A Bayle

F Plouraboué

Spectral properties of Fluid Structure Interaction pressure/stress waves in liquid filled pipes

Keywords: Fluid Structure Interaction (FSI), Liquid filled pipes, Junction coupling, Operator's spectrum, Time domain solution, Method of characteristics (MOC)

We hereby develop a theoretical framework for analyzing Fluid Structure Interaction (FSI) waves propagation occurring in liquid filled pipes to manage a large family set of boundary conditions (e.g. junctions coupling effects). A self-adjoint operator theory framework leads to the analytical derivation of a transcendental equations for operator's spectrum. The latter provides the system's natural resonant frequencies as well as permit to find the discrete mode orthogonal basis decomposition. This theoretical framework also permits to demonstrate that the spectrum is uniquely composed into simple eigenvalues enabling explicit time-domain solutions from inverse-Laplace transform. The analysis is directly conducted in the time-domain but the obtained spectrum also applies to Fourier transformed frequency analysis. The obtained analytical solutions are successfully confronted with numerical simulation obtained using the Method of characteristic (MOC) for the same four equations (FSI) model on the very same configurations. The spectrum sensitivity matrix is also explicitly evaluated.

Introduction

Wave propagation in liquid-filled pipe systems have been investigated for a long time [START_REF] Joukowsky | Uber den hydraulischen stoss in wasserleitungsro hren.(on the hydraulic hammer in water supply pipes) mé moires de l'académie impériale des sciences de st.-petersbourg[END_REF][START_REF] Korteweg | Ueber die fortpflanzungsgeschwindigkeit des schalles in elastischen rohren, on the speed of sound propagation in elastic tubes[END_REF][START_REF] Skalak | An extension of the theory of waterhammer[END_REF][START_REF] Holmboe | The Effect of Viscous Shear on Transients in Liquid Lines[END_REF][START_REF] Burmann | Water hammer in coaxial pipe systems[END_REF][START_REF] Budny | The Influence of Structural Damping on Internal Pressure During a Transient Pipe Flow[END_REF][START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF] to cite only a few, possibly seminal, contributers. The phenomenology is now well understood as fully discussed in exhaustive and sagacious review papers [START_REF] Tijsseling | Fluid structure interaction in liquid filled pipe systems: a review[END_REF][START_REF] Li | FSI research in pipeline systems -A review of the literature[END_REF][START_REF] Ferras | One-Dimensional Fluid-Structure Interaction Models in Pressurized Fluid-Filled Pipes: A Review[END_REF]. Fluid Structure Interaction (FSI) arising between pressure/stress propagation have been recognized as one major modeling pathway, leading to four coupled hyperbolic equations in the case of axi-symmetric compressive planar waves modes propagation [START_REF] Burmann | Water hammer in coaxial pipe systems[END_REF][START_REF] Čanić | Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow[END_REF][START_REF] Plona | Axisymmetric wave propagation in fluid-loaded cylindrical shells. II: Theory versus experiment[END_REF][START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF][START_REF] Sinha | Axisymmetric wave propagation in fluid-loaded cylindrical shells. i: Theory[END_REF]. The long wavelength approximation is a widely established and validated framework [START_REF] Skalak | An extension of the theory of waterhammer[END_REF][START_REF] Walker | Pulse Propagation in Fluid-Filled Tubes[END_REF][START_REF] Kizilova | Pressure wave propagation in liquid-filled tubes of viscoelastic material[END_REF][START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF]. It permits to neglect secondary FSI effects associated with rotatory vibration modes or radial inertia (e.g. bending, twisting, etc..), the analysis of which needs a more complex set of equations. Considering an averaged formulation both in solid and fluid for mass and momentum conservation equations, [START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF] derive a set of four coupled hyperbolic equations highlighting the overriding role of the Poisson's coupling effects, namely the axial transmission of the radial stresses and strains via the Poisson's modulus, on the whole dynamic. In [START_REF] Tijsseling | Waterhammer with fluid-structure interaction[END_REF] two other major coupling mechanisms were spelled out: (i) the junction occurring at edge conditions and, (ii) the friction coupling resulting from viscous effects in boundary layers and/or pipe's supports. Whereas (i) is precisely analyzed in this study, on the contrary, (ii) is not considered. Pipe's support coupling effects has nevertheless been thoroughly analyzed in [START_REF] Budny | The Influence of Structural Damping on Internal Pressure During a Transient Pipe Flow[END_REF][START_REF] Liu | Vibration analysis of liquid-filled pipelines with elastic constraints[END_REF][START_REF] Keramat | Experimental investigation of transients-induced fluid-structure interaction in a pipeline with multiple-axial supports[END_REF].

Recent and active motivations to analyze the FSI vibrations in pipes lies from the use of water-hammer waves in defect/leak detection and localization [START_REF] Wang | Matched-field processing for leak localization in a viscoelastic pipe: An experimental study[END_REF][START_REF] Wang | Spectral-based methods for pipeline leakage localization[END_REF][START_REF] Keramat | Transient-based leak detection in the frequency domain considering fluid-structure interaction and viscoelasticity[END_REF][START_REF] Keramat | Spectral based pipeline leak detection using a single spatial measurement[END_REF][START_REF] Che | Transient wave-based methods for anomaly detection in fluid pipes: A review[END_REF]. Since in common practice "localized" pipe anomaly, such as a leak and a discrete blockage leads to a modification of the Fourier peaks of the signal, a spectral-based diagnostic signal processing has been sough for [START_REF] Wang | Matched-field processing for leak localization in a viscoelastic pipe: An experimental study[END_REF][START_REF] Keramat | Transient-based leak detection in the frequency domain considering fluid-structure interaction and viscoelasticity[END_REF][START_REF] Keramat | Spectral based pipeline leak detection using a single spatial measurement[END_REF][START_REF] Che | Transient wave-based methods for anomaly detection in fluid pipes: A review[END_REF]. In this context, the ability to obtain an explicit derivation of the traveling waves system spectral properties can pave the way to elaborate spectral diagnostic signal processing strategies among which the spectral sensitivity matrix is a central one [START_REF] Duan | Transient-based frequency domain method for dead-end side branch detection in reservoir pipeline-valve systems[END_REF][START_REF] Duan | Accuracy and sensitivity evaluation of tfr method for leak detection in multiple-pipeline water supply systems[END_REF]. As spectral sensitivity matrix is a time-consuming and noise-sensitive quantity, a purely numerical estimate of this quantity, e.g. based upon finite-difference estimate, is sometimes not precise enough or too demanding (in case of high-dimensionality of parameter space). This is why it is either interesting to lower the parameter space dimensions and/or to find analytical estimate of this spectral sensitivity matrix, as performed here. Many contributions relying on the Laplace/frequency-domain numerical resolution of the FSI four equations using transfer matrix method (TMM) can be found [START_REF] Li | FSI research in pipeline systems -A review of the literature[END_REF][START_REF] Lesmez | Modal Analysis of Vibrations in Liquid-Filled Piping Systems[END_REF][START_REF] Tijsseling | Exact solution of linear hyperbolic four-equation system in axial liquid-pipe vibration[END_REF][START_REF] Zhang | FSI Analysis of Liquid-Filled Pipes[END_REF][START_REF] Li | Frequency domain analysis of fluid-structure interaction in liquid-filled pipe systems by transfer matrix method[END_REF][START_REF] Li | Vibration analysis of pipes conveying fluid by transfer matrix method[END_REF][START_REF] Wang | Spectral-based methods for pipeline leakage localization[END_REF]. Nevertheless, to our knowledge, no fully explicit time-domain solutions nor explicit spectrum have been previously obtained except in [START_REF] Mei | Pressure and wall shear stress in blood hammer -Analytical theory[END_REF], which has ignored Poisson's coupling and therefore (FSI) interactions. In the following, we thereby focus on developing a new analytical framework using operator's theory, for pressure/stress wave operator. We derive a straightforward real transcendental equation for the spectrum and successfully spelled out an orthogonal projection basis for the uncoupled diagonalized wave operator. The separation of variables technique is used to handle the derivation of a pressure-stress solution in time-domain (a Fourier transform may thereafter easily be managed, if required, to find the corresponding frequency domain solution). The paper is organized as follow. Section 2.1 describes the dimensionless constitutive (FSI) 4-equations model, boundary conditions sets, parametric description, proper wave dimensional and dimensionless velocity propagation, and the resulting diagonalized 2-waves equations reminiscent of [START_REF] Zhang | FSI Analysis of Liquid-Filled Pipes[END_REF]'s solution strategy. Section 3 provides the theoretical framework defining the self-adjoint operator for the separable waves solutions decomposition. Section 4 illustrates the comparison between the obtained analytical solutions and previously published numerical or theoretical results for specific sets of boundary conditions. Section 5 then provides the spectrum sensitivity matrix for each boundary condition set.

Governing equations

Dimensionless constitutive model

Let us consider a cylindrical tube having inner radius R 0 , wall thickness e, and length L, which defines the following aspect and geometrical parameters

α = e R 0 , and, = R 0 L . ( 1
)
The tube is supposed to be entirely filled with a fluid having density ρ f , bulk modulus K, perturbed pressure P * and velocity W * . Considering low-Mach waves, the fluid density is considered as constant and equal to the reference density ρ f as in [START_REF] Tijsseling | Fluid-structure interaction in case of waterhammer with cavitation[END_REF][START_REF] Chaudhry | Applied Hydraulic Transients[END_REF]. The elastic solid response is associated with Young's modulus, E, Poisson's modulus ν s , perturbed axial stress, σ * , perturbed axial strain ζ * , and density ρ s supposed constant. [START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF] derived the classical pulse wave speed within the fluid, c p , distinct from the elastic pulse wave speed within the solid, c s , the ratio of which is denoted

C s c 2 p = K ρ f 1 + 2K αE 2(1-ν 2 s ) 2+α + α(1 + ν s ) , c 2 s = E ρ s , C s = c s c p . (2)
The coupled system is furthermore described through the dimensionless density ratio

D = ρ f ρ s , (3)
so that the dimensionless FSI four-equations derived in [START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF] achieves as follows

∂ τ W = -∂ Z P, (4)
∂ τ P + ∂ Z W = 2αν s ∂ Z ζ, (5)
∂ τ ζ = D α ∂ Z σ, (6)
∂ τ σ - αC 2 s D ∂ Z ζ = 2ν s α(2 + α) ∂ τ P. (7)
where W , P , ζ, σ are dimensionless quantities referring to the fluid longitudinal velocity, the fluid pressure, the longitudinal solid deformation velocity and the longitudinal stress, respectively. The physical time t is scaled on the advective fluid pulse one, i.e. τ ≡ cp L t, whilst the longitudinal coordinate z is scaled on the tube's length, i.e. z ≡ LZ. More details on the hereby dimensionless derivation is provided in Appendix B. ( 4)-( 7) represents a set of two coupled hyperbolic equations. While the first part (4)-( 5) is associated with the acoustic waves propagation in the fluid, the second part ( 6)- [START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF] describes the propagation of axial compressible waves in the solid tube. Poisson's coupling is highlighted by the presence of the Poisson's modulus in both source terms of ( 5) and [START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF]. It is noteworthy to point-out that (4)- [START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF] are the leading order contributions regarding small parameter [START_REF] Li | FSI research in pipeline systems -a review of the literature[END_REF]. [START_REF] Skalak | An extension of the theory of waterhammer[END_REF][START_REF] Lin | Wave Propagation through Fluid Contained in a Cylindrical, Elastic Shell[END_REF][START_REF] Walker | Pulse Propagation in Fluid-Filled Tubes[END_REF] in-depth analyzed the secondary (FSI) interactions occurring in liquid filled pipe systems, revealing their significant impact at very high-frequencies only, the cut-off of which f Kc , is known as the Korteweg's stop band, [START_REF] Gaultier | Wave propagation in a fluid filled rubber tube: Theoretical and experimental results for korteweg's wave[END_REF]

f Kc = C s f 0 , with, f 0 = c p 2πL . (8)
For f Kc > f > f 0 , the axial dynamics prevails over the radial one and despite simplifications (4)-( 7) is relevant to investigate several configurations [START_REF] Ghidaoui | A review of water hammer theory and practice[END_REF][START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF][START_REF] Keramat | Fluid-structure interaction with pipe-wall viscoelasticity during water hammer[END_REF][START_REF] Aliabadi | Frequency response of water hammer with fluid-structure interaction in a viscoelastic pipe[END_REF]. This frequency cut-off f Kc nevertheless stands as a frequency limitation of the proposed analysis and will be thereafter discussed. Last but not least, it is known from [START_REF] Mei | Pressure and wall shear stress in blood hammer -Analytical theory[END_REF][START_REF] Mei | Effects of thin plaque on blood hammer-An asymptotic theory[END_REF] that viscous shear dissipation occurring at the fluid and solid interface may have a significant impact on the coupling dynamic. Even though of physical interest, this issue is herein discarded but in-depth analyzed in [START_REF] Bayle | Low Mach number theory of pressure waves inside an elastic tube[END_REF]. Considering the acoustic framework for the fluid whilst using the linearity of the FSI-governing equations, only the perturbed component of the physical field are investigated so that the initial conditions are

Y(Z, 0) = ∂ τ Y(Z, 0) = 0, (9)
with Y(Z, τ ) a four column vector,

Y(Z, τ ) = P (Z, τ ), σ(Z, τ ), W (Z, τ ), ζ(Z, τ ) T , (10)
where subscript T holds for the conjugate transpose.

Vectorial two-waves system of FSI four-equations

The four coupled hyperbolic ( 4)-( 7) are hereby re-organized to bring-up a d'Alembert operator upon the time-space dependent variable Y

∂ 2 τ -C 2 Y ∂ 2 Z Y(Z, τ ) = 0, (11)
where,

C 2 P = 1 2ν s D 2νs α(2+α) 4ν 2 s D α(2+α) + C 2 s , C 2 W = 1 -2αν s -2νsD α 2 (2+α) 4ν 2 s D α(2+α) + C 2 s ( 12
)
and

C 2 Y = C 2 P 0 0 C 2 W . (13)
Eigenvalues of the C 2 P and C 2 W matrices correspond to coupled vibrating modes wave speeds propagation. Both matrices have identical eigenvalues, c 2 ± , the solutions of the polynomial characteristic problem

c 4 ± -1 + C 2 s + 4ν 2 s D α(2 + α) c 2 ± + C 2 s = 0, (14)
then achieves as follows

c 2 ± = 1 + C 2 s + 4ν 2 s D α(2+α) ± 1 + C 2 s + 4ν 2 s D α(2+α) 2 -4C 2 s 2 . ( 15
)
(15) stands for the dimensionless version of coupled wave speed modes derived in [START_REF] Budny | The Influence of Structural Damping on Internal Pressure During a Transient Pipe Flow[END_REF][START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF][START_REF] Zhang | FSI Analysis of Liquid-Filled Pipes[END_REF]. The choice for denoting C 2 P and C 2 W matrices with a square as well as its eigenvalues c 2 ± , now becomes clear since c ± describes the wave speed of each propagating mode, each governed by their specific D'Alembert operator. Dimensionless wave speed, c -, has a value close to one, whilst c + is found close to C s . In Appendix E a systematic asymptotic analysis provides the Poisson coupling corrections to these quantities as ν s 1. The mode c -is thus associated with the fluid pulse mode while, c + is associated with the solid elastic one. Furthermore, it is noteworthy to point-out that the negative mode of ( 15) is always real in the Korteweg's frequency range since for real

C 2 s parameter 1 -C 2 s 2 > - 4ν 2 s D α(2 + α) 4ν 2 s D α(2 + α) + 2 1 + C 2 s , (16)
is always satisfied. This property nevertheless vanishes as f > f Kc , in which case the radial contributions leads to a dispersive waves, the propagating velocity of which can be complex [START_REF] Skalak | An extension of the theory of waterhammer[END_REF][START_REF] Gaultier | Wave propagation in a fluid filled rubber tube: Theoretical and experimental results for korteweg's wave[END_REF]. The D'Alembert wave propagation operator is hereby regarded within the diagonal base of C 2 Y as classically performed in coupled hyperbolic systems [START_REF] Keramat | Transient-based leak detection in the frequency domain considering fluid-structure interaction and viscoelasticity[END_REF][START_REF] Zhang | FSI Analysis of Liquid-Filled Pipes[END_REF][START_REF] Li | Frequency domain analysis of fluid-structure interaction in liquid-filled pipe systems by transfer matrix method[END_REF]. The transition matrices Π Y of the diagonal base change can easily be deduced from

C 2 P and C 2 W eigenvectors Π Y = Π P 0 0 Π W , Π P = 2νsD c 2 --1 2νsD c 2 + -1 1 1 , Π W = 2ανs c 2 --1 2ανs c 2 + -1 1 1 , (17)
such as the transition relations

C 2 Y = Π -1 Y C 2 Y Π Y = C 2 P 0 0 C 2 W , and, Y = Π -1 Y Y, (18)
where

C 2 P = Π -1 P C 2 P Π P = c 2 - 0 0 c 2 + , and, C 2 W = Π -1 W C 2 W Π W = c 2 - 0 0 c 2 + . ( 19
)
The dimensionless wave-equations system [START_REF] Čanić | Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow[END_REF] and its initial conditions [START_REF] Li | FSI research in pipeline systems -A review of the literature[END_REF] expressed in the eigenvector base finally reads

∂ 2 τ -C 2 Y ∂ 2 Z Y = 0, (20)
Y(Z, 0) = ∂ τ Y(Z, 0) = 0. (21)
The operator [START_REF] Čanić | Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow[END_REF] diagonalization has indeed simplified the mode propagation now described by two independent D'Alembert propagating waves but obviously not suppressed their coupling. The coupling is now recast in the resulting upstream and downstream boundary conditions. Boundary conditions for Y can be deduced from the mechanical boundary conditions associated with Y using change-basis relations [START_REF] Wang | Matched-field processing for leak localization in a viscoelastic pipe: An experimental study[END_REF]. For the sake of notation simplification let us introduce four 4 × 4 matrices N , M, Q, R and S(τ ) a eight-dimensional column vector. Boundary conditions can formally be written as a rectangular 8 × 16 linear system

N M 0 0 0 0 Q R (8×16)     Y(0, τ ) ∂ Z Y(0, τ ) Y(1, τ ) ∂ Z Y(1, τ )     (16×1) = S (8×1) (τ ). ( 22
)
Specifics set of boundary conditions are later on considered in section 4. The resolution of this vector waves equation is usually handled by Laplace transform, combined with usual transfer matrix method [START_REF] Zhang | FSI Analysis of Liquid-Filled Pipes[END_REF][START_REF] Li | FSI research in pipeline systems -A review of the literature[END_REF][START_REF] Li | Frequency domain analysis of fluid-structure interaction in liquid-filled pipe systems by transfer matrix method[END_REF]. Some analytical difficulties are sometimes nevertheless encountered when performing the inverse Laplace transform. Thereafter a new analytical derivation for solution of [START_REF] Wang | Spectral-based methods for pipeline leakage localization[END_REF], having initial conditions [START_REF] Keramat | Transient-based leak detection in the frequency domain considering fluid-structure interaction and viscoelasticity[END_REF] and spatial boundary conditions [START_REF] Keramat | Spectral based pipeline leak detection using a single spatial measurement[END_REF], relying on variable separation, spectral analysis and self-adjoint operator theory is proposed.

Analytical framework

Self-adjoint operator theory

Let us define the operator H, acting on the square-integrable real four dimensional column vector field Ψ(Z),

∀Ψ(Z) ∈ L 4 (R) × L 4 (R), Ψ(Z) → HΨ(Z) = C 2 Y • ∂ 2 Z Ψ(Z), (23)
with the homogeneous associated set of spatial boundary conditions,

N M 0 0 0 0 Q R (8×16)     Ψ(0) ∂ Z Ψ(0) Ψ(1) ∂ Z Ψ(1)     (16×1) = 0. ( 24
)
Let us set up the general scalar product,

∀Ψ, Ψ ∈ L 4 (R) × L 4 (R), Ψ , Ψ = 4 j=1 η j 1 0 Ψ j (Z)Ψ j (Z)dZ, (25)
with j = 1, 2, 3, 4 referring to the j th components of vector η ≡ (η 1 , η 2 , η 3 , η 4 ) ∈ R 4 , a yet unknown real vector which is adapted to each specific problem. Invoking the definition of H in [START_REF] Che | Transient wave-based methods for anomaly detection in fluid pipes: A review[END_REF], the search for self-adjoint condition for operator H, equipped with scalar product [START_REF] Duan | Accuracy and sensitivity evaluation of tfr method for leak detection in multiple-pipeline water supply systems[END_REF], performing a double integration by parts leads to

HΨ, Ψ = C 2 Y • ∂ 2 Z Ψ, Ψ = Ψ, C 2 Y • ∂ 2 Z Ψ + 4 j=1 η j c 2 j ∂ Z Ψ j (Z)Ψ j (Z) -Ψ j (Z)∂ Z Ψ j (Z) 1 0 , (26)
where c 2 j are the j th diagonal terms of C 2 Y . Self-adjoint property HΨ, Ψ = Ψ, HΨ , is thus obtained from condition

4 j=1 η j c 2 j ∂ Z Ψ j (Z)Ψ j (Z) -Ψ j (Z)∂ Z Ψ j (Z) 1 0 = 0. ( 27
)
In the following, the scalar-product weight parameter η is adjusted with given boundary conditions set so as to ensure relation eq [START_REF] Tijsseling | Exact solution of linear hyperbolic four-equation system in axial liquid-pipe vibration[END_REF], and thus selfadjointness.

Eigenfunction base and spectrum condition

The self-adjoint operator H spectrum S P , is composed of real discrete eigenvalues having a related discrete orthogonal basis [START_REF] Lewin | Eléments de théorie spectrale : le laplacien sur un ouvert borné[END_REF]. This property obviously remains for frequency up to the Korteweg's band stop. Denoting -λ 2 k the k th eigenvalue, then being real negative in accordance with the well known eigenvalues of the Laplacian , Φ k its related eigen-function, the eigenvalue problem reads

HΦ k (Z) = -λ 2 k Φ k (Z), (28)
so that using [START_REF] Che | Transient wave-based methods for anomaly detection in fluid pipes: A review[END_REF], one gets

∂ 2 Z Φ k (Z) = -λ 2 k C -2 Y Φ k (Z), (29)
where we have introduced notation

C -2 Y ≡ [C 2 Y ] -1 for the inverse of matrix C 2
Y defined in [START_REF] Keramat | Experimental investigation of transients-induced fluid-structure interaction in a pipeline with multiple-axial supports[END_REF]. The solution of ( 29) then achieves as follows

Φ k (Z) ∂ Z Φ k (Z) = T k (Z) Φ k (0) ∂ Z Φ k (0) (8×1) , (30)
with

T k (Z) = ∂ Z T(Z) T(Z) ∂ 2 Z T(Z) ∂ Z T(Z) , T(Z) = T s (Z) 0 0 T s (Z) , (31)
and

T s (Z, λ k ) = 1 λ k   c -sin λ k Z c - 0 0 c + sin λ k Z c +   . ( 32
)
This formulation is a modal time-domain version of the popular transfer matrix method, (TMM), developed in the frequency domain for coupled hyperbolic problems in [START_REF] Li | FSI research in pipeline systems -A review of the literature[END_REF][START_REF] Keramat | Transient-based leak detection in the frequency domain considering fluid-structure interaction and viscoelasticity[END_REF][START_REF] Keramat | Spectral based pipeline leak detection using a single spatial measurement[END_REF][START_REF] Lesmez | Modal Analysis of Vibrations in Liquid-Filled Piping Systems[END_REF][START_REF] Tijsseling | Exact solution of linear hyperbolic four-equation system in axial liquid-pipe vibration[END_REF][START_REF] Zhang | FSI Analysis of Liquid-Filled Pipes[END_REF][START_REF] Li | Frequency domain analysis of fluid-structure interaction in liquid-filled pipe systems by transfer matrix method[END_REF][START_REF] Li | Vibration analysis of pipes conveying fluid by transfer matrix method[END_REF] but for the adaption to the need for a Laplacian operator of two boundary conditions to be specified. Combining [START_REF] Li | Vibration analysis of pipes conveying fluid by transfer matrix method[END_REF] and [START_REF] Tijsseling | Fluid-structure interaction in case of waterhammer with cavitation[END_REF] in boundary condition set [START_REF] Duan | Transient-based frequency domain method for dead-end side branch detection in reservoir pipeline-valve systems[END_REF] provides the following linear condition to be fulfilled by mode amplitudes

[Φ k (0), ∂ Z Φ k (0)] N M Q∂ Z T(1) + R∂ 2 Z T(1) QT(1) + R∂ Z T(1) Φ k (0) ∂ Z Φ k (0) = 0. (33)
The trivial zero solution of (33) being irrelevant, the non-trivial solution necessitates a one-dimensional non-empty kernel of (33)'s matrix, i.e. a zero eigenvalue of the matrix acting on [Φ k (0), ∂ Z Φ k (0)] vector. This condition is equivalent to set a zero determinant of (33)'s matrix, i.e.

N M Q∂ Z T(1) + R∂ 2 Z T(1) QT(1) + R∂ Z T(1) = 0. (34)
(34) is met for specific values of λ k , providing the spectrum S P of operator H. It leads to a transcendental equation for λ k specific to each boundary condition set, to be computed numerically, as done in section 4. The spectrum provides each resonant frequency of the system, which should lie in the frequency range given in (8

), i.e. λ k ∈ [-f Kc /f 0 , f Kc /f 0 ] . It remains to determine the modal-dependent amplitudes of Φ k (Z).
From [START_REF] Li | FSI research in pipeline systems -a review of the literature[END_REF] one can realize that the amplitude vector [Φ k (0), ∂ Z Φ k (0)] is defined up to any arbitrary multiplicative constant, as the kernel of (33)'s matrix is non-empty.

Hence, among the eight amplitude parameters of eight-dimensional vector [Φ k (0), ∂ Z Φ k (0)], one can be kept to any arbitrary value, which is equivalent to chose a unitary eigenfunction Φ k (Z) such as

Φ k (Z) ∂ Z Φ k (Z) = 1 Φk (Z) Φk (Z) ∂ Z Φk (Z) , (35)
where Φk (Z) stands as the generator of the solution space associated with the linear system (33).

Solution for 2D-vector homogeneous wave equation

The solution of the two-waves equations ( 20) associated with the initial boundary conditions [START_REF] Keramat | Transient-based leak detection in the frequency domain considering fluid-structure interaction and viscoelasticity[END_REF] are searched under decomposition

Y(Z, τ ) = Y h (Z, τ ) + Y p (Z, τ ), ( 36
)
where subscript h refers to homogeneous solution whilst subscript p refers to particular one associated with non-homogeneous boundary conditions. The latter is regarded as a separated variable time-space function. The Z behavior is furthermore decomposed into the first order polynomial

Y p (Z, τ ) = ZY 1 p (τ ) + Y 0 p (τ ). ( 37
)
Since the previous expression should ensures the boundary condition system spelled out in [START_REF] Keramat | Spectral based pipeline leak detection using a single spatial measurement[END_REF], it consequently results

N M Q Q + R (8×8) Y 0 p (τ ) Y 1 p (τ ) (8×1) = S (8×1) (τ ). ( 38
)
The resolution of ( 38) then provides Y p . The homogeneous component, Y h , is hereby decomposed over the eigenvector base of the self-adjoint operator

H so that Y = Sp a k (τ )Φ k (Z) + Y p (Z, τ ), (39)
or, using Φk (Z) in ( 35)

Y(Z, τ ) = Sp a k (τ ) Φk (Z) Φk (Z) + Y p (Z, τ ), ( 40
)
where a k (τ ) is the k th mode time-dependent amplitude. Invoking the initial rest conditions [START_REF] Keramat | Transient-based leak detection in the frequency domain considering fluid-structure interaction and viscoelasticity[END_REF], with definition of Φk (Z) in [START_REF] Lin | Wave Propagation through Fluid Contained in a Cylindrical, Elastic Shell[END_REF], leads to,

a k (0) = - Y p (Z, 0), Φk (Z) Φk (Z) , and, ∂ τ a k (0) = - ∂ τ Y p (Z, 0), Φk (Z) Φk (Z) . (41)
The proposed decomposition of Y(Z, τ ) in ( 40) must hereby satisfy the wave equation system [START_REF] Wang | Spectral-based methods for pipeline leakage localization[END_REF]. Regarding the definition of H in [START_REF] Che | Transient wave-based methods for anomaly detection in fluid pipes: A review[END_REF] with the spatial polynomial decomposition of Y p in (37) it achieves as follows

HY p = 0, (42)
so that

∂ 2 τ -H S P a k (τ ) Φk (Z) Φk (Z) = -∂ 2 τ Y p . (43)
Now projecting over Φk (Z) leads to,

∂ 2 τ + λ 2 k a k (τ ) = -∂ 2 τ Y p , Φk (Z) Φk (Z) . (44)
This ordinary differential equation, having initial conditions [START_REF] Bayle | Low Mach number theory of pressure waves inside an elastic tube[END_REF], is solved leading to (Further details provided in Appendix C)

a k (τ ) = λ k τ 0 Y p , Φk (Z) (t) Φk (Z) sin (λ k (τ -t)) dt - Y p , Φk (Z) Φk (Z) . ( 45
)
Finally, combining the (45) with the Y's definition in [START_REF] Mei | Effects of thin plaque on blood hammer-An asymptotic theory[END_REF] results in

Y(Z, τ ) = S P λ k τ 0 Y p , Φk (Z) (t) Φk (Z) 2 sin (λ k (τ -t)) dt Φk (Z) - S P Y p , Φk (Z) Φk (Z) 2 Φk (Z) + Y p (Z, τ ). ( 46
)
In the following, (46) is used to cross-check/validate predictions of two wellknown configurations of liquid filled pipes problem: (i) a constant pressure tank with a fixed instantaneous closing valve (Cf Fig. 1a), (ii) a constant pressure tank with a free instantaneous closing valve (Cf Fig. 1b). The related upstream and downstream boundary conditions are analyzed in depth in [START_REF] Tijsseling | Fluid-structure interaction in case of waterhammer with cavitation[END_REF]. It is important to mention that, in the special case of boundary conditions having no pressure/velocity coupling, the previous general framework associated with four coupled waves equations boils down into two decoupled two-waves propagation, not only sharing the same spectrum, but also the same modes and amplitudes. In other words, in the absence of coupling between pressure and velocity in the boundary conditions, the four-waves equations set degenerates into a two-waves one. In the next section, we consider this simplified degenerate case for two classes of boundary conditions.

ρ f = 1000 kg • m 3 ρ s = 7900 kg • m -3 K = 2.1 GP a E = 210 GP a ν f = 9.493 • 10 -7 m 2 s -1 ν s = 0.3 R 0 = 0.395 m e = 0.008 m L = 20 m
Table 1: Physical and geometrical properties for the analysis of the reservoir-pipe-valve system (anchored and free). The parameter values are extracted from [START_REF] Tijsseling | Exact solution of linear hyperbolic four-equation system in axial liquid-pipe vibration[END_REF].

All comparisons and analysis are based upon the parameter set introduced in Table 1. The natural fluid-pulse frequency of both configurations is found equal to f 0 ≈ 8Hz, whereas the cutoff frequency is f Kc ≈ 2060Hz. The λ k should then lie in the range λ k ∈ [-247.6, 247.6] in order to fulfill the modeling assumptions (Cf. section 2.1).

Analytical solution

In the first configuration (Cf. Figure 1a), the pipe is supposed perfectly anchored, both upstream and downstream. Thus, no solid axial movement occurs at both pipe dead-ends which can be modeled as a Dirichlet-Dirichlet condition on the axial solid displacement acceleration. Furthermore, the reservoir does not impede any upstream pressure fluctuation which can be interpreted as an homogeneous Dirichlet condition for the pressure. Finally, an instantaneous valve closure downstream is modeled by a Dirac distribution, δ(τ ) (i.e. singular perturbation), acting on the axial fluid acceleration. The four boundary condition thereby achieves as follows

P (0, τ ) = 0 , ∂ τ W (1, τ ) = -δ(τ ), ∂ τ ζ(0, τ ) = ∂ τ ζ(1, τ ) = 0, ( 47
)
or otherwise regarding ( 4)&( 6)

P (0, τ ) = 0 , ∂ Z P (1, τ ) = δ(τ ), ∂ Z σ(0, τ ) = ∂ Z σ(1, τ ) = 0. ( 48
)
The initial Dirichlet conditions upon both fluid and solid acceleration quantities thus turn into Neumann conditions for the fluid pressure and solid stress fields. Invoking base-change ( 17)-( 19), whilst introducing

β = c + c - c 2 --1 c 2 + -1 , (49)
the matrices introduced in ( 22) describing boundary conditions in the diagonalized basis can be found explicitly

N = 1 βc - c + 0 0 , M = 0 0 1 1 , Q = 0, R = N + M, (50)
and,

S(τ ) = c 2 --1 δ(τ ) 2ν s D (0, 0, 1, 0) T , (51)
The determination of the scalar-product weight parameter η ≡ (η 1 , η 2 ), introduced in (27), is hereby overcame. Injecting boundary conditions (48) within self-adjointness one [START_REF] Tijsseling | Exact solution of linear hyperbolic four-equation system in axial liquid-pipe vibration[END_REF] for the couple of two 2D unknown column vectors

Ψ(Z), Ψ (Z) = (ψ 1 (Z), ψ 2 (Z)) T , ψ 1 (Z), ψ 2 (Z) T , (52)
yields

∂ Z Ψ 1 (0)Ψ 1 (0) -Ψ 1 (0)∂ Z Ψ 1 (0) η 1 + η 2 c 3 + βc 3 - = 0, (53)
so that the relation between η 1 & η 2 can be found

η 2 = -η 1 β c - c + 3 . ( 54
)
Scalar product [START_REF] Duan | Accuracy and sensitivity evaluation of tfr method for leak detection in multiple-pipeline water supply systems[END_REF] then results in

Ψ, Ψ = η 1 1 0 Ψ 1 (z)Ψ 1 (z) -β c - c + 3 Ψ 2 (z)Ψ 2 (z) dz. ( 55
)
The λ k are hence solutions of the formal relation [START_REF] Li | FSI research in pipeline systems -a review of the literature[END_REF]. Using C 2 P in ( 19), boundary condition matrices (50) and T s (Z) in [START_REF] Tijsseling | Fluid-structure interaction in case of waterhammer with cavitation[END_REF], leads to the explicit (and simplified) transcendental equation

β sin λ k c - cos λ k c + -sin λ k c + cos λ k c - = 0. ( 56
)
The particular component of P(Z, τ ), P p (Z, τ ), follows from the spatial polynomial decomposition [START_REF] Ghidaoui | A review of water hammer theory and practice[END_REF] and the resolution of the linear system [START_REF] Keramat | Fluid-structure interaction with pipe-wall viscoelasticity during water hammer[END_REF], whilst Φk (Z) determination is found combining transfer matrix formulation [START_REF] Li | Vibration analysis of pipes conveying fluid by transfer matrix method[END_REF], linear boundary condition system [START_REF] Chaudhry | Applied Hydraulic Transients[END_REF] and the associated boundary condition matrices ( 50)

Φk (Z) =   cos λ k c -Z + tan λ k c -sin λ k c -Z -c + βc -cos λ k c + Z + tan λ k c + sin λ k c + Z   , (57)
P p (Z, τ ) = δ(τ )Z det (Π P ) 1 -1 . ( 58
)
Further details on the derivation are provided in Appendix D. The combination of Φk (Z) and P p (Z, τ ) vectors in (57)-(58) fulfill the expression P(Z, τ ) in ( 46)

P(Z, τ ) -P p (Z, τ ) = S P Z 1 -1 , Φk (Z) det (Π P ) Φk (Z) 2 Φk (Z) [λ k sin (λ k τ ) -δ(τ )] , (59)
with,

Φk (Z) 2 = c + β cos 2 λ k c + -c -cos 2 λ k c - 2c + β cos 2 λ k c + cos 2 λ k c - , (60)
Z 1 -1 , Φk (Z) = c 2 -cos λ k c + -cos λ k c - λ 2 k cos λ k c -cos λ k c + . (61)
4.1.2. ν s parametric analysis Since the Poisson coupling, relies on the Poisson modulus, it is relevant to investigate the ν s dependency of the previous analytical expressions. By performing an asymptotic analysis, without considering FSI effects, [START_REF] Mei | Pressure and wall shear stress in blood hammer -Analytical theory[END_REF][START_REF] Kuiken | Amplification of pressure fluctuations due to fluidstructure interaction[END_REF] found the spectrum associated with the pressure waves in the fluid. It is interesting to point out that the solution (59) converges toward the [START_REF] Mei | Pressure and wall shear stress in blood hammer -Analytical theory[END_REF]'s leading-order one, as ν s → 0. The [START_REF] Mei | Pressure and wall shear stress in blood hammer -Analytical theory[END_REF]'s leading-order solution is

λ k = π 1 + k 2 , ∀k ∈ N, (62)
P (Z, τ ) = 2 ∞ k=0 (-1) k sin (λ k Z) sin (λ k τ ) λ k . (63)
A more detailed derivation of this result is provided in Appendix E. The ν s dependence of both spectrum and pressure waves compared with no-FSI solutions (i.e ν s = 0 case) is further illustrated in Figure 2. Figure 2b shows that the spectrum of the first configuration can be interpreted as the union between the fluid vibration modes (continuous blue lines) and the pure elastic ones (red squared lines). The solid and fluid contributions to the spectrum are easily identified from the analysis of the no-(FSI) configuration, as theoretically detailed in Appendix E (Cf. (E.10)). In Figure 2a the (59) solution for ν s = 0 is confronted with the [START_REF] Mei | Pressure and wall shear stress in blood hammer -Analytical theory[END_REF]'s leading order one and, for comparison with the ν s = 0.3 case, all other parameters being equals. One can observed that high-frequency pressure oscillations progressively growing in time are revealed by the FSI coupling. These high-frequency oscillations are nevertheless expected to be damped by the viscous fluid friction or some structural energetic losses. In the ν s → 0 limit, [START_REF] Mei | Pressure and wall shear stress in blood hammer -Analytical theory[END_REF]'s pointed-out that fluid viscous friction exponentially damps each resonant mode. This feature remains when 1a). In (a), the [START_REF] Mei | Pressure and wall shear stress in blood hammer -Analytical theory[END_REF]'s leading order solution is provided, in black dashed lines, as to point out the convergence of the models. In (b), as ν s vary, some eigenvalues come close one-another, but a careful inspection shows no cross-over between the depicted eigenvalues.

considering (FSI) interactions, as analyzed in [START_REF] Bayle | Low Mach number theory of pressure waves inside an elastic tube[END_REF]. When dealing with timescale of the order of the advective pulse wave speed one, i.e. O (L/c p ), the energetic losses are shown [START_REF] Budny | The Influence of Structural Damping on Internal Pressure During a Transient Pipe Flow[END_REF][START_REF] Mei | Pressure and wall shear stress in blood hammer -Analytical theory[END_REF] to have a negligible impact on the overall coupled dynamic.

Constant pressure tank with free instantaneous closing valve

A second configuration, depicted in Figure 1b, is hereby analyzed whereby the downstream valve is free to move axially. Hence, upstream, the same conditions as in (48) are applied, with homogeneous Neumann condition for the axial stress and homogeneous Dirichlet condition for the pressure field set as P (0, τ ) = 0 , and, ∂ Z σ(0, τ ) = 0. (64) Downstream, the boundary condition strongly differs from the previous configuration from valve longitudinal motion. Since α(2+α) is equal to the ratio of solid surface (i.e. πeR 0 (2 + α)) to the fluid one (i.e. πR 2 0 ), the static equilibrium of forces at valve location, in the absence of valve's inertia, reduces to α(2

+ α)σ(1, τ ) = P (1, τ ). ( 65
)
The axial solid and fluid acceleration matching at the valve further imposes

∂ τ W (1, τ ) -α∂ τ ζ(1, τ ) = -δ(τ ), (66)
or otherwise regarding relations ( 4) & ( 6)

∂ Z P (1, τ ) + D∂ Z σ(1, τ ) = δ(τ ). ( 67
)
The α constant in (66) arises from dimensionless arguments and can be obtained regarding the Appendix B. As the valve is now free to move, the downstream boundary condition slightly differs form the one provided in (47). Indeed, the acceleration perturbation is now applied to the relative acceleration of the fluid with respect to the tube motion, [START_REF] Tijsseling | Fluid-structure interaction in case of waterhammer with cavitation[END_REF]. Considering change-basis ( 17)-( 18) and introducing parameters

κ ± = D + 2ν s D c 2 ± -1
, and,

βc -κ - c + κ + = 1 - 2νsD α(2+α)(c 2 + -1) 1 - 2νsD α(2+α)(c 2 --1) , (68)
yields

N = 1 βc - c + 0 0 , M = 0 0 1 1 , Q = 0 0 1 βc -κ - c + κ + , R = κ -κ + 0 0 , (69)
with the forcing term S(τ ) = δ(τ ) (0, 0, 1, 0) T (70)

The non-trivial parametric relation (68) is established in Appendix F. The same footsteps are hereby applied to overcome the resolution of P(Z, τ ). The self-adjoint condition (27) leads to

η 1 + c + c - 3 η 2 β ∂ Z Ψ 1 (Z)Ψ 1 (Z) -Ψ 1 (Z)∂ Z Ψ 1 (Z) 1 0 = 0, (71)
so that

η 2 = -η 1 β c - c + 3 . (72)
Thereby, scalar product defined in (25) remains identical to (55). A combination boundary condition matrices (69), the spectrum condition (34) and transfer matrices ( 31)-( 32) expressions, leads to the following (simplified) transcendental equation

1 + κ - κ + 2 cos λ k c + cos λ k c - + 1 β 1 + βκ - κ + 2 sin λ k c + sin λ k c - = 2κ - κ + . (73)
Finally, one can find Φk (Z) and P p (Z, τ ) vectors fields using boundary condition system [START_REF] Keramat | Spectral based pipeline leak detection using a single spatial measurement[END_REF] and boundary condition matrices (69), leading to (detailed derivation provided in Appendix G),

Φk (Z) =   cos λ k c -Z + ξ k sin λ k c -Z -c + βc -cos λ k c + Z + βξ k sin λ k c + Z   , (74)
P p (Z, τ ) = δ(τ ) κ --κ + 1 -1 Z -γ 1 -c + βc - , (75) with,
ξ k = sin λ k c --κ + βκ -sin λ k c + cos λ k c --κ + κ -cos λ k c + , and, γ = κ -c -β -c + κ + c + (κ --κ + ) . (76)
Since, according to the scalar product (55) and ( 72)

Φk (Z) 2 = - c -+ c -β 2 ξ 2 k -c + β (ξ 2 k + 1)) 2c + β + c -ξ k cos 2 λ k c + -cos 2 λ k c - λ k + c - β 2 ξ 2 k -1 β sin 2λ k c + -(ξ 2 k -1) sin 2λ k c - 4λ k , (77)
and

P p (Z, τ ), Φk (Z) = c -δ(τ ) κ --κ + F -- c -F + c + -γ [G + -G -] (78)
F ± = cos λ k c± [c ± -θ ± λ k ξ k ] + sin λ k c± [λ k + c ± θ ± ξ k ] -c ± λ 2 k , (79)
G ± = θ ± ξ k cos λ k c± -1 -sin λ k c± θ ± λ k , (80)
θ -= 1, θ + = β, (81)
the expression of P(Z, τ ) in (46) pursues

P(Z, τ ) -P p (Z, τ ) = c - S P F --c -F + c + -γ [G + -G -] (κ --κ + ) Φk (Z) 2 Φk (Z) [λ k sin (λ k τ ) -δ(τ )] . (82)
The above expressions (77)-( 81) have been cross-checked using formal calculus softwares.

Comparisons and illustrations

We now discuss quantitative illustrations of the provided analytical solutions so as to demonstrate their matching with previously published numerical results. Since many previous contributions have considered Fourier representation of test cases, our theoretical predictions for the discrete spectrum associated with discrete frequency peaks are first analyzed. The spectrum formally given by general relation [START_REF] Li | FSI research in pipeline systems -a review of the literature[END_REF] and more specifically by explicit simple transcendent relations (56) and ( 73) are compared to the Fourier transform peaks in Figure 3. The determinant's zeros, λ k , are classically related to the natural frequencies accordingly to the following linear law, [START_REF] Zhang | FSI Analysis of Liquid-Filled Pipes[END_REF]

f k = λ k f 0 , (83)
where f k is the k th natural frequency expressed in Hz and f 0 is the natural acoustic fluid frequency introduced in (8). A perfect matching of the predicted peaks (dotted lines) with the numerical one can be observed in Figure 3 . More precisely, Figure 3a compares [START_REF] Yang | Longitudinal vibration analysis of multispan liquid-filled pipelines with rigid constraints[END_REF]'s results with our prediction for boundary conditions (i) (Cf Fig. 1a) whereas Figure 3b consider [START_REF] Zhang | FSI Analysis of Liquid-Filled Pipes[END_REF]'s prediction for boundary conditions (ii) (Cf Fig. 1b). These theoretical predictions are matched to numerical results without any adjusted parameters. More quantitative comparison between the numerical peaks and their theoretical predictions are provided in table 2 with excellent agreement. Temporal predictions are also compared with MOC numerical results so as to test every details of the analytical solutions, i.e. not only the spectrum but also the eigenmodes and their amplitudes. Figure 4 provides this comparison with [START_REF] Tijsseling | Exact solution of linear hyperbolic four-equation system in axial liquid-pipe vibration[END_REF]'s MOC solutions for the set of parameters presented in Table 1. In each case a nearly perfect matching between the theoretical prediction and the numerical computation can be observed. Zoom are provided for highfrequency fidelity check. The resulting mismatch might be attributable to Comparison between [START_REF] Yang | Longitudinal vibration analysis of multispan liquid-filled pipelines with rigid constraints[END_REF]'s prediction at the valve location with eigenvalues obtained from transcendental equation ( 56). (b) comparison between [START_REF] Zhang | FSI Analysis of Liquid-Filled Pipes[END_REF]'s pressure prediction at the valve location with prediction from transcendental equation (73).

First boundary condition (i) (Cf Fig. 1a) Second boundary condition (ii) (Cf Fig. 1b) [START_REF] Yang | Longitudinal vibration analysis of multispan liquid-filled pipelines with rigid constraints[END_REF] (Hz) Theorical (Hz) ∆ (%) [START_REF] Zhang | FSI Analysis of Liquid-Filled Pipes[END_REF] (Hz) Theorical (Hz) ∆ (%) the inaccuracy of data collection from [START_REF] Tijsseling | Exact solution of linear hyperbolic four-equation system in axial liquid-pipe vibration[END_REF]. No parameter adjustment have been used.

Sensitivity matrix evaluation

Wave speed, c ± , deviations

From the analytical expressions of c ± in [START_REF] Kizilova | Pressure wave propagation in liquid-filled tubes of viscoelastic material[END_REF], the respective derivation of the dimensionless wave speeds are deduced upon dimensionless variables describing the fluid structure interactions. The parametric dependence of c ± E K , ν s , D, α is known so that any derivative ∂c ± ∂X (X being any structural parameter embedded in parameters E K , ν s , D in (3) and α) in ( 1) is possible to compute analytically. In order to simplify the algebraic expressions the squared wave speeds [START_REF] Kizilova | Pressure wave propagation in liquid-filled tubes of viscoelastic material[END_REF] derivatives are evaluated

∂c 2 ± ∂α = µ c 1 + α α (2 + α)   1 + ± 2ν 2 s -1 + 2ν s D + µ a (c 2 + + c 2 -) 2 -4C 2 s   , ( 84
)
∂c 2 ± ∂ E K = D 2   1 ± µ a + 2ν s D -1 (c 2 + + c 2 -) 2 -4C 2 s   (85) ∂c 2 ± ∂ν s = 1 2   2D ± 4ν s (D 2 -µ c ) + 4D µa-1 2 -1 (c 2 + + c 2 -) 2 -4C 2 s   , ( 86
)
∂c 2 ± ∂ E K = D 2   1 ± µ a + 2ν s D -1 (c 2 + + c 2 -) 2 -4C 2 s   , ( 87
)
∂c 2 ± ∂D = 1 2D   µ a + 2ν s D ± (1 + µ a + 2ν s D) (µ a + 2ν s D) -2C 2 s (c 2 + + c 2 -) 2 -4C 2 s   ,( 88
)
where µ a = D 2 + E K + 4 α(2+α) and, µ c = -4D α(2+α) are defined in (E.2)-(E.4). From (84)-(88) the velocity derivatives can easily be deduced from

∂ X c ± = 1 2c ± ∂ X c 2 ± . (89)

Sensitivity matrix for boundary condition (i)

From (56) it is possible to find an analytical expression of the sensitivity of the eigenvalues λ k with respect to the parameters set (ν s α, E/K, D). Using (49) one gets

∂β ∂X = ∂β ∂c - ∂c - ∂X + ∂β ∂c + ∂c + ∂X , (90)
∂β ∂c ± = ∓ β c ± c 2 ± + 1 c 2 ± -1 , (91)
whilst from (56) one gets

∂λ k ∂X = -c -∂ X β tan λ k c - + λ k c - β∂ X c -- c - c + 2 ∂ X c + + tan λ k c - tan λ k c + ∂ X c --β c - c + 2 ∂ X c + β - c - c + + tan λ k c - tan λ k c + 1 -β c - c + . (92)
The spectrum sensitivity of the first configuration (Cf. Fig. 1a) is depicted in Figures 5a-5d, for the first five eigenmodes. The sensitivity analysis of the first configuration reveals an increased parametric dependence for higher modes. The higher the mode, the more sensitive it is to the dimensionless parameters variation. Furthermore, this analysis highlights a high sensitivity of the first five eigenmodes with respect to the density ratio D as depicted on Figure 5d, whilst the system is found weakly dependent on the E/K ratio, as illustrated in Figure 5c.

Sensitivity matrix for boundary condition (ii)

A similar footpath is hereby provided for the reservoir-pipe-free valve configuration. The spectrum transcendental equation derived in (73), which holds for the second configuration (Cf Fig. 1b), is found dependent upon a set of four dimensionless parameters namely c ± , β and

κ(c ± , ν s ) = κ - κ + ≡ c + βc - c 2 --(1 -2ν s ) c 2 + -(1 -2ν s ) . ( 93
)
The derivative of κ with respect to the dimensionless quantity X ≡ (ν s , α, E/K, D) thus achieves as follows

∂ X κ = c + c -β ∂ X κ r + κ r 1 c + ∂ X c + - 1 c - ∂ X c -- 1 β ∂ X β (94)
κ r = c 2 --(1 -2ν s ) c 2 + -(1 -2ν s ) , (95)
∂ X κ r = 2 c 2 + -(1 -2ν s ) c -c + 1 c + ∂ X c -- κ r c - ∂ X c + + ∂ X ν s (1 -κ r ) , (96)
whereas from (73) one gets

∂ X λ k = - 2κ∂ X κ tan λ k c+ 1+(βκ) 2 βc- -1+κ 2 c+ + tan λ k c- 1+(βκ) 2 βc+ -1+κ 2 c- + tan λ k c+ tan λ k c- 1+(βκ) 2 β 2 ∂ X β -2κ [β∂ X κ + κ∂ X β] tan λ k c+ 1+(βκ) 2 βc- -1+κ 2 c+ + tan λ k c- 1+(βκ) 2 βc+ -1+κ 2 c- + 2 ∂ X κ cos λ k c-cos λ k c+ tan λ k c+ 1+(βκ) 2 βc- -1+κ 2 c+ + tan λ k c- 1+(βκ) 2 βc+ -1+κ 2 c- +λ k tan λ k c+ 1 c-∂ X c - 1+(βκ) 2 βc- -1 c+ ∂ X c + 1+κ 2 c+ + tan λ k c- 1 c+ ∂ X c + 1+(βκ) 2 βc+ -1 c-∂ X c - 1+κ 2 c- tan λ k c+ 1+(βκ) 2 βc- -1+κ 2 c+ + tan λ k c- 1+(βκ) 2 βc+ -1+κ 2 c- . ( 97
)
The derivative of the five first eigenmodes is once again investigated and their related variations with respect to (ν s , α, E/K, D) are depicted in Figure 6a-6d. A very same sensitivity ordering is found for the second configuration. The density ratio variations remain the most sensitive parameter whereas the E K ratio has little impact on the coupled dynamic. It is interesting to point-out that the higher eigenmodes are no longer the most sensitive in this second case, as illustrated in Figures 6a-6b. The boundary condition couplings, or junction couplings, occurring in the second configuration, then reshape the eigenmode structure and their sensitivity.

Conclusion

This contribution provide a theoretical framework for the analysis of analytical solutions for FSI pulsed waves propagation inside liquid filled tubes. This framework leads to an explicit spectrum derivation and analytical eigenmode decomposition. It favorably compares with previously published results. The spectrum sensitivity matrix has been computed explicitly with respect to (dimensionless) parameter derivatives, as well as wave velocity derivatives. The provided solutions might be used in various contexts associated with signal processing interpretation, parameter identification or boundary condition de-convolution inside liquid filled pipes.

Constant ratio

Appendix B. Dimensionless FSI four equations [START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF] provides a derivation of well-known FSI four-equations system. From averaged mass and momentum conservation equations within fluid and solid, [START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF] founds the coupled hyperbolic equations for longitudinal fluid velocity W * , pressure P * , longitudinal solid deformation velocity ζ * and longitudinal stress σ * ,

∂ t W * = - 1 ρ f ∂ z P * , (B.1) ∂ z W * + 1 ρ f c 2 p ∂ t P * = 2ν s ∂ z ζ * , (B.2) ∂ t ζ * = 1 ρ s ∂ z σ * , (B.3) ∂ t σ * -E∂ z ζ * = 2ν s α (2 + α) ∂ t P * , (B.4)
where subscript * refers to dimensional quantities. The velocity perturbation W * reference amplitude is set as W 0 , so that as W * = W 0 W . The pressure perturbation P * , is known from [START_REF] Joukowsky | Uber den hydraulischen stoss in wasserleitungsro hren.(on the hydraulic hammer in water supply pipes) mé moires de l'académie impériale des sciences de st.-petersbourg[END_REF]'s theory to match the dynamic pulse overpressure so that P * = ρ f c p W 0 P . From stress continuity at the tube wall, the axial perturbed stress, σ * is prescribed having the same order of magnitude as P * , i.e. σ * = ρ f c p W 0 σ. The axial displacement field, ζ * , is set as ζ * = α M R 0 ζ in order to ensure both axial velocity matching at the tube's wall and the small strain and small displacement hypothesis framework. Furthermore, the physical time, t, is scaled on the advection time, t = L cp τ whilst axial longitudinal scale is set as z = LZ. Using these scaling within (B.1)-(B.4), leads to [START_REF] Tijsseling | Water hammer with fluid-structure interaction in thickwalled pipes[END_REF].

Appendix C. Resolution of the constitutive ODE in a k (τ ) Let's consider the ODE ∂ 2 τ + λ 2 k a k (τ ) = - ∂ 2 τ Y p , Φk (Z) Φk , (C.1)
and its initial conditions

a k (0) = - Y p (Z, 0), Φk (Z) Φk , and, ∂ τ a k (0) = - ∂ τ Y p (Z, 0), Φk (Z) Φk . (C.2)
A Laplace transform, hereby denoted L, approach is employed to solve (C.1). Introducing s, the conjugate variable of t and setting up hat notation for Laplace variables yields

s 2 + λ 2 k â(s) -sa k (0) -∂ τ a k (0) = -s 2 Ŷp , Φk (Z) Φk + s Y p (Z, 0), Φk (Z) + ∂ τ Y p (Z, 0), Φk (Z) Φk . (C.3) Inovking (C.2) reads to âk (s) = λ 2 k Ŷp , Φk (Z) (s 2 + λ 2 k ) Φk - Ŷp , Φk (Z) Φk . (C.4) Since L (sin (λ k τ )) = λ k s 2 +λ 2 k
, the use of convolution theorem finally results in

a k (τ ) = λ k τ 0 Y p , Φk (Z) (t) sin (λ k [τ -t]) dt Φk - Y p , Φk (Z) Φk . (C.5)
Appendix D. Derivation of Φk (Z) in configuration (i)

Let's focus on the first configuration boundary matrices (50) along with the homogeneous system upon (Φ k (0), ∂ Z Φ k (0)) T in [START_REF] Chaudhry | Applied Hydraulic Transients[END_REF]. Introducing the fourth unknown column vector

Φ k (0) ∂ Z Φ k (0) = φ - k , φ + k , ∂ Z φ - k , ∂ Z φ + k T (D.1) yields         1 βc - c + 0 0 0 0 1 c - 1 c + - λ k sin λ k c - c - - λ k βc -sin λ k c + c 2 + cos λ k c - c - βc -cos λ k c + c 2 + - λ k sin λ k c - c - - λ k sin λ k c + c + cos λ k c - c - cos λ k c + c +             φ - k φ + k ∂ Z φ - k ∂ Z φ + k     = 0. (D.2)
A non homogeneous solution of this linear system follows from re-organizing the last two lines of (D.2) and achieves in

  φ + k ∂ Z φ - k ∂ Z φ + k   = φ - k     -c - βc - λ k c -tan λ k c - -c + βc - λ k c + tan λ k c +     . (D.3)
The use of (D.3) within the equations set (30)- [START_REF] Tijsseling | Fluid-structure interaction in case of waterhammer with cavitation[END_REF] in case of a separate pressure -velocity boundary conditions (i.e. T ≡ T s such as transfer matrices order are divided by two) results in

Φ k (Z) = φ - k   cos λ k Z c - + tan λ k c -sin λ k Z c - -c + βc -cos λ k Z c + + tan λ k c + sin λ k Z c +   , (D.4)
so that the reduced eigenfunction form Φk (Z) in [START_REF] Lin | Wave Propagation through Fluid Contained in a Cylindrical, Elastic Shell[END_REF], follows

Φk (Z) =   cos λ k Z c - + tan λ k c -sin λ k Z c - -c + βc -cos λ k Z c + + tan λ k c + sin λ k Z c +   . (D.5)
Appendix E.3. Pressure solution of configuration (i) in the ν s = 0 limit The pressure field for configuration (i) is obtained combining (59) with the change of basis ( 17)-( 19) and achieves in

P (Z, τ ) = c 2 + -1 (c 2 + -c 2 -) S P Z 1 -1 , Φk (Z) Φk (Z) 2 Φk (Z) • e 1 λ k sin (λ k τ ) + c 2 --1 (c 2 + -c 2 -) S P Z 1 -1 , Φk (Z) Φk (Z) 2 Φk (Z) • e 2 λ k sin (λ k τ ) . (E.11)
Using β in (49), Φk (Z) in (57), Φk (Z) 2 in (60) and Z 1 -1 , Φk (Z) in (61) leads to

c 2 ± -1 (c 2 + -c 2 -) Z 1 -1 , Φk (Z) Φk (Z) 2 Φk (Z) • e1 2 = ± c 2 --1 (c 2 + -c 2 -) 2c 2 + cos λ k c + cos λ k c - cos λ k c + -cos λ k c - λ 2 k c 2 + c 2 - c 2 --1 c 2 + -1 cos 2 λ k c + -cos 2 λ k c - cos λ k c ∓ Z ± c 2 --1 (c 2 + -c 2 -) 2c 2 + cos λ k c ± cos λ k c + -cos λ k c - λ 2 k c 2 + c 2 - c 2 --1 c 2 + -1 cos 2 λ k c + -cos 2 λ k c - sin λ k c ∓ sin λ k c ∓ Z , (E.12)
where upper/lower symbols ± & ∓ are combined with upper e 1 ≡ 1 0 /lower e 2 ≡ 0 1 symbol e1 2 . The ν s = 0 spectrum limit (E.10) is the union of two distinct subsets, the contribution of which in (E.11) is now discussed separately.

• Fluid spectrum contribution When ν s = 0 the fluid spectrum is λ kF = π 1 2 + k . Furthermore, when ν s 1, using (E.8), one finds cos 2 λ kF • Solid spectrum contribution When ν s = 0 the solid spectrum is λ kS = √ µ a π (1 + k). In this case, (E.12)'s denominators do not cancel each other so that the ν s 1 behavior is trivial, Reorganising (F.2) then follows

c 2 ± -1 (c 2 + -c 2 -) Z 1 -1 , Φk (Z) Φk (Z) 2 Φk (Z) • e1
κ - χκ + = c 2 + -1 c 2 --1 2 c 2 --1 + 2ν s c 2 + -1 + 2ν s c 2 --1 -2νsD α(2+α) c 2 + -1 -2νsD α(2+α)
.

(F.3)

Developping the term

c 2 ± -1 + 2ν s c 2 ± -1 - 2ν s D α(2 + α) = c 4 ± -c 2 ± 1 + 2ν s D α(2 + α) + 1 -2ν s + 1 + 2ν s D α(2 + α) -2ν s - 4ν 2 s D α(2 + α)
, (F.4) whilst using relation [START_REF] Walker | Pulse Propagation in Fluid-Filled Tubes[END_REF] for dimensionless wave speeds in (F.4) leads to

c 2 ± -1 + 2ν s • c 2 ± -1 - 2ν s D α(2 + α) = -c 2 ± -1 1 + 2ν s D α(2 + α) -2ν s -C 2 s - 4ν 2 s D α(2 + α)
, (F.5) so that using β definition (49) within (F.3) simplifies to

κ - χκ + = c 2 + -1 c 2 --1 = c + βc - . (F.6)
Finally, one finds

c -βκ - c + κ + = χ = 1 - 2νsD α(2+α)(c 2 + -1) 1 - 2νsD α(2+α)(c 2 - -1)
.

(F.7)

Appendix G. Derivation of Φk (Z) for second configuration (Cf. Fig. 1b)

Let's focus on the second configuration boundary matrices (69) along with the homogeneous system upon (Φ k (0), ∂ Z Φ k (0)) T in [START_REF] Chaudhry | Applied Hydraulic Transients[END_REF]. Using (D.1), it follows

       1 βc - c + 0 0 0 0 1 c - 1 c + - λ k κ -sin λ k c - c - - λ k κ + sin λ k c + c + κ -cos λ k c - κ + cos λ k c + cos λ k c - βc -κ - c + κ + cos λ k c + c -sin λ k c - λ k βc -κ -sin λ k c + κ +           φ - k φ + k ∂ Z φ - k ∂ Z φ + k    = 0. (G.1)
Using the first two lines simplifies to

-λ k c -sin λ k c --κ + βκ -sin λ k c + cos λ k c --κ + κ -cos λ k c + cos λ k c --κ - κ + cos λ k c + c - λ k sin λ k c --βκ - κ + sin λ k c + φ - k ∂ Z φ - k = 0, (G.2) and φ + k = - c + βc - φ - k , and, ∂ Z φ + k = - c + c - ∂ Z φ - k . (G.3)
Let's check out the following relation

sin λ k c --κ + βκ -sin λ k c + cos λ k c --κ + κ -cos λ k c + = - cos λ k c --κ - κ + cos λ k c + sin λ k c --βκ - κ + sin λ k c + . (G.4)
The (G.

∂ Z φ - k = λ k ξ k c - φ - k . (G.8)
Finally, using the expression of φ + k , ∂ Z φ + k , ∂ Z φ - k versus φ - k in (G.3)-(G.8) and the equations set (30)- [START_REF] Tijsseling | Fluid-structure interaction in case of waterhammer with cavitation[END_REF] in case of a separate pressure -velocity boundary conditions (i.e T ≡ T s such as transfer matrices order are divided by two) results in

Φ k (Z) = φ - k     cos λ k c -Z -c + βc -cos λ k c + Z   + ξ k   sin λ k c -Z -c + c -sin λ k c + Z   
 , (G.9) so that the reduced form of Φ k (Z), Φk (Z) is consequently

Φk (Z) =   cos λ k c -Z + ξ k sin λ k c - -c + βc -cos λ k c + Z + βξ k sin λ k Z c +   .
(G.10)

4 .

 4 Application to specific boundary conditions 4.1. Constant pressure tank with a fixed instantaneous closing valve (a) First configuration : anchored valve (b) Second configuration : axially free valve

Figure 1 :

 1 Figure 1: Boundary condition set investigated for the liquid filled pipe problem

  no-(FSI) to (FSI) Solid contribution Fluid contribution (b) Spectrum (28 first eigenvalues) versus the Poisson modulus νs in the Korteweg's frequency range.

Figure 2 :

 2 Figure2: Impact of the (FSI) on the expected: (a) pressure response and, (b) spectrum for the first case configuration (Cf. Fig.1a). In (a), the[START_REF] Mei | Pressure and wall shear stress in blood hammer -Analytical theory[END_REF]'s leading order solution is provided, in black dashed lines, as to point out the convergence of the models. In (b), as ν s vary, some eigenvalues come close one-another, but a careful inspection shows no cross-over between the depicted eigenvalues.

  Second configuration (Cf Fig.1b).

Figure 3 :

 3 Figure 3: Continuous black-lines : Fourier transform of the pressure field at valve location within the pipe versus frequency f . Blue dotted lines : spectrum eigenvalues. (a)Comparison between[START_REF] Yang | Longitudinal vibration analysis of multispan liquid-filled pipelines with rigid constraints[END_REF]'s prediction at the valve location with eigenvalues obtained from transcendental equation (56). (b) comparison between[START_REF] Zhang | FSI Analysis of Liquid-Filled Pipes[END_REF]'s pressure prediction at the valve location with prediction from transcendental equation (73).

  Second configuration (Cf Fig.1b).

Figure 4 :

 4 Figure 4: Comparison between pressure field analytical solution at middle pipe location (continuous blue lines) with MOC solutions (dotted black lines). Non-FSI solutions (i.e ν s = 0) are provided for illustration in brown dotted lines. Insets provide a zoom for careful check.

Figure 5 :

 5 Figure 5: Sensitivity analysis of the first five resonnant eigenmodes associated with the spectrum equation (56). (a) ∂ νs λ k . (b) ∂ α λ k . (c) ∂ (E/K) λ k . (d) ∂ D λ k . The numerical gradients are depicted in continuous lines whereas the analytical expressions are represented by dotted lines.

Figure 6 :

 6 Figure 6: Same conventions as Figure 5 for spectrum equation (73). (a) ∂ νs λ k . (b) ∂ α λ k . (c) ∂ (E/K) λ k . (d) ∂ D λ k .The numerical gradients are depicted in continuous lines whereas the analytical expressions are represented by dotted lines.

2 k

 2 Order of magnitude of the steady velocity m • s -1 Caracteristic velocities c p Modified Korteweg's wave speed / Fluid pulse wave speed m•s -1 Ratio of the inner tube's radius by the tube's length α Ratio of the tube's thickness by the inner tube's C s Ratio of the solid elastic wave speed by the fluid pulse wave speed D Ratio of fluid density by the solid density Theorical elements W (Z, τ ) Perturbed axial fluid velocity P (Z, τ ) Perturbed fluid pressure σ(Z, τ ) Perturbed axial fluid stress tensor ζ(Z, τ ) Perturbed axial solid displacement velocity H Self auto adjoint operator -λ Eigenvalue of H a k (τ ) Temporal amplitude of Y h f k Natural frequencies Characteristic matrices Y(Z, τ ) Perturbed dimensionless fluid and solid pressure-velocity vector P(Z, τ ) Perturbed dimensionless pressure -stress vector C P Speed matrix of the pressure-stress wave equation C W Speed matrix of the velocity-displacement velocity wave equation C Y Speed matrix coupled fluid and solid pressure-velocity wave equation Y Perturbed dimensionless fluid and solid pressure-velocity vector within the diagonalisation basis Y p (Z, τ ) Particular part of Y Y h (Z, τ ) Homogeneous part of Y P Perturbed dimensionless pressure -stress vector within the diagonalisation basis P h (Z, τ ) Homogeneous part of P P p (Z, τ ) Particular part of P C P Speed matrix within the diagonalisation basis C W Speed matrix within the diagonalisation basis C Y Speed matrix coupled fluid and solid pressure-velocity wave equation N , M, Q, R, S Boundary condition matrices within the diagonalisation basis Φ k ± , γ, ξ k Constants κ = κ - κ +

c 2 + - 1 2 + -1 c 2 + -c 2 - Z 1 - 1 , 2 kF+ 1 - 1 ,

 2122211211 cos 2 λ kF c + ∼ O (ν s ). The ν s 1, asymptotic behaviour of (E.12) can be obtained using (E.6), (E.7) and (E.8) and results in c Φk (Z) Φk (Z) 2 Φk (Z) • e 1 = (-1) k 2 sin (λ kF Z) λ O (ν s ) , (E.13) Φk (Z) Φk (Z) 2 Φk (Z) • e 2 = O (ν s ) . (E.14)

2 = 2 ∞ k=0 (- 1 )

 22k=01 O (ν s ) . (E.[START_REF] Kizilova | Pressure wave propagation in liquid-filled tubes of viscoelastic material[END_REF] Combining (E.8)-(E.15) in (E.11), finally leads to[START_REF] Mei | Pressure and wall shear stress in blood hammer -Analytical theory[END_REF]'s leading order solution lim νs→0 (P (Z, τ )) = k sin (λ kF Z) sin (λ kF τ ) the definition of κ ± in (68), it leads to κ -

Table 2 :

 2 Comparative analysis of natural frequencies for the two study configurations. For each encountered frequency, the relative error ∆(%) is estimated.

	13.1	13.00	0.8	12	12.4	3.2
	38.5	38.3	0.5	32	31.8	0.6
	64.0	63.8	0.3	56	55.5	0.9
	89.6	89.3	0.3	73	72.9	0.1
	115.1	114.6	0.4	97	96.6	0.4
	131.8	131.7	0.08	116	115.8	0.2
	141.3	140.8	0.3	141	140.5	0.4
	166.6	165.9	0.4	161	160.0	0.6
	192.1	191.4	0.4	185	183.9	0.6
				202	201.4	0.3
				226	224.9	0.5
				245	243.9	0.5

  which is the transcendent spectrum (73) satisfied by λ k . Introducing ξ k

		4) holds only if
	sin 2 λ k c -	-	βκ -κ +	+	κ + βκ -	sin	λ k c -	sin	λ k c +	+ sin 2 λ k c +	=
		-cos 2 λ k c -	+	κ -κ +	+	κ + κ -	cos	λ k c -	cos	λ k c +		-cos 2 λ k c +	, (G.5)
	or otherwise
	1 β	1 +	κ -β κ +	2	sin	λ k c -		sin	λ k c +	+ 1 +	κ -κ +	2	cos	λ k c -	cos	λ k c +
																=	2κ -κ +	, (G.6)
						ξ k =	sin λ k c --κ + βκ -sin λ k c + cos λ k c --κ + c + κ -cos λ k	,	(G.7)
	thus leads to

Acknowledgement

This work was supported by the collaborative ANRT Grant CIFRE 2019/1453 co-funded by SETOM, dedicated society of Veolia for the public drinking water service of Toulouse Métropole operating under the brand Eau de Toulouse Métropole. The authors have no competing interests to declare.

Appendix E. ν s = 0 limit of the first configuration (Cf. Fig. 1a)

We hereby consider the ν s = 0 limit of our solution and compare it with the one of [START_REF] Mei | Pressure and wall shear stress in blood hammer -Analytical theory[END_REF] (only its leading order).

Appendix E.1. Wave speed mode c ± for ν s 1 Using c 2 p , c 2 s and C 2 s definitions (2), one can find

with

Furthermore, accordingly to [START_REF] Kizilova | Pressure wave propagation in liquid-filled tubes of viscoelastic material[END_REF], the wave speed c ± reads

, (E.5) so that Taylor expanding (E.5) in the ν s 1 results in

Appendix E.2. Spectrum of configuration (i) in the ν s = 0 limit Using β definition (49) and regarding (E.7)-(E.8) in the limit ν s → 0, the spectrum transcendental equation (56) reduces to,

The solution of which are [START_REF] Mei | Pressure and wall shear stress in blood hammer -Analytical theory[END_REF]'s spectrum union pure elastic-wave eigenvalues,