-c shows the 390 wavenumber-frequency content of the primary field at the driving frequencies, reconstructed from lower driving voltages to avoid saturation biases, and time windowed to select only the incident train. Theoretical dispersion relations are superimposed in thin colored circles to facilitate interpretation. Then, Figs. 6-d and -e show the late field (i.e. for t > 120 µs to exclude saturation) filtered at the ZGV frequency. When both piezos are active, the observed field is indeed dominated by the expected ZGV mode with an almost purely space-harmonic content, and is maximal at the location where the interaction occurred. 400 3.2. Experiment #2: A 0 -A 0 → ZGV We report in this section a second experiment that is similar to the previous one except that it explores the A 0 -A 0 → ZGV interaction.

Methodology. Comb type heads consisting of a smooth (top)

405 and a machined (bottom) faces are manufactured in brass disks (see Fig. 7-a). Piezoelectric disks are glued on the smooth faces and operated near a thickness resonance. The machined faces are glued to the plate and transmit an out-of-plane excitation distributed in a pattern that matches a A 0 planar wavefront 410 (2.6 mm at 1 MHz), thus selectively exciting this mode (see Fig. 7-b,c). The heads are oriented to form an angle of 155 • between the preferential directions of the waves being generated, according to the predictions of Section 2.3. The rear surroundings of each transducer are covered with adhesive paste with the twofold purpose to dampen the strong specular rays coming from the neighboring edges, and to reduce the resonant behavior of the heads. Relying on the good promises of the subtraction processing (see Fig. 5-g), the driving frequencies are set equal: f 1 = f 2 = 0.99 MHz.

420

Results. Figure 7 shows (x, y) scans of the wave field (4×4 cm 2 sampled in 80 × 80 points). For details in the interpretation, the reader is referred to the previous section and to the very similar Fig. 6. Here again, the detection of the ZGV mode resulting from the non-collinear interaction is demonstrated.

Measured non-linear acoustic parameters

The theoretical framework presented in Sec. 2.4 and Appendix A is restricted to plane waves. However, the experiments involve beams of finite width. The S 1 S 2 ZGV mode is quite dispersive, and hence prone to diffraction attenuation.

430

When generated by a brief point or line source such as a laser impact, diffraction dominates over material damping at early times, resulting in a fast t -1/2 decay [START_REF] Prada | Power law decay of zero group velocity lamb modes[END_REF]. Here the context is somewhat different in that the interaction zone acts as a distributed source with a moving phase, but we do notice a depen-435 dence on the time window used to extract the ZGV amplitude. This dependence can be well observed in Fig. 8-f on the part labeled free decay -see Sec. 3.4.

There are two known ways to cope with diffraction and damping. The first and best way is to fit the observed space-440 time variations with a propagation model. It is well documented for collinear generation (2), however, methodological developments are still needed for non-collinear mixing. The second way is to use only the slope of the variations, taken as close to the source as possible. This is the approach retained here. Go-445 ing back to Fig. 8-f, and also to Fig. 5-e, variations appear well linear over time segments of at least 50 µs, as the plane wave model predicts it. In a collinear setup this is a good indication that attenuation biases are limited on these scales.

Methodology. The ZGV amplitude is measured by time-450 windowing over 50 µs just after the interaction has ceased. For instance, the black window drawn in Fig. 5-a is replaced with a window going from 80 µs to 130 µs. The signals are then 3D-Fourier-transformed and |A 3 | is read at the ZGV frequency as the max value in the (k x , k y ) space. The amplitudes |A 1 | 455 and |A 2 | are obtained similarly by time-windowing to select the interaction (e.g. from 30 µs to 70 µs in Fig. 5-a), 3D-Fourier-transforming, and reading the max values at the driving frequencies f 1 and f 2 . Finally, Eq. ( 7) is applied by taking c l = 6270 m/s, t = 50 µs as excitation duration; β th A 0 ,A 0 ,ZGV = -0.2 to 1.4 (see Tab. 3), the agreement is very good for the A 0 -A 0 → ZGV experiment while it is bad for the S 0 -S 0 → ZGV one. We currently lack a satisfactory explanation, although finite-beam effects could have been naively discarded despite the justification given above, and although this comparison is only indicative because the model material constants (l, m, n) represent other aluminum alloys.

f 2 f 1 2 f 2 2 f 1 f ZGV 3 f 2 3 f 1 Out-of-plane displacement (nm)
3.4. Experiment #3: generation at low amplitudes Methodology. We go back to the S 0 -S 0 → ZGV interaction and use the same setup as for Exp. #1, except that the amplifier is now removed (see Fig. 8-a). The outputs of the signal generator are directly connected to the piezoelectric disks (with 50 Ω loads) glued on the wedges. The driving frequencies f 1 and f 2 are set with a small difference f 2f 1 = 10 kHz (i.e. ±0.5%) around half the ZGV frequency. The generator is able to deliver up to 20 Vpp into 50 Ω on each channel. Compared to Exp. #1, the input voltages are lower by a factor of roughly 10. Non-linearly generated amplitudes are then expected to be reduced by a factor of ≈ 100. To compensate these lower levels, we increase the number of cycles from a few tens to 2000 (i.e. ≈ ×40). This clearly does not balance the drop entirely, but proves to generate amplitudes high enough to be unambiguously detected with a few tens averages.

Results and discussion. The results presented in Fig. 8 were recorded with 50 averages. Because of the lower amplitudes, the interferometer no longer saturates during excitation. This pattern progressively appears until t = 2000 µs, that is, when the excitation stops, and then vanishes. Figure 8-c shows a typical signal recorded at the center of the observation window, with a peak amplitude about 0.8 nm when both transducers are emitting. The beatings in the first 2000 µs are due to the small 505 difference between the driving frequencies. The spectrum of this entire signal is shown in Figure 8-d: as emphasized by the inset around 2 MHz no peaks are measured at 2 f 1 or 2 f 2 , meaning that the non-linearity of the receiving system is now indeed too small to be observed. The peak at f 1 + f 2 (1.8 10 -3 nm peak amplitude) has, in turn, a SNR of about 20 dB. The same signal is then shown in Fig. 8-e in a time-frequency representation, with a focus on the ZGV frequency -the magnitude at the driving frequencies is also plotted in gray lines. A tendency of linear increase with time can indeed be observed during forcing, followed by a slow, free decay. Finally, Fig. 8-f shows the same time-frequency analysis applied to the data transformed into the wavenumber domain (x, y) → (k x , k y ). The black line represents the magnitude of the point ( f ZGV , k ZGV ) as a function of time, while gray lines show the values at neighboring points (the solid/dashed curves stand for minus/plus a given percentage in the wavenumber k). The time dependence can this way be observed more accurately -the Fourier transform on 64 × 64 points can improve the SNR by up to +36 dB for a pure harmonic pattern. One can now observe a trend towards saturation 525 in the growth. It is also noteworthy that the decay rate results faster for greater (dashed lines) than for smaller (solid lines) wavenumbers. An explication could be that greater wavenumbers are indeed more dispersive than smaller ones, although it is not clear to us whether viscous attenuation also contributes 530 quantitatively to this difference. Before closing this section, we should not disregard the fact that the very attractive prospects offered by this low power setup come at the cost of technical difficulties that are common to resonance experiments. Indeed, the price to pay is that the driv-535 ing frequencies must be adjusted with extreme care, because a target error of one part over 2000 (the number of cycles) now significantly degrades the phase matching condition. Such required accuracy starts to be sensitive to temperature effects: this was not an issue for scans running over a couple hours, but we 540 did observe variations with overnight fluctuations (about ±2 • in the room). Another limitation could be attenuation. Here the large number of cycles was still well below the quality factor of the ZGV mode, which often exceeds 10 4 in metal plates, so that even more cycles would have yielded higher amplitudes.

545

But in more dissipative media, compensating low amplitudes with long excitations is obviously a more limited strategy.

Conclusion

The non-linear generation of the S 1 S 2 ZGV Lamb mode at the intersection of two high amplitude beams of lower fre-550 quency has been analyzed theoretically and experimentally in an aluminum plate. The methodology followed can be transposed to other materials, to the difference interaction, and to other targeted modes. A new derivation of the coupling coefficients was given that is suited to an observation of the interac-555 tion as a function of time, and is thereby adapted to zero group velocity modes. Theoretical values of these coefficients were discussed for the most canonical cases. The experimental realizations involved the fundamental symmetric and antisymmetric Lamb modes, which are relatively easy to excite selectively 560 and efficiently. A complete space-time observation of the phenomenon was achieved in an experiment operated at low power, from the emergence of the mode during forcing to its free decay. The main features of the interaction were observed, such as non-trivial interaction angle, quadratic dependence with the 565 driving voltage, or linear dependence with the number of excitation cycles.

In the context of harmonic generation, the specificity of zero group velocity modes is that energy can be accumulated by simply launching longer primary waves into an intersection area 570 whose diameter can be as small as a few plate thicknesses. In contrast, in configurations that produce propagating modes, accumulating energy requires to increase the interaction area, which dilutes information and soon faces trivial limitations related to the geometrical constraints of the actual system. Taking 575 advantage of this appealing property, we showed that targeting such a mode enables to generate detectable non-linear effects t im e -fr e q .

S 0 A 0 b. c. #1 #2 #1 & #2 π rad/mm #1 #2 (#1 & #2) -#1 -#2 π rad/mm
#1 and #2 turned on #1 and #2 turned off using input powers up to two orders of magnitude lower than when targeting other modes. Zero group velocity modes appear as promising tools based on which novel non-destructive characterization methods can be developed to assess locally the non-linear properties of a plate. Among possible applications, we believe that they could be relevant to evaluate the bonding quality of an adhesive joint. They could also be a route to achieve harmonic generation with a fully non-contact system using a time-modulated laser excitation.

pendix B. The driving sources are composed of two parts, labeled with superscript ± to indicate respectively sum and difference interactions: f vol = Re 1 2 A 1 A 2 F ± vol e i[(ω 1 ±ω 2 )t-(k 1 ±k 2 ) T x] (A 2 must be replaced by its conjugate A * 2 for the "-" case), and similarly for f sur f . We recall (see e.g. (53; 54)) that when Fourier-Appendix B. On the choice of normalization of the modeshapes Amplitude. In scalar models of harmonic generation, the amplitudes of the components of the displacement field u are generally defined as u = i A i cos ω i (tx/c). If we stretch this representation into a transverse space dimension z (call it depth) over a distance h (call it thickness), we can introduce a trivial modeshape U i (z) = 1 without changing the definition of A i . The time-average and z-integrated mechanical energy carried by such harmonic wave with unit amplitude A i = 1 is

E i = 1 2 ρω 2 i h 0 |U i (z)| 2 dz = 1 2 ρω 2 i h
, where ρ is the density of the medium. Now, in the case of an elastic plate of density ρ(z), with the purpose of dealing with coupling constants that have a similar meaning, it seems natural to require the modeshapes U i (z) to be normalized such that their mechanical energy extends the scalar case: Phase. Defining the phase of U i is also necessary, although the choice is somewhat more arbitrary. In an isotropic plate, the in-plane and out-of-plane components of the modeshapes have a phase difference of ±π/2, and either of them can be chosen z = 0 is convenient in that they can be simultaneously null only for k = 0 -this is not the case for U i,z which is null for SH modes and cancels at ω/k = c l for P-SV modes. We then decide to require U i,x | z=0 and U i,y | z=0 to be non-negative in the radial and azimuthal directions, i.e. U i,x | z=0 ≥ 0 and U i,y | z=0 ≥ 0 for k/||k|| = (1 0 0) T .

E i = 1 2 ω 2
An example. Applied to the SH modes of an isotropic and homogeneous plate in the k/||k|| = (1 0 0) T phase direction, this definition gives U 0 = (0 1 0) T and U i = √ 2 cos(i z π/h) (0 1 0) T for i > 0. 700 Scaling of the coupling constants γ (t) 1,2,3 and β 1,2,3 with the plate thickness. Normalized as described upper, U i does not depend on h. Then, one can see from Eq. (A.9) that E i /ω i scales as ω i h, that is, does not depend on h either. Furthermore, by changing to dimensionless coordinates x → x = x/h and ∇ → 1 h ∇ x, we

Figure 1 :

 1 Figure 1: (a) Schematic of the non-collinear interaction. (b) Photograph of the experimental setup. (c) Snapshot of the primary field arriving in the interaction region. (d) Fourier-transform of the field remaining after the interaction at the ZGV frequency. (e) Spectrum for piezos #1 and #2 activated alone or simultaneously.

Figure 2 :

 2 Figure 2: Space of pairs (( f 1 , k 1 ); ( f 2 , k 2 )) solutions to the phase matching condition for S 0 -S 0 → ZGV, SH 0 -SH 0 → ZGV and A 0 -A 0 → ZGV interactions. (a) Admissible frequency ranges (thick lines) superimposed on dispersion curves (thin lines). (b) Admissible wavevectors. Examples of pairs (k 1 , k 2 ) S 0 are drawn and the reflection across the point 1 2 k ZGV relating them is represented by a thin dotted line. (c) Interaction angle as a function of the driving frequency of one of the two modes.

Figure 3 :

 3 Figure 3: Normalized non-linear coefficients for the three canonical combinations. The non-linear acoustic parameter β 1,2,3 can be obtained from a set of curves by applying Eq. (5).

  bits at a 15 MHz sampling rate and signals are averaged 40 (for 2D scans) to 80 times (at single points) to improve the 300 signal to noise ratio (SNR). The piezoelectric disks are operated near a thickness resonance and can hence be considered as narrow-band transducers. Experiments #1 and #2 operate at voltages about 200 Vpp (i.e. 100 W/channel) on 50 cycles, while Exp. #3 does not use the amplifier and operates at 20 Vpp 305 (i.e. 1 W/channel) on 2000 cycles.

3. 1 .

 1 Experiment #1: S 0 -S 0 → ZGV We report in this section a first experiment in which the ZGV mode is generated using the S 0 -S 0 → ZGV interaction. Methodology. Wedges made of Nylon (type PA6) are manu-310 factured to selectively excite the S 0 mode. Snell-Descartes' law Primary modes Configuration

Figure 4 :

 4 Figure 4: Schematic of the experimental setup common to Exps. #1 and #2.The primary field is generated by two contact transducers fed with high voltage. The surface out-of-plane displacement is recorded as a point detection over a 2D grid.

  425

ω 1 =

 1 ω 2 = 460 2π × 0.99 MHz and |α 1,2,3 | = 0.99 for the A 0 -A 0 → ZGV case; ω 1 = 2π × 0.89 MHz, ω 2 = 2π × 1.09 MHz and |α 1,2,3 | = 0.39 for the S 0 -S 0 → ZGV case. Results. We obtained |A 1 | = 8.9 nm, |A 2 | = 5.4 nm and |A 3 | = 13.4 × 10 -3 nm for the S 0 -S 0 → ZGV experiment, giving 465 |β exp S 0 ,S 0 ,ZGV | = 0.7, and |A 1 | = 11.1 nm, |A 2 | = 10.9 nm and |A 3 | = 19.8 × 10 -3 nm for the A 0 -A 0 → ZGV experiment, giving |β exp A 0 ,A 0 ,ZGV | = 1.1. Compared to the plausible values estimated in the theoretical analysis β th S 0 ,S 0 ,ZGV = 6.4 to 11.9 and

Figure 5 :Figure 6 :

 56 Figure 5: Typical signals at the center of the crossing region for the S 0 -S 0 → ZGV interaction. (a) Time signal. (b) Time-frequency analysis. (c) Spectrum of the non-saturated part (t > 120 µs) for piezos #1 and #2 activated alone or simultaneously. (d, e, f, g) Peak amplitude at the sum frequency when varying (d) the driving voltage of both piezos, (e) the number of cycles of the tonebursts, (f) the sum frequency f 1 + f 2 while keeping constant ( f 1f 2 )/2 = 0.1 MHz, (g) the difference | f 1f 2 |/2 while keeping constant f 1 + f 2 = 1.98 MHz.

Figure 7 :

 7 Figure 7: 2D view of the A 0 -A 0 → ZGV interaction. (a) Photograph of the setup showing the brass heads with A 0 -like footprints, with red arrows representing outgoing waves. (b) Snapshot of the primary field arriving in the interaction region. (c) Wavenumber content of the primary field at the driving frequencies. (d) Real part of the field at the ZGV frequency (windowed to t > 120 µs to exclude saturation, see Fig. 5-a). (e) Wavenumber content of (d).

Figure 8 :

 8 Figure 8: Low power generation of the S 0 -S 0 → ZGV. (a) Schematic of the experimental setup (Exp. #3). (b) Snapshots of the field (real part) at the ZGV frequency processed with a time-frequency filter. (c) Typical signal (out-of-plane displacement). (d) Spectrum of (c) (entire signal). (e) Time-frequency analysis of (c) at the ZGV frequency (blue) and at the driving frequencies (grays). (f) Time-frequency analysis in the wavenumber domain at the ZGV frequency: at k = k ZGV (black), k = k ZGV (1pct) (solid grays) and k = k ZGV (1 + pct) (dashed grays).

00

  ρdz is the average density. Note that if the medium is homogeneous ρ(z) = ρ, Eq. (B.1) simplifies to re-||U i (z)|| 2 dz = 1.

Table 1 :

 1 Material constants representative of several aluminum alloys. The third order moduli were determined by R. T. Smith et al. (47) from acoustoelastic measurements.

	Density	Lamé's		Murnaghan's
	ρ	λ	µ	l	m	n
	2.7 g/cm 3 58 24 -311 -401 -408 GPa
				-224 -237 -276
				-202 -305 -300
				-388 -358 -320
				-337 -395 -436

Note that there could in principle be more than one mode satisfying the phase-matching condition simultaneously. Such exceptional cases are not discussed here.

ric considerations -it equals 2 arccos(c 1 /c 3 ), with c 1 and c 3 the phase velocities of primary and secondary waves. The two ticks (blue line) correspond to f 1 ,f 2 = 1 2 f ZGV ± 20%.Figure 2-b represents the complete set of admissible primary wavevectors -it

Note that several definitions coexist in the literature and often include a minus sign (i.e. measuring softening for positive strains) and/or a ×2 factor.

with weakest transfer rate, because it depends on m and n with similar magnitudes but opposite signs and has zero component along l. On the other hand, the S 0 -S 0 → ZGV interaction sums contributions of l and m while it depends weakly on n, and is therefore the strongest one. The A 0 -A 0 → ZGV interaction, 260 although somewhat weaker, is in turn interesting in that its relative sensitivities to the three constants change significantly with the frequency, such that, in principle, it seems possible to infer

Acknowledgments

P. Mora acknowledges funding from the EUR-IAGS postdoctoral program. We thank H. Méziere, J. Marudai and E.

Egon for manufacturing the transducers. Thanks to J. Blondeau and S. Letourneur for their support in setting up the experiments, and to R. Hodé for running preliminary characterization measurements.

Codes availability

Python programs for finding solutions to the phase-matching condition and for obtaining theoretical values of the non-linear parameter are available upon request.

is actually the raw output of the contour finder. To help undermakes this early stage fully observable and enables a more complete picture.

Figure 8-b shows a few snapshots of the field (real part) during and after excitation with a time-frequency processing centered at the ZGV frequency. One can clearly see that a wave

Appendix A. Derivation of the interaction coefficients

The interaction coefficients appearing in Eq. (3b) are usually derived in the literature in the frequency domain, by expanding in a modal series the small wave field resulting from the non-linear interaction. Here, we take a slightly different approach in that the modal series is expressed in the wavenumber domain to allow for an analytical treatment of the time variable. Indeed, this time-domain resolution is necessary because Zero Group Velocity modes yield singularities when viewed in the frequency domain -their forced response grows unbounded with time as energy accumulates without traveling.

We start by briefly recalling the equations of motion for finite amplitudes in a waveguide as they can be established in the perturbative approximation. More details can be found in e.g. (50; 51; 52) for the collinear case, and in (9; 6) for a formalism more suited to the non-collinear case. Let x = (x, y, z) T refer to the so called "natural" coordinates, i.e. attached to the system at rest. The system under interest is a traction-free plate that is unbounded in the (x, y) plane and comprised between z = 0 and z = h (see Fig. 2-d). Two wave trains of finite amplitude, labeled #1 and #2 further on, are generated within the plate. We call u(x) the total displacement field produced by these waves and their interaction. Perturbation theory starts by expanding u as a sum of terms of different order:

where u 1 and u 2 (order 1) denote the primary fields, and u 11 , u 22 and u 12 (order 2) denote fields resulting from non-linear interactions: u 11 refers to the self-interaction of wave #1, u 22 to the self-interaction of wave #2, and u 12 to the mutual interaction of waves #1 and #2. Note that u 12 is the same as u 3 in Sec.

2.1 when the phase matching condition is satisfied and a single 620 mode dominates. We shall focus only on this latter term further on. For u 12 , the balance of forces and boundary conditions read, in absence of external load:

In Eqs. (A.2) n z is the unit vector along the z axis, oriented upwards, and S L (u 12 ) = C : ∇u 12 is the linear part of 625 the first Piola-Kirchhoff stress tensor referring to u 12 , with C the 4th-rank stiffness tensor. The source terms appearing in the right hand side are

2) the non-linear part of the first Piola-Kirchhoff stress tensor of the total field u that 630 gathers products between ∇u 1 and ∇u 2 (the last argument "2" emphasizes the truncation order).

We assume that the material is hyperelastic, with a classical, quadratic non-linear component. In the case of an isotropic medium, the elastic energy density function can be written as:

In Eq. (A.

3) E = 1 2 [∇u + ∇u T + ∇u T ∇u] is the Lagrangian strain tensor, λ and µ are Lamé's constants, and A, B and C are Landau and Lifshitz's third order elastic constants -we recall the relations with Murnaghan's constants: l = B+C, m = 1 2 A+B and n = A. In this context, it can be established [START_REF] Chillara | Interaction of guided wave modes in isotropic weakly nonlinear elastic plates: Higher harmonic generation[END_REF] that:

in which I is the identity tensor of order 2 and rank 3. Equation (A.4) and the results below are also valid for arbitrary variations of the elastic constants with depth. The case of an anisotropic medium requires to modify Eqs. (A.3) and (A.4) accordingly, but the further derivation still stands.

645

We now take the primary fields to be monochromatic and composed of a single mode each:

in which we assume ω 1 ≥ ω 2 > 0. In Eqs. (A.5) the amplitudes A 1 and A 2 are defined depending on the normalization chosen for the modeshapes U 1 and U 2 . This choice is postponed to Aptransformed into the wavenumber domain (x, y) → (k x , k y ), the causal Green's tensor of Eqs. (A.2) reads:

where z is the depth location of the impulsive source, H is the hermitian (or conjugate-transpose) operator, H is the Heaviside unitstep function, and U n (k, z) is the modeshape of the n th Lamb mode, associated with the eigen-angular frequency ω n (k) ≥ 0. Expressed over the Lamb modes of the plate, the resulting field is then of the form:

. The superscripts ± and the substitution A 2 → A * 2 for the "-" case are from now on considered as implicit. The modal participation factors divided by the primary 660 amplitudes are:

As we are not interested in rigid body motions ω n = 0, the normalizing term in Eq. (A.6a) can be more conveniently written as the mean modal energy by multiplying by the halved squared angular frequency:

The temporal dependence is obtained by convolving the driving function e i(ω 1 ±ω 2 )t (assumed to be zero for t < 0) with the modal propagator g n (t). Combining terms, we arrive at:

The sine function can be written as a sum of two exponentials, a form which exhibits two separate contributions: a first one associated with -e -iω n (t-τ) /2i which oscillates rapidly and cannot grow significantly, and a second one associated with e iω n (t-τ) /2i which can grow without limit if the resonance condition ω n = ω 1 ± ω 2 is matched. We shall now neglect the former and approximate A n (t) by keeping only the latter:

where we defined:

Finally, if we focus only on the resonance condition and slightly change the definition of A n (t) by pulling out the time-harmonic term e i(ω 1 ±ω 2 )t (i.e. we adopt the definition of Eq. (3a)), then Eq. (A.11) becomes Eq. (3b).

665

The coefficients γ (t) 1,2,n express how fast in time two given primary modes transfer energy into a third mode, and are defined up to an arbitrary factor that reflects the normalization chosen for the modes. They are analogous to what is often called "mixing power" in the literature, although referring to a transfer rate 670 measured in space, along the propagation axis. The link between both forms is discussed below.

Correspondence with a representation in space. Let us compare the coupling coefficients as they appear in time and space representations. We now write the secondary field as:

When the phase-matching condition is satisfied, it can be established (9; 6) that: .15) and q = (k 1 ±k 2 )/||k 1 ±k 2 || is the phase direction. In Eq. (A.15) P nn is the mean power flux of mode n along q:

U n the velocity of the mode. Using the identity h 0 P nn dz = c e,n E n q e,n , where c e,n is the energy (or group) velocity, q e,n is a unit vector in the direction of the energy flux, and E n is the time-average mechanical energy defined in Eq. (A.9), the ratio between both rates of transfer of energy can be obtained:

1,2,n = P nn E n = c e,n q T q e,n . (A.17 to be pure real -this property applies to any layering of materials belonging to a class of anisotropy that includes orthotropy [START_REF] Kausel | Thin-layer method: Formulation in the time domain[END_REF]. We then chose to require the in-plane components U i,x and U i,y to be pure-real. As for the ± sign that remains to be set, imposing a condition on the U i,x and U i,y components at see from Eqs. (A.4) and (A.8) that F n is proportional to 1/h 2 . We then see from Eqs. (A.12) and (4) that γ (t) 1,2,3 scales as 1/h 2 while β 1,2,3 is a constant that only depends on the mode triplet.


Francesca Ronchini

Romain Serizel

Nicolas Turpault

Samuele Cornell

Detection and Classification of Acoustic Scenes and Events 2021

Keywords: Sound event detection, synthetic soundscapes, open-source datasets, deep learning

Detection and Classification Acoustic Scene and Events Challenge 2021 Task 4 uses a heterogeneous dataset that includes both recorded and synthetic soundscapes. Until recently only target sound events were considered when synthesizing the soundscapes. However, recorded soundscapes often contain a substantial amount of non-target events that may affect the performance. In this paper, we focus on the impact of these non-target events in the synthetic soundscapes. Firstly, we investigate to what extent using non-target events alternatively during the training or validation phase (or none of them) helps the system to correctly detect target events. Secondly, we analyze to what extend adjusting the signal-to-noise ratio between target and non-target events at training improves the sound event detection performance. The results show that using both target and non-target events for only one of the phases (validation or training) helps the system to properly detect sound events, outperforming the baseline (which uses non-target events in both phases). The paper also reports the results of a preliminary study on evaluating the system on clips that contain only non-target events. This opens questions for future work on non-target subset and acoustic similarity between target and non-target events which might confuse the system.

INTRODUCTION

The main goal of ambient sound and scene analysis is to automatically extract information from sounds that surround us and analyze them for different purposes and applications. Between the different area of interest, ambient sound analysis have a considerable impact on applications such as noise monitoring in smart cities [START_REF] Bello | Sonyc: A system for the monitoring, analysis and mitigation of urban noise pollution[END_REF][START_REF] Bello | Sound analysis in smart cities[END_REF], domestic applications such as smart homes and home security solutions [START_REF] Serizel | Large-scale weakly labeled semi-supervised sound event detection in domestic environments[END_REF][START_REF] Debes | Monitoring activities of daily living in smart homes: Understanding human behavior[END_REF], health monitoring systems [START_REF] Zigel | A method for automatic fall detection of elderly people using floor vibrations and sound-proof of concept on human mimicking doll falls[END_REF], multimedia information retrieval [START_REF] Jin | Event-based video retrieval using audio[END_REF] and bioacoustics domain [START_REF] Morfi | Deep perceptual embeddings for unlabelled animal sound events[END_REF]. Sound Event Detection (SED) aims to identify the onset and offset of the sound events present in a soundscape and to correctly classify them, labeling the events according to the target sound classes that they belong to. Nowadays, deep learning is the main method used to approach the problem. However, one of the main limitations of deep learning models is the requirement of large amounts of labeled training data to reach good performance. The process of labeling data is time-consuming and bias-prone mainly due to human errors and disagreement given the subjectivity in the perception of some sound event onsets and offsets [START_REF] Turpault | Sound event detection in domestic environments with weakly labeled data and soundscape synthesis[END_REF]. To overcome these limitations, recent works are investigating alternatives to train deep neural networks with a small amount of labeled data together with a bigger set of unlabeled data [START_REF] Serizel | Large-scale weakly labeled semi-supervised sound event detection in domestic environments[END_REF][START_REF] Serizel | Sound event detection from partially annotated data: Trends and challenges[END_REF][START_REF] Shah | A closer look at weak label learning for audio events[END_REF][START_REF] Turpault | Sound event detection in domestic environments with weakly labeled data and soundscape synthesis[END_REF][START_REF] Mcfee | Adaptive pooling operators for weakly labeled sound event detection[END_REF]. Among them, Detection and Classification Acoustic Scenes and Events Challenge (DCASE) 2021 Task 4 uses an heterogeneous dataset that includes both recorded and synthetic soundscapes [START_REF] Turpault | Sound event detection in domestic environments with weakly labeled data and soundscape synthesis[END_REF]. This latter soundscapes provide a cheap way to obtain strongly labeled data. Until recently, synthesized soundscapes were generated considering only target sound events. However, recorded soundscapes also contain a considerable amount of non-target events that might influence the performance of the system.

The purpose of this paper is to focus on the impact on the system's performance when non-target events are included in the synthetic soundscapes of the training dataset. The study has been mainly divided into three stages. Firstly, we investigate to what extent using non-target events alternatively during training or validation helps the system to correctly detect the target sound events. Mainly motivated from the results of the first experiment, in the second part of the study, we focus on understanding to what extend adjusting the target to non-target signal-to-noise ratio (TNTSNR) at training improves the sound event detection performance. Results regarding a preliminary study on the evaluation of the system using clips containing only non-target events are also reported, opening questions for future studies on possible acoustic similarity between target and non-target sound events which might confuse the SED system. 1 .

PROBLEM DEFINITION AND DATASET GENERATION

Problem definition

The primary goal of the DCASE 2021 Challenge Task 4 is the development of a semi-supervised system for SED, exploiting an heterogeneous and unbalanced training dataset. The goal of the system is to correctly classify the sound event classes and to localize the different target sound events present in an audio clip in terms of timing. Each audio recording can contain more than one event. Some of those could also be overlapped. The use of a larger amount of unlabeled recorded clips is motivated by the limitations related to annotating a SED dataset (human-error-prone and timeconsuming). Alternatively, synthesized soundscapes are an easy way to have strongly annotated data. In fact, the user can easily generate the soundscapes starting from isolated sound events. On the other hand, in most of the recorded soundscapes the target sound classes are almost never present alone. For this reason, one of the main novelties of the DCASE 2021 Challenge Task 4 is the introduction of non-target isolated events in the synthetic soundscapes 2 . This paper explores the impact of the non-target sound events on the baseline system performance, with the final goal of understanding and highlighting how to correctly exploit them to generate realistic soundscapes.

Dataset generation

The dataset used in this paper is the DESED dataset3 [START_REF] Serizel | Sound event detection in synthetic domestic environments[END_REF][START_REF] Turpault | Sound event detection in domestic environments with weakly labeled data and soundscape synthesis[END_REF], which is the same provided for the DCASE 2021 Challenge Task 4. It is composed of 10 seconds length audio clips either recorded in a domestic environment or synthesized to reproduce such an environment 4 . The synthetic part of the dataset is generated with Scaper [START_REF] Salamon | Scaper: A library for soundscape synthesis and augmentation[END_REF], a Python library for soundscape synthesis and augmentation, which allows to control audio parameters. The recorded soundscapes are taken from AudioSet [START_REF] Gemmeke | Audio set: An ontology and human-labeled dataset for audio events[END_REF]. The foreground events (both target and non-target) are obtained from the Freesound Dataset (FSD50k) [START_REF] Fonseca | Fsd50k: an open dataset of human-labeled sound events[END_REF], while the background sounds are obtained from the SINS dataset (activity class "other") [START_REF] Dekkers | The sins database for detection of daily activities in a home environment using an acoustic sensor network[END_REF] and TUT scenes 2016 development dataset [START_REF] Mesaros | Tut database for acoustic scene classification and sound event detection[END_REF]. In particular, non-target events are the intersection of FUSS dataset [START_REF] Wisdom | What's all the fuss about free universal sound separation data?[END_REF] and FSD50k dataset in order to have compatibilty with the source separation baseline system.

In this article, we modify only the synthetic subset of the dataset. Starting from the synthetic part of the DESED dataset, we generated different versions of it in order to investigate how non-target events impact the system performance and to what extent their relationship with the target events affects the training phase of the system. The following subsections describe the different subsets used for the experiments, which have been generated using Scaper.

Synthetic training set

The synthetic training set is the same set of data released for the DCASE 2021 Challenge Task 4. It includes 10000 audio clips where both target and non-target sound events could be present in each clip. The distribution of the sound events among the files have been determined considering the co-occurrences between the different sound events. The co-occurrences have been calculate considering the strong annotations released for the AudioSet dataset [START_REF] Hershey | The benefit of temporally-strong labels in audio event classification[END_REF] 5 . A second version of this dataset has been generated where only target events are present. The datasets will be hereafter referred as synth tg ntg (used by the official baseline system) and synth tg for the synthetic subset including target and non-target events and the synthetic subset including only target events, respectively.

Synthetic validation set

The synthetic validation set is the same as the synthetic validation dataset supplied for the DCASE 2021 Challenge Task 4. It includes 3000 audio clips including target and non-target events, which distribution has been defined calculating the co-occurrences between sound events. We generated a second version of the dataset containing only target events. The datasets will be referred to as synth tg ntg val (used by the baseline system) and synth tg val (only target sound events).

Synthetic evaluation set

The synthetic 2021 evaluation set is composed by 1000 audio clips. In the context of the challenge, this subset is used for analysis purposes. We will refer to it as synth tg ntg eval. It contains target and non-target events distributed between the different audio clips according to the pre-calculated co-occurrences. Two different versions of the synth tg ntg eval set have been generated, synth tg eval (only target sound events) and synth ntg eval (only non-target sound events).

Varying TNTSNR training and validation set

With the aim of studying what would be the impact of varying the TNTSNR on the system performance, different versions of synth tg ntg and synth tg ntg val have been generated. In particular, for each of them, three versions have been created. The SNR of the non-target events have been decreased by 5 dB, 10 dB and 15 dB compared to their original value. The original SNR of the sound events is randomly selected between 6 dB and 30 dB, so the more we decrease the SNR, the less the sound will be audible, with some of the events that will not be audible at all. These subsets will be subsequently referred to as synth 5dB, synth 10dB, synth 15dB for the training subsets and synth 5dB val, synth 10dB val, synth 15dB val for the validation subsets.

Public evaluation set

The public evaluation set is composed of recorded audio clips extracted from Youtube videos that are under creative common licenses. This is part of the evaluation dataset released for the evaluation phase of the DCASE 2021 Challenge Task 4 and considered for ranking. The set will be referred to as public.

EXPERIMENTS TASK SETUP

In order to compare the results with the official baseline, we used the same SED mean-teacher system released for this year challenge. More information regarding the system can be found at Turpault et al. [START_REF] Turpault | Sound event detection in domestic environments with weakly labeled data and soundscape synthesis[END_REF] and on the official webpage of the DCASE Challenge Task 4. All the different models have been trained 5 times. This paper reports the average of the scores and the confidence intervals related to those. Only for the baseline model we do no report the confidence intervals because we have considered the results using the checkpoint made available for it 6 . The metrics considered for the study are the two polyphonic sound detection score (PSDS) [START_REF] Bilen | A framework for the robust evaluation of sound event detection[END_REF] scenarios defined for the DCASE 2021 Challenge Task 4, since these are the official metrics used in the challenge. The scope of these experiments is twofold: understand the impact of non-target events on the system performance and investigate to what extend the TNTSNR helps the network to correctly predict the sound events in both matched and mismatched conditions. In order to do so, we divided the experiment into three stages. The first part of the study is focused on understanding the influence of training the system with non-target events. This experiment is described and discussed in Section 4. Section 5 reports the results and the relative discussion of the second part of the experiment where we investigate if a mismatch in terms of TNTSNR between datasets could have an impact on the output of the system. Section 6 reports preliminary results of the last stage of the experiment, regarding the evaluation of the system on the synth ntg eval dataset, formed by only non-target sound events, in order to investigate if some classes could get acoustically confused at training, having a negative impact on the performance. The last stage has been motivated by the results of the second part of the experiment.

Non

USING TARGET/NON-TARGET AT TRAINING

In the first experiment we concentrate on training the system with different combinations of the training dataset. Table 1 reports the results of the experiment evaluating the system on the public set. We check-marked the columns NT Train or/and NT Val according to if the non-target sound events are present or not in the synthetic sounscapes. From the results it is possible to observe that using nontarget sound events during training and validation improves the performance by a large margin with relaxed segmentation constraints (PSDS2) but only marginally with strict segmentation constraints (PSDS1). In this latter case what matters the most is the use of nontarget sound events during the validation. A possible explanation is that synthetic soundscapes with non-target sound events are actually too difficult and confuse the systems when used during the training but they still help reducing the mismatch with recorded soundscapes during model selection (validation).

Table 2 reports the results considering the synth tg ntg eval and synth tg eval evaluation sets. In all cases the best performance is obtained in matched training/evaluation conditions. The performance obtained on synth tg ntg eval are lower than the performance obtained on synth tg eval even in matched conditions. Not surprisingly, this confirm that including non-target sound events makes the SED task more difficult. Interestingly, as opposed to the previous experiment, the most important here is to have matched conditions during training and to a lesser extent during validation. In order to verify the low impact of non-target sound events at training when evaluating on recorded soundscapes, in the next experiment we investigate a possible mismatch in terms in TNTSNR. Evaluating with public set.

Non-target

VARYING TNTSNR AT TRAINING

The second part of the study focuses on understanding the impact of varying the TNTSNR at training and validation aiming at finding a TNTSNR condition that could match better the recorded soundscapes. For each TNTSNR, we use similar combinations as the ones used in Section 4, replacing the set without non-target sound events by a set with adjusted TNTSNR. For example, considering the 5 dB case, the combinations considered would be:

• training using the synth tg ntg set and validating with synth 5dB val; • training with synth 5dB and validating with synth tg ntg val;

• training and validating with synth 5dB and synth 5dB val.

The fourth combination is the official DCASE Task 4 baseline. Repeating the experiment with all the varying TNTSNR, allow us to analyse to what extend the loudness of the non-target events helps matching the evaluation conditions on recorded clips. Table 3, 4 and 5 report the performance on the public set when using a TNTSNR of 5 dB, 10 dB and 15 dB, respectively. When the TNTSNR is 5 dB or 10 dB, the performance changes only marginally between configurations. Increasing the TNTSNR to 15 dB leads to a behaviour more similar to the one obtained in Table 1. The best performance is obtained when training with TNTSNR is 15 dB and validating on synth tg ntg val. This could be explained by the fact TNTSNR 15 dB is a condition closer to that of the recorded soundscapes and the fact that it allows for selecting models that will be more robust towards non-target events at test time.

In the last experiment, we investigate the impact of varying the TNTSNR during validation phase, while using the synt tg for training. Results are reported on Table 6, where it is possible to observe that all of them overcome the baseline or are comparable with it, with the best performance obtained for 10 dB TNTSNR. These experiments could indicate that recorded soundscapes in public in general have a TNTSNR of about 10 -15 dB which should be confirmed by complementary experiments.

EVALUATING ON NON-TARGET EVENTS ONLY

Based on the previous experiments, TNTSNR could be one reason of mismatch between the synthetic soundscapes and the recorded soundscapes. But this could not explain all the performance differences observed here. In particular why in general having lower TNTSNR during training is decreasing the performance regardless of the validation. One possibility is that the system gets acoustically confused by a possible similarity in sound between events when soundscapes tend to be less dominated by target events. So we evaluated the system using the synth ntg eval, where only nontarget events are considered, to see for which classes the system would output false positives. We evaluated the system on the public set; considering the systems trained for the first experiment (see Table 1). Results show that some sound events are detected more than others. For some classes as Speech, this could be explained by the original event distribution (indicated in the first column) but for some other classes as Dishes there is a discrepancy between the original distribution and the amount of false alarms. Interestingly the amount of false alarms is decreased sensibly for most of the classes when including non-target sound events during training.

CONCLUSIONS AND FUTURE WORK

This paper analyzes the impact of including non-target sound events in the synthetic soundscapes of the training dataset for SED systems trained on heterogeneous dataset. In particular, the experiments are divided into three stages: in the first part, we explore to what extend using non-target sound events at training has an impact on the system's performance, secondly we investigate the impact of varying TNTSNR and we conclude the study by analyzing a possible confusion of the SED model in case of false alarms triggered by non-target sound events.

From the results reported on this paper, we can conclude that using non-target sound events can help the SED system to better detect the target sound events, but it is not clear to what extend and what would be the best way to generate the soundscapes. Results show that the final SED performance could depend on mismatches between synthetic and recorded soundscapes, part of which could be due to the TNTSNR but not only. Results on the last experiment show that using non-target events at training decreases the amount of false alarms at test but from this experiment it is not possible to conclude on the impact of non-target sound events on the confusion between the target sound events. This is a first track for future investigation on the topic. Additionally, the impact of the non-target sound events at training on the ability of the system to better segment the target sound events in noisy soundscapes would have to be investigated. A final open question is the impact of the per class distribution of the sound events (both target and non-target) and their co-occurrence distribution on the SED performance.

Table 1 :

 1 Evaluation results for the public set, considering the different combinations of using target and non-target sound events at training and validation.

	-target Train Val	PSDS1	PSDS2
		33.81 (0.36) 52.62 (0.19)
		35.92 (0.49) 54.85 (0.29)
		34.90 (0.82) 53.07 (1.22)
		36.40	58.00

Table 2 :

 2 Evaluation results for the synth tg ntg eval set and synth tg eval set, considering the different combination of using target and non-target sound events at training and validation.

	Train	Val	Eval set		PSDS1	PSDS2
			synth tg ntg eval	23.22 (1.33)	36.44 (2.62)
			synth tg ntg eval	20.08 (0.39)	31.33 (1.29)
			synth tg ntg eval	20.13 (0.35)	30.99 (1.07)
			synth tg ntg eval	25.14	40.12
			synth tg eval	42.82 (2.42)	58.26 (2.08)
			synth tg eval	46.92 (1.02)	62.79 (0.55)
			synth tg eval	47.73 (0.33)	62.54 (1.00)
			synth tg eval	43.22	61.09
		Non-target Train Val	PSDS1	PSDS2
		Original	5 dB	35.57 (0.28) 56.68 (1.77)
		5 dB	Original 36.25 (1.26) 57.53 (1.06)
		5 dB	5 dB	35.46 (0.46) 58.09 (0.74)
		Original Original	36.40	58.00

Table 3 :

 3 Evaluation results for the second part of the experiment, varying TNTSNR by 5 dB (synth 5dB and synth 5dB val). Evaluating with public set.

	Non-target Train Val	PSDS1	PSDS2
	Original	10 db	36.23 (1.11) 57.82 (1.37)
	10 db	Original 36.42 (0.77) 58.94 (0.89)
	10 db	10 db	36.20 (1.14) 57.92 (1.04)
	Original Original	36.40	58.00

Table 4 :

 4 Evaluation results for the second part of the experiment, varying TNTSNR by 10 dB (synth 10dB and synth 10dB val). Evaluating with public set.

	Non-target Train Val	PSDS1	PSDS2
	Original	15 dB	36.08 (1.13) 57.78 (1.33)
	15 dB	Original 37.37 (0.70) 58.64 (1.34)
	15 dB	15 dB	36.10 (0.50) 57.36 (0.89)
	Original Original	36.40	58.00

Table 5 :

 5 Evaluation results for the second part of the experiment, varying TNTSNR by 15 dB (synth 15dB and synth 15dB val).

Table 6 :

 6 Evaluation results of the SED system, training with synth tg, validating with varying TNTNSNR set and evaluating with public set.

	Validation set	PSDS1	PSDS2
	synth 5dB val	38.68 (1.07) 60.57 (0.78)
	synth 10dB val 39.07 (0.75) 60.75 (0.80)
	synth 15dB val 37.95 (0.53) 59.99 (1.14)

Table 7 :

 7 Preliminary evaluation results by classes, evaluating the system with synth ntg eval. Nsys (A): training with synth tg, validating with synth tg val; Nsys (B): training with synth tg ntg, validating with synth tg val; Nsys (C): training with synth tg, validating with synth tg ntg val; Base: baseline using target and nontarget events for training and validation.

	Classes	Nref	A	B	Nsys	C	Base
	Dog	197	135	126	146	79
	Vacuum cleaner	127	31	42		44	47
	Alarm bell	191	47	50		52	59
	Running water	116	34	41		61	30
	Dishes	405	1478 395 1270	305
	Blender	100	63	32		55	19
	Frying	156	70	41		60	33
	Speech	1686	206	181	180	201
	Cat	141	99	103	98	73
	Electric shaver	103	21	18		18	7

http://dcase.community/challenge2021

https://project.inria.fr/desed/

For a detailed description of the DESED dataset and how it is generated the reader is referred to the original DESED article[START_REF] Turpault | Sound event detection in domestic environments with weakly labeled data and soundscape synthesis[END_REF] and DCASE 2021 task 4 webpage: http://dcase.community/challenge2021

The co-occurrences distribution and the code used to compute them will be distributed.

https://zenodo.org/record/4639817

ACKNOWLEDGEMENTS

We would like to thank all the other organizers of DCASE 2021 Challenge Task 4. In particular, we thank Eduardo Fonseca and Daniel P. W. Ellis for their help with the strong labels of the Au-dioSet dataset used to compute the events co-occurrences, and Justin Salamon and Prem Seetharaman for their help with Scaper.

This work was made with the support of the French National Research Agency, in the framework of the project LEAUDS Learning to understand audio scenes (ANR-18-CE23-0020), the project CPS4EU Cyber Physical Systems for Europe (Grant Agreement number: 826276) and the French region Grand-Est. Experiments presented in this paper were carried out using the Grid5000 testbed, supported by a scientific interest group hosted by Inria and including CNRS, RENATER and several Universities as well as other organizations (see https://www.grid5000).


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The CHIMERE 21 project 27 septembre 2021

The context

CNRM-CM5 ALADIN53

Couple 2

IPSL-CM5A IPSL-INERIS-WRF

Couple 3

CNRM-CM5 RCA4

Couple 4

IPSL-CM5A RCA4

Couple 5

MPI-ESM RCA4

•RCP 4.5 Temperatures Precipitation

Near future +0,8 to +1,5 °C +1 to +35 %

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  Figure from Lemaitre-Basset (2020)

  Figure fromLemaitre-Basset (2020)

  5: moderate increase (only for the far future) RCP 8.5: important increase, especially downstream Need to answer the following questions: « What is the probability that an indicator change of X % under climate change? » et « At which temporal horizon could this trend emerge? » Pourcentage of all simulations indicating a trend of at least 10% of low flows(VCN3) , especially for RCP 8.5 and far future -Heterogeneous precipitation evolution: increase for winter, large uncertainty for summer -Projections of CHIMERE 21 warmer and wetter than previous works Hydrology: -Large uncertainties for summer, possible decrease of streamflows for upstream -Increase during winter, especially downstream -Climate brings most of uncertainties on Meuse streamflows Hydroclimate evolutions study gave thoughts for food regarding adaptation strategies -Evolutions of streamflows question strong stakes: -Increase of streamflow downstream, which is already identified as a territory with high risk of floods -Possible decrease of low flows upstream Explore 2: a France-wide project aiming at evaluating impacts of climate change using Drias 2020 (started recently) LIFE Eau&Climat (https://www.gesteau.fr/life-eau-climat): a project aiming at helping local water stakeholders to evaluate climate change impacts, to take them into account for planning and to undertake adaptation strategies

The CHIMERE 21 team