Rayleigh wave transport along a surface with random impedance
É. SavinONERA – The French Aerospace Lab, Châtillon cedex, France
ABSTRACT: In this paper we derive a radiative transfer equation for Rayleigh surface waves on the flat surfaceof an elastic half-space with a random boundary impedance. This transport equation applies to the surface waveenergy density resolved in phase space. The analysis uses some classical pseudo-differential calculus appliedto a steady-state formulation. Explicit expressions of the scattering cross-section for Rayleigh waves and theirleakage rate to bulk waves within the half-space are obtained as functions of the power spectrum of the randomimpedance. The influence of the latter on the scattering and radiation mean free paths is studied numerically.This problem has interest for applications to non-destructive monitoring and evaluation of structural components,seismic site effects in geophysics, passive imaging, or civil engineering among several other significant examples.
1 INTRODUCTION
The propagation of wave energy in random mediacan be analyzed with a radiative transfer equationin the case of weak random fluctuations and prop-agation distances/times that are long with respectto the small wavelength/period of the signal (high-frequency limit). A systematic and efficient way toderive the transport regime from random wave equa-tions is presented by Papanicolaou & Ryzhik (1999),among others. Here we focus on the case of elastic(Rayleigh) surface waves on a randomly inhomoge-neous surface in this very limit. This issue is a typicalexample of high-frequency wave propagation in atwo-dimensional randomly heterogeneous medium. Aclassical result in quantum physics is to consider thatall states in two-dimensional disordered systems arelocalized (Imry 1997). However it is observed for clas-sical waves that the localization length can be muchlarger than the scattering mean free path, such thattransport dominates over localization in a broad rangeof distances. The basic objective of this research isto derive such transport properties, considering moreparticularly elastic surface waves. Multiple scatteringof the Rayleigh waves propagating on the flat sur-face of an elastic half-space with a random boundaryimpedance may be described by a transport modelas outlined above. In this respect, the methodologyof Bal et al. (2000) or Bal and Ryzhik (2002) canbe extended to the case of vector surface waves inorder to derive a radiative transfer equation for theiramplitude. The latter applies to the associated energydensity resolved in phase space. The analysis usessome classical pseudo-differential calculus and con-siders a steady-state formulation. It also yields explicitexpressions of the scattering cross-section for thosesurface waves and their leakage rate to bulk waveswithin the half-space. These parameters classicallydescribe the so-called mesoscopic regime of wave
propagation phenomena; they have a direct physicalinterpretation in terms of scattering and radiation meanfree paths. For surface waves they can be obtainedas functions of the power spectrum of the randomimpedance. A formal computation of the scatteringparameters for Rayleigh waves has however not beenreported so far, to the author’s knowledge. The mainpurpose of this paper is thus to propose a possi-ble theoretical derivation of such properties. Theirstudy is of broad interest for applications to seismicsite effects in earthquake engineering (Lombaert andClouteau 2006), SAW imaging techniques (White &Voltmer 1965), passive imaging techniques (Shapiroet al. 2005), or civil engineering and non-destructiveevaluation of structural components at large, amongseveral other significant examples.
In the following section, we first introduce thebasic equations describing the propagation of high-frequency Rayleigh waves and the notations usedthroughout the paper. The short Sect. 3 introduces themain mathematical tools used in the subsequent anal-ysis of the contemplated problem, which is formalizedas a two-scale boundary condition for the amplitudevector of the Rayleigh waves. The main results of thisresearch are outlined in the Sect. 4, where a detailedderivation of the corresponding radiative transportequation is provided together with a short discus-sion of the influence of the random impedance powerspectrum on the scattering parameters (scattering andradiation mean free paths).
2 HIGH-FREQUENCY RAYLEIGH WAVESSETTING
2.1 Notations and hypotheses
A Rayleigh wave u = (ux, uy, uz) propagating at the fre-quency ω > 0 in an homogeneous, isotropic elastic
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half-space (x, z) ∈ R2 × R+ of which surface � is
{z = 0} reads as:
where:
i = √−1, and k = kk with k =√
k2x + k2
y is the hor-izontal wave vector such that k · n = 0 if n is theoutward unit normal to the half-space. Note that theunit wave vector k lies on the unit circle S1 of R
2.The vertical wavenumbers φα(k) for either com-pressional (α = p) or transverse (α = s) motions aredefined by:
Here kα = ωcα
and wave celerities cα are given as func-tions of the half-space material density � and Lamé’smodulii λ, µ by:
Coefficients Ap, As1 and As2 are derived from thetraction-free boundary condition on �:
which also reads as:
where:
The solvability condition for the above linear systemyields Rayleigh’s secular equation:
which has only one positive solution k2R such that kR >
ks > kp. Introducing notations φp(k) = ip and φs(k) =iq for k > ks, the above equation is also:
Instead of the usual traction-free condition t(u) = 0 on�, we shall consider a more general boundary condi-tion of the form (see e.g. Bal et al. 2000, Garova et al.1999):
where (Z(x) , x ∈ R2) is a second-order, mean-square
homogeneous and centered stochastic process withreal values and correlation function R:
and E stands for the ensemble average (mathemati-cal expectation). It is assumed that its fluctuations areweak, that is:
and that the wavelength and correlation length are ofthe same order ε. Then the boundary condition readsin the rescaled variables x → x
ε, z → z
εas:
The amplitude√
ε of the random impedance is con-sistent with Eq. (6) above. We also assume thatsteady-state solutions uε of Eq. (7) have the form:
where:
Aε ∈ C, and matrix Q is given by Eq. (1). Introduc-ing the functions Q(k) = (Q(k, 0)b(k), n) and P(k) =(P(k)b(k), n) given for k = |k| by:
and:
respectively, with P(k) defined by Eq. (2), the bound-ary condition of Eq. (7) finally reads:
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In the above P(D) and Q(D) are the pseudo-differentialoperators on S(R2) (the Schwartz space of all C∞ func-tions on R
2 which are rapidly decreasing toward 0 atinfinity as well as all their derivatives) with symbolsP(k) and Q(k), respectively. Here the standard formof the Fourier transform in L2(R2) is:
and the pseudo-differential operator K(x, D) of symbolK(x, k) (a smooth complex-valued function of botharguments) is:
3 MATHEMATICAL TOOLS
This short section introduces the main mathemat-ical tools used in the subsequent analysis for thederivation of the transport equation of Rayleigh wavesfrom the two-scale boundary condition (10) definingtheir propagation at the surface of an homogeneoushalf-space.
3.1 Wigner transform and Wigner measure
The transport equation for the energy density of a high-frequency Rayleigh wave is derived starting from theWigner distribution of its amplitude Aε. It is a functionof position x on the surface and wave vector k and it isscaled by the small parameter ε. The Wigner transformof f , g ∈ S ′(R2) is defined by:
where f stands for the complex conjugate of f . Pro-vided that the sequence Aε lies in a bounded subsetof L2(R2), the complex-valued sequence Wε[Aε] :=Wε[Aε, Aε] has (up to an extracted subsequence) a realweak-* limit in S ′(T ∗
R2) as ε → 0 which is also a non-
negative measure, the so-called Wigner measure W ofAε (Gérard et al. 1997).
3.2 Some pseudo-differential calculus
In this section we give without proofs some usefulresults on pseudo-differential operators (see Bal 2005,Gérard et al. 1997 for some examples in the same con-text) which we shall need in the subsequent section forthe derivation of the transport properties of the Wigner
measure W . First, we will have to compute the Wignertransform:
for an homogeneous operator K(D). This relation canbe expanded in the form (Bal 2005, Gérard et al. 1997):
for a non-homogeneous operator K(x, D), where{K , W } stands for the usual Poisson bracket:
To account for the highly oscillatory fluctuations ofthe random impedance we shall also use the relation(Bal 2005):
where γ is the Fourier transform of γ with respect tothe fast scale x
ε.
4 TRANSPORT EQUATION FOR RAYLEIGHWAVES
This section now presents the main results of thisresearch: the derivation of the radiative transport equa-tion for the amplitude vector Aε of the Rayleigh waves.The analysis starts from the Wigner-Liouville equationassociated to the two-scale boundary condition (10)and is carried on using a multi-scale expansion ofthe Wigner transform of the sequence (Aε) about itsWigner measure (the limit Wigner transform as ε goesto 0). Equating like powers of ε yields the dispersionproperties of the Wigner measure, then the explicitexpression of the lower-order corrector of the Wignertransform, and finally the radiative transfer equationitself. These developments basically follow the deriva-tion in (Bal 2005, Bal et al. 2000, Bal and Ryzhik 2002,Papanicolaou and Ryzhik 1999, Powell and Vanneste2005).
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4.1 The Wigner-Liouville equation
Premultiplying Eq. (10) by A∗ε , where f ∗ means com-
plex conjugation and transposition if need be, thentaking the Wigner transform and summing with theadjoint equation, one obtains the so-called Wigner-Liouville equation:
where y := xε. The Wigner transform Wε[Aε] of the
sequence Aε is now expanded in two scales x (the slowone) and y (the fast one) as (Papanicolaou & Ryzhik1999):
and plugged into Eq. (15), which becomes1 owingto Eq. (12):
since εD = εDx + Dy whenever y = xε.
4.2 Dispersion properties of the Wigner measure
The above Eq. (17) yields to the leading order O(ε0)the following dispersion equation:
Thus the Wigner measure W (x, k) reads:
where w is a scalar positive measure on S∗R
2 and is defined by Eq. (9). Here ′ := d
dk and (k) = 0 isnothing but Rayleigh’s dispersion equation (3).
1 Notation W (x, k)P∗(k − εD) should be interpretedas the inverse Fourier transform of the functionW (k′, k)P(k − εk′), in accordance with the definition (11).
4.3 The O(√
ε) corrector
To the order O(√
ε) one has owing to Eq. (14):
since W does not depend on the fast scale y, or, ifone considers the Fourier transform W 1
2of W 1
2with
respect to that fast scale:
Here 0 < θ � 1 is a small arbitrary parameter intro-duced for regularization that will be sent to 0 at theend of the derivation (Papanicolaou & Ryzhik 1999).Thus:
with:
4.4 Derivation of the radiative transfer equation
To the first order O(ε) one now has owing to Eq. (13)and again Eq. (14):
where:
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We then consider the ensemble average of the aboveequation. First it is assumed that W1 is orthogonalto W in order to justify the expansion (16), so thatE{PW1 + W1P∗}= 0. Next, using Eq. (20) one has:
But E{Z(k′ )Z(k′′)}= (2π)2R(k′)δ(k′ + k′′) by defini-tion of the power spectral density R(k) of the mean-square stationary stochastic process (Z(x) , x ∈ R
2), sothat taking the ensemble average of Iθ yields:
In the above calculation a crucial (mixing) assumptionhas been used: indeed it is expected that E{ZZW } �E{ZZ}E{W } since both quantities Z and W vary ondifferent scales; this assumption may be justified rig-orously for the Schrödinger equation (Erdös & Yau2000) or the scalar wave equation in a discrete set-ting (Lukkarinen & Spohn 2007). Note also that in theabove equation and below E{W } is still denoted by Wfor convenience. Then we obtain:
where:
The above integrals are split in four parts, for |k′| ≥ kR,ks ≤ |k′| < kR, kp ≤ |k′| < ks, and |k′| < kp.As k is suchthat |k| = kR because the Wigner measure W (x, k) has
support on that circle, (k) is real and Q(k) is purelyimaginary. Q(k′) is purely imaginary whenever |k′| >kp, and (k′) is real whenever |k′| ≥ ks or |k′| ≤ kp;thus the second integral reduces to:
in the limit θ → 0. Also W (x, k′) = 0 whenever |k′| <kR, so that the first integral reduces to:
as θ → 0. Since the left-hand-side in Eq. (21) is:
and Q(k) = Q(k′) if |k| = |k′|, one finally gets thetransport equation:
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Figure 1. Normalized phase () and transport velocities ofRayleigh waves as functions of Poisson’s coefficient ν.
for the Wigner measure W of Rayleigh waves ampli-tude, with:
The right-hand-side of Eq. (22) is the scattering opera-tor of Rayleigh surface waves into themselves with thesame wave numbers |k| = |k′| = kR but different direc-tions. The second term on the left-hand-side accountsfor leakage of Rayleigh waves into bulk waves whichpropagate away from the surface in the lower half-space. It is a purely lossy term because the half-spaceis homogeneous and thus bulk waves do not return tothe interface. Indeed, one can easily check from (9)and (8) that:
whenever |k′| ≤ ks. As (k) = 0 on the support ofW (x, k), the loss factor is:
where |k| = kR. At last, the first term on the left-hand-side is the streaming flow of energy along straight rayson the surface at the speed ′(k) for |k| = kR. Thisnormalized transport velocity is plotted on Fig. (1) asa function of the half-space Poisson’s ratio −1 ≤ ν ≤ 1
2 .Rayleigh waves phase speed cR as given by Eq. (3) isalso displayed (normalized by cs).
4.5 Influence of the impedance power spectrum
The strength of scattering and leakage of Rayleighsurface waves may be compared as in Bal et al.
Figure 2. The ratio �/�rad as function of Poisson’s coeffi-cient ν for k0 = 0.
(2000) introducing the scattering and radiation lengthsdefined by:
and �−1rad := η, respectively. For a statistically isotropic
random impedance (Z(x) , x ∈ R2), such that its power
spectral density R(k) depends on |k| solely, theseparameters are:
where |k| = kR, and:
where S(k , φ) = R(√
k2 + k2R − 2kkR cos φ). The ratio
�/�rad is plotted on Fig. (2) through Fig. (4) as afunction of ν for a Gaussian model of correlation:
where√
σ is the correlation length of the randomimpedance and k0 is its central frequency. The caseσ = 0 corresponds to a delta-correlated impedance.One observes that radiation dominates over scatteringfor positive values of Poisson’s coefficient. An effi-cient means to increase scattering over radiation is to
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Figure 3. The ratio �/�rad as function of Poisson’s coeffi-cient ν for k0 = kR.
Figure 4. The ratio �/�rad as function of Poisson’s coeffi-cient ν for k0 = 3kR.
increase the central frequency of the spectrum (25).Similar conclusions may be reached for an exponentialmodel of correlation, for example, corresponding toa mean-square non-differentiable random impedanceprocess.
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