Andrey Sarychev

Alexander Shuvalov

Marco Spadini

SURFACE SHEAR WAVES IN A HALF-PLANE WITH DEPTH-VARIANT STRUCTURE

Keywords: 2000 Mathematics Subject Classification. 74J15, 34B08, 34B24 functionally graded medium and surface shear waves and parametric Sturm-Liouville problem

We consider the propagation of surface shear waves in a halfplane, whose shear modulus and density depend continuously on the depth coordinate. The problem amounts to studying the parametric Sturm-Liouville equation on a half-line with frequency and wave number as the parameters. The Neumann (traction-free) boundary condition and the requirement of decay at infinity are imposed. The condition of solvability of the boundary value problem determines the dispersion spectrum in the wave number/frequency plane for the corresponding surface wave. We establish the criteria for nonexistence of surface waves and for the existence of a finite number of surface wave solutions; the number grows and tends to infinity with the wave number. The most intriguing result is a possibility of the existence of infinite number of solutions for any given wave number. These three options are conditioned by the asymptotic behaviour of the shear modulus and density close to infinite depth.

Introduction

In the physical context, the present paper is concerned with the problem of existence of surface shear waves in functionally graded semi-infinite media with a traction-free boundary. Surface acoustic waves find numerous applications in various fields extending from geophysics to microelectronics. Their localization near the surface (decay into the depth) makes them extremely advantageous in nondestructive material testing for detection of surface and subsurface defects (surface wave sensors). Small wavelength of surface waves enables their application in filters and transducers used in modern miniature devices [START_REF] Biryukov | Surface Acoustic Waves in Inhomogeneous Media[END_REF]. Functionally graded materials are the materials with coordinate-dependent properties. They may be of natural origin (e.g. bones), they may occur due to material aging, or they may be specially manufactured to realize desired physical behaviour [START_REF] Birman | Modeling and analysis of functionally graded materials and structures[END_REF][START_REF] Kennett | Seismic wave propagation in stratified media[END_REF]. The model of a functionally graded medium is widely used in seismology.

In the framework of our study, we consider shear acoustic waves in a functionally graded isotropic half-space with the shear modulus µ(y) and density ρ(y) depending on the depth coordinate y. We assume both functions ρ(y) and µ(y) to be continuous and positive; further assumptions are introduced in Sections 3,4.

It is known that, for generic couple (ω, k) of frequency and wave number, there are no surface wave solutions which satisfy both boundary traction-free (Neumann) condition and the decay condition at infinite depth. For many bi-parametric problems the set of admissible ω, k is known to be a union of a number of eigencurves [START_REF] Atkinson | Multiparameter Eigenvalue Problems. Sturm-Liouville Theory[END_REF]Ch.6]) in ωk-plane, called in the physical context dispersion curves. Our goal is to characterize the pairs (ω, k), for which the solutions of the described boundary value problems exist.

The situation is elementary, when ρ(y), µ(y) are constants, and is relatively uncomplicated, when ρ(y), µ(y) become constants on an interval [y s , +∞[. In Section 5 we briefly consider the latter homogeneous substrate case as a particular instance of our general treatment. There has been a number of studies, which either treat the problem asymptotically for large ω, k, or assume particular choice of functions ρ(y), µ(y), or their periodicity [START_REF] Achenbach | Antiplane surface waves on a half-space with depth-dependent properties[END_REF][START_REF] Ting | Existence of anti-plane shear surface waves in anisotropic elastic half-space with depth-dependent material properties[END_REF][START_REF] Xiaoshan | Transverse shear surface wave in a functionally graded material infinite half space[END_REF][START_REF] Shuvalov | Existence and spectral properties of shear horizontal surface acoustic waves in vertically periodic half-spaces[END_REF][START_REF] Shuvalov | Love waves in a coated vertically periodic substrate[END_REF]. We address the general case of unrestricted ω, k and any continuous ρ(y), µ(y), for which explicit solution of wave equation is unavailable.

The paper is organized as follows. Section 3 contains the auxiliary results. In Section 4 we formulate the corresponding parametric Sturm-Liouville problem on a half-line and introduce the assumptions for the material coefficients ρ(y), µ(y). Section 5 contains the formulations of the main results, which are the criteria for non-existence of surface waves (Theorem 5.1) and for the existence of N (k) surface wave solutions, with N (k) → ∞ as k → ∞ (Theorem 5.2). The most intriguing result is a possibility of the existence of infinite number of solutions, N (k) = ∞, for any given k (Theorem 5.3). These three options are conditioned by the properties of µ(y) and ρ(y). Section 6 contains the proofs of the above theorems.

Mathematical Setting of the Problem

We consider the 2D wave equation We make an assumption of M, ρ depending only on y and of M being a scalar matrix M = µ(y)Id.

Under the adopted assumptions equation (1) reads as

(4) ρ(y)û tt = µ(y)û xx + ∂ y (µ(y)∂ y û).
We will seek solutions of the form û(x, y) = u(y)e i(kx-ωt) .

Substituting û(x, y) into (4) and cancelling e i(kx-ωt) , one gets for u(y) the equation

ρ(y)u(y)(-ω 2 ) = µ(y)u(y)(-k 2 ) + ∂ y (µ(y)∂ y u(y)).
We denote the (total) derivative ∂ y by ′ arriving at the second-order linear differential equation

(5) (µ(y)u ′ (y)) ′ + (ω 2 ρ(y) -k 2 µ(y))u = 0.
The boundary conditions (2) and (3) formulated for u(y) become

u ′ (0) = 0, (6)
lim y→+∞ u(y) = 0. (7)
We assume both functions ρ(y) and µ(y) to be continuous and positive on [0, +∞[; further assumptions are introduced in Sections 3,4.

As it was already mentioned, for generic ω, k there are no solutions of (5), which satisfy both boundary conditions ( 6) and [START_REF] Xiaoshan | Transverse shear surface wave in a functionally graded material infinite half space[END_REF]. Our goal is to characterize the pairs (ω, k), for which the solutions of the boundary value problems ( 5)-( 6)-( 7) exist.

Second-order Linear Ordinary Differential Equation on a

Half-Line: Auxiliary Results

3.1.

Second-Order Linear Equation. Equation ( 5) is a particular type of the second-order linear differential equation

d 2 ū dτ 2 + γ(τ )ū(τ ) = 0, where ū(τ ) = u(y(τ )) and γ(τ ) = µ(y(τ ))γ(y(τ )).
Another form of ( 8) is its representation as a system of first-order differential equations for the variables u(y), w(y) = µ(y)u ′ (y):

u ′ (y) = w(y) µ(y)
, w ′ (y) = -γ(y)u(y), [START_REF] Bellman | Stability theory of differential equations[END_REF] or in the matrix form for Z = w u :

(12) Z ′ (y) = dZ dy = C(y)Z(y), C(y) = 0 -γ(y) (µ(y)) -1 0 .
Performing substitution (9), we transform (12) into the system for the function

Z(τ ) = Z(y(τ )) (13) d Z dτ = C(τ ) Z(τ ), C(τ ) = 0 -γ(τ ) 1 0 .
We concentrate for a moment on the asymptotic properties of the solutions of ( 8), ( 10), ( 12), (13) at infinity.

Asymptotic Properties of Solutions for y → +∞.

The matrix of the coefficients C(y) of the system [START_REF] Hartman | Ordinary differential equations[END_REF] for each y is traceless, hence, by the Liouville formula, the Wronskian of a fundamental system of solutions is constant in y. This precludes a possibility of having two independent solutions, which would both tend to zero at infinity. Important characteristics of the asymptotics of the system at infinity are determined by the limit of the coefficient matrix for y → +∞ (whenever it exists):

C ∞ = lim y→+∞ C(y) = 0 -γ ∞ (µ ∞ ) -1 0 ,
where µ ∞ = lim y→+∞ µ(y), γ ∞ = lim y→+∞ γ(y).

Whenever det C ∞ = γ ∞ (µ ∞ ) -1 > 0, or, equivalently, γ ∞ > 0, the eigenvalues of C ∞ are purely imaginary and one can conclude (see Proposition 3.4 below) the non-existence of a solution of system [START_REF] Hartman | Ordinary differential equations[END_REF] with lim y→+∞ u(y) = 0.

If on the contrary det C ∞ < 0, then the eigenvalues of C ∞ are real numbers of opposite signs and the existence of a solution of [START_REF] Hartman | Ordinary differential equations[END_REF] with lim y→∞ u(y) = 0 is guaranteed under some additional conditions on the functions µ(y), γ(y).

Note that det C∞ = µ 2 ∞ det C ∞ and therefore a similar conclusion holds for the solutions of system [START_REF] Arnold | The Sturm theorems and symplectic geometry[END_REF].

Later on we use a number of results which follow the quasi-classical or WKBapproximation paradigm ([10, Ch.2]). We formulate the results for equations [START_REF] Shuvalov | Existence and spectral properties of shear horizontal surface acoustic waves in vertically periodic half-spaces[END_REF] or [START_REF] Fedoryuk | Asymptotic Analysis[END_REF].

Let us introduce linear space G of the coefficients γ(y) of equations ( 8) as a space of functions γ(y) = γ ∞ + β(y), with γ ∞ being a constant and β(y) a continuous function on [0, +∞[ such that:

lim y→+∞ β(y) = 0, ( 14
) +∞ 0 |β(y)|dy < ∞. ( 15
)
Evidently

lim y→∞ γ(y) = γ ∞ .
Introduce in G the norm ( 16)

γ(•) 01 = |γ ∞ | + β(•) C 0 + β(•) L1 .
For each The first classical result regards the so called non-elliptic case for equation [START_REF] Fedoryuk | Asymptotic Analysis[END_REF], where the coefficient γ(•) ∈ G -(τ 0 ). Proposition 3.1 (see [11, §6.12]). . Consider the equation

y 0 ∈ [0, +∞[ we introduce a subset G -(y 0 ) ⊂ G, consisting of the functions γ(y) = γ ∞ + β(y), for which γ ∞ < 0 and γ ∞ + β(y) < 0 on [y 0 , +∞[. Similarly we define G + (y 0 ) ⊂ G, consisting of the functions γ(y) = γ ∞ + β(y),
(17) u ′′ (τ ) + γ(τ )u = u ′′ (τ ) + -λ 2 + β(τ ) u = 0, λ > 0.
Assume β(τ ) to be continuous and to satisfy [START_REF] Destrade | Seismic Rayleigh waves on an exponentially graded, orthotropic half-space[END_REF]. Then, for equation [START_REF] Salah | A theoretical study of propagation of Rayleigh surface waves in functionally graded piezoelectric (fgpm) half-space[END_REF]

there exist τ 0 ≥ 0, constants c 1 , c 2 , d 1 , c ′ 1 , c ′ 2 , d ′ 1 and two solutions u λ (τ ), u -λ (τ ) such that ∀τ ≥ τ 0 : c ′ 2 exp λτ -d ′ 1 τ τ0 |β(θ)| dθ ≤ u λ (τ ) ≤ c ′ 1 exp λτ + d ′ 1 τ τ0 |β(θ)| dθ , ( 18
)
c 2 exp -λτ -d 1 τ τ0 |β(θ)| dθ ≤ u -λ (τ ) ≤ c 1 exp -λτ + d 1 τ τ0 |β(θ)| dθ . ( 19
)
Corollary 3.2 (see [START_REF] Hartman | Ordinary differential equations[END_REF]§XI.9]). . Assume the assumptions of Proposition 3.1 to hold and β(•) to satisfy [START_REF] Shuvalov | The high-frequency dispersion coefficient for the Rayleigh velocity in a vertically inhomogeneous anisotropic halfspace[END_REF]. Then the solutions u λ , u -λ satisfy

u λ ∼ u ′ λ λ ∼ e λτ , u -λ ∼ - u ′ -λ λ ∼ e -λτ
as τ → +∞. Next we pass on to the elliptic case (see [START_REF] Hartman | Ordinary differential equations[END_REF]§XI.8]; Corollary 8.1), where the coefficient γ(•) ∈ G + (y 0 ). Proposition 3.4. Consider the equation

(20) u ′′ (τ ) + γ(τ )u = u ′′ (τ ) + λ 2 + β(τ ) u = 0, λ > 0 with γ(•) ∈ G + (y 0 ).
Then, for any real a, b there is a unique solution of equation [START_REF] Cerveny | Seismic ray theory[END_REF] with the asymptotics

u(τ ) = (a + o(1)) cos λτ + (b + o(1)) sin λτ, (21) u ′ (τ ) = (-λa + o(1)) sin λτ + (λb + o(1)) cos λτ,
as τ → +∞.

Prüfer's Coordinates.

We consider Prüfer's coordinates (see [START_REF] Hartman | Ordinary differential equations[END_REF][START_REF] Atkinson | Multiparameter Eigenvalue Problems. Sturm-Liouville Theory[END_REF]):

(22) r = (u 2 + µ 2 u ′2 ) 1 2 = (u 2 + w 2 ) 1 2 , φ = Arctg u w ,
where again w = µu ′ . For the vector function Z = w u we denote φ by Arg Z (the choice of a continuous branch is done in a standard way). In coordinates ( 22) system ( 8) takes the form:

(23) r ′ = µ -1 (y) -γ(y) r sin φ cos φ, φ ′ = γ(y) sin 2 φ + µ -1 (y) cos 2 φ;
note that the second equation is decoupled from the first one. We list some facts concerning the evolution of Arg Z(y). Recall that µ(y) in equation ( 8) meets Assumption 3.1.

Proposition 3.5.

i) If γ(y) ≥ 0 (respectively, γ(y) > 0) on an interval, then for a solution Z(y) of (11) Prüfer's angle variable φ = Arg Z is non-decreasing (increasing) on the interval. ii) If γ(y) < 0 on an interval I, then the first and the third quadrants -Arg Z ∈]0, π /2[ and Arg Z ∈]π, 3π /2[ -are invariant for system [START_REF] Bellman | Stability theory of differential equations[END_REF] on I. iii) For any γ(y) there is a kind of weakened monotonicity for Arg Z: if Arg Z(ỹ) > mπ, then Arg Z(y) > mπ for any y > ỹ.

Property i) follows from (23). So does property ii), since, according to (23), φ ′ (πm) > 0 and φ ′ ( π /2 + πm) < 0 for negative γ. Property iii) follows from the fact that in (23) φ ′ (mπ) = µ -1 (mπ) > 0.

Oscillatory Equations.

Second-order linear differential equation is oscillatory ([12, §XI.5 ]) on [0, +∞[ when its every solution has infinite number of zeros on [0, +∞[, or equivalently the set of zeros of any solution has no upper limit, or equivalently for every solution its Prufer's coordinate Arg Z (see the previous Subsection) satisfies

lim sup y→+∞ Arg Z(y) = +∞.
An obvious example of oscillatory equation is [START_REF] Cerveny | Seismic ray theory[END_REF], when the assumptions of Proposition 3.4 are met.

We are interested in conditions, under which the same equation is oscillatory for vanishing λ. We formulate the result (see [START_REF] Hartman | Ordinary differential equations[END_REF]§XI.5], [10, Ch.2, §6]) for equation [START_REF] Fedoryuk | Asymptotic Analysis[END_REF]. Proposition 3.6. Let γ(•) in [START_REF] Fedoryuk | Asymptotic Analysis[END_REF] be continuous of bounded variation on every interval [0, T ], γ(τ ) > 0 on some interval [τ 0 , +∞[, and

+∞ τ0 (γ(τ )) 1/2 dτ = +∞, ( 24
) T τ0 (γ(τ )) -1 |dγ(τ )| = o T τ0 (γ(τ )) 1/2 dτ , as T → +∞. (25)
Then, equation [START_REF] Fedoryuk | Asymptotic Analysis[END_REF] is oscillatory.

3.5. Hamiltonian Form. One can rewrite the system [START_REF] Hartman | Ordinary differential equations[END_REF] in the Hamiltonian form

(26) u ′ = ∂H ∂w = w µ(y) , w ′ = - ∂H ∂u = -γ(y)u
with the Hamiltonian

H = 1 2 w 2 µ(y) + γ(y)u 2 .
We denote by -→ h the (Hamiltonian) vector field at the right-hand side of (26). As it is well known, equation ( 8) follows from a variational principle, i.e. ( 8) is the Euler-Lagrange equation, which represents necessary minimality condition for a variational problem

+∞ 0 µ(y) (u ′ (y)) 2 -γ(y) (u(y)) 2 dy → min
with appropriate boundary conditions. The Hamiltonian form of the minimality condition for the same variational problem is precisely (26). For Prüfer's angle φ = Arctan u w there holds

φ ′ = -w ′ u + wu ′ u 2 + w 2 = γu 2 + w 2 /µ u 2 + w 2 = 2H u 2 + w 2 .
The last equation is equivalent to the differential equations (23) for Prüfer's coordinate φ. Remark 3.1. A simple but relevant (see [START_REF] Arnold | The Sturm theorems and symplectic geometry[END_REF]) computation is provided by derivation of u(y)w(y) along the trajectories of Hamiltonian system (26):

(27) d dy (uw) = ∂-→ h (uw) = ∂-→ h u w + u ∂-→ h w = -γu 2 + w 2 µ ,
wherefrom it follows, among other things, that uw is nondecreasing (respectively increasing) on the intervals where γ(y) ≤ 0 (respectively γ(y) < 0). Proposition 3.5 and Remark 3.1 allow us to arrive at a conclusion on qualitative behaviour of solutions on an interval, where γ(τ ) < 0 in [START_REF] Salah | A theoretical study of propagation of Rayleigh surface waves in functionally graded piezoelectric (fgpm) half-space[END_REF].

According to Proposition 3.1, there is a decaying solution, along which (according to Remark 3.1) uw grows. Hence the solution approaches the origin either in the second or in the fourth quadrant, where uw < 0. Proposition 3.7. Let γ(τ ) meet the assumptions of Proposition 3.1 with γ(τ ) < 0 for τ ∈ [τ 0 , +∞[. Then, the decaying solutions ±u(τ ) of (17) correspond to the solutions ±Z(τ ) of [START_REF] Bellman | Stability theory of differential equations[END_REF]

with Arg Z(τ ) ∈ [ π /2, π] and Arg (-Z)(τ ) ∈ [ 3π /2, 2π] for τ ∈ [τ 0 , +∞[.
Other solutions, which start in the same quadrants, escape to either the first or the third quadrant, which, according to Proposition 3.5, are invariant for [START_REF] Salah | A theoretical study of propagation of Rayleigh surface waves in functionally graded piezoelectric (fgpm) half-space[END_REF] whenever γ(τ ) < 0. According to Remark 3.1, the product uw (positive in these quadrants) grows along the respective trajectories, which tend to infinity. We provide analogue of the comparison result (in particular, of relation ( 29)) for the decaying solutions of (28), when y 1 = +∞. We were not able to trace it in the literature and provide a (short) proof. Proposition 3.9 (comparison result for decaying solutions on a half-line). Consider the pair of second-order equations (28) with the coefficient µ(y) meeting Assumption 3.1 and with γ(y), γ(y) belonging to G -(y 0 ). Let Proof. Without loss of generality we may assume µ(y) ≡ 1; otherwise we perform substitution (9) of the independent variable, which preserves relation (30) for the coefficients. By (30) and ( 27), the functions uw and ũ w are increasing on [y 0 , +∞[. As long as the limits of these functions at +∞ are null, we conclude that (uw)(y) < 0, (ũ w)(y) < 0 on [y 0 , +∞[ and then without loss of generality we may assume that u(y), ũ(y) are positive, while w(y), w(y) are negative on [y 0 , +∞[. Denote γ(y) -γ(y) by ∆γ(y) and represent the second one of equations ( 28 Note that Arg Z = Arccot w(y) u(y) , Arg Z = Arccot w(y) ũ(y) and since the function z → Arccot z is Lipschitzian with constant 1:

Arg Z(y) -Arg Z(y) = Arccot w(y) u(y) -Arccot w(y) ũ(y) ≤ w(y) u(y) - w(y) ũ(y)
and the left-hand side tends to 0 as ∆γ(τ ) 01 → 0. □

Existence of Surface Waves and Parametric Sturm-Liouville Problem

We come back to equation ( 5) and simplify the notations putting

Ω = ω 2 , K = k 2 , A = (K, Ω), (37) γ A (y) = Ωρ(y) -Kµ(y),
thus arriving at the equation

(38) (µ(y)u ′ ) ′ (y) + (Ωρ(y) -Kµ(y)) γ A (y) u(y) = 0
with the parameter A.

Performing the substitution of the independent variable the way it is done in (9), we get the equation: In equations ( 38) and (39) the dependence of the coefficients on the parameters Ω, K is linear; the functions ρ(y), µ(y), ρ(τ ), μ(τ ) are positive. Note that μ(0) = µ(0), ρ(0) = ρ(0) and

(39) d 2 ū dτ 2 + μ(τ ) (Ωρ(τ ) -K μ(τ )) γA (τ ) ū(τ ) = 0,
μ(+∞) = µ(+∞), ρ(+∞) = ρ(+∞), γA (+∞) = Ωµ ∞ ρ ∞ -Kµ 2
∞ . We know from the previous Section that if equation (39) meets the assumptions of Proposition 3.1, then it has a solution, which satisfies the boundary condition at infinity [START_REF] Xiaoshan | Transverse shear surface wave in a functionally graded material infinite half space[END_REF]. We are interested, though, in the solutions, which satisfy at the same time the boundary condition ( 6), and it is not possible for generic combinations of ρ(y), μ(y), Ω, K, which enter (39) via the coefficient γA (•). In other words we get parametric Sturm-Liouville problem on a half-line for equation (39) (or (38)) with the boundary conditions ( 6)- [START_REF] Xiaoshan | Transverse shear surface wave in a functionally graded material infinite half space[END_REF].

Let us introduce the vector-function a(y) = (ρ(y), µ(y)), which characterizes our medium, and formulate the assumptions for the medium in terms of a(y).

Assumption 4.1 (Lipschitz continuity). The function a(y) = (ρ(y), µ(y)) is Lipschitz continuous on [0, +∞[. There exists a finite limit

lim y→+∞ a(y) = a ∞ , a ∞ = (ρ ∞ , µ ∞ ) , ρ ∞ > 0, µ ∞ > 0.
. The values a(y) have both positive coordinates; hence the values of Arg a(y) lie in [0, π /2]. As long as a ∞ = 0, Arg a ∞ is properly defined.

Assumption 4.2 (integral boundedness). The function

â(y) = (ρ(y), μ(y)) = a(y) -a(+∞) = (ρ(y) -ρ ∞ , µ(y) -µ ∞ ) is integrable on [0, +∞[: ∞ 0 |â(y)| dy < ∞. ρ µ (ρ∞, µ∞) (ρ0, µ0) (K, Ω) (ρ∞, µ∞) (ρ0, µ0) (ρ∞, µ∞) (ρ0, µ0) (3) (2) (1)

Assumption 4.3 (monotonicity at infinity).

There exists an interval Ī =]ȳ, +∞[ such that either: i) Arg a(y) < Arg a ∞ Ī -positive monotonicity at infinity, or ii) Arg a(y) > Arg a ∞ on Ī -negative monotonicity at infinity.

Examples of the curves a(y) = (ρ(y), µ(y)) are drawn in Figure 1 together with the vector A = (K, Ω). The curves (1) and ( 2) are negatively monotonous at infinity, while the curve (3) is positively monotonous at infinity.

Assume the vector of parameters A = (K, Ω) to belong to (the positive quadrant of) the oriented plane, in which the curve y → a(y), y ∈ [0, +∞[ is contained. We define Arg A = Arctan Ω K . The following remarks are important. Remark 4.1. Fix an admissible A. If Arg a(y) < Arg A (respectively, Arg a(y) > Arg A) for some y, then γ A (y) defined by (37) is positive (respectively, negative). □

Remark 4.2. Under Assumptions 4.1 and 4.2, for any admissible A and γ A (•)

defined by (37) there holds: Proof. By Assumption 3.1, τ (y) defined by ( 9) is Lipschitzian homeomorphism of [0, +∞[ onto itself. Hence the functions μ, ρ defined by (40) are bounded, Lipschitzian, with finite limits at infinity, i.e. Assumption 4.1 is valid for them.

(1) γ A (y) -γ A (+∞) y→+∞ -----→ 0; (2) ∞ 0 |γ A (y) -γ A (+∞)|dy < ∞; (3) if γ A (y)
Under substitution (9), the vector-function a(y) = (ρ(y), µ(y)) is transformed into ā(τ ) = μ(τ ) (ρ(τ ), μ(τ )). Hence Arg a(y) = Arg ā(τ (y)) and all the monotonicity properties listed in Assumption 4.3 are maintained.

Regarding Assumption 4.2 we perform substitution ( 9) and obtain:

+∞ 0 |γ A (τ ) -γA (+∞)| dτ = +∞ 0 |µ(y)γ A (y) -µ(+∞)γ A (+∞)| µ(y) dy = = +∞ 0 (γ A (y) -γ A (+∞)) + γ A (+∞) (µ(y) -µ(+∞)) (µ(y)) -1 dy < ∞, since (µ(y)) -1 is bounded on [0, +∞[. □ For the limit case, in which A ∞ = (K ∞ , Ω ∞ ) = βa ∞ , β > 0, or in other words Arg A ∞ = Arg a ∞ , we get γ A∞ (y) = Ω ∞ ρ(y) -K ∞ µ(y) = β(µ ∞ (ρ ∞ + ρ(y)) -ρ ∞ (µ ∞ + μ(y)) = βγ ∞ (y), where (41) γ∞ (y) = µ ∞ ρ(y) -ρ ∞ μ(y).
Remark 4.3.

(1) Under Assumption 4.1, for each A = (K, Ω) with Arg A < Arg a ∞ , there exists an interval, [y -, +∞[, on which γ A (y) < 0.

(2) Under Assumption 4.3i) (respectively 4.3ii)), there is an interval [ȳ, +∞[, on which γ ∞ (y) is positive (respectively negative).

Results

Key information for our treatment is provided by the limit-case equation, which corresponds to the vectors of parameters with γ∞ (y) as in (41).

A ∞ = (K ∞ , Ω ∞ ) = βa ∞ , β > 0.
We formulate here main results of the paper; the proofs are provided in the next Section. Our first result establishes non-existence of solutions under a kind of global negative monotonicity of a(y) at infinity. Then, there are no admissible values of parameters K, Ω, for which solutions of (38)-( 6)-( 7) exist.

The assumptions of the theorem are met by curve (1) in Fig. 1. If (43) does not hold, then one can guarantee existence of solutions at least for sufficiently large K, Ω.

Theorem 5.2. Let assumptions 4.1-4.2-4.3 hold and in addition Arg a(y) <

Arg a ∞ for y ∈ I -a non-null sub-interval of [0, +∞[. Then for each N > 0 ∃K N such that ∀K > K N there are at least N values Ω j ∈ μ ρ K, µ∞ ρ∞ K , j = 1, . . . , N , such that for each (K, Ω j ) the solution of (38)-( 6)-( 7) exists. (ρ∞, µ∞) (ρ0, µ0) Figure 2. The functions ρ(y) and µ(y) in the homogeneous substrate example become constant when y ≥ y s , as illustrated by the curve in black. Note that its projection (in gray) on the (ρ, µ)-plane is a curve, which exhibits negative monotonicity at infinity. Finally there is a case, in which for each K > 0 one finds a numerable set of

Ω j ∈ μ ρ K, µ∞
ρ∞ K such that the solution exists for (K, Ω j ). It happens when the limit-case equation ( 42) is oscillatory (see Subsection 3.4). Theorem 5.3. Let assumptions 4.1-4.2-4.3i) hold and the limit-case equation (42) be oscillatory 1 .

Then for each K > 0 there exists a numerable set of Ω m ∈ μ ρ K, µ∞ ρ∞ K , m = 1, . . . , such that: i) for A m = ( K, Ω m ) the solution of (38)-( 6)-( 7) exists; ii) Ω m increase with m and accumulate (only) to Ω = µ∞ ρ∞ K; iii) for the vector solutions Z(y; A m ) there holds Arg Z(y; A m ) ∈ [(m -1 /2) π, mπ] for y sufficiently large. Remark 5.2. Assumptions 4.1-4.2-4.3i) hold for curve [START_REF] Kennett | Seismic wave propagation in stratified media[END_REF] in Fig. 1, but the oscillatory property for the limit-case equation can not be concluded from the curve only, since it also depends on its parametrization. □

Homogeneous Substrate

Example. This is a particular case, in which the properties of the medium become depth-independent starting from some depth. For the model under discussion this means existence of y s such that µ(y) and ρ(y) are constant on the interval [y s , +∞[: µ(y) ≡ µ s , ρ(y) ≡ ρ s on [y s , +∞[ (see Fig. 2). We denote

a s = (ρ s , µ s ). Then, a ∞ = lim y→∞ a(y) = a s and â(y) = a(y) -a ∞ vanishes on [y s , +∞[. If ∀y ∈ [0, +∞[: Arg a(y) ≥ Arg a s or, the same ∀y ∈ [0, +∞[: µ(y) ρ(y) ≥ µ s ρ s ,
then we are under assumptions of Theorem 5.1 and solutions of (38)-( 6)-( 7) do not exist.

If µ(y) ρ(y) < µs ρs on some non-null subinterval of [0, +∞[, then we fall under assumptions of Theorem 5.2 and hence its claim holds. 1 We assume (24) and (25) to hold

Proofs

Since substitution (9) transforms parametric equation (38) into its standard form (39) and Assumptions 4.1,4.2,4.3 are maintained under (9), we may take, without loss of generality, µ(y) ≡ 1 in (38).

The proof of Theorem 5.1 is easy. Pick some A = (K, Ω). There are two options: Arg a ∞ ≤ Arg A or Arg a ∞ > Arg A.

In the first case, by monotonicity and continuity assumptions, the coefficient γ A (y) in equation ( 38) is non-negative on some interval [y 0 , +∞[. Then, by Proposition 3.4, there exists a fundamental system of solutions of the form [START_REF] Tanaka | Surface acoustic waves in two-dimensional periodic elastic structures[END_REF] and none of the solutions of (38) tend to the origin as y → +∞.

If otherwise Arg A < Arg a ∞ ≤ Arg a(y) ∀y ∈ [0, +∞[, or the same, 27), for a solution Z(y, A) = w u there holds d dy (u(y)w(y)) > 0. This enters in contradiction with the boundary conditions ( 6)-( 7), according to which u(0)w(0) = 0 and lim y→+∞ (u(y)w(y)) = 0.

Ω ≤ K min y∈[0,+∞[ µ(y) ρ(y) = µ(y) ρ(y) = μ ρ , then γ A (y) < 0 on [0, +∞[. By (
Proof of Theorem 5.3. We start with a sketch of the proof. Take a vector of parameters A ∞ = ( K, Ω) collinear to a ∞ = (ρ ∞ , µ ∞ ) and consider its perturbation A ∞,s = ( K, Ω -s). It is immediate to see that for each s > 0 equation (38) with A = A ∞,s and the coefficient

(44) γ A∞,s (y) = γ A∞ (y) -sρ(y) = -sρ ∞ + ( Ω -s)ρ(y) -K μ(y)
meets the assumptions of Proposition 3.1, and hence the equation (45) u ′′ + γ A∞,s (y)u = 0 possesses a decaying solution Z + (y, A ∞,s ). Simultaneously, we consider the solutions Z 0 (y, A ∞,s ) of the same equation with the boundary condition [START_REF] Ting | Existence of anti-plane shear surface waves in anisotropic elastic half-space with depth-dependent material properties[END_REF]. The goal is to detect the values s > 0, for which the solutions Z 0 (y, A ∞,s ) and Z + (y, A ∞,s ) meet at some intermediate point ȳ ∈ [0, +∞[, i.e admit at ȳ the same value (mod π). In such a case they (or their opposites) can be concatenated into solutions of (38)-( 6)- [START_REF] Xiaoshan | Transverse shear surface wave in a functionally graded material infinite half space[END_REF]. The possibility of such meeting follows from Propositions 3.8 and 3.9, according to which for a sufficiently large intermediate point ȳ ∈ [0, +∞[ the vectors Z 0 (ȳ, A ∞,s ) and Z + (ȳ, A ∞,s ) rotate in opposite directions as s grows from some s > 0.

Increasing ȳ if necessary, one can assume that ∀s ≥ s one has γ A∞,s (y) < 0 on (ȳ, +∞[ and Arg Z + (ȳ, A ∞,s ) ∈] π /2, π[. On the other hand, for small s > 0, Arg Z 0 (ȳ, A ∞,s ) is close to Arg Z 0 (ȳ, A ∞ ), which, due to the oscillation property of the limit-case equation, tends to +∞ as ȳ → +∞. Therefore for each natural m one can find (again increasing ȳ when necessary) small s > 0 such that Arg Z 0 (ȳ, A ∞,s ) > πm. As s will grow from s to Ω, Arg Z 0 (ȳ, A ∞,s ) will decrease from the value greater than πm to the value less than π and during this evolution it becomes equal (mod π) to Arg Z + (ȳ, A ∞,s ) for m distinct values of s. Now we provide the detailed proofs of the statements i)-iii) of the Theorem. i) Pick K > 0 and take Ω = µ∞ ρ∞ K, so that A ∞ = K, Ω is collinear with a ∞ . Consider the limit-case equation (42) with the parameter A ∞ and choose the solution Z 0 (•; A ∞ ), which satisfies the boundary condition [START_REF] Ting | Existence of anti-plane shear surface waves in anisotropic elastic half-space with depth-dependent material properties[END_REF]. As long as equation ( 42) is oscillatory, Arg Z 0 (y; A ∞ ) tends to infinity as y → +∞. Hence, for each natural m ∃y m ∈ [0, +∞[ such that Arg Z 0 (y m ; A ∞ ) > πm.

By the continuity of the trajectories of (38) with respect to the parameter A, one can find s > 0 such that for any s ∈]0, s] and for A ∞,s = ( K, Ω -s) there holds Arg Z 0 (y m ; A ∞,s ) > πm. For the function γ A∞,s (y) defined by (44) one can find ȳ ≥ y m such that γ A∞,s (y) < 0 on [ȳ, +∞[. It follows from Remark 3.5iii) that Arg Z 0 (ȳ; A ∞,s ) > πm.

The second-order equation ( 45) for s = s meets the assumptions of Proposition 3.1 and hence has the decaying solution Z + (y; A ∞,s ). By Proposition 3.7, there holds:

∀y ≥ ȳ : Arg Z + (y; A ∞,s ) ∈ ] π /2, π[ (mod π).
Letting s grow from s towards Ω, we note that the values of γ A∞,s (y) = γ A∞ (y) -sρ(y) on [0, +∞[ diminish; in particular, γ A∞,s (y) < 0 for y ∈ [ȳ, +∞[ for all s ≥ s.

According to Proposition 3.8, the function s → Arg Z 0 (ȳ; A ∞,s ) decreases monotonously from the value Arg Z 0 (ȳ; A ∞,s ) > πm to the value Arg Z 0 (ȳ; A ∞, Ω) ∈ ]0, π[.

Consider now the decaying solutions Z + (y; A ∞,s ). Proposition 3.9 implies that for chosen ȳ Arg Z + (ȳ; A ∞,s ) grows with the growth of s, remaining (mod π) in the interval ] π /2, π[. During the evolution there occur (at least) m values of s j , j = 1, . . . , m, for which

Arg Z + (ȳ; A ∞,sj ) = Arg Z 0 (ȳ; A ∞,sj ) -πn (n -integer). Then the concatenations (46) Z(y; A ∞,sj ) = Z 0 (y; A ∞,sj ), y ≤ ȳ, (-1) n Z + (y; A ∞,sj ), y ≥ ȳ,
are the decaying solutions of the corresponding equations

u ′′ + ( Ω -s j )ρ(y) -Kµ(y) u = 0,
and (46) satisfies the boundary condition ( 6)- [START_REF] Xiaoshan | Transverse shear surface wave in a functionally graded material infinite half space[END_REF]. ii) Let Ω ∈]0, Ω) be a limit point of Ω n = Ω-s n , n = 1, . . .. Then Ω = Ωs < Ω.

Consider γ A∞,s . There exists ỹ, such that γ A∞,s < 0 on [ỹ, +∞[. Pick the decaying solution Z + (y; A ∞,s ).

According to the aforesaid ∀y ∈ [ỹ, +∞[ :

Arg Z + (y; A ∞,s ) ∈] π /2, π[ (mod π).
Consider the solution Z 0 (y; A ∞,s ), which meets the initial condition [START_REF] Ting | Existence of anti-plane shear surface waves in anisotropic elastic half-space with depth-dependent material properties[END_REF]. If Arg Z + (ỹ; A ∞,s ) = Arg Z 0 (ỹ; A ∞,s ) (mod π), then the inequality holds for values of s close to s, and in particular for all s n , but finite number of them, and this results in a contradiction.

Let Arg Z 0 (ỹ; A ∞,s ) -Arg Z + (ỹ; A ∞,s ) = πm. Since the function Arg Z 0 (ỹ; A ∞,s ) -Arg Z + (ỹ; A ∞,s ) decreases with the growth of s, one concludes:

Arg Z + (ỹ; A ∞,s ) = Arg Z 0 (ỹ; A ∞,s ) (mod π)
for all s = s from a sufficiently small neighborhood of s and hence for all s n but a finite number of them, which leads us to the same contradiction.

iii) By the construction provided in i), for each natural m, there exist A m = ( K, Ω m ) and the decaying solution Z(y, A m ) of ( 38)-( 6)- [START_REF] Xiaoshan | Transverse shear surface wave in a functionally graded material infinite half space[END_REF], which converges to the origin in such a way that Arg Z(y, A m ) ∈ [π(m -1 /2), πm] for sufficiently large y.

To prove its uniqueness, we assume on the contrary that there exists another A ′ = ( K, Ω ′ ) and a decaying solution of (38)-( 6)-( 7) such that for y ∈ [y 0 , +∞[ γ Am (y) < 0, γ A ′ (y) < 0 and both Arg Z(y, A ′ ) and Arg Z(y, A m ) belong to [π(m -1 /2), πm] for y ∈ [y 0 , +∞[.

Let for example Ω ′ > Ω m . Then γ Am (y) < γ A ′ (y) and hence Arg Z(y 0 , A m ) < Arg Z(y 0 , A ′ ). This enters in contradiction with the result of Proposition 3.9. □ Proof of Theorem 5.2. Let us pick K > 0 and take Ω = Ω = µ∞ ρ∞ K, so that A ∞ = K, Ω is collinear with a ∞ . By assumptions of the Theorem, the function γ A∞ (y) admits positive values on some non-null subinterval ]c, c[⊂ [0, +∞[. The same holds true for γ βA∞ with βA ∞ = (β K, β Ω), β > 0.

Our proof can be accomplished along the lines of the proof of Theorem 5.3 if one proves that for any N there exists β N > 0, for which the solution Z 0 (y, β N A ∞ ) with initial condition [START_REF] Ting | Existence of anti-plane shear surface waves in anisotropic elastic half-space with depth-dependent material properties[END_REF] The rest of the proof follows the proof of Theorem 5.3. One can also conclude from (47) that N (k) ∼ k as k → ∞, where N (k) is the number of surface wave solutions with a given wave number k = K 1/2 . □

Research Outlooks

Our study can be generalized in several directions. One possible direction is applying the same approach to more complex models of functionally graded materials, such as the anysotropic elastic media, or media with piezoelectric and/or piezomagnetic properties, where the wave propagation is described by a vector equation.

There is a considerable amount of research of surface waves in the above mentioned cases, see, e.g., [START_REF] Destrade | Seismic Rayleigh waves on an exponentially graded, orthotropic half-space[END_REF][START_REF] Shuvalov | The high-frequency dispersion coefficient for the Rayleigh velocity in a vertically inhomogeneous anisotropic halfspace[END_REF][START_REF] Collet | Bleustein-Gulyaev waves in some functionally graded materials[END_REF][START_REF] Salah | A theoretical study of propagation of Rayleigh surface waves in functionally graded piezoelectric (fgpm) half-space[END_REF][START_REF] Shodja | Shear horizonal surface acoustic waves in functionally graded magneto-electro-elastic half-space[END_REF][START_REF] Gasparyan | Shear waves in functionally graded electro-magneto-elastic media[END_REF]. However the treatment of the problem therein has been restricted either to the asymptotic for large k, ω in application to a certain fundamental branch, or to the particular dependence of the material coefficients on the depth coordinate.

Another generalization is considering surface waves in a medium with material coefficients depending on more than one spatial variable, which invokes partial differential equations. This setting is also addressed in ample literature, see e.g. [START_REF] Cerveny | Seismic ray theory[END_REF][START_REF] Tanaka | Surface acoustic waves in two-dimensional periodic elastic structures[END_REF][START_REF] Kutsenko | Shear surface waves in phononic crystals[END_REF], where it is tackled either in the framework of the high-frequency asymptotical ray method, or under the assumption of periodicity.

One more aspect worthy of further study concerns the case of semi-infinite multilayered aperiodic medium, whose material properties are piecewise continuous. Note that the results on oscillation and asymptotics of second-order differential equations, which we exploited in the present paper, required continuity of a(y). We intend to address the case of the multilayered aperiodic medium elsewhere.

Conclusions

We provided criteria for non-existence/existence of surface waves in terms of the vector function a(y) = (ρ(y), µ(y)), which characterizes the depth-dependence of the material parameters of the medium. If for all y the vectors a(y) lie counterclockwise with respect to the vector a(+∞) = lim y→+∞ a(y), then no surface waves exist in the half-space. If, otherwise, a(y) lie clockwise with respect to the vector a(+∞) for y ∈]y 0 , y 1 [, then surface waves do exist for sufficiently large k and their number N (k) grows with the growth of k.

An apparently more interesting result is a possibility of existence of an infinite number of surface waves for any given k. For this to occur, the latter condition for existence of surface waves must be supplemented by a certain monotonicity property for the vector function a(y) along with the oscillation property for the particular limit case second-order equation (see Section 3 and (42)). Email address: marco.spadini@unifi.it

( 1 )

 1 ρû tt -∇ (M ∇û) = 0 in a half-plane {(x, y) : y > 0}. One imposes the Neumann boundary condition[START_REF] Birman | Modeling and analysis of functionally graded materials and structures[END_REF] û′ y | y=0 = 0. We seek the solutions, which decay at infinity:

( 8 )Assumption 3 . 1 .

 831 (µ(y)u ′ (y)) ′ + γ(y)u = 0 defined on a half line [0, +∞[. We assume from now on that the function µ(s) ≥ µ > 0 on [0, +∞[, is continuous on [0, +∞[ and admits a finite limit lims→∞ µ(s) = µ ∞ > 0. □The following substitution of the independent variable is invertible (τ (y) is strictly growing) and satisfies the relation: d dτ = µ d dy . By Assumption 3.1, the functions µ(s), (µ(s))-1 are both bounded on [0, +∞[ and therefore the function τ (y) and its inverse y(τ ) are Lipschitzian. Besides +∞ 0 (µ(s)) -1 ds = ∞, i.e. τ (y) is Lipschitzian homeomorphism of [0, +∞[ onto [0, +∞[. This substitution transforms (8) into the standard form (10)

  for which γ ∞ > 0 and γ ∞ + β(y) > 0 on [y 0 , +∞[. Both G -(y 0 ) and G + (y 0 ) are open subsets of G in the above introduced norm. It is easy to verify that substitution (9) transforms the space G into itself and the sets G -(y 0 ), G + (y 0 ) into G -(τ (y 0 )), G + (τ (y 0 )), correspondingly.

Corollary 3 . 3 .

 33 For each γ(•) sufficiently close to γ(•) in the norm (16) the equation u ′′ (τ ) + γ(τ )u(τ ) = 0 has a decaying solution.

3. 6 .

 6 Sturmian Properties of Trajectories. We provide few results from the Sturm theory. First result is classical ([START_REF] Fedoryuk | Asymptotic Analysis[END_REF],[START_REF] Atkinson | Multiparameter Eigenvalue Problems. Sturm-Liouville Theory[END_REF], [12, Ch. X,XI]) and follows directly from the second equation (23).

Proposition 3 . 8 (

 38 comparison result). Consider a pair of second-order equations (28) (µ(y)u ′ (y)) ′ + γ(y)u = 0, (µ(y)u ′ (y)) ′ + γ(y)u(y) = 0, where µ(y) meets Assumption 3.1 and γ(y) ≥ γ(y), ∀y ∈ [y 0 , +∞[. If for y 1 ≥ y 0 and a pair of vector solutions Z = w u , Z = w ũ of the first and the second equations (28) Arg Z(y 1 ) = Arg Z(y 1 ), then ∀y ≥ y 1 : Arg Z(y) ≥ Arg Z(y) and (29) ∀y ∈ [y 0 , y 1 ] : Arg Z(y) ≤ Arg Z(y).

  (30) 0 > γ(y) ≥ γ(y), ∀y ∈ [y 0 , +∞[. If Z, Z are the decaying solutions of equations (28), then (31) Arg Z(y) ≤ Arg Z(y), ∀y ≥ y 0 .

  ũ′′ + γ(y)ũ = -∆γ(y)ũ; ∆γ(y) > 0 by(30).Applying the integral form of the Lagrange identity (or Green's formula, see[START_REF] Hartman | Ordinary differential equations[END_REF] §XI.2]) to the respective vector solutions Z = , of which the second one is written as (32), we conclude:∀y ≥ y 0 : (u w -wũ)| +∞ y = +∞ y -∆γ(s)ũ(s)u(s)ds < 0.Given that (u w -wũ) vanishes at +∞, we obtain:(33) ∀y ≥ y 0 : -u(y) w(y) + w(y)ũ(y) = +∞ y -∆γ(s)ũ(s)u(s)ds < 0.Dividing the inequality in (33) by the positive value w(y) w(y), we get ∀y ≥ y 0 : ũfollows. □ We establish the continuous dependence of decaying solutions on the coefficient γ(•) in • 01 -norm. Proposition 3.10 (continuous dependence of decaying solutions on the right-hand side). Consider equations (28). Let γ(•) = -λ 2 + β(•) ∈ G - y0 for some y 0 ∈ [0, +∞[. Then, for any γ(•) = -λ2 + β(•) sufficiently close to γ(•) in • 01 -norm: i) both equations (28) possess the decaying vector solutions Z(•), Z(•) with Arg Z, Arg Z ∈ [ π /2, π]; ii) for each y ∈ [y 0 , +∞[ Arg Z(y) -Arg Z(y) → 0, as γ(•) -γ(•) 01 → 0.Proof. Again we may proceed assuming µ(y) ≡ 1. i) Any γ(•) sufficiently close to γ(•) in • 01 -norm belongs to G - y0 ,which is open with respect to the norm. The existence of the decaying solutions Z(y), Z(y) follows from Corollary 3.3. Since both γ and γ are negative on [y 0 , +∞[, we conclude by Proposition 3.7 that Arg Z(y) and Arg Z(y) lie in [ π /2, π] for y ∈ [y 0 , +∞[. This implies that for s ∈ [y 0 , +∞[, w(s), w(s) are negative, while u(s), ũ(τ ) are positive and by (11) decrease. ii) Recall that ∆γ(•) = γ(•) -γ(•). Invoking the equality in (33) and dividing it by -u(y)ũ(y), )ν(s)ν(s)dτ, where ν(s) = ũ(s) ũ(y) , ν(s) = u(s) u(y) are the solutions of the first and second equation (28), which are normalized by the condition: ν(y) = ν(y) = 1.By the aforesaid ν(s), ν(s) decrease; hence(35) ν(s) ≤ 1, ν(s) ≤ 1, for s ≥ y.According to Proposition 3.1, there existc 1 , d 1 > 0, s 0 > y such that (36) ν(s) ≤ c 1 exp -λs + d 1 s s0 |β(σ)|dσ , ∀s > s 0 .From the proof of the Proposition (see[11, §6.12, §2.6]) it follows that one can choose in (36) any c 1 > 1, a sufficiently large d 1 and then choose s 0 such that d 1 sup s≥s0 |β(s)| < λ. The same holds for the second one of equations (28).For each γ from a small neighborhood of γ in • 01 -norm, λ and λ as well as sup τ ≥τ0 |β(τ )| and sup τ ≥τ0 | β(τ )| are close. Thus, one can choose common c 1 , d 1 , τ 0 for all the equations with the coefficient γ from the neighborhood. Besides there is a common upper bound B for the corresponding norms β(•) L1 . Then, by (34),(35) and (36), +∞ y ∆γ(s)ν(s)ν(s)ds ≤ s0 y |∆γ(s)| ds + c 2 1 e 2d1B ∞ s0 e -λs |∆γ(s)| ds with the right-hand side tending to 0 as ∆γ(s) 01 → 0.

  ) = ρ(y(τ )), μ(τ ) = µ(y(τ )), ū(τ ) = u(y(τ )).

Figure 1 .

 1 Figure 1. Curves parameterized by y, which exhibit different types of monotonicity at infinity

Proposition 4 . 1 .

 41 Let Assumption 3.1 hold and let equation (38) meet Assumptions 4.1, 4.2, 4.3 for any admissible A. Then equation (39) meets the same Assumptions.

  For such choice of parameters equation (38) takes the form (42) (µ(y)u ′ ) ′ + βγ ∞ (y)u = 0

Theorem 5 . 1 .

 51 Let assumptions 4.1-4.2 hold and (43) ∀y ∈ [0, +∞[: Arg a(y) ≥ Arg a ∞ .

Remark 5 . 1 .

 51 The curves[START_REF] Birman | Modeling and analysis of functionally graded materials and structures[END_REF] and[START_REF] Kennett | Seismic wave propagation in stratified media[END_REF] inFig. 1 meet assumptions of the Theorem. □ y ρ µ (ρ∞, µ∞, ys)

  satisfies Arg Z 0 (c, β N A ∞ ) > πN . Consider the equation u ′′ (y) + γ βA∞ (y)u = u ′′ (y) + βγ A∞ (y)u = 0 on the interval [0, c]. It is known ([4, §A.3, §A.5]) that the number of zeros of the solution u(y, γ βA∞ (•)), or, the same, the increment of Prüfer's angle Arg Z(y, γ βA∞(•)) -Arg Z(0, γ βA∞ (•)) γ A∞ (η), 0)) 1/2 dη + O(β 1 /3 )as β → +∞. Hence choosing sufficiently large β > 0, we can get a solution Z 0 (y; βA ∞ satisfying the boundary condition[START_REF] Ting | Existence of anti-plane shear surface waves in anisotropic elastic half-space with depth-dependent material properties[END_REF] with the property that Arg Z 0 (c; βA ∞ ) > N π.Proposition 3.5 yields Z 0 (y; βA ∞ ) > N π, ∀y > c.

(

  A. Sarychev) A. Sarychev, DiMaI, Università di Firenze,, v. delle Pandette 9, Firenze, 50127 Italy, Tel.: +39 055 2759659 Email address: asarychev@unifi.it (A. Shuvalov) A. Shuvalov, Université de Bordeaux, UMR 5295,, Talence 33405 France, Tel.: +33 5 40 00 31 37 Email address: alexander.shuvalov@u-bordeaux.fr (M. Spadini) M. Spadini, DiMaI, Università di Firenze,, via S.Marta 3, Firenze 50139 Italy, Tel.: +39 055 2758923

  admits positive values, then so does γ A ′ (y) with any A ′ such that Arg A ′ is sufficiently close to Arg A. □

	We wish to check what occurs with Assumptions 4.1-4.2-4.3 after substitution
	(9).


Timothy Claeys

Franc ¸ois-Xavier Molina

Mališa Vučinić

Thomas Watteyne

Emmanuel Baccelli

RIOT and OpenWSN 6TiSCH: Happy Together

Keywords: Internet of Things, 6TiSCH, OpenWSN, RTOS, RIOT

Short development cycles, application-field diversity, and requirements on network size or reliability put an ever increasing strain on Internet of Things (IoT) application developers. Real-time embedded operating systems (RTOS) aim to provide a key set of features, abstractions and services that enable faster development. To fulfill the promise of wire-like communication reliability, wireless standards such as WirelessHART, ISA100.11a and 6TiSCH have been developed and are used in the industry. Keeping these networks synchronized requires precise timing information from the underlying hardware. However, the hardware abstractions of an RTOS do come with an overhead, and the question arises on how these abstractions impact the performance of a complex network stack. To study this, we integrated Open-WSN, a standards-compliant open-source implementation of the 6TiSCH network stack, with RIOT, a prominent open-source RTOS. We compare the minimalistic "bare metal" approach of OpenWSN with RIOT's full-fledged RTOS environment. We study the impact on network performance, power consumption and real-time application properties. On the one hand, we show that using RIOT to execute a 6TiSCH stack does not degrade power consumption or network performance. On the other hand, we demonstrate how RIOT brings improvements on the time it takes to execute application tasks.

I. INTRODUCTION

The evolution of Wireless Sensor Networks (WSNs) into the Internet of Things (IoT) has triggered the proliferation of available hardware meeting the low-cost and low-power requirements. As the applications of this new technology have evolved, so have the operating systems (OSs). From TinyOS [START_REF] Levis | TinyOS: An Operating System for Sensor Networks[END_REF] and Contiki OS [START_REF] Dunkels | Contiki -A Lightweight and Flexible Operating System for Tiny Networked Sensors[END_REF] in the early 2000s, to more recent real-time approaches like RIOT [START_REF] Baccelli | RIOT: An Open Source Operating System for Low-End Embedded Devices in the IoT[END_REF], we have seen a constant drive towards open-source code and communitydriven development. In parallel with the development efforts on OSs, standardization bodies have defined open communication specifications like the Internet Engineering Task Force (IETF) 6TiSCH [START_REF] Vilajosana | IETF 6TiSCH: A Tutorial[END_REF] to bridge the performance of industrial low-power wireless networking with the ease-of-use of the Internet Protocol (IP).

Keeping 6TiSCH networks synchronized requires precise timing information from the underlying hardware. The abstractions provided by the OS do come with an overhead. The complexity introduced by a full-fledged OS in order to execute what is often a single application task is not always deemed necessary. The alternative approach is to operate closer to the hardware and directly make use of the available resources. With less code to run, one may expect a better performance at the cost of code extensibility. With this in mind, we answer the question: How does an OS affect the performance of a 6TiSCH communication stack?

We do so by making use of the reference implementation of the 6TiSCH protocol stack, OpenWSN [START_REF] Watteyne | OpenWSN: a standards-based lowpower wireless development environment[END_REF]. In its vanilla mode, OpenWSN employs a minimalistic approach and runs over a thin hardware abstraction layer with a basic task scheduler. We integrate OpenWSN stack into RIOT, a real-time, full-fledged OS. We abstract OpenWSN as a RIOT thread, while allowing time critical operations to be executed in interrupt mode.

The contribution of our work is twofold. First, we run OpenWSN in its vanilla mode alongside its integration with RIOT, creating heterogeneous networks. We execute over 70 experiments totaling 140 hours on a testbed deployed in Inria-Paris, France. We show that the overhead introduced by the OS does not influence the performance of the network in terms of reliability and duty cycle. When differences in base system configurations are accounted for, we also observe similar power consumption. We study the real-time properties of both approaches and demonstrate improvements when using RIOT to schedule application tasks running on top of the networking stack. Second, we release the developments related to this paper in open-source form for the benefit of OpenWSN and RIOT communities.

The remainder of this paper is organized as follows. Section II gives a primer on OpenWSN and RIOT, necessary for the comprehension of the results. Section III describes the technical details of the OpenWSN and RIOT integration. Section IV summarizes the technical results. Section V discusses the technical challenges encountered. Finally, Section VI concludes this paper.

II. BACKGROUND A. OpenWSN

OpenWSN is a free and open-source implementation of the IETF 6TiSCH specification. It supports a wide range of commercial off-the-shelf (COTS) low-power networking hardware, including CC2538, SAMR21-XPRO, IoTLAB-M3 and nRF52840. The project consists of six statically-linked software libraries, see Fig. 1: board support package (BSP), kernel, drivers, stack, web, app.

1) Board Support Package: The BSP library provides a Hardware Abstraction Layer (HAL). It groups the different functions that directly interact with the underlying hardware, i.e. writing into registers. Each supported board has its distinct implementation details, but they all expose the same Application Programming Interface (API). This approach allows the hardware platforms to share the rest of the libraries.

2) Kernel: OpenWSN's kernel (OpenOS) consists of a simple non-preemptive task scheduler. It queues different task contexts created by the various layers of the networking stack and the applications. All tasks have an associated priority. In general, tasks scheduled by the lower layers of the stack have a higher priority. When a task gets executed, the scheduler calls the callback in the context. The run-to-completion scheduler regains control when the callback finishes its execution.

3) Drivers: The driver library implements a generic interface to UART (OpenSerial), the timer infrastructure (OpenTimers), and sensors (OpenSensors), e.g., temperature, humidity, light, infrared, etc. The nodes in the Open-WSN 6TiSCH network primarily use serial communication for logging. Additionally, the border router uses the serial interface to transfer packets from the mesh network to the gateway. Different hardware timers are abstracted by the OpenTimers code. The API can create many virtual timers that run concurrently.

4) Stack:

The OpenWSN stack (OpenStack) is built around the Time-Slotted Channel Hopping (TSCH) protocol, introduced in the 2015 IEEE802.15.4e amendment. The TSCH implementation requires micro-second accuracy by the underlying hardware and is entirely interrupt-driven to minimize radio duty-cycle (below 1%) and CPU activity. The 6TiSCH specification extended TSCH with the 6TiSCH Operation Sublayer (6top) and Minimal Scheduling Function (MSF) to provide IP-compliance. To optimize the limited payload space and to carry large IPv6 datagrams, OpenWSN implements 6LoWPAN fragmentation and header compression. Applications can interact with the stack through the UDP socket API.

5) Web & Applications:

The web library (OpenWeb) implements CoAP and security protocols such as OSCORE. Applications (OpenApps) can either directly interact with the stack using sockets or use the CoAP code to expose resources or make RESTful requests to other CoAP servers.

B. RIOT

RIOT is a general-purpose OS designed for small IoT devices based on micro-controllers.

1) Kernel: RIOT mimicks a micro-kernel architecture [START_REF] Baccelli | RIOT: An Open Source Operating System for Low-End Embedded Devices in the IoT[END_REF]. Various software modules are aggregated around a minimal kernel providing:

• Multi-threading: thread synchronization with mutexes, semaphores, small and fast inter-process communication. • A tickless O(1) scheduler with priorities and preemption, which can be used to enforce real-time properties. The kernel employs separate thread context and memory stack, with a small Thread Control Block (TCB) enabling minimized stack usage globally.

2) Cross-Hardware Support & Unifed APIs: Crosshardware support is ingrained using a multi-level hardware abstraction including:

• cpu which implements and exposes basic microcontroller functionalities.

• periph which implements and exposes a unified API to access to microcontroller peripherals, leveraged by drivers.

• board which bundles, configures, and maps the selection of CPU and drivers required on a specific IoT device. 3) Network Stack APIs: RIOT provides a number of higherlevel APIs including North-and South-bound interfaces to facilitate drop-in-replacement of the network stack.

A generic South-bound interface consisting of a radio Hardware Abstraction Layer (HAL) or radios compliant with IEEE802.15.4, called ieee802154_dev. It provides a well defined hardware access that allows implementing agnostic PHY and MAC layers on top. Additionally, it allows for direct radio access and configuration when a network stack needs it.

A generic North-bound interface called sock provides a socket-like API for network stacks to offer connections establishment for sending/receiving datagrams on behalf of higher-level logic (i.e. application logic, or another library). Several types are defined, depending on the considered network layer: sock_ip for raw IP layer services, sock_udp or sock_tcp for transport layer services, and sock_dtls for DTLS services. Both synchronous and asynchronous network access are possible. Using these abstractions, RIOT applications can run unchanged on a large variety of network stacks supported in RIOT, including but not limited to the following 6LoWPAN stacks: GNRC, OpenThread, lwIP.

4) Power Management: RIOT's power management system is based on two modules: periph_pm and pm_layered. periph_pm defines for each CPU a series of power modes which define different run or sleep modes. Every power mode entails different power savings at the expense of running peripherals, clock speed, wake up sources, RAM retention. However, different applications, drivers, activities have different requirements. When pm_layered is used, modules do not explicitly set a power mode but may block and unblock specific power modes according to their functioning requirements. If a power mode is blocked, the blocker requires that power mode or lower to run. Then, every time a context switch occurs, the lowest possible power mode is selected in "cascade".

5) RIOT Ecosystem & External Library Support: RIOT aggregates a large ecosystem of external libraries, facilitated by the pkg system that streamlines the seamless integration at compile-time of a variety of third-party software modules.

For instance, the OpenThread [START_REF] Kim | Thread/OpenThread: A Compromise in Low-Power Wireless Multihop Network Architecture for the Internet of Things[END_REF] and lwIP [START_REF] Dunkels | Design and Implementation of the lwIP TCP/IP Stack[END_REF] network stacks are integrated in RIOT using the pkg system. The diversity of libraries integrated with the pkg system includes drivers, cryptographic libraries, graphics libraries, machine learning libraries, scripting support [START_REF] Baccelli | Scripting over-the-air: Towards Containers on Low-End Devices in the Internet of Things[END_REF] etc.

III. INTEGRATING OPENWSN INTO RIOT

We integrated the OpenWSN project into RIOT using RIOT's pkg system, see Fig. 2 OpenOS is replaced by a RIOT event loop. sock provides the Northbound interface and the radio and sctimer module interface with RIOT's ieee802154_hal and ztimer.

A. Hardware Abstractions 1) Timer: In a 6TiSCH mesh, nodes periodically resynchronize, which is a critical operation for the network to function. Improved time synchronization between individual nodes translates directly into lower energy consumption. In practice, nodes use stable (±30 ppm) low-power crystals (32 kHz) as a time reference to minimize synchronization traffic in the network. In OpenWSN, the sctimer BSP module abstracts the low-power timers for use by the OpenTimers API.

When porting to RIOT, we replaces the sctimer with RIOT's ztimer module. The ztimer module wraps hardware timers clocked from stable low-power oscillators. In addition, it provides a continuous time reference, even when the device enters a low-power mode (LPM), shutting down most of its other peripherals.

2) Radio: The OpenWSN's TSCH MAC-layer requires direct access to the radio BSP module. The TSCH protocol instructs with great time precision the radio to turn on, turn off, transmit, and receive. Any delay in the execution of these state changes would lead to energy waste and the danger of desynchronizing the node from the network. When integrating OpenWSN into RIOT, we mapped the radio BSP module to RIOT's ieee802154_hal module.

An initial attempt resulted in a conflict between the two finite state machine (FSM) implementations: TSCH state machine of OpenWSN and the RIOT's radio state machine (Fig. 3). RIOT's ieee802154_hal enforces a standard radio behavior independent from the capabilities of the underlying radio hardware. For example, the ieee802154_hal explicitly requires that the radio resides in the OFF state before changing the radio frequency. In some scenarios, the ieee802154_hal FSM conflicted with OpenWSN's implementation of the TSCH FSM. In OpenWSN's TSCH, when the radio receives a frame, an Interrupt Service Routine (ISR) executes. The ISR turns off the radio and reads the contents from the frame buffer. However, when mapped to an ieee802154_hal, the getReceivedFrame function returns garbage because turning off the radio invalidated the contents of the frame buffer. Both FSMs were later updated to resolve the conflicts.

OpenWSN

RIOT's IEEE802154_HAL

Fig. 3: Conflicting state machine implementations.

B. Network Socket

During the integration process, the OpenWSN socket API was updated to closely resemble the generic RIOT socket interface, i.e., sock. By unifying the API, future applications and libraries can be easily shared.

Due to its non-preemptive scheduler, and thus the singlethread nature of OpenWSN, only asynchronous sockets are supported. With OpenWSN integrated into RIOT, the stack and application code is no longer restricted by the non-blocking requirement. The port extends the OpenWSN socket API with blocking sockets and inter-process communication (IPC) between different threads.

C. Scheduler

OpenOS, OpenWSN's task scheduler, is replaced by an event loop, using RIOT's Event Queue library. The loop iterates over several queues, each with a specific priority. The OpenWSN tasks are mapped to RIOT events and pushed in a queue with corresponding priority.

IV. EVALUATION

To evaluate the effect of RIOT on the performance of the 6TiSCH/OpenWSN implementation, we set up four types of experiments to study: network performance, power consumption, the effect of the kernel on task queue occupancy, and the real-time properties of the scheduling mechanisms involved. In the experiments, we compare a vanilla OpenWSN (OpenWSN V ) implementation with our OpenWSN-RIOT port (OpenWSN R ). Table I

A. Network Performance Evaluation

To study the impact of the RIOT's kernel on the 6TiSCH network performance, we perform experiments using the OpenTestbed [START_REF] Muñoz | OpenTestBed: Poor Man's IoT Testbed[END_REF], a testbed consisting of OpenMote-B devices deployed throughout the Inria-Paris office buildings. We form 30-node networks. Using the methodology of Ko et al. [START_REF] Ko | ContikiRPL and TinyRPL: Happy together[END_REF], we start with a fully homogeneous network of OpenWSN V nodes. In subsequent runs we mix in, in steps of five, OpenWSN R nodes until we reach a fully homogeneous OpenWSN R network. Each node transmits 60 UDP datagrams per hour, on average. For each network configuration, we run the experiment 10 times. Each experiment run lasts 2 h. All results are presented with a 95% confidence interval. The average of each run is denoted by a circle, as well as the average over all runs denoted by an asterisk.

1) Reliability: We assess the end-to-end network reliability in terms of the Packet Delivery Ratio (PDR). PDR is computed as the ratio between the number of datagrams received and the number of datagrams generated by the application. Fig. 4 plots the obtained PDR. We can see that in most cases PDR is above 99%, often reaching 100%. Observed PDR drops are explained by network topologies that force the traffic in the network to pass only through a couple of nodes, as it was being routed towards the border router. For instance, in one iteration in the full OpenWSN V configuration, roughly 3/4 of the nodes in the network routed traffic through a single node. The latter results in drops at that node due to an overflown internal packet queue. Other PDR drops are caused by routing parent changes and the time it takes for the network to stabilize. We do not observe degradation in terms of the end-to-end PDR when introducing OpenWSN R nodes in the network.

All OpenWSN

Vanilla All OpenWSN On RIOT 0.96 0.97 0.98 0.99

1.00

End-to-end PDR Fig. 4: End-to-end reliability shows no degradation when introducing OpenWSN R nodes, in steps of five, in the network.

2) Radio Duty Cycle:

The radio duty cycle refers to the ratio between the cumulative time that the radio chip is powered on, and the measurement period, as seen by the networking stack. Fig. 5 plots the obtained radio duty cycle. We observe consistency across experiments and configurations. Nodes in the network stay consistently synchronized across experiments, independent of the OS heterogeneity. Synchronization losses in the network would have increased the duty cycle, because desynchronized nodes keep the radio constantly on, listening, while attempting to resynchronize. Minimal differences are due to the randomness in the topology formed for each experiment run. This result is expected since duty cycle values depend on the networking stack operation and are handled from the ISR context.

All

B. Power Consumption

We measure the power consumption by connecting an STM32 Power shield 1 , to the nodes in the network. We create a simple star network with one border router, one OpenWSN V and one OpenWSN R node. There is no application running on top of the stack. We attach the power shield in turn to the OpenWSN V and to the OpenWSN R node. We configure the power shield as specified in Table II. We measure the power consumption once the network is stable. We use the lowest CPU LPM in both cases. We are interested in the following metrics: 1) Minimum Current (I min ): the consumption of the node when the board operates in LPM. 2) Peak Current (I max ): the maximal observed current, typically during radio Tx/Rx. 3) Average Current (I avg ): average current consumption. 4) Energy (E): the energy consumed is measured as V * I avg * t, where V is the supply voltage and t is the acquisition time. 5) Normalized Energy (E N ):

E N = E -V * I min * t.
We remove from E the component due to I min . I min depends on the base system configuration and is independent of the OS or the power management system. Table III summarizes the power consumption measurements. We can observe a difference in I avg , where OpenWSN V outperforms OpenWSN R . The difference is due to the significantly higher I min in the OpenWSN R case, which comes from a different base configuration of the system when the board is in LPM. If we consider the normalized energy, E N , we observe that OpenWSN R consumes 0.08% less than OpenWSN R when only running the networking stack. The result is surprising, considering that OpenWSN R introduces overhead due to the mapping of the BSP radio and timer modules, as described in Section III. OpenWSN R also combines two different radio state machines, as depicted in Fig. 3.

C. Scheduler Performance

We study the impact of the kernel on the performance of the task scheduler. Recall that in case of OpenWSN V , the task 1 https://www.st.com/en/evaluation-tools/x-nucleo-lpm01a.html scheduler is OpenOS, while in the OpenWSN R configuration, OpenOS was replaced with RIOT's Event Library. Although the RIOT documentation uses the term event to denote elements of the event queue, in this section we use the term task for both OpenWSN tasks and RIOT events.

1) Impact of the kernel: The first experiment simultaneously monitors the task queue of an OpenWSN V and OpenWSN R node over a period of 30 min. Both nodes are synchronized to a border router and have no children. There is no application running on top of the network stack. Once the network is stable, we generate a timestamp for every task being pushed to the queue and a second timestamp when that task is removed from the queue just before being executed. We are interested in the following metrics:

• Task Wait Time (T W ): Calculated as T W = T D -T A .
T A is the moment when an task arrives in the queue. T D is the instant when it is removed from the queue. • Arrival Rate (λ): the average number of tasks generated by the task over a given period of time. Interestingly, we notice that the task arrival rate (λ) is higher for the OpenWSN R configuration. At the time of writing, we don't have a clear explanation for this phenomenon. Since there are more tasks being generated per second, the value for T W is also greater. Another factor that could influence the T W value is the context switching behavior of RIOT's kernel in the OpenWSN R node. When the task queue is empty, the RIOT kernel performs a context switch, looking for other threads that need attending. If no other active thread is found, the CPU executes the WFI() instruction2 . On OpenWSN V nodes, OpenOS runs directly on top of the hardware. OpenOS is mono-threaded so no context switches need to occur. The longterm average queue occupancy is derived from the T W and λ, which explains the discrepancy.

2) Real-time Properties: To study the real-time properties of the scheduling mechanisms involved, we set up an experiment running two simple applications with different priorities on top of the stack. On the OpenWSN V node, applications are executed by OpenOS, while on OpenWSN R , applications live in separate RIOT threads and are thus executed by the RIOT kernel. The applications have real-time requirements, meaning that the schedulers should execute the tasks of the application as soon as possible when they are scheduled. The first application has a task (T A ) which takes 100 ms to execute. The task is scheduled every 60 s. The second application has a task (T B ) that requires 10 ms to execute and it is scheduled every 100 ms. Using OpenOS, we measure the average wait time T W of 8.1 ms for (T A ) and 8.6 ms for (T A ). Using the RIOT kernel, we measure the average wait time T W of 68 µs for (T A ) and 69 µs for (T A ). The results clearly show the advantage of RIOT's RTOS kernel over the simplistic OpenOS approach when executing several, more advanced apps with specific real-time requirements.

V. TECHNICAL CHALLENGES

Issues related to hardware and the BSP. During the initial integration work, we noticed that the networks containing OpenWSN R nodes, performed significantly worse than the homogeneous OpenWSN V networks. Further investigation uncovered that the problem was caused by different calculation methods for radio signal indicators RSSI and LQI. This difference affected the formation of the network and its performance in general.

Layering abstractions. The integration work provided an incentive for both communities to verify and improve the different APIs and layering abstractions. OpenWSN developed a new socket API which is closely aligned with RIOT's sock interface. The new API provides a better separation between the applications and the CoAP library, and the network stack. For RIOT, the integration of the OpenWSN socket layer, was a valuable test for the design of their sock interface. At the lower layers, the mapping of OpenWSN TSCH module to RIOT's ieee802154_hal exposed bugs and undefined behavior in the state machines of both implementations, which were fixed.

Role of the operating system. RIOT's multi-threaded kernel allows the applications to execute CPU-intensive functions without the risk of blocking other critical tasks. A relevant example are the applications using application-layer security protocols (e.g. EDHOC and OSCORE). Not all devices have hardware acceleration for all the mandatory cryptographic primitives. This is an issue because software implementation of elliptic curve cryptography operations on an Arm Cortex-M3 typically takes between 1 s and 2 s to complete [START_REF] Zandberg | Secure Firmware Updates for Constrained IoT Devices Using Open Standards: A Reality Check[END_REF]. The OS plays an important role in balancing the execution time and real-time requirements of the different tasks.

VI. CONCLUSIONS & PERSPECTIVES

In this paper, we study the effect of the OS on the performance of a time-sensitive 6TiSCH communication stack. We integrate the OpenWSN project, the reference open-source implementation of 6TiSCH, into RIOT, a full-fledged realtime OS. When executing the 6TiSCH networking stack in the RIOT context, we do not observe degradation in terms of the end-to-end reliability and radio duty cycle. When measuring power consumption, we observe that RIOT's power management benefits the stack and results in a slight decrease in consumption, once the base system configuration in LPM mode is accounted for. We also demonstrate benefits in scheduling delays from the point of view of applications with real-time requirements. All the code developed in this paper have been published in the open-source form, for the benefit of OpenWSN and RIOT communities.

Using the unified APIs we propose in this paper, our current work includes supporting hardware and libraries (or applications) that can be compatible as-is, with both RIOT and OpenWSN. On-going efforts towards shared OpenWSN/RIOT libraries include implementations of the IETF's Software Updates for Internet of Things (SUIT) and the IETF's Ephemeral Diffie-Hellman Over COSE (EDHOC) key exchange protocol.

Fig. 1 :

 1 Fig. 1: An overview of the different parts of the OpenWSN.

Fig. 2 :

 2 Fig. 2: The OpenWSN stack runs inside a RIOT thread.OpenOS is replaced by a RIOT event loop. sock provides the Northbound interface and the radio and sctimer module interface with RIOT's ieee802154_hal and ztimer.

Fig. 5 :

 5 Fig. 5: Radio Duty Cycle stays consistently low when we add OpenWSN R nodes in steps of five. The Radio Duty Cycle is independent from the OS heterogeneity.

Drivers OpenStack 6TiSCH MSF IEEE 802.15.4e (L2 security) UDP Socket API 6TiSCH 6top 6LoWPAN RPL ICMPV6 OpenOS OpenApps OpenWeb CoAP OSCORE CoAP Apps UDP Apps Kernel OpenTimers OpenSerial OpenSensors Board Support Package (BSP) CC2538 NRF52840 SAMR21-XPRO ... IoT-LAB_M3

  . Our implementation is opensource, published in[START_REF] Molina | OpenWSN port to RIOT[END_REF]. From the point of view of RIOT, OpenWSN operates as any other networking stack supported by the OS. The North-bound interface of the OpenStack library exposes the UDP socket API. The South-bound interface interacts with RIOT's HAL. From the point of view of OpenWSN, RIOT acts as a board support package. It implements all the BSP functions required for the correct operation of the OpenStack: sctimer and radio modules.

	RIOT Applications	OpenWSN Applications
		Socket API	RIOT
			UDP	Thread
		6TiSCH MSF	RPL ICMPv6
		6TiSCH 6top	6LoWPAN
		IEEE 802.15.4e
		(L2 security)
	OpenTimers OpenSerial	OpenStack	Kernel (scheduler)
	Drivers	radio	sctimer	RIOT Event loop
		BSP
	IEEE802154_hal	ztimer ztimer
		Hardware Platforms

TSCH FSM ISR: Process Received Frame

	turn off	get received
	radio	fram
	radio off state +	returns
	invalidates	garbage from
	frame buffer	frame buffer

TABLE I :

 I lists the different configuration parameters of the OpenWSN stack. OpenWSN configuration parameters.

	Parameter	Value
	App. traffic load	Avg. 1 pkt/min
	RPL DIO period	10 s
	RPL DAO period	60 s
	TSCH slotframe length	101
	TSCH slot duration	20 ms
	TSCH max retransmission	15
	Frequency channels	16

TABLE II :

 II STM32 Power shield parameters.

	Parameter	Value
	Sampling frequency	100 kHz
	Acquisition time	100 s
	Supply voltage	3.3 V

TABLE III :

 III Power consumption measurements.

	Config	I min	I max	I avg	E	E N
	OpenWSN V	0.26 mA 38.3 mA 1.10 mA 0.36 J 0.28 J
	OpenWSN R	0.44 mA 37.1 mA 1.20 mA 0.40 J 0.25 J

•

  Long-Term Average Queue Occupancy (L): the longterm average number of tasks in the queue. Calculated using Little's Law.• Maximum Queue Occupancy T max : maximum number of tasks in the queue at any given moment.

TABLE IV :

 IV Scheduler performance measurements.

	Config	T W [µs]	λ task s	L	T max
	OpenWSN V	111.92	13.99	1.56e-3	5
	OpenWSN R	143.65	16.45	2.36e-3	4

WFI() is part of the Arm Cortex instruction set, other CPU platforms have similar instructions.

Acknowledgements: This work is partly funded by the European Commission through H2020 SPARTA, www.sparta.eu.