Antoine Villefer
email: antoinevillefer@edf.fr

Michel Benoit

Damien Violeau

Christopher Luneau

Hubert Branger

Influence of following, regular and irregular long waves on wind-wave growth with fetch: an experimental study

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The infinite diversity of waves in the ocean makes the understanding of the complex sea-states dynamics rather difficult. Numerous experimental studies both in the field and in laboratories have been carried out over the last decades to improve the knowledge on the particular case of bimodal sea-states or more precisely on wind-wave and swell combination. Many field measurements [START_REF] Donelan | The Air-Sea Momentum Flux in Conditions of Wind Sea and Swell[END_REF][START_REF] Hwang | Observations of Wind Wave Development in Mixed Seas and Unsteady Wind Forcing[END_REF]Vincent et al. 2019) show evidence of a modified wind-wave generation in the presence of swell. However, field experiments can hardly be generalized due to the specific wind and atmospheric conditions varying from one experiment to another. In wind-wave tank facilities, wind-wave and swell combination can be studied in controlled conditions using a wave-maker (e.g. a mechanically actuated paddle) for representing swell and a closed-loop air circulation to produce a quality airflow creating short wind-waves above the long mechanically-generated waves (paddle-waves). Thereby, both wave systems are sufficiently separated in frequency to observe a bimodal sea-state. A series of experiments was conducted in a well-controlled laboratory environment in Marseilles (France) with two objectives. The first one was to identify the mechanisms at stake in the interaction between the two wave systems. Assessing the extent at which those mechanisms can be transposed to wind-wave growth in the ocean was the second objective.

The wind-wave generation over a train of long monochromatic paddle-waves propagating along the wind direction is a well-documented experimental configuration [START_REF] Mitsuyasu | Interactions between Water Waves and Wind (1)[END_REF][START_REF] Phillips | Wave breaking in the presence of wind drift and swell[END_REF][START_REF] Donelan | The effect of swell on the growth of wind waves[END_REF]. In these conditions, a drastic reduction on the energy density of the wind-wave component is unanimously observed. The intensity of this reduction increases with the long-wave steepness ak (where a is the wave amplitude and k the wave number). [START_REF] Donelan | The effect of swell on the growth of wind waves[END_REF] found a windsea elevation variance reduced by a factor of about 2.5 in presence of long waves. A similar experiment by [START_REF] Mitsuyasu | The Effect of Swell on the Growth of Wind Waves[END_REF] with long waves propagating opposite to the wind direction revealed wind-wave height intensification. More recently, [START_REF] Benetazzo | Analysis of the effect of fish oil on wind waves and implications for air-water interaction studies[END_REF] and [START_REF] Bailey | Predicting Wind Wave Suppression on Irregular Long Waves[END_REF] carried out experiments (in an open-air basin without a closed-loop airflow in the case of Bailey et al.) on wind over irregular (JONSWAP-type) paddle-waves. These latter experiments also showed a reduced wind-wave growth.

Variance density spectrum E( f , θ, x, t), with f and θ are respectively the wave frequency and direction, is a powerful manner to describe wind-wave growth. For infinite depth, the evolution of E( f , θ, x, t) is governed by the spectral energy balance involving source/sink terms S( f , θ, x, t):

dE dt = S in + S ds + S nl (1)
The left hand side is the kinematic part of the equation while the right hand side includes three source/sink terms corresponding to wind energy input S in , dissipation through white-capping S ds and nonlinear four-wave interactions S nl .

Experiments of wind over monochromatic mechanically-generated waves propagating in the same direction have received a lot of attention to provide a physical explanation to wind-wave reduction and to consider at what extent this effect could appear at natural scale. First, the hypothesis of an enhanced wave breaking of the windsea, represented by S ds , due to the addition of wind drift and long-wave orbital velocities, was suggested by [START_REF] Phillips | Wave breaking in the presence of wind drift and swell[END_REF]. This suggestion was later shaded by Wright (1976) arguing that enhanced wave breaking could not account for the magnitude of wind-wave reduction in the case of high wind velocities. According to [START_REF] Masson | On the Nonlinear Coupling between Swell and Wind Waves[END_REF], nonlinear four-wave interactions (i.e. S nl ) is also involved in the physical evolution when the ratio of long waves over short-waves frequencies is greater than 0.6. This ratio discriminates the interaction between wind-waves and swell in the ocean (ratio lower than 0.6) and in most laboratory experiments (ratio greater than 0.6). [START_REF] Chen | Effects of Long Waves on Wind-Generated Waves[END_REF] developed a model based on a sheltering effect of the long waves absorbing momentum from the wind, thus reducing the available momentum for wind-wave generation (i.e. S in ). Their model delivered results in accordance with laboratory experiments but closely depending on the growth rate coefficient β formulated by [START_REF] Miles | On the generation of surface waves by shear flows[END_REF] as S in = βE, which is known to be highly variable, especially at laboratory scale. The growth rate coefficient itself depends on the long-wave age, C/u , with C the long-wave (phase) celerity, u = (τ/ρ) 1/2 the friction velocity, τ the total turbulent shear stress and ρ the air density. Chen and Belcher's results suggest that the wind-wave reduction observed in laboratory environment with young waves (C/u << 20)

would probably not occur with older ocean swell for which C/u > 20. More directly, Chen and

Belcher's model predicts that the sheltering effect occurs when the long waves are, to a good extent, slower than the wind. Lately, [START_REF] Donelan | Modulation of short wind waves by long waves[END_REF] attributed the wind-wave reduction to both the modified airflow (i.e. S in ) and the orbital velocity gradients in the presence of the long waves.

In our experiment, short waves were generated by the wind in a large closed-loop wind/wave tank facility, and long waves were generated with a wave-maker, with both cases of monochromatic waves and irregular (JONSWAP-type) waves. All the measurements were carried out on a fetchlimited wave field. The fetch limited case is a generic case of spatial wave evolution consisting in a steady wind forcing over a statistically stationary wave-field. The statistical wave-field can be related to the wave-spectrum notion.

In order to compare our experimental results with other laboratory experiments and field observations, an appropriate scaling has to be applied. Following [START_REF] Kitaigorodskii | Application of the theory of similarity to the analysis of wind generated wave motion as a stochastic process[END_REF], the parameters describing wind-wave growth are the total variance of the free-surface elevation m 0 = η 2 and the peak frequency f p as a function of the fetch X. Those variables are made dimensionless using the friction velocity u and the acceleration due to gravity g:

f * p = f p u g (2) m * 0 = m 0 g 2 u 4
(3)

X * = Xg u 2 (4)
Although the wind speed 10 m above the mean free surface U 10 = U(z = 10 m) is usually employed as a scaling wind speed, the friction velocity is chosen here since it does not depend on height and it better represents the momentum transfer at the air/water interface [START_REF] Janssen | The Interaction of Ocean Waves and Wind[END_REF].

The facility and the data processing methods are introduced in Section 2. The wind profiles determined in presence and in absence of paddle-waves, used as a scaling tool for the subsequent analyses, are presented in section 3. The growth of wind-waves is measured and processed in absence of long waves in section 4, then in presence of monochromatic waves in section 5 and finally with irregular waves in section 6. A comparison of these different cases and their relation with the prototype scale are discussed in section 7. Section 8 gives a summary of results and their application, followed by a further discussion on future studies on the topic.

Experimental Facility and Data Processing

a. The Facility

The IRPHÉ/Pythéas wind-wave tank, sketched in Figure 1, consists of a closed-loop tunnel for air circulation with a 1.5-m-high air cavity above a basin with a test section 40-m-long and 2.60-m-wide. The water depth h, set to 0.80 m, was kept constant during all the experiments. For a complete description of the facility, one can refer to [START_REF] Coantic | Wind-water tunnel simulation of small-scale ocean-atmosphere interactions[END_REF]. The wind is generated by an air blower in the upper part of the tunnel and guided through a settling chamber with a honeycomb and a converging nozzle ; this system produces a uniform airflow at the entrance of the water basin. At the upwind end of the basin, an immersed paddle wave-maker (piston-type) can be used for generating either monochromatic or irregular waves propagating along the wind direction. At the downwind end, a permeable absorbing beach with a 6-degrees-slope was installed to minimize wave reflection. The basin side walls are punctuated with windows to observe the water surface. A larger 5-m-long clear glass is situated at the 25-m-fetch to perform measurements that require a sight on the water surface.

For these experiments, eight resistance-type and four capacitance-type wave gauges were distributed along the basin to measure the instantaneous free surface elevation η (Figure 1). Two anemometers were situated in front of the clear glass at a 25-m-fetch (Figure 1). Wind reference velocity U ref was measured at approximately 0.5 m above the water surface with a Gill 2D sonic

anemometer. An additional hot film anemometer from E+E Electronik supported by a vertical telescopic pole allowed to measure the vertical profile of the mean (i.e. time averaged) horizontal velocity U(z). Friction velocity and aerodynamic roughness z 0 are determined by fitting the logarithmic profile (see Section 3) to the vertical time-averaged wind profile in neutral atmospheric conditions for all the experiments (air and water were roughly at the same temperature). The friction velocity is considered constant along the test section because of the slightly divergence of the air section along the wind-wave tank allowing a zero pressure longitudinal gradient.

Apart from the air blower controlled with a graduated potentiometer, the control of the wavemaker and all the gauges output signals were gathered on a computer using a LabView program.

Wind velocity could be generated between 2 and 14 m.s -1 (U ref ) by regulating the potentiometer.

In order to generate waves, the required temporal free surface elevation was convoluted with the wave-maker transfer function, and then transferred as a voltage signal from the computer to the piston wave-maker, converting the input signal to actual waves. Both regular (Stokes-type) and irregular (JONSWAP-type) wave trains were generated. The JONSWAP spectrum is given by:

E( f ) = α p H s 2 f 4 p f 5 exp - 5 4 f p f 4 γ exp - ( f -fp ) 2 2σ 2 fp 2 (5)
where, using the JONSWAP's recommendations [START_REF] Hasselmann | Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP)[END_REF], σ = 0.07 for f < f p and σ = 0.09 for f > f p , γ = 3.3 is the peak enhancement factor and α p = 0.2 is the Phillips constant.

The significant wave height H s and the peak frequency f p are the parameters to adjust in order to obtain the desired sea-state.

To generate the time series of irregular waves, inverse FFT (Fast Fourier Transform) transforms of the complex amplitudes from the JONSWAP spectra (using random phases) were used over the frequency range [0.4-4.0] Hz. This wide range of frequency, which is the recommended mechanical range for the wavemaker, was necessary in order to obtain a smooth transition between the JONSWAP peak and the high frequency waves. However, the wave-maker was able to accurately generate a sea-state corresponding to a JONSWAP-type spectrum over the frequency range [0.4-2.0] Hz. Above 2 Hz the energy distribution could slightly deviate from the f -5 desired tail.

Acquisition of wave gauges signals were launched from the computer for a 20 minutes duration at the sampling rate 256 Hz. Before every acquisition, it was verified that the waves in the wave tank were in stationary fetch-limited conditions by waiting a sufficient long time before acquiring data. During the experiments with wind, the instantaneous horizontal wind velocity was measured using the hot film anemometer from the lower point above the water surface (i.e. slightly above the highest wave) at about ten vertically distributed points over a range of 20 cm and a duration of 240 s to obtain the vertical profile of the mean horizontal velocity. In order to be able to differentiate swell from wind-waves, the mechanically-generated wave frequency (respectively the peak frequency for irregular waves) was chosen equal to 0.6 Hz. This choice accounts for the wind-waves lower peak frequency reaching 1 Hz at maximum fetch and maximum wind velocity.

Using the dispersion relation for the long waves, the non-dimensional depth k h = 1.33 corresponds to intermediate water depth. Thus, bottom friction slightly dissipates the paddle-waves as they propagate along the tank. Bottom friction and additionnal dissipation mechanisms, such as sidewalls friction, that occur in laboratory experiments have been quantified using Dorn's formula [START_REF] Dorn | Boundary dissipation of oscillatory waves[END_REF]. At the downwind end of the tank, more than 95 % of the long wave energy is conserved. Wind-waves, however, are short enough (k h > 3) to be considered as propagating in deep water and unaffected by bottom friction. Wind-wave dissipation by sidewalls remains weak since their wavelenghts is much smaller than the channel width [START_REF] Shemer | On Evolution of Young Wind Waves in Time and Space[END_REF]. Overall, the dissipation due to bottom and sidewalls friction is relatively low and is not further considered in the present study.

Table 1 gathers the characteristics of all the tests presented in this paper.

b. Spectral Separation Methods

A spectral approach was chosen to characterize the interaction between wave systems. Spectra were calculated using the Welch method by partitioning the surface elevation time records into shorter records of 16384 points with a 50 % overlap and a Hanning windowing. Depending on the sea-states, the resulting spectra were unimodal (wind-waves only, irregular swell only) or multimodal (monochromatic swell with harmonics, swell and wind-waves combination). Separating wave systems in multi-modal sea-states is necessary to have access to their individual characteristics (e.g. significant wave height, peak period, spectral width) in order to compare windsea growth with and without swell for instance. In the literature, this separation is commonly performed using a transition frequency that divides the wave spectrum in two parts: swell and windsea. That method works while the peaks of both wave systems are sufficiently separated. In some cases, as it will be shown in this paper, windsea and swell components are overlapping, making difficult the separation with the former method. Two methods to separate the mechanically-generated wave part from the wind-wave part of multi-modal spectra are introduced in this section. The first method deals with the multi-modal sea-states with a monochromatic swell. The second method refers to multi-modal sea-states with a JONSWAP-type swell. Both methods are based on curve-fitting using a nonlinear least square method, and are described below.

The generation of monochromatic long waves with the wave-maker leads to the generation of additional super-harmonic waves in intermediate depth conditions. These harmonics occupy the same frequency range as the windsea making difficult the distinction between the proper energies and peak frequencies of these overlapping systems. The first stage of the method consists in the removal of the spectral bands containing the frequencies of the monochromatic wave harmonics into the initially measured multi-modal spectrum. The removed number of points is arbitrary and depends upon the width of the corresponding harmonic peaks. Thus, although the wind-wave peak then becomes discontinuous at this stage, its shape is conserved. The next stage is to fit the discontinuous wind-wave spectral peak with a JONSWAP-type spectrum initialized with the triplet ( f p , H s , γ = 3.3). The initial value of the significant wave height H s is equal to the integral of the spectrum and the initial peak frequency f p is equal to the maximum value of the spectrum.

The fitted JONSWAP-type spectrum enables to access the energy and the peak frequency of the wind-wave part of the spectrum. Then, the harmonic-only part of the spectrum can be obtained by subtracting the wind-wave part from the initially measured multi-modal spectrum.

A spectrum combining irregular-swell and wind-waves can be decomposed using the following procedure inspired from [START_REF] Mackay | Modelling and Description of Omnidirectional Wave Spectra[END_REF]. First, a unimodal JONSWAP-type spectrum E + (see ( 5)) is fitted on the bimodal measured spectrum E m using the triplet ( f + p , H + s , γ + = 3.3) as initial values: H + s is determined using the integral of E m and f + p is equal to the frequency of the higher peak. For this first fit, the peak frequency is a constant parameter in order to fit the higher peak only. The resulting fitted triplet is

( f + p , H + s , γ + ) (the prime denoting the fitted parameters). A second triplet ( f - p , H - s , γ -= 3.
3) is used as initial value for the second (i.e. the lower) peak with H - s from the integral of E -= E m -E + and f - p as the peak frequency of E -. The sum of two JONSWAP-type spectra (i.e. bimodal) E -+ E + is fitted on the bimodal measured spectrum E m using ( f - p , H - s , γ -) and ( f + p , H + s , γ + ) as initial values. Thus, the measured spectrum is decomposed in two JONSWAP-type spectra permitting to determine the energy and peak frequency of each wave system.

An illustration of the spectral separation of a measured bimodal spectrum is given in Figure 2. The figure shows that the spectra resulting from the spectral separation are an accurate fit for both long-wave and wind-wave peaks but slightly deviate from the measured spectrum in the high frequency range. It is also interesting to note the slight reduction of wind-wave energy at the peak of the wind-wave spectrum whether the spectrum is separated or included in the bimodal spectrum. This last observation proves the necessity of a proper separation of each wave system in a multi-modal measured spectrum so that they can be compared to the wave systems observed individually in unimodal measured spectra as it will be performed in the following.

Estimation of the Vertical Wind Profile

Among the numerous methods for estimating wind fluxes, the profile method was used for this experiment. Thus, the airflow above water waves is considered analogous to that above a stationary rough surface. Assuming a constant flux layer and using the [START_REF] Monin | Basic laws of turbulent mixing in the surface layer of the atmosphere[END_REF] turbulence similarity theory, the vertical velocity wind profile is assumed logarithmic and may be written as:

U(z) = u κ ln z z 0 ( 6
)
where κ = 0.41 is the von Karman constant and z 0 is the aerodynamic roughness. The measured vertical evolution of the mean horizontal wind velocity is fitted with the logarithmic turbulent velocity profile (6). In that way, the friction velocity and the aerodynamic roughness are found for cases with and without paddle-waves.

Figure 3 displays the vertical profiles of the averaged horizontal wind velocity with and without paddle-waves, for monochromatic and irregular waves. The measured wind velocities have an overall tendency to increase when paddle-waves are present. Regarding the fitted logarithmic profiles, paddle-waves have a trend to increase the friction velocity, thus inducing an increase in the total shear stress τ at the air/water interface. Those results are in disagreement with [START_REF] Chen | Effects of Long Waves on Wind-Generated Waves[END_REF] who designed a model for the reduction of wind-waves due to long monochromatic waves based on the absence of variation of the total stress with long-wave steepness.

However, the large error-bars corresponding to the standard deviation in the calculation of the mean horizontal velocity depicted in Figure 3 show the uncertainties associated with the logarithmic fit. Thus, there is a high degree of uncertainty in the estimation of u . Regarding the measurements, two evaluations of the friction velocity in the same experimental conditions could lead to a 20 % variation in the results. Additionally, water sprays at high wind speed (e.g. U ref = 14 m.s -1 ) could invalidate the measurements by reaching the hot film and cooling down its temperature, thus measuring abnormally high velocities. Overall, the logarithmic profile is prone to be criticized for a wind-over-long-wave use.

Due to the uncertainties previously mentioned, the measured data will be scaled using a unique friction velocity for each wind speed. This friction velocity is determined using the profile method on pure windsea cases (see Figure 3). Thus, the measurement uncertainties added by the presence of long waves are avoided, but possible effects of such waves on the wind friction velocity will be ignored. [START_REF] Shemer | On Evolution of Young Wind Waves in Time and Space[END_REF] observed that the friction velocity does not depend notably on the fetch in a similar fetch-limited case. Using this latter observation in our study, the friction velocity is assumed to be constant along the fetch with an impact of the surface drift velocity considered as negligible [START_REF] Shemer | On Evolution of Young Wind Waves in Time and Space[END_REF].

Wind-Wave Growth

This section aims to verify the similarities on fetch-limited wind-wave growth between our results, anterior laboratory experiments and ocean measurements. In this section, experimental data with wind only (i.e. without mechanically-generated waves) are analyzed.

Laboratory experiments are, on one hand, necessary to facilitate the understanding of wind-wave growth through controlled conditions. On the other hand, limitations are present due the limited size and the peculiarities from a wave tank to another. To quantify these limitations, fetch-limited cases of wind-wave growth under four increasing wind velocities have been carried out in the wind-wave tank. In Figure 4, the evolution of the measured dimensionless wind-wave energy and peak frequency are plotted and compared to two selected empirical laws. The first one is [START_REF] Mitsuyasu | The growth of duration-limited wind waves[END_REF] law exclusively based on laboratory experiments:

m * 0 = 4.49 × 10 -5 (X * ) 1.282 and f * p = 1.19(X * ) -0.357 (7)
The second one is [START_REF] Kahma | Reconciling discrepancies in the observed growth of wind-generated waves[END_REF] law for stable stratification (adapted to friction velocity scaling from [START_REF] Komen | Dynamics and Modelling of Ocean Waves[END_REF]) obtained from a collection of laboratory and ocean data:

m * 0 = 7.3 × 10 -4 (X * ) 0.85 and f * p = 0.477(X * ) -0.26 (8)
Figure 4 shows the differences in wind-wave growth depending on the wind speed in our experiments. At high wind speed, U ref ≥ 10 m.s -1 , energy and peak frequency exhibit a linear evolution with fetch in log-log coordinates confirming a power law behavior. Additionally, this power law appears to be in accordance with [START_REF] Mitsuyasu | The growth of duration-limited wind waves[END_REF]'s law.

At lower wind speed, water surface tension affects the evolution of wind-wave energy, especially because most of the waves are short, hence in the capillary-gravity range, where surface tension strongly modifies the wave dynamics. The case U ref = 6 m.s -1 is characterized by a two-stage evolution that clearly appears in the energy variation: the first stage, X * < 10 3 , shows a strong slope which is attenuated, in a second stage, at larger fetches. Surface tension impact thus decreases gradually as the waves grow. The second stage, X * > 10 3 , is linear and parallel to Mitsuyasu and Rikiishi's power law indicating that the surface tension effect on waves is lower. Wind-waves generated by a 4 m.s -1 wind deviate markedly from both the experimental curves for higher wind speeds and empirical laws: surface tension seems to have a significant effect in that case.

Moreover, both dimensionless energy and peak frequency magnitudes agree reasonably well with Kahma and Calkoen's law. The differences with this latter law lie in the variations of the variables with fetch. Since Mitsuyasu and Rikiishi's law is a good fit for our experimental data variations, absolute value of the power coefficients from Kahma and Calkoen's law (0.85 for the energy, 0.26 for the peak frequency in (8)) appear somewhat too low to accurately describe laboratory data (corresponding exponents are 1.282 for the energy, 0.357 for the peak frequency in ( 7)). Recently, Shemer (2019) also fitted power laws to a set of small-scale experiments, and found exponents in rather good agreement with the values obtained here, namely 1.012 for the energy and 0.27 for the peak frequency.

Another relevant comparison with empirical law can be done using Toba's 3/2 law (Toba 1997):

H * s = 0.062( f * p ) 3/2 (9)
or in term of energy formulation

m * 0 = 2.4 × 10 -4 ( f * p ) 3 (10)
This law is well adapted to ocean measurements. Contrary to power law previously mentioned the fetch does not appear in this relation. This is a benefit since the fetch is difficult to define in the real ocean because of the wind changing speed and direction. Note that the previous two sets of laws ( 7) and ( 8) disagree with (10) in the sense that the ratio of their power coefficients is slightly above the value of 3 as claimed by Toba.

Figure 5 shows the agreement between our experimental results and Toba's law. In line with

Figure 4, once wind-waves are long enough to be free from surface tension effect (i.e. below a certain peak frequency or above a certain wind speed) Toba's law is well adapted to describe our experimental data set.

This section proved that laboratory experiments are, to a good extent, appropriate to describe short fetch evolution of a growing windsea in the ocean. From this observation, long mechanicallygenerated waves representing swell are now added to the system in order to observe their effect on the growing windsea.

Effect of Monochromatic Paddle-Waves on Wind-Wave Growth

To assess the effect of (following) swell on wind-wave growth we repeated the same wind forcing conditions of the previous section with now long monochromatic paddle-waves generated at X = 0 m with the wave-maker. Two values of steepness of these long waves were considered: ak = 2.7 % and ak = 5.6 %, with a = H/2 = √ 2m 0 the wave amplitude and k the wavenumber, corresponding to wave frequency f = 0.6 Hz. In Figure 6, those wave conditions are added on the diagram proposed by Le [START_REF] Méhauté | An Introduction to Hydrodynamics and Water Waves[END_REF]. This permits to visualize that the long waves have moderate level of nonlinearity, lying in the validity domain of Stokes second order wave theory. As will be seen hereafter, harmonic modes of the fundamental forcing frequency will be present in the analyzed spectra.

In the wave tank, the addition of these regular long waves had the direct and visible effect of attenuating the wind-waves. Figure 7 illustrates this interaction through the spatial evolution of the variance density spectrum for the wind speed U re f = 10 m.s -1 . For that purpose, the pure wind-wave spectrum is compared to the spectrum combining wind-waves and paddle-waves. The semi-log scale in Figure 7 emphasizes the generation of wind-waves at high frequencies. At short fetch (from 0 to 10 m), wind-wave peak frequency in the presence of long waves deviates from the pure wind-wave peak frequency: the wind-wave peak frequency becomes lower as the steepness of the long-wave increases. This downshift is even more pronounced at a lower wind speed (not shown here). The frequency deviation from the pure windsea growth is also depicted in Figure 8 using the spectral decomposition presented in Section 2.b. A possible explanation for this downshift is the presence of high frequency harmonics initiating wind-wave growth at frequencies lower than in the absence of paddle-waves. The induced frequency shift reduces at larger fetch and eventually vanishes. Thus, the progression of the wind-wave peak towards low frequencies is slowed down in the presence of long monochromatic waves. [START_REF] Donelan | The effect of swell on the growth of wind waves[END_REF] drew attention to the slowed rate of progression of the paddle-modified windsea peak towards low frequencies with increasing fetch. In his 100-m-long tank, the introduction of long waves upshifted the windsea peak frequency at large fetch (X ∼ 70 m). This tendency of the pure windsea peak to overtake the paddle-modified one in its progression towards low frequencies could be imagined as a scenario in our case with an extended fetch: in Figure 8, an hypothetical spatial extension of the ak = 5.6 % peak frequency evolution towards higher fetch would probably lead to the observation of Donelan. According to him, the frequency shift previously described could be due to an alteration of the dispersion relation when a paddle-wave passes through a group of wind-waves, resulting in a detuning of the resonance leading to a modified nonlinear quartet interaction. This hypothesis is also relevant to explain the broadening of the paddle-modified wind-wave peak in its forward face (low-frequency).

Regarding the relative energy of wind-wave, paddle-waves broaden and flatten the corresponding spectral peak (Figure 7) resulting in a total energy (m 0 , see Figure 8) smaller than the total energy of pure windsea. This observation is in line with previous studies [START_REF] Mitsuyasu | Interactions between Water Waves and Wind (1)[END_REF][START_REF] Phillips | Wave breaking in the presence of wind drift and swell[END_REF][START_REF] Donelan | The effect of swell on the growth of wind waves[END_REF]. At fixed wind speed and for a given fetch, wind-wave total energy decreases as the steepness of the paddle-wave increases. This dependence on steepness has been related to a sheltering effect by [START_REF] Chen | Effects of Long Waves on Wind-Generated Waves[END_REF].

In the high frequency tail of the spectra in Figure 7, a small wind-wave harmonic can be observed at about 2 times the wind-wave peak frequency, for the pure wind-wave case, and to a lesser extent with paddle-waves of steepness ak = 2.7 %. It is hardly noticeable with paddles-waves of steepness ak = 5.6 %. The decrease of this second harmonic peak as the long-wave steepness increases is a consequence of the reduction of the amplitude of the main wind-wave spectral peak with increasing long-wave steepness, resulting in a lower nonlinearity level of the wind wave component. Except for this difference, the high frequency tail is not altered in the presence of paddle-waves. This last observation is in disagreement with [START_REF] Donelan | The effect of swell on the growth of wind waves[END_REF] who noticed a higher tail for the cases including paddle-waves.

A semi log-scale is necessary to observe the high frequency behaviour of wave-spectra but differences around the spectral peaks are usually less visible. Removing the semi log-scale permits to discern the location of the significant energy in the spectra. Figure 9 shows the energy transfers differentiating so called "pure" paddle-waves (a) and "pure" wind-waves (b) cases from the paddlewave-plus-wind-wave combination (c) using a direct subtraction between the measured spectra.

The pink area is a good illustration of the interaction between long monochromatic waves and short wind-waves. The wind-wave reduction previously observed appears in the negative pink area between 1.25 and 2 Hz. The pink area under the first harmonic shows the amplification of the monochromatic paddle-waves. A striking effect illustrated in Figure 9 is the amplification of the second harmonic of the paddle-waves by the wind: negligible in Figure 9.a, the second harmonic is prominent and even more amplified than the first harmonic in Figure 9.c. This amplification is a consequence of the frequency of the second harmonic of the fundamental mode of the paddlewaves lying in the frequency band of the wind wave peak. Though less marked, this effect also manifests on the third harmonic at 1.8 Hz. In the same vein, it is observed in Figure 7 that higherorder harmonics of the paddle-waves are significantly amplified (see for instance the spectra with paddle-waves of steepness ak = 5.6 % for fetches from 10 to 20 m, for which the third harmonic is highly enhanced so has to become higher than the second harmonic. Thus, higher-order harmonics plays an important role in the interaction between both wave systems. These ideal waves enable a separation of the physical phenomena taking part into wave systems interactions. But adverse effects such as the amplification of wave harmonics can alter the direct comparison with ocean waves.

Effect of Irregular Paddle-Waves on Wind-Wave Growth

In order to consider a more realistic long-wave system representing swell, a JONSWAP-type sea-state was generated in the wave tank. The spatial evolution of the bimodal sea-states with fetch is compared to the pure windsea case using a spectral representation in Figure 10. In this section, the steepness ak of the irregular waves is defined as a = H s /(2 √ 2) = √ 2m 0 and k = k p . Using this definition, monochromatic and irregular waves with the same steepness have the same energy.

As in Figure 7, the spatial evolution of spectra with fetch is shown in Figure 10. A striking effect is the downshift of the wind-wave peak frequency in the presence of long irregular waves. The downshift appears at the earliest stage of wind-wave growth (fetch X = 1.99 m in Figure 10) and has the tendency to increase with increasing paddle-wave steepness. It then slightly reduces with fetch but remains significant even at the maximal fetch (i.e. X = 29.98 m).

Figure 11 depicts different slopes in the non-dimensional peak-frequency evolution depending on the presence of paddle-waves and their relative steepness: the progression towards low-frequency is slowed down with increasing paddle-waves steepness. A likely cause for the downshift is the presence of paddle-wave energy at relatively high frequencies initiating wind-wave growth at lower frequencies than for the pure windsea case.

Regarding the wind-wave energy, a small amplification effect due to long irregular waves is shown in Figure 11 for the wind speed U ref = 14 m.s -1 . This amplification, more significant at short fetch, has a tendency to reduce with increasing fetch. This reflects a slower energy variation with fetch in the presence of irregular paddle-waves. At lower wind speed, this small amplification becomes a small reduction at the maximal fetch (see table 1).

Overall, the effect of irregular waves on wind-wave energy appears weaker compared to monochromatic waves of the same energy. In the present study, the absence of a clear reduction of the windsea energy due to the presence of irregular paddle-waves is in disagreement with [START_REF] Benetazzo | Analysis of the effect of fish oil on wind waves and implications for air-water interaction studies[END_REF] and [START_REF] Bailey | Predicting Wind Wave Suppression on Irregular Long Waves[END_REF]. A plausible explanation for these diverging observations, based on [START_REF] Benetazzo | Analysis of the effect of fish oil on wind waves and implications for air-water interaction studies[END_REF], is an insufficient separation between the wave systems leading to a misinterpretation of the spectra combining wind-waves and paddle-waves. Using

Figure 10 to illustrate this latter remark, if one only considers the energy at the peak frequency of the pure windsea, the presence of paddle-waves indeed causes a wind-wave reduction at this particular frequency. Thus, it is only by considering the downshift that one can observe wind-wave energy has not been reduced but downshifted towards lower frequencies. Figure 10 in the paper of [START_REF] Benetazzo | Analysis of the effect of fish oil on wind waves and implications for air-water interaction studies[END_REF] shows that the wind-wave peak frequency downshift cannot be observed in their case due to the presence of the paddle-wave peak at 1 Hz. A lower paddle-wave frequency, in our case, permits the observation of this downshift.

Discussion

a. Wind-Wave Generation on Long Mechanically-Generated Waves

The effect of both monochromatic and irregular long waves on windsea have been studied

experimentally in a wind-wave flume. Although these long-wave systems are very different in term of spectral shape, their comparison in the presence of wind provides new insight about the physical processes at stake in the interaction between long waves and short wind-waves.

In the wind-wave generation process, a first impact of paddle-waves lies in the interaction with the airflow. It has been shown in Figure 3 that the vertical profile of wind speed is modified in the presence of paddle-waves. This observation is not surprising since the wind stress is mainly supported by waves traveling slower than the wind which is the case for both wind-waves and paddle-waves in our experiments. Additionally, the use of the logarithmic profile method to determine the friction velocity and the aerodynamic roughness seems questionable since the presence of long waves propagating slower than the wind is prone to invalidate the analogy with the airflow above a stationary rough plate. A method free from the logarithmic profile hypothesis, such as the eddy-correlation method using a thin cross-X hot wire anemometer, would be more adapted for the characterization of the airflow in the presence of paddle-waves; this is left for further work.

The other facet of this interaction lies in the amplification of paddle-waves that absorb momentum from the wind.

The physical process, named 'sheltering effect' by [START_REF] Chen | Effects of Long Waves on Wind-Generated Waves[END_REF], mostly affects the wind-wave energy and occurs especially with monochromatic paddle-waves. The main character-istic differentiating these latter waves from the irregular waves is the distribution of wave energy over a range of frequencies. The wave trough between two consecutive crests of the dominant wave mode is much deeper with monochromatic paddle-waves than with irregular ones. In the case of irregular paddle-waves, the troughs are often "filled" with other wave modes causing lower height differences between crests and troughs than in the monochromatic case.

Regarding the wind-wave evolution in the frequency domain, irregular and monochromatic long waves have the similar property to downshift the early generation frequency of the wind-waves as it is visible at short fetch in Figures 7 and10. This effect is even more pronounced with irregular waves. The paddle-waves bring some initial energy at high frequencies with the high frequency harmonics for the monochromatic case and the high frequency tail in the irregular case. This initial energy seems to facilitate wind-wave growth at a lower frequencies than in absence of paddle-waves where no initial energy is present. With irregular paddle-waves, the energy continuum brought by the high frequency tail of the spectrum amplifies this property, thus initiating wind-wave growth with a higher energy and a lower frequency than without paddle-waves. Thus, the wind energy input at the early stage of wind-wave growth is somehow shifted towards lower frequencies and added to existing waves when an initial sea state is present.

As it has been seen in Figure 8 and 11, the evolution of wind-wave energy and peak-frequencies with fetch using log-scaled axis can be approximated by straight lines. Thus, in the same manner of the previously used empirical power laws, the wind-wave growth observed in our tank can be described using power laws. Table 1 gathers the power laws parameters providing a concise summary of our results on wind-wave growth with and without paddle-waves. In Table 1, the absolute values of coefficients c and e globally decrease with paddle-waves steepness, showing that both the variations of wind-wave energy and peak-frequencies with fetch become milder.

Additionally, these variations are even more softened in the presence of irregular paddle-waves.

with fetch. At this point, the reduced wind-wave energy evolution with monochromatic paddlewaves must be distinguished from the slightly amplified wind-wave energy with irregular paddlewaves in comparison with the pure windsea (see Figures 8 and11). The slowed increase of wind-wave energy with monochromatic paddle-waves, that has been extensively studied in the past, has been related to wind-wave enhanced dissipation [START_REF] Phillips | Wave breaking in the presence of wind drift and swell[END_REF], long-wave sheltering effect [START_REF] Chen | Effects of Long Waves on Wind-Generated Waves[END_REF] and nonlinear wave-wave interactions [START_REF] Masson | On the Nonlinear Coupling between Swell and Wind Waves[END_REF]).

However, this latter process is only significant when the ratio of long waves over short-waves frequencies is less than 0.6 according to [START_REF] Masson | On the Nonlinear Coupling between Swell and Wind Waves[END_REF]. This condition is not satisfied in our case with the first harmonic of the monochromatic waves, yet wind-waves could be interacting with the second harmonic closer to the wind wave peak (Figure 9). If an analogous analysis is performed on the slowed increase of wind-wave energy with fetch in the presence of irregular waves, since the sheltering effect is certainly reduced due to the spectral property, only enhanced dissipation and nonlinear wave-wave interaction are possible candidates. In this latter case, Masson's ratio is 0.6 at maximal fetch implying a weak nonlinear interaction between wave systems. The remaining candidate would be the enhanced wave breaking due to wind-wave interaction with the addition of long waves orbital velocities and wind drift as formulated by [START_REF] Phillips | Wave breaking in the presence of wind drift and swell[END_REF].

The slowed rate of progression of the windsea peak towards lower frequencies with increasing fetch can be related to nonlinear interactions. With monochromatic paddle-waves, [START_REF] Donelan | The effect of swell on the growth of wind waves[END_REF] suggested that the wind-wave energy transfer towards low frequencies might be reduced due to wind-waves reduced steepness, thus nonlinearity, compared to pure windsea. This latter suggestion can be illustrated in Figure 12 where the wind-wave energy is expressed as a function of the windwave peak frequency. In this figure, wind-wave growth in the presence of paddle-waves (blue and red dots respectively for monochromatic and irregular paddle-waves) are situated below the pure wind-wave growth (black dots). Thus, at a fixed wind-wave energy, the corresponding wind-wave peak frequencies in the presence of paddle-waves are lower than in the pure wind sea case. Since a lower wave frequency relates to a lower wavenumber k (using the dispersion relation) and the wave energy is proportional to the squared wave amplitude a, it appears that the wind-wave steepness is reduced in the presence of paddle-waves. Additionally, Figure 12 shows that the wind-wave steepness is even more lowered when irregular paddle-waves are present. This latter observation suggests that nonlinear interactions are a good candidate to explain the slowed decrease of wind wave peak frequency with fetch in the case with irregular waves.

b. Relation to Wind Waves Growth at Prototype Scale

The extent at witch our experimental results, can be generalized at natural scale is discussed in the following. As mentioned in Introduction, the main difference between laboratory and natural scale measurements lies in the wave age of the swell. While the wind velocity conditions produced during the experiment are close to that observed in natural conditions, the mechanically-generated waves representing swell are significantly shorter, thus slower, than the swell conditions encountered in the ocean. Thus, an appropriate scaling was performed to compare our experiments with equivalent natural scale observations. Being aware of the main differences in term of wave age, qualitative comparisons permitted to draw tendencies about the physical processes involved in the interaction between swell and wind-waves at natural scale.

First, an effort was devoted to the determination of the wind friction velocity in order to perform an accurate scaling. Some doubts were formulated regarding the measuring device and the pertinence of the logarithmic profile method to determine the friction velocity in the presence of paddle-waves.

Using ocean buoys measurements, Vincent et al. ( 2019) showed that the friction velocity tends to increase with swell steepness especially at low wind speeds. A similar trend is observed in Figure both scales, it would be necessary to use a method free from the logarithmic profile analogy for the wind profile characterization.

As it was highlighted in section 4, wind-wave growth (without paddle-waves) in laboratories shows similarities with that at natural scale. To observe these similarities, laboratory wind-waves must be long enough to be free from surface tension effects. Then, the wind-wave energy and peak frequency (Figure 4) were close to Kahma and Calkoen's empirical relations, representing wind-wave growth at natural scale, but slightly deviated in terms of variations with increasing fetch.

The introduction of paddle-waves, representing swell, significantly modifies the wind-wave field.

First, monochromatic paddle-waves cause a reduction of wind-wave energy. This reduction has received numerous interpretations that reached an agreement on the fact that the physical processes responsible for that reduction are proper to steep short waves (equivalent to paddle-waves in natural reservoirs) traveling slower than the wind (i.e. C << U 10 ). This wind-wave reduction phenomenon was used to design a reduced wind input model for high frequencies waves (i.e. waves with a wave age equivalent to paddle-wave) and high winds in [START_REF] Ardhuin | Semiempirical Dissipation Source Functions for Ocean Waves. Part I: Definition, Calibration, and Validation[END_REF]. Some gray areas remain on the role played by the monochromatic wave harmonics in the modification of the wind-wave growth. The prominence of these harmonics is principally inherent in laboratory steep monochromatic paddle-waves and might damage the relation with wind-waves at natural scale.

Secondly, the wind-wave peak frequency and its variations with increasing fetch are modified in the presence of either irregular or, to a lower extent, monochromatic paddle-waves. In Figures 8 and11, the wind-wave peak frequency variations with fetch are closer to Kahma and Calkoen's than to Mitsuyasu and Rikiishi's power laws although the latter is specific for laboratory wind-wave growth. In Table 1, the same tendency can be observed by comparing the power law coefficient e in the presence of paddle waves with the corresponding coefficient in Kahma and Calkoen's law (e = -0.26). Furthermore, in the presence of irregular paddle-waves, wind-wave energy evolution with increasing fetch follows this agreement with Kahma and Calkoen relations. This accordance can be observed in Figure 11 or by comparing the power law coefficient c in Table 1 with the coefficient c = 0.85 of Kahma and Calkoen. Finally, irregular paddle-waves seem to restore wind-wave growth as it is observed at natural scale.

Conclusion

A series of experiments were carried out at IRPHÉ/Pythéas wind-wave tank in Marseilles. The interactions between long paddle-waves, representing swell, and short wind-waves were investigated with both monochromatic and irregular paddle-waves. An analysis was performed in order to identify the interaction mechanisms that can be transported at prototype scale.

Wind-wave growth was studied in absence of paddle-waves to assess the similarities with existing experiments. It was found that, after the young wind-waves emancipate from the gravity-capillary state, wind-wave growth is in agreement with laboratory based empirical laws but differ from more general laws (i.e. laws valid at prototype scale) in terms of variations with fetch.

By calculating the vertical profile of the wind horizontal velocity using the logarithmic profile method, the friction velocity had a tendency to increase in the presence of paddle-waves. However, the measurements uncertainties added to the limited validity of the logarithmic profile method to characterize the airflow in the presence of long paddle-waves prevented a more quantitative analysis. The use of a more accurate method, such as the eddy correlation method, is let for further work.

The interactions between wind and paddle-waves have repercussions on the wave systems. The paddle-waves are amplified by the wind which echoes back to the sheltering effect highlighted by wind-wave growth by reducing the quantity of momentum available for wind-wave generation.

In this latter case, the wind mostly impacts the wind-wave energy. Thus, the wind-wave peak frequency evolution with fetch is similar to wind only conditions, resulting in a reduced wind-wave steepness in the presence of monochromatic paddle-waves.

The effect of irregular paddle-waves on wind-wave growth is quite different since the windwave parameter affected by this interaction is mostly the wind-wave peak frequency. The energy continuum brought by the paddle-waves high frequency tail initiates wind-wave growth at a lower frequency than in the absence of paddle-waves. Thus, it was found that the irregular low frequency waves downshift wind-wave peak frequency in a significant manner. The wind-wave energy being only slightly impacted by comparison with the peak-frequency, irregular paddle-waves reduces wind-wave steepness even more than in the monochromatic paddle-wave case.

The similarities between the action of monochromatic and irregular paddle-waves on wind-wave growth lie in the wind-wave energy and peak frequencies reduced variations with fetch. It is interesting to note that most laboratory measurements, for example [START_REF] Mitsuyasu | The growth of duration-limited wind waves[END_REF], display an accelerated wind-wave growth (i.e. wind-wave energy increases faster and the peak frequency reduces faster) by comparison with ocean measurements, illustrated here by [START_REF] Kahma | Reconciling discrepancies in the observed growth of wind-generated waves[END_REF]'s empirical law. Especially with irregular paddle-waves, which is a more faithful reproduction of a typical ocean sea-state, wind-wave growth is somehow in accordance with [START_REF] Kahma | Reconciling discrepancies in the observed growth of wind-generated waves[END_REF]

LIST OF FIGURES

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .

 123456789101112 Fig. 1. Sketch of the IRPHÉ/Pythéas wind-wave tank facility showing the location of the measuring devices. The vertical exaggeration is × 5. . . . . . . . . . . . . . . . 35 Fig. 2. Example of spectral separation for a spectrum obtained at a 30-m-fetch with a wind of 10 m.s -1 in presence of irregular long waves of 2.7 % steepness. E + and E -are respectively the JONSWAP fit of the higher and the lower peaks. . . . . . . . . . . . . 36 Fig. 3. Vertical profiles of the averaged horizontal wind velocity with and without paddle-waves associated logarithmic fits. The corresponding friction velocities values (mean slope of the fitted lines) are indicated for each wind only cases. . . . . . . . . . . . . . 37 Fig. 4. Variations of dimensionless wave energy and peak frequency with dimensionless fetch for different reference wind speeds (U ref ) compared to empirical laws. . . . . . . . . 38 Fig. 5. Variations of the wind-wave dimensionless energy with the dimensionless peak frequency for different equivalent wind speed (U ref ) compared to variations according to Toba's 3/2 law (10). . . . . . . . . . . . . . . . . . . . . . . . . . 39 Fig. 6. Representation of regular paddle-wave conditions on Le Méhauté's diagram (Le Méhauté 1976). . . . . . . . . . . . . . . . . . . . . . . . . 40 Fig. 7. The spatial evolution of the variance density spectrum with and without regular paddle-waves of steepness ak and peak frequency 0.6 Hz for a wind speed U ref = 10 m.s -1 . . . . . . 41 Fig. 8. The spatial evolution of wind-wave energy and peak frequency with and without regular paddle-waves of steepness ak and peak frequency 0.6 Hz for a wind speed U ref = 10 m.s -1 . . . 42 Fig. 9. Pure paddle-wave spectrum E p (a) and pure wind-wave spectra E w (b) by comparison with the bimodal spectrum E w+p (c) with monochromatic paddle-waves of steepness ak = 2.7 % and peak frequency 0.6 Hz at fetch of 30 m for a wind speed U ref = 10 m.s -1 . The pink area is the difference between the bimodal spectrum E w+p and the sum of the pure spectra E w + E p . . 43 Fig. 10. The spatial evolution of the variance density frequency spectrum with and without irregular paddle-waves of steepness ak and peak frequency 0.6 Hz for a wind speed U ref = 14 m.s -1 . . . 44 Fig. 11. The spatial evolution of wind-wave energy and peak frequency with and without irregular paddle-waves of steepness ak and peak frequency 0.6 Hz for a wind speed U ref = 14 m.s -1 . . . 45 Fig. 12. Variations of the wind-waves dimensionless energy with the dimensionless peak frequency for different reference wind speeds (squares: U ref = 6 m.s -1 , circles: U ref = 10 m.s -1 , triangles: U ref = 14 m.s -1 ) compared to variations according to Toba's 3/2 law. . . . . . 46

F

  . 5. Variations of the wind-wave dimensionless energy with the dimensionless peak frequency for different equivalent wind speed (U ref ) compared to variations according to Toba's 3/2 law (10).

  Representation of regular paddle-wave conditions on Le Méhauté's diagram (Le Méhauté 1976). spatial evolution of the variance density spectrum with and without regular paddle-waves of steepness ak and peak frequency 0.6 Hz for a wind speed U ref = 10 m.s -1 . spatial evolution of wind-wave energy and peak frequency with and without regular paddle-waves of steepness ak and peak frequency 0.6 Hz for a wind speed U ref = 10 m.s -1 . Pure paddle-wave spectrum E p (a) and pure wind-wave spectra E w (b) by comparison with the bimodal spectrum E w+p (c) with monochromatic paddle-waves of steepness ak = 2.7 % and peak frequency 0.6 Hz at fetch of 30 m for a wind speed U ref = 10 m.s -1 . The pink area is the difference between the bimodal spectrum E w+p and the sum of the pure spectra E w + E p . . The spatial evolution of the variance density frequency spectrum with and without irregular paddlewaves of steepness ak and peak frequency 0.6 Hz for a wind speed U ref = 14 m.s -1 . . The spatial evolution of wind-wave energy and peak frequency with and without irregular paddlewaves of steepness ak and peak frequency 0.6 Hz for a wind speed U ref = 14 m.s -1 . . Variations of the wind-waves dimensionless energy with the dimensionless peak frequency for different reference wind speeds (squares: U ref = 6 m.s -1 , circles: U ref = 10 m.s -1 , triangles: U ref = 14 m.s -1 ) compared to variations according to Toba's 3/2 law.

  and wave forcing. Using the separation of the physical mechanisms at the root of wind-wave generation in the spectral energy balance equation (1), the ability of the spectral wave model in reproducing wind-wave growth with and without paddle-waves might enable to identify the mechanisms at the origin of the wave systems interactions observed in laboratory. Ultimately, the use of a spectral wave model may help to evaluate the extent at which the mechanisms observed in laboratories can be transposed to prototype scale. Acknowledgments. Antoine Villefer acknowledges the financial support of his PhD research program provided by the French ANRT (Association Nationale de la Recherche et de la Technologie) with CIFRE grant number 2019-1257. The authors thanks Maria-Joao Teles (EDF R&D LNHE) and Jeffrey Harris (ENPC/LHSV) for many fruitful discussions in the course of this study.

	U ref (m.s -1 )		6	10	14
	U 10 (m.s -1 )		9	13	20
	u (m.s -1 )		0.31	0.48	0.88
	z 0 × 10 4 (m)		1.4	1.6	9.3
	Swell type	ak (%)		(b × 10 4 , c)
	Monochromatic 5.6	(0.13, 1.28) (0.59, 1.05)	-
			2.7	(2.00, 0.99) (0.31, 1.29) (4.61, 0.82)
			0.0	(0.02, 1.64) (0.11, 1.50) (0.52, 1.30)
	Irregular	2.7	(2.16, 1.01) (0.96, 1.17) (1.27, 1.18)
			4.1	(2.68, 1.01) (7.56, 0.90) (5.47, 0.91)
	Swell type	ak (%)		(d, e)
	Monochromatic 5.6 (0.60, -0.28) (0.64, -0.30)	-
			2.7 (1.68, -0.42) (1.27, -0.38) (0.86, -0.33)
			0.0 (3.46, -0.50) (1.36, -0.39) (0.89, -0.33)
	Irregular	2.7 (0.06, -0.01) (0.28, -0.20) (0.34, -0.22)
			4.1 (0.11, -0.11) (0.07, -0.02) (0.21, -0.14)
	T	1. Wind characteristics and corresponding power law coefficients representing the fetch dependence of
	windsea peak frequency and energy (m * 0 = b(X
					's law. A result from this latter observation is that wind-wave growth
	in laboratories seems closer to wind-wave growth at prototype scale when irregular paddle-waves
	are present.

Finally, our experiments provide a valuable database for validating a spectral wave model. From this database, spectral wave model performances can be assessed under generic conditions for both wind * ) c and f * p = d(X * ) e ). The lines with bold characters refer to the pure windsea cases.

Data availability statement.

LIST OF TABLES


C Pérez Arroyo

G Daviller

G Puigt

C Airiau

Hydrodynamic -Acoustic Filtering of a Supersonic Under-expanded Jet

Introduction

The noise perceived in the aft-cabin for an aircraft at cruise conditions is mainly due to the turbofan jet. The pressure mismatch between the ambient air and the secondary stream of a turbofan engine leads to the formation of (diamond-shaped) shock-cells. These series of expansion and compression waves interact with the vortical structures developing in the mixing layer of the jet. This interaction process generates intense noise components on top of the turbulent mixing noise, which makes supersonic jets noisier than their subsonic counterparts [START_REF] Tam | Supersonic jet noise[END_REF]. The result is a broadband shock-cell associated noise (BBSAN), radiated mainly in the forward direction, which impinges on the aircraft fuselage and it is then transmitted to the cabin.

In this paper, the shock-cell noise generated by an axisymmetric under-expanded single jet is investigated using Large-Eddy Simulations (LES). The paper addresses in a first term, the code characteristics and simulation setup and procedure. The second part of the paper focuses on the application of acoustic-hydrodynamic filtering and its impact on the cross-correlation in the near-field.

Numerical Formulation

The full compressible Navier-Stokes equations in skew-symmetric formulation are solved using the Finite Volume multi-block structured solver elsA (Onera's software [START_REF] Cambier | The Onera elsA CFD software: input from research and feedback from industry[END_REF]). The spatial scheme is based on the well-known implicit compact finite difference scheme of 6th order of Lele [START_REF] Lele | Compact finite difference schemes with spectral-like resolution[END_REF], extended to Finite Volumes by Fosso et al. [START_REF] Fosso-Pouangué | Curvilinear Finite Volume Schemes using High Order Compact Interpolation[END_REF]. The above scheme is stabilized by the compact filter of Visbal & Gaitonde [START_REF] Visbal | On the Use of Higher-Order Finite-Difference Schemes on Curvilinear and Deforming Meshes[END_REF] that is also used as an implicit subgrid-scale model for the present LES. Time integration is performed by a six-step 3rd order Runge-Kutta DRP scheme of Bogey and Bailly [START_REF] Bogey | A family of low dispersive and low dissipative explicit schemes for flow and noise computations[END_REF].

Simulation Setup and Procedure

Time-dependent simulations are presented of a contoured convergent nozzle with exit diameter D = 38.0mm and a modeled nozzle lip thickness of t = 0.125D. The nozzle is operated under-expanded at the stagnation to ambient pressure ratio p s /p ∞ = 2.27. The modeled exit and ambient conditions match those in the experimental set-up of André [START_REF] André | Etude expérimentale de l'effet du vol sur le bruit de choc de jets supersoniques sous-détendus[END_REF]. The Reynolds number, Re, based on the jet exit diameter is 1.2 × 10 6 and the fully expanded jet Mach number is M j = 1.15.

The numerical computation is initialized by a RANS simulation using the Spalart-Allmaras turbulence model [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF]. The RANS solution is wall resolved in the inner and outer sections of the nozzle with a y + < 1. Once mesh convergence is achieved, the LES run is then initialized from the RANS simulation. The inner part of the nozzle is removed from the LES simulation and the RANS nozzle exit conservative variables are imposed, as [START_REF] Suzuki | Shock leakage through an unsteady vortex-laden mixing layer: application to jet screech[END_REF][START_REF] Viswanathan | Numerical Prediction of Noise from Round and Beveled Nozzles[END_REF][START_REF] Shur | Noise prediction for increasingly complex jets. Part I: Methods and tests[END_REF][START_REF] Shur | Further steps in LES-based noise prediction for complex jets[END_REF][START_REF] Schaupp | On a method for direct numerical simulation of shear layer/compression wave interaction for aeroacoustic investigations[END_REF][START_REF] Schulze | Numerical simulation of supersonic jet noise[END_REF]. In addition, no inflow forcing is used.

The computational domain used for the LES simulation extends 40D in the axial direction and 7D in the radial direction. The mesh consists in 75 × 10 6 cells with (1052 × 270 × 256) cells in the axial, radial and azimuthal directions respectively. The maximum expansion ratio between adjacent cells achieved in the mesh is less than 4%.

Non-reflective boundary conditions of Tam and Dong [START_REF] Tam | Radiation and outflow boundary conditions for direct computation of acoustic and flow disturbances in a nonuniform mean flow[END_REF] extended to three dimensions by Bogey and Bailly [START_REF] Bogey | Three-dimensional non-reflective boundary conditions for acoustic simulations: far field formulation and validation test cases[END_REF] are used in the exterior inlet as well as in the lateral boundaries. The exit condition is based on the characteristic formulation of Poinsot and Lele [START_REF] Poinsot | Boundary Conditions for Direct Simulations of Compressible Viscous Flows[END_REF]. Furthermore, sponge layers are coupled around the domain to attenuate exiting vorticity waves. Due to the fact that the interior of the nozzle is not modeled, no inflow forcing is applied at the exit of the nozzle to avoid parasite noise.

The simulation runs for 120 non-dimensional time units (t = tD/C ∞ ) in order to reach statistically independent results.

Hydrodynamic -Acoustic filtering

The acoustic component of a jet can be filtered from the hydrodynamic one in the near-field, where the mean velocity is zero using the methodology shown by Tinney and Jordan [START_REF] Tinney | others: The near pressure field of co-axial subsonic jets[END_REF]. The original pressure signal p(x p ,t) is transformed into the wavelength -frequency domain by

p(k x , ω) = p(x p ,t)W (x p )e -j(k x x p +ωt) dx p dt, (1)
where the reference coordinate x p is such that it is located outside the jet verifying a null mean velocity throughout the new coordinate. W (x p ) is a weight function used to smooth the signal at the ends of the axis. An acoustic fluctuation generated by a noise source inside the jet will be propagated outside the jet at the ambient sound speed c ∞ . However, the acoustic perturbation will be propagated at a supersonic speed c 2 x = c ∞ (2d f + c ∞ ) in the x p coordinate system, where d is the perpendicular distance from the axis to the noise source and f is its frequency. The subsonic and supersonic components, i.e. the hydrodynamic and acoustic components can be recovered from the transformed signal obtained by Eq. 1 using only the ranges p(k x > ω, ω < k x c ∞ ) and p(k x < ω, ω > k x c ∞ ) respectively as:

p sub (x p ,t) = p(k x > ω, ω < k x c ∞ )e -j(k x x p +ωt) dk x dω, (2)
p super (x p ,t) = p(k x < ω, ω > k x c ∞ )e -j(k x x p +ωt) dk x dω. (3)
The filtering has been applied to the supersonic under-expanded jet near-field over several lines in order to obtain a two-dimensional field of the pressure perturbation. Figure 1a shows the original pressure perturbations of a snapshot where the jet is represented by the axial velocity. It can be compared with the filtered hydrodynamic and acoustic components in figures 1b and 1c respectively. The filtering allows the hydrodynamic component to be clearly visible and the acoustic component to gain in detail specially near the jet, where the hydrodynamic perturbations are more intense.

The importance of the hydrodynamic-acoustic filtering is further noted when spatio-temporal cross-correlations of the filtered pressure are carried out in the nearfield. Figures 2a, 2b and2c show the cross-correlation between one point located at [x/D = 0, y/D = 3] and an array located at y/D = 1 with an inclination of 5 • , following the expansion of the shear layer.

The shock-cell noise is found around x/D = 4 with a cross-correlation of 0.38. When the filtering is applied, the cross-correlation peak increases up to 0.41 and it becomes clearer downstream of the end of the potential core (x/D > 9). The filtering improves the cross-correlation specially downstream where the hydrodynamic components have the same general direction as the acoustic waves. Figures 3a,3b the above-mentioned array. At this position, the original cross-correlation shown in figure 3a differs completely from the acoustic component shown in figure 3c. The original signal is clearly dominated by the hydrodynamic component shown in 3b adopting the same shape. The mixing noise of the large turbulent structures is mostly generated at the end of the potential core and radiated at 30 • [START_REF] Tam | Supersonic jet noise[END_REF]. The cross-correlation of the acoustic component agrees with the theory showing a high correlation up to this location, with a convective velocity at the speed of sound.

Conclusions

The hydrodynamic -acoustic filtering of the near-field of a supersonic underexpanded axisymmetric jet has been carried out. Post-processing techniques such as spatio-temporal cross-correlations are affected by the filtering achieving different conclusions depending on whether or not the filtering is used. The shock-cell noise and the mixing noise of the large turbulent structures have been identified in figures 2c and 3c respectively. This approach looks very promising to study in depth the shock-cell noise mechanism. The simulation of a co-axial nozzle where the secondary jet is under-expanded and other filtering techniques will be applied the future.

Fig. 1a FigFig. 1c

 1a1c Fig.1aPressure perturbations of the original flow, -150Pa < p ≤ 150Pa.

Fig. 2a Fig. 3a Fig. 3c

 2a3a3c Fig. 2a Spatio-temporal pressure cross-correlation of the complete original signal at x/D = 0.0.

  Pérez Arroyo • G. Puigt Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique, 42 Avenue Gaspard Coriolis, 31057 Toulouse Cedex 01, France, e-mail: {cparroyo,guillaume.puigt}@ cerfacs.fr

	G. Daviller • C. Airiau
	Institut de Mcanique des Fluides de Toulouse, UMR 5502 CNRS/INPT-UPS, Alle du professeur
	Camille Soula, 31400 Toulouse, France, e-mail: {guillaume.daviller,christophe.
	airiau}@imft.fr
	1

C.

Acknowledgments

This research project has been supported by a Marie Curie Initial Training Networks (ITN) AeroTraNet 2 of the European Community Seventh Framework Programme (FP7) under contract number PITN-GA-2012-317142 that aims to generate a ready to use model for shock-cell noise characterization.

This work was performed using HPC resources from GENCI-[CCRT/CINES/IDRIS] (Grant 2015-[x20152a6074]).


Annie-Claude Pérez
email: annie-claude.perez@univ-tln.fr

Claude Jauffret
email: jauffret@univ-tln.fr

Observability in Target Motion Analysis from the Sum or the Difference of Ranges with Two Stationary Sensors

Keywords: Target motion analysis, tracking, range-only, TDOA, bistatic, observability, Fisher information I

We address in this paper the problem of observability in target motion analyses (TMA), when the measurements are a sum or a difference of ranges between a target and two motionless observers. Necessary and sufficient conditions of observability are given in time difference of arrival (TDOA) situation and in a bistatic situation.

INTRODUCTION

In any problem of estimation, the first step consists of analyzing observability of the parameter of interest. When the noise-free measurements depend linearly on the parameter (one deal with a linear system), the task is easy: the parameter is observable when the matrix linking the measurement vector to the parameter vector is full-ranked. In a large class of probability laws of additive noise, this is equivalent to the regularity of the Fisher information matrix (FIM) [START_REF] Jauffret | Observability and Fisher Information Matrix in Nonlinear Regression[END_REF]. When the system is nonlinear, the definition of observability must be extended: the parameter can be either locally observable, if it is observable in an open vicinity, or globally observable if this open vicinity is the whole space in which it is defined. Analyzing observability requires more mathematical tools. Systems encountered in target motion analysis (TMA) problems are nonlinear in most cases. The observability is hence a hard task unless the noise-free system can be transformed into an equivalent linear system, as in bearingsonly TMA [START_REF] Jauffret | Observability in Passive Target Motion Analysis[END_REF][START_REF] Fogel | Nth-order Dynamics Target Observability from Angle Measurements[END_REF][START_REF] Nardone | Observability Criteria for Bearings-Only Target Motion Analysis[END_REF][START_REF] Le Cadre | Discrete-Time Observability and Estimability Analysis for Bearings-Only Target Motion Analysis[END_REF]. In the TMA literature, two main ways are used for this purpose: the rank of the FIM can be studied as in [START_REF] Arnold | Target Parameter Estimation Using Measurements Acquired with a Small Number of Sensors[END_REF] [START_REF] Xiao | Observability and Performance Analysis of Bi/Multi-Static Doppler-Only Radar[END_REF], or we can use analytical tools as in [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF][START_REF] Pignol | Properties of range-only target motion analysis[END_REF][START_REF] Jauffret | Observability: Range-Only Versus Bearings-Only Target Motion Analysis When the Observer maneuvers Smoothly[END_REF]. When the FIM rank serves as observability criterion, some precaution must be taken: the rank must be constant in an open subspace containing the parameter [START_REF] Rothenberg | Identification in Parametric Models[END_REF]. If not, the conclusion can be wrong. For example, in range-only TMA, the FIM can be singular at an isolated point and still this point is observable. Conversely, the FIM can be full ranked, and the parameter remains unobservable [START_REF] Jauffret | Observability and Fisher Information Matrix in Nonlinear Regression[END_REF]. We claim that using the determinant of the FIM (or its rank) to conclude observability can mislead. In this paper, we intend analyzing observability of the trajectory of a target in constant velocity (CV) motion, in two TMA problems: when the measurement are time differences of arrival (TDAO) and when the measurements are those collected in a bistatic configuration (as claimed in [START_REF] Jauffret | Doppler-Only Target Motion Analysis in a High Duty Cycle Sonar System[END_REF]). Our main tool comes from the polynomial algebra. The analysis will be made in continuous time. In section II, the notations are given.

Section III is devoted to the case of range difference measurements. In this part, we give conclusions about observability in TDOA-TMA. The bistatic configuration is addressed in section IV. A conclusion follows.

II. NOTATIONS AND FUNDAMENTAL LEMMAS

A. Notations

A target (T) and two observers (O1) and (O2) are in the same plane, given a Cartesian system. The target on in CV motion all along the scenario, while the observers are stationary. The scenario starts at time 0  t and finishes at time The position of the target is  

T )] ( ) ( [ t y t x t P T T T  at time t . Its velocity is   T ] [ T T T T y x dt t dP V    
. The ranges relatively to the two observers are

    t P O t r T 1 1  and     t P O t r T 2 2



, respectively. Figure 1 illustrates the used notations.

    t r t r 2 1 is a constant function or a polynomial a  a S   t r 1   t r 2 O 1 O O 2 function of degree 2 if, and only if       2 2 1 t r t r   with 1    is a polynomial function. Proof:               t r t r t r t r t r t r 2 1 2 2 2 1 2 2 1 2                                t y a t x t y a t x t y a t x a t x T T T T T T T 2 2 2 2 2 2 2 2 2            If       2 2 1 t r t r   is a polynomial function, then               t r t r t r t r t r t r 2 1 2 2 2 1 2 2 1 2      is a polynomial function too. Let us compute its degree: Since                     t y a t x t y a t x t r t r T T T T 2 2 2 2 2 2 2 1      , the highest degree is 4. The coefficient of 4 t is 2 2 T T x y  & & . If 2 2 0 T T x y   & & , then     t r t r 2 2 2 1
is a polynomial function of degree 4 ; therefore the degree of    

t r t r 2 1 is 2. Otherwise, 0 T T xy  && ,     t r t r 2 2 2 1 is a constant function, and     t r t r 2 1
too.

The converse is obvious.

QED

In the coming study, we will distinguish three cases:

    t r t r 2 1 is a constant     t r t r 2 1 is a polynomial function.     t r t r 2 1
is neither a constant, nor a polynomial function.

Lemma 2:

    t r t r 2 1
is a constant function if, and only if the target is stationary.

Proof Suppose that     t r t r 2 1 hence       2 2 1 t r t r is a constant function. Since                       t y a t x t y a t x t r t r T T T T 2 2 2 2 2 2 1     
, all the coefficients of the non-null powers of t are null, in particular the coefficient of 4 t , which is

2 2 T T y x    . Hence the target is fixed.
The converse is obvious.

QED

Remark 1:

In the proofs of the following lemmas and propositions, we will use the following property:

If the target is moving in the (Ox) axis, then

 

  Proof

If     t r t r 2 1
is a polynomial function of degree 2, then, three numbers  ,  ,  exist such that

             t t t r t r 2 1
. The numbers  ,  can be real or not, whereas  is a real number. a) If  and  are two real numbers, then Let us examine two subcases :

                    2 2 2 2 2 2 2           t t t y a t x t y a t x T T T T Consequently,                 0 2 2 2 2         
If    , then                   4 2 2 2 2 2         t t y a t x t y a t x T T T T . As a consequence                            2 2 2 2 2 1 2 2     t t
       * 2 1       t t t r t r
. We get

                    2 2 2 2 2 2 2 *           t t t y a t x t y a t x T T T T . Necessarily, two real numbers 1  exist 2  such that                                            2 1 2 2 2 1 2 2 * * t t t y a t x t t t y a t x T T T T We have       0 2 2      T T y a x , and       0 2 2      T T y a x . We deduce that         2 2 a x a x T T      , hence that   0 4  a x T  . But   t x T is a polynomial function of degree 0 or 1. We conclude that   0  t x T , . t  The target is moving in the (Oy) axis. Conversely, if the target is moving in the line   2 1 , O O , then                     a t x a t x a t x a t x t r t r T T T T        2 2 2 1 , with 1    . If the target is moving in the perpendicular bisector of the segment   2 1 , O O , then         2 2 2 2 2 1 2 T T r t r t a y t a y t             .

QED III. RANGE DIFFERENCE

A. Analysis

In this section, the available measurement is the difference of ranges between each observer and the target. The observability analysis will be conducted with

      2 2 1 t r t r 
and its sign.

Proposition 1:

The range difference is null if, and only if the target is in the perpendicular bisector of the segment  

2 1 , O O .
In this case, the trajectory of the target is not observable.

Proof

Suppose that the difference of range is null.

        2 2 2 1 2 1 t r t r t r t r    , that is,             t y a t x t y a t x T T T T 2 2 2 2      . We have hence   0  a t x T , that is   0  t x T . The target is in the (0y) axis.
Conversely, if the target is in the (0y) axis, the ranges are equal.

QED Proposition 2:

The range difference is a non-null constant if, and only if the target is motionless out of the perpendicular bisector of the segment  

Proof

If the difference of range is a non-null constant, its square is a non-null constant, as well:

        cste 2 2 1 2 2 2 1    t r t r t r t r . Since     2 2 2 1 t r t r  is a polynomial function,     t r t r 2 1 is a constant or a polynomial of degree 2. From Lemma 2, if     t r t r 2 1
is a constant, then the target is stationary.

From Lemma 3, if     t r t r 2 1
is a polynomial of degree 2, then the target is moving in the axis (Ox) or in the axis (Oy). The last case must be discarded since the range difference is non-null. Hence the target is moving in the axis (Ox). Remark 1 helps us to conclude.

The converse is obvious.

QED Proposition 3:

The range difference is a polynomial function of degree 1 if, and only if the target is moving between the two sensors. In this case, the trajectory of the target is observable: their initial abscissa is its half of the first difference of range, and its velocity is half of the time coefficient.

Proof

If the range difference is a polynomial function of degree 1, then necessarily    

t r t r 2 1
is a polynomial of degree 2. Hence the target is moving in the line   . From Remark 1, the target is moving between the two sensors.

The converse is obvious.

QED Proposition 4:

The square of the range difference is neither a polynomial function nor a constant function if, and only if the target is in CV motion out of the line   Proof:

Let a ghost-target G detected by the same measurements:

          f G G T t t r t r t r t r , 0 , 2 1 2 1                    f G G T t t r t r t r t r , 0 , 2 2 1 2 2 1                       t r t r t r t r t r t r t r t r G G G G 2 1 2 2 2 1 2 1 2 2 2 1 2 2                       t r t r t r t r t r t r t r t r G G G G 2 1 2 1 2 2 2 1 2 2 2 1 2 2       (1) Hence,         t r t r t r t r G G 2 1 2 1  is a polynomial function of degree 2.
But, by assumption,

    t r t r 2 1
is not a polynomial function; hence

    t r t r G G 2 1
as well.

Consequently, it exists two polynomial functions of degree 2, say

  t P 2 and   t Q 2 , and a non-polynomial function   t g such that         t g t P t r t r   2 2 1 , (2)
        t g t Q t r t r G G   2 2 1
.

(

) Since     t r t r 2 2 2 1 and     t r t r G G 2 2 2 1 3
are polynomial functions of degree 4, the functions

      t g t g t P 2 2 2  and       t g t g t Q 2 2 2 
are polynomial functions of degree 4.

We deduce that

                    t g t Q t P t Q t P t r t r t r t r G G 2 2 2 2 2 2 2 2 2 1 2 2 2 1 2      .
At this point, we have to discuss about the nature of

    t Q t P 2 2  : 1) If     0 2 2   t Q t P                     t Q t P t Q t P t r t r t r t r t g G G 2 2 2 2 2 2 2 2 2 1 2 2 2 1 2     
is a rational fraction (the ratio of two polynomial functions), say

      t S t R t g 4 4



(irreducible fraction).

Now, eq (1) implies that

                    2 2 1 2 1 2 2 2 2 1 2 2 2 1 4 t r t r t r t r t r t r t r t r G G G G                                  2 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 1 2 1 2 1 4 8 t r t r t r t r t r t r t r t r t r t r t r t r G G G G G G        which means that         t r t r t r t r G G 2 1 2 1
is a polynomial function of degree 4.

From ( 2) and (3), the assumption of (4). This case must be discarded.

                      t g t Q t P t g t Q t P t r t r t r t r G G     2 2 2 2 2 1 2 1                         t g t Q P t S t R t Q t P t r t r t r t r G G      2 2 4 4 2 2 2 1 2 1                             t Q t P t R t Q t P t r t r t r t r t S t g t R G G 2 2 4 2 2 2 1 2 1 4 4      . Hence,     t g t R 4 is a polynomial function, which is equal to     t S t R

2) Hence

    0 2 2   t Q t P From (1), we get readily         t r t r t r t r G G 2 2 2 1 2 2 2 1    (5)         t r t r t r t r G G 2 1 2 1 2 2  (6)
Let us develop the left term of (5), then its the right terms :

        t y t x a t r t r T T 2 2 2 2 2 2 1 2 2 2     ,         t y t x a t r t r G G G 2 2 2 2 2 2 1 2 2 2     . Consequently, (5) is equivalent to           f G G T T T t t y t x t y t x , 0 , 2 2 2 2      (7)
which means that the two targets are in the same circle whose center is the middle point of  

                        t x t y a t y t x t x t y a t y t x G G G G T T T T 2 2 2 2 2 2 2 2 2 2 2 2 2 2        Using (7), we get         t x t y t x t y G G T T 2 2 2 2    (8)
Eq. ( 7) and ( 8

B. Application: TDOA [6]

In a TDOA, the measurement is the difference of ranges up to a multiplicative constant that is the speed of the wave (in the medium of interest). So, the previous analysis is directly applicable. Proposition 3 and Proposition 5 give us necessary and sufficient conditions of observability. Moreover, if a third observer is added to the scenario, the trajectory of any target in CV motion is observable from TDOA.

IV. RANGE SUM

A. Analysis

Here also, the observability analysis will be based upon

the square of     t r t r 2 1  :               t r t r t r t r t r t r 2 1 2 2 2 1 2 2 1 2     .
Note that the sum is always positive.

We can straightforwardly give the following result:

Proposition 6:

The square of the range sum is a polynomial function of degree 2 if, and only if the target is either in the line  

  0  t y T . Consequently,       t y a t r t r T 2 2 2 1 2   
, and

          t y a t r t r T 2 2 2 2 1 4    .
The identification of the coefficients of the polynomial function 

      2 2 1 t r t r 
If the target is moving in   2 1 ,O O , then     a t r t r t 2 2 1   
. This case must be discarded. 

If the target is moving in the line   2 1 O O , but out of   2 1 ,O O , then       1 2 2 , T r t r t x t t     .
Conversely, if the target is either in the line   The square of the range sum is a polynomial function of degree 2 and the range sum is not a polynomial function of degree 1 if, and only if the target is in the perpendicular bisector of the segment   The proof is the part a) of the proof of Proposition 6.

Proposition 8:

The range sum is a polynomial function of degree 1 if, and only if the target is in the line  

          f G G T t t r t r t r t r , 0 , 2 1 2 1                    f G G T t t r t r t r t r , 0 , 2 2 1 2 2 1                       t r t r t r t r t r t r t r t r G G G G 2 1 2 2 2 1 2 1 2 2 2 1 2 2                       t r t r t r t r t r t r t r t r G G G G 2 1 2 1 2 2 2 1 2 2 2 1 2 2       
Since this last equation is eq. ( 1) up to the sign of the right term, the rest of the proof is the same the proof of Proposition 4: We end up with the two following equalities:

                         t y t x t y t x t y t x t y t x G G T T G G T T 2 2 2 2 2 2 2 2
As a consequence,    

t x t x T G   , and     t y t y T G   . Conversely, if     t x t x T G   , and     t y t y T G   , then           f G G T t t r t r t r t r , 0 , 2 1 2 1      . QED.

Proposition 10:

The range sum is a constant function if, and only if the target is either in the segment   2 1 , O O , or stationary. In this case, the trajectory of the target is not observable. B. Application: the passive bistatic radar [START_REF] Jauffret | Doppler-Only Target Motion Analysis in a High Duty Cycle Sonar System[END_REF] In a bistatic configuration, O1 is the transmitter, and O2 is the receiver. The transmitter sends a signal continuously and the target plays the role of mirror: it reflects the signal emitted by the transmitter toward the receiver. If the signal  

t s E is a single tone, that is         t f a t s E 0 2 sin
, the signal reflected by the target is the received signal with with i=1,2. It follows that the instantaneous frequency at time

t is                 c t r t r f t f 2 1 0 1   for 0, f t T      
The emitted frequency 0 f and the wave speed c are assumed to be known.

We have             t r t P t r t P t r t r 2 2 1 1 2 1      , with         T T T T y t y x a t x t P      1 , and        y t y x a t x t P T T T      2
, that are two polynomial functions of degree 1.

If a ghost-target

G exists, far from the transmitter and receiver

  t r G1 and   t r G2 respectively, then                c t r t r f t f G G 2 1 0 1   . Note that                                 2 2 2 1 1 2 2 2 1 1 2 2 1 1 2 2 1 1 
                  t r t P t r t P t r t P t r t P t r t P t r t P t r t P t r t P G G G G G G G G                                 2 2 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 rational fraction rational fraction 2 2 G G G G G G G G

P t P t P t P t P t P t P t P t r t r t r t r t r t r t r t r t

                                      1 4
If   t f is a constant function, then          t t r t r 2 1 . Hence,       2 2 2 2 2 1 2         t t t r t r . Therefore,     t r t r 2 1
is a constant function or a polynomial function of degree 2.

If     t r t r 2 1
is a constant function, then the target is stationary (see Lemma 2). We have no way to know its location.

If     t r t r 2 1
is a polynomial function of degree 2, then T is moving in the line   The first statement is the converse of Proposition 11. We only have to prove that the trajectory of the target is observable.

Two cases must be distinguished:

1) The target is moving along the perpendicular bisector of the segment  

    t r t r 2 1 is a polynomial function of degree 2. If a ghost- target G exists,     t r t r G G 2 1
is necessarily a polynomial function of degree 2. Therefore, G is moving in the perpendicular bisector of the segment  

2 1 , O O ,       t y a t r t r G G G 2 2 2 1    . Since         t r t r t r t r G G 2 1 2 1        , we have         t y a y t y t y a y t y T T T G G G 2 2 2 2      (9)         t y a t y a y t y y t y G T G G T T 2 2 2 2       .         t y a t y a y t y y t y G T G G T T 2 2 2 2 2                         t y a y t y t y a y t y T G G G T T 2 2 2 2 2 2 2 2                  2 2 2 2 2 2 2 2 2 T G G T G G T T y y t y t y y t y y t y a                    0 2 2 2 2 2 2 2 2 2      G G T T G T G T y t y y t y a y y t y t y     (10)
This polynomial function of degree 4 is equal to zero. Then all its coefficients are null, in particular the one of 4 t , that is

  0 2 2   G T G T y y y y     . Because 0  T y  and 0  G y  , we have 2 2 T G y y    . Introducing this equality in (14), we get     t y t y T G 2 2  . Eq implies that   T T y t y  et   G G y t y  have the same sign. Hence     t y t y T G   with 1    .
2) The target is moving out of the perpendicular bisector of the segment  

                            1
G G G G G G G G                                                       2 2 2 1 2 1 2 1 2 12 1 2 1 2 1 2 1 2 rational fraction 4 4 4 G G G G G G G G P P P P P P PP F r r r r r r r r                 1 4 4 4 4 2 4 4 4 43 Consequently, 2 1 2 1 G G r r r r is a polynomial function. Since neither 2 1 r r nor 2 1 G G r r are polynomial functions, a positive real number  exists such that 2 1 2 1 r r r r G G   . Hence, 2 2 2 1 2 2 2 2 1 r r r r G G   . Consequently, either      2 2 2 2 2 1 2 1 r r r r G G   or      2 1 2 2 2 2 2 1 r r r r G G   , with 2 et 0 , 0         . Let us study each case:                                                  t y a t
        1 1 2 2 G G r t r t r t r t          . The equality         t r t r t r t r G G 2 1 2 1        implies that 1   :     t r t r

QED

Remark 3: In [START_REF] Xiao | Observability and Performance Analysis of Bi/Multi-Static Doppler-Only Radar[END_REF], the rank of the FIM was used as observability criterion. The conclusion of the authors of [START_REF] Xiao | Observability and Performance Analysis of Bi/Multi-Static Doppler-Only Radar[END_REF] is that the trajectory is observable if the target does not travel toward the transmitter or toward the receiver. But, the authors did not care if the rank of the FIM was constant in an open vicinity of the parameter of interest. It is not the case. 12 contradicts their conclusion.

V. CONCLUSION

In this paper, we addressed the question about observability of the trajectory of a target in CV motion, when the sum of range (between the target and two stationary observers) or their difference is available. Through an analysis based on the algebraic nature of the noise-free we end up with the following nonambiguous answers:

1) A motionless target is unobservable from the knowlegde of the sum or the difference of ranges.

2) A moving target is observable under the following conditions:  In TDOA (two passive sensors), the trajectory of the target is observable if, and only if the target does not travel in the perpendicular bisector of the segment  

Fig. 1 :

 1 Fig. 1: typical scenario

  the target is in the line spanned by the two sensors, out of the segment   2 1 , O O . In both cases, the trajectory of the target is not observable. Nevertheless, we are able to know if the target is on the left or on the right of the segment  

  3). Since the range difference is non-null, the target is in the line   2 1 , O O

  case, its trajectory is observable up to the axial symmetry around the line  

  roots. This is not

  allows us to recover the initial position and the velocity of the target. b) If the target is in the line   us to finish the proof :

  case, the trajectory of the target is observable up to the symmetry around the center of  

..

  In this case, the trajectory of the target is observable up to the symmetry around the center ofThe proof is the part b) of the proof of Proposition 6 (second indent).Proposition 9:a) The square of the range sum is neither a constant function nor a polynomial function if, and only if the target is in CV motion, out of the line   target is observable up to the axial symmetries around these two lines.ProofThe part a) is a consequence of Proposition 7 and Proposition 8. Only, the part b) has to be proved.Let a ghost-target G detected by the same measurements:

  the proof.

  In a bistatic situation (a transmitter and a receiver), the observability is guaranteed if and only if the target is moving out of the line  

  ) allows us to conclude that

	  x t T		  x t G	, and		G y	  t			y	S	  t .
	QED.
	The knowledge of the sign of the range difference
	removes the symmetry relatively to the perpendicular
	bisector of the segment 	2 1 , O O		, and can state the
	following property:
	Proposition 5:
	The range difference is neither a constant, nor a
	polynomial function of degree 1 if, and only if the target is
	out of the line  O 1 O 2		and out of the perpendicular bisector
	of the segment 	2 1 , O O		.
	In this case, its trajectory is observable up to the axial
	symmetry around the line  O 1 O 2		.
	Remark 2: with noisy measurements, a statistical test
	allows us to decide if the difference of range is always null,
	constant and non-null, linear or non-linear. Depending on
	the response of the test, we can know, before tenting to
	estimate the trajectory, if it is observable or not.