Scattering of Elastic Waves

In Volume 1, the propagation medium, assumed to be homogeneous, was either infinite (Chapter 1), divided into two parts by a plane interface (Chapters 2 and 3), or limited in one or two dimensions (Chapter 4), the waves propagating freely in this or other dimensions. Here, the propagation medium is locally inhomogeneous; these heterogeneities may be inherent to the structure or may result from manufacturing defects (inclusions) or from the aging of the parts (cracks). Waves scattered by an obstacle propagate away from it, possibly in privileged directions, even if the medium is isotropic. Several applications in underwater acoustics are based on target scattering: detecting mines or ships by an active sonar, locating shoals of fish, or mapping ocean floors. In non-destructive testing by ultrasound, the information carried by the scattered waves is used to characterize these heterogeneities or to prevent damage, such as the delamination of composite materials. Scatterers can be deliberately introduced, for instance, to attenuate backward waves emitted by a piezoelectric transducer and thus dampen its responses (section 3.1). Recently, the development of periodic structures (phononic crystals) or heterostructures made of resonant inclusions dispersed in a matrix (acoustic metamaterials) opened the way to the filtering and the control of elastic wave propagation.

The interaction of a (plane) elastic wave with a particle embedded in a matrix gives rise to many phenomena (Figure 2.1), such as:

- – the reflection at the particle–matrix interface;
- – the transmission inside the particle with a possible mode conversion, and then internal reflections and refraction;
- – the conversion into waves propagating at the interface with the matrix, and then re-emission into bulk waves;
- – the diffraction by the edges of the object;
- – the absorption of part of the energy within the particle.

All of these phenomena, which can give rise to resonances involving bulk, surface or guided waves, have been studied separately in Volume 1. The scattering theory accounts for these phenomena in a global way. The approach consists of defining, in the harmonic case, an incident field and a scattered field that satisfy the propagation equations in each of the two media. The boundary conditions at the interface between the scatterer and the matrix are applied to the total field.

Formally, the displacement field * u* is written as the superposition of an incident displacement field

In a perfect fluid or an isotropic solid, the acoustic field associated with longitudinal waves is determined through the knowledge of a scalar potential Φ, while the field associated with transverse waves in the solid derives from a vector potential * ψ*. For obstacles having a canonical form (cylinder or sphere), the method of analysis consists of:

- – expanding the potentials of the incident wave on the basis of harmonic solutions (cylindrical or spherical) to the Helmholtz equation (see Appendix 1);
- – postulating a similar development, with unknown coefficients, for the scattered waves;
- – determining these
*scattering coefficients*by applying, for both fields, the boundary conditions at the fluid–solid interface.

Due to the orthogonality of cylindrical and spherical harmonics, these conditions can be applied independently to each harmonic of order *n*. The problem can be reduced to the resolution of a linear system having, *a priori*, an infinite number of equations. Finally, the scattered field is obtained as an expansion of cylindrical or spherical harmonics. Fortunately, the number of terms can be reduced according to the convergence of the series of harmonic functions.

The interest in this model is to predict the evolution of the acoustic intensity scattered in each direction, according to the elastic and geometrical characteristics of the scatterer. The complexity of the calculation will quickly convince the reader of the advantage of considering the symmetry of the “incident wave + scatterer” configuration in order to reduce the number of modes to take into account in the developments. From this point of view, it is very simple to use the principle stated by Pierre Curie in 1894: “When certain causes produce certain effects, the elements of symmetry of the causes must be found in the effects produced.”

As an illustration, let us study two cases that will be examined in sections 2.1 and 2.3.2.

1) A longitudinal plane wave propagating along the

*x*axis encounters a cylinder infinitely long along*z*(Figure 2.2(a)). The configuration is symmetric with respect to the*xy*plane: the scattered waves (longitudinal or transverse) are necessarily polarized in the*xy*plane:=__u__(__u__*r, θ*) with*u*= 0._{z}2) A longitudinal plane wave propagating along the

*z*axis encounters a sphere (Figure 2.2(b)). The configuration is axisymmetric around the*z*axis: the scattered field is independent of the polar angle*ϕ*:=__u__(__u__*r, θ*) with*u*_{ϕ}= 0.

SCATTERING CROSS-SECTION.– One of the results of this analysis is the existence of three scattering regimes depending on the value of the wavelength with respect to the characteristic dimensions of the heterogeneities:

- – the
*Rayleigh scattering*regime, when the wavelength is large compared to the size of the hetereogeneity; - – the
*resonant*regime, when the wavelength is comparable to the size of the hetereogeneity; - – the
*geometric*regime, when the wavelength is small compared to the size of the hetereogeneity.

These regimes are characterized by a different dependence of the scattered power (as a function of frequency) and acoustic intensity (depending on the direction). These variations are expressed by two scattering cross-sections, which we define first in the case of a sphere. Using *I _{inc}* to denote the acoustic intensity (power per unit area) transported by the incident wave, the

This one is obtained by integrating the scattered intensity *I _{d}*(

*σ* is homogeneous to a surface and independent of *r*, because, in the far field, the intensity of the scattered waves decreases as 1*/r*^{2}. This quantity is usually normalized with respect to the dimensions of the scatterer. In the case of a rigid (in acoustics) or opaque (in optics) sphere, whose radius *R* is very large compared to the wavelength *λ*, and for a directional beam, it is natural to assume that, in the absence of absorption, the scattered power corresponds to the rays intercepted by the particle, that is, *πR*^{2}*I _{inc}*. Indeed, since

When the radius *R* is of the order of the wavelength, resonances and/or interference between the incident wave and the forward scattered waves considerably increase the efficiency of the scattering. This regime, described by the Mie theory in optics, is called *resonant*. Let us note that in acoustics the phenomena are more complex, especially in the case of an elastic scatterer buried in a solid matrix.

In order to characterize the scattering anisotropy, it is necessary to introduce a differential scattering *section σ _{d}*(

Given the relation [2.2] and the 1*/r*^{2} decay of the scattered intensity, *σ _{d}* only depends on the direction defined by the polar angles

In the case of a *cylinder, infinite and homogeneous* along the *z* axis, the scattered acoustic intensity *I _{d}*(

where:

is the differential cross-section per unit length of the cylinder.

In the first section, entitled *acoustic scattering*, the scatterer is a cylinder immersed in a perfect fluid, therefore, the incident wave is longitudinal. Two cases are studied: a rigid cylinder and an elastic cylinder. In the second section, the elastic scattering by a cylinder embedded in an isotropic solid matrix is illustrated by a practical example for a longitudinal and a transverse incident wave. The scattering by a rigid or an elastic sphere buried in an isotropic solid matrix is treated in the third section. For both types of incident waves, the scattering cross-sections are calculated and their variations are illustrated in typical configurations. The phenomena of multiple scattering by a set of particles are discussed at the end of the chapter, using only a scalar approach.

In this section, we study the scattering of an acoustic wave by an infinitely long cylinder, of radius *R*, immersed in a fluid. The propagation is analyzed for time-harmonic waves of angular frequency *ω* and the time dependence in *e*^{−iωt} is deliberately omitted. Let *R _{c}* be the cylindrical coordinate system (

where *J _{n}* is the Bessel function of the first kind and where

where is the Hankel function of the first kind, associated with a divergent wave. The amplitudes *T _{n}* of the cylindrical harmonics are to be determined from the boundary conditions. The total pressure and the radial velocity in the fluid are deduced from formulae [1.3] and [1.4]:

and:

where the derivatives of the Bessel and Hankel functions are given by relations [A1.14].

The separation of variables method thus leads to express the acoustic fields in the form of modal developments. These developments are convergent series. Apart from the monopolar mode (*n* = 0) and the dipolar mode (*n* = 1), the contribution of the *n*th order mode to this series is preponderant, as long as *n* is less than the reduced wave number *kR*. For numerical reasons, the series may therefore be truncated and the number of modes to consider for a satisfactory convergence depends on the frequency domain. At low frequencies, that is, when *kR <* 1, only the monopolar and dipolar modes have a significant influence. It is, therefore, usual to retain only these two modes, which leads to simpler analytical developments.

If the cylinder is immobile and rigid, the radial component of the particle velocity cancels on its surface: *v _{r}*(

Given the orthogonality of the cos(*nθ*) functions, for each cylindrical harmonic we get:

Given the expression [A1.14] for the derivative of Bessel and Hankel functions, the scattering coefficients *T _{n}* can be written as:

In order to calculate the scattering cross-section, the solutions must be expressed in the far field. Since the Hankel function of the first kind and its derivative have the following asymptotic behavior:

the radial velocity and acoustic pressure associated with the scattered wave are given by the approximate expressions:

where the *far-field scattering function f* (*θ*) is equal to:

The scattered acoustic intensity is the time-average of the radial component of the Poynting vector:

In the far-field, *I _{d}* is proportional to the intensity of the incident plane wave

and where:

is the *differential scattering cross-section*, defined by equation [2.5]. The scattering cross-section corresponds to the average of *σ _{d}*(

Since the cylinder is uniform and infinitely long, the integration along *z* is limited to the unit length. Upon normalizing with respect to the geometric cross-section 2*R* of the cylinder of unit length, the substitution of the expression [2.15] of the far-field scattering function *f* (*θ*) into relation [2.19] leads to:

In the long wavelength regime, that is for the summation appearing in the far-field scattering function *f* (*θ*) can be restricted to the first two modes. In this case, the scattering coefficients are given by the approximate expressions:

and the normalized differential cross-section *σ _{d}*/2

At high frequencies the normalized differential cross-section can be approximated by:

where the first term is a geometrical term representing the reflection by the cylinder, and the second one is a corrective term on the main lobe (Morse and Feshbach 1953).

The differential cross-section is plotted in Figure 2.3 as a function of angle *θ* for different values of the product In each figure, the arrow indicates the propagation direction of the incident wave. At low frequencies, the differential scattering cross-section is small for *θ* = 0 and maximum for *θ* = *π*: the scattering takes place backward. Conversely, at high frequencies, the differential scattering cross-section is small for *θ* = *π* and maximum for *θ* = 0: the scattering takes place forward. This result, which can be surprising, *a priori*, is explained in the following paragraph.

Figure 2.4 depicts the interaction, at high frequency of a longitudinal plane wave with a rigid cylinder. The map of the scattered pressure field (Figure 2.4(a)) illustrates the magnitude of the forward emission lobe shown in Figure 2.3(c). On the map of the total pressure field (Figure 2.4(b)), there is clearly a shadow area behind the cylinder. It results from the destructive interference between the incident and scattered fields, which have almost the same amplitude, and are opposite in phase. The power scattered toward the front is, therefore, equal to 2*RI _{inc}*. The second part of the scattered power comes from the reflection (clearly visible in Figure 2.4(a)), which corresponds to the geometric term in expression [2.23]. The integration over

The scattering by an isotropic elastic cylinder involves calculations similar to those developed for a rigid cylinder, but taking into account the longitudinal and transverse waves transmitted inside the cylinder. The cylinder material is characterized by its mass density *ρ* and its Lamé constants *λ* and *μ*, so that the longitudinal and transverse wave velocities are: The solution to the propagation equation is searched using the Helmholtz decomposition (equation [1.77] of Volume 1):

where the scalar potentials Φ_{L} and Φ_{T} for longitudinal and transverse waves, polarized in the *xy* plane, are solutions to the Helmholtz equations (see section 1.2.6.1, Volume 1):

The displacement components are expressed in terms of Φ_{L} and Φ_{T} by the relations [1.126] in Volume 1:

Only stresses are involved in the boundary conditions. It is therefore necessary to express them, using formulae [A1.9] of Volume 1, as functions of the scalar potentials Φ_{L} and Φ_{T} :

During the calculation, and in order to limit the order of the derivatives to one, the second derivatives of the potentials, with respect to *r*, have been eliminated using the Helmholtz equations.

In the fluid, the velocity potential Φ_{i} is defined by the development [2.6]. Since * v* = −

The potential of the transverse wave, transmitted in the cylinder, is expressed as a function of sin(*nθ*), while the potential of the longitudinal waves (incident, scattered, and transmitted) is expressed as a function of cos(*nθ*). This difference can be explained by the derivative operation according to *θ* and applied to the potential Φ_{T} in relations [2.26] and [2.27] giving the displacement and the stress From the expressions for the potentials, the normal displacement and the stresses and take the following forms at *r* = *R*:

with:

and where

The boundary conditions on the cylinder surface *r* = *R* are:

The substitution in the boundary conditions of the expressions of the pressure [2.8] and the radial velocity [2.9] in the fluid, as well as the displacements and stresses [2.29] in the solid, leads to a system of three equations with an infinite number of unknowns: the scattering coefficients *A _{n}*,

and for each harmonic of order *n >* 0:

The resolution of this system provides the scattering coefficient *T _{n}* for the waves in the fluid and, as in the previous section, the scattering cross-section. Its normalized value

In the harmonic case, these waves are the solutions, finite at the origin, to the Helmholtz equations [2.25], propagating on the surface of the cylinder. The potentials Φ_{L} and Φ_{T} are given by equation [A1.17], by replacing *R _{n}* with Bessel functions of the first kind

These exponential solutions represent waves propagating anticlockwise at the phase velocity *V* = *ωR/n*. Assuming that the surface of the cylinder is traction free, the stresses *σ _{rr}* and

By taking into account expressions [2.30] of coefficients the characteristic equation takes the form:

and the ratio of the amplitudes of the potentials, for *n* ≠ 0, is given by:

The characteristic equation [2.36] is solved by denoting:

For each value of *n*, there exists an infinite number of solutions *x _{m}*, to which correspond the angular frequencies

Figure 2.6(a) shows the variations in the normalized frequency *ωR/V _{T}*

The position of the first circumferential modes (*n,m*), indicated in Figure 2.5, is in good agreement with the resonances of the scattering cross-section in the steel cylinder. Given its very small acoustic impedance, the presence of water causes little disturbance in the propagation of waves on the surface of the steel cylinder. This is no longer true for the plexiglass cylinder, and the characteristic equation of the interface waves are thus obtained by canceling the determinant of the system [2.33].

In this section, the scatterer is an elastic cylinder embedded in an isotropic solid matrix. The displacements in the matrix and cylinder are calculated using the Helmholtz decomposition [2.24]. Let *a _{I}* (I = L, T) be the amplitude of the displacement potential for the incident plane wave (longitudinal or transverse) propagating in the

The potential associated with a scattered wave takes the form:

where the scattering coefficient represents the amplitude of the cylindrical harmonic *n* of the scattered mode M for an incident wave of type I. The calculation of the scattering coefficients involves the boundary conditions on the cylinder surface *r* = *R*. This requires the calculation of displacements and stresses, given by the expressions [2.26] and [2.27], respectively.

For an incident wave of type I (L or T), the differential scattering cross-section is the sum of two terms: *σ _{d}* =

Given the orthogonality of the trigonometric functions, the scattering cross-sections normalized by the geometric cross-section of the scatterer can be written as:

Scattering by a glass cylinder buried in an epoxy matrix is illustrated in Figure 2.7. The differential scattering cross-sections into longitudinal and transverse waves are plotted in the case of a *longitudinal incident wave* and for two values of the dimensionless wave number The behavior of the scattered longitudinal wave is comparable with that of the acoustic wave scattered by an immersed cylinder. At low frequency the scattering is weak and mainly takes place backward (decreasing *x*). At medium frequency backward and forward (increasing *x*) scattering are comparable. The converted transverse waves behave similarly, but their scattering is oriented at ±40^{◦} with respect to the propagation direction. Further, it is interesting to note that at low frequency, the differential scattering cross-section of the transverse wave is twice that of the longitudinal wave. More generally, the conversion of a longitudinal wave into a transverse wave is significant at low frequencies, as long as the cylinder is more rigid than the matrix.

The differential scattering cross-sections into longitudinal and transverse waves are plotted in Figure 2.8 in the case of a *transverse incident wave* for two values of the dimensionless wave number The transverse wave is scattered along three main directions: backward and in two directions close to the *y* axis, perpendicular to the propagation direction. The converted longitudinal waves have the same behavior as the converted transverse waves presented in Figure 2.7, but with much smaller amplitudes (Liu *et al.* 2000). Wave conversion is thus smaller in the case of the transverse incident wave.

The variation of the scattering cross-sections *versus* frequency is shown in Figure 2.9 for a longitudinal and a transverse incident wave. As previously indicated, the wave conversion in the case of the longitudinal wave is significant at low frequency, that is, for At higher frequencies, the scattering cross-section *γ _{T}* presents an oscillatory behavior whose period is (Figure 2.9(b)).

ORDER OF MAGNITUDE.– For a glass fiber of diameter 10 *μ*m and a frequency *f* = 5 MHz, the reduced wave numbers are _{L} = 0.13 and _{T} = 0.29. In the case of an epoxy resin matrix and a longitudinal incident wave, the normalized scattering cross-sections are small: *γ _{L}* = 0.004 and

In the first section, we give the general expressions of the displacement components and mechanical stresses in terms of the scalar potentials. In the second and third sections, we analyze successively the interaction of a longitudinal and a transverse elastic wave with a spherical particle embedded in an isotropic matrix. The scattering by a rigid particle is first investigated and then the scattering by a gaseous or a solid elastic inclusion.

Let *R _{s}* be the spherical coordinate system (

Considering expressions [A1.11] and [A1.15] in Volume 1 for the operators __grad__ and __rot__ in spherical coordinates, the components of the mechanical displacement are expressed as

Only the stresses *σ _{rr}*,

with:

In the following section, the potentials of the incident and scattered waves are developed over spherical harmonics. In Appendix 1, these harmonics are expressed using spherical Bessel functions and Legendre polynomials.

The case of a rigid sphere is treated first, in order to illustrate the decomposition of the incident and scattered fields on the spherical harmonics, as well as the use of orthogonality relations to calculate the scattering coefficients. The scattering by an elastic particle is then studied in the low-frequency domain, where monopolar and dipolar resonances appear. In the final section, the scattering cross-sections and the far-field scattering functions are calculated using approximations that are valid in the far field.

The incident wave is a longitudinal plane wave propagating in the *z* direction with the scalar potential and the displacement * u^{i}* =

where *j _{n}* and

where are the scattering coefficients for longitudinal and transverse waves, respectively. Note that is the spherical Hankel function of the first kind and order *n* corresponding to a divergent wave. For the transverse wave, the sum must necessarily start from the term *n* = 1 because there is no monopolar transverse wave (Volume 1, section 1.2.6.2). By writing _{L} = *k _{L}*

with and:

For the incident wave (*p* = *i*), the function *f _{n}* is the spherical Bessel function of the first kind

The orthogonality relation [A1.41] for the Legendre functions allows us to decouple this system into one system for each spherical harmonic:

The scattering coefficients are then given by the relations (Gaunaurd and Überall 1976):

At low frequencies, the scattering coefficients of the monopolar (*n* = 0) and dipolar (*n* = 1) modes are dominant in the modal series. By performing a limited development of spherical Bessel and Hankel functions for _{L} *<<* 1, the scattering coefficients of these modes are approximately written as (Valier-Brasier *et al.* 2015):

In the case of a particle with mass density *ρ _{p}* and Lamé constants

The transmission coefficients of the waves in the particle and the scattering coefficients in the matrix, (see equation [2.47]), are determined by the boundary conditions.

At *low frequency*, that is, for _{L} *<<* 1, the scattering coefficients for the first two modes are given by Valier-Brasier *et al.* (2015):

The monopolar mode (*n* = 0) depends directly on the difference between the bulk moduli of the two media while the dipolar (*n* = 1) mode depends on the density ratio = *ρ _{p}*

This resonance occurs when the angular frequency *ω* cancels the denominator of the scattering coefficient

Assuming a rheology of Kelvin–Voigt type (equation [1.299] of Volume 1) for the shear modulus: *μ* = *μ*_{0} − *iωη* and by writing we get:

where:

with:

The terms Γ_{vis} and Γ_{rad} express the viscosity and radiation losses, respectively. The resonance frequency *f*_{0} = *ω*_{0}/2*π*, therefore, depends on the competition between the bulk modulus of the particle and the shear modulus of the host medium.

The complex resonance frequency, that is, the solution to equation [2.57], is given by:

and the quality factor can be written as:

If *the particle is gaseous*, its bulk modulus is small. For the air, *K _{p}* ≈ 140 kPa and the resonance is driven by the shear modulus of the solid. If the latter is too rigid, the resonance angular frequency

When the host medium is a viscous liquid, the static shear modulus *μ*_{0} is zero and the modulus *η* corresponds to the shear viscosity (section 1.5.5, Volume 1). The resonance frequency then takes the form:

The resonance of a gas bubble in water, discovered by Minnaert, is very intense since its quality factor is very high:

ORDER OF MAGNITUDE.– For a gas bubble in water, the typical value of the product *f*_{0}*R* is 3.26 m/s. The product _{L} = 2*πf*_{0}*R/V _{f}* ≈ 0.014 is very small in front of unity: the Minnaert resonance occurs at a very low frequency.

The dipolar resonance of an elastic particle occurs when the denominator of the corresponding scattering coefficients cancels out:

Assuming that the complex root of this equation is expressed by:

The resonance frequency and the quality factor [2.61] are thus given by Duranteau *et al.* (2016):

For large arguments, the approximate expression for the spherical Hankel function of the first kind is given by [A1.40]. The substitution of the potentials [2.47] in equations [2.43] and [2.44] allows us to express the displacements and the stresses of the scattered wave in the far field:

with:

The scattered acoustic intensity, defined by:

is written as follows in the far field, with

According to equation [2.2], the scattered power is the average intensity over a sphere of arbitrary large radius *r*:

The scattering cross-section *σ* = *P _{d}*

or again, by normalizing by the geometric cross-section of the scatterer *S* = *πR*^{2}:

The scattering of a longitudinal plane wave by an elastic sphere buried in an epoxy resin is shown in Figure 2.10. The normalized cross-sections *γ _{L}* and

The moduli of the far-field scattering functions, *f _{r}* and

The analysis of the scattering of a transverse wave by a spherical particle is more complicated than the scattering of an incident longitudinal wave, because of the polarization perpendicular to the propagation direction. The calculation of a transverse plane wave in the spherical coordinate basis is first carried out using Helmholtz decomposition, which requires two scalar potentials. The scattering by a rigid sphere is then examined; the decomposition of the incident and scattered fields on the spherical harmonics shows that one of the transverse waves is coupled with the converted longitudinal wave, while the other is decoupled. In the last section, the scattering cross-sections and far-field scattering functions are calculated and the resonances arising from the propagation of the Stoneley wave on the surface of a lead sphere are highlighted.

We are interested in the scattering by a sphere of a plane transverse wave propagating in the *z* direction and polarized along *x*. A vector potential parallel to *y*:

imposes a particle displacement * u* oriented in the

Given the expression for the vector * e_{y}*:

the components of the vector * ψ* are defined in the spherical coordinate system through the relations:

The vector potential * ψ* is expressed in terms of two scalar potentials

that is:

By analogy with the development [A1.48] of a plane wave, the scalar potentials *φ _{T}* and

where the amplitudes *A _{n}* and

the radial component of the potential * ψ* takes the form:

Given the spherical harmonic expansion [A1.48] of the plane wave and since *P*_{0}(cos *θ*) = 1 and we get, after derivation with respect to *θ*:

Comparing expressions [2.77] and [2.82] for the component *ψ _{r}* leads to:

Given the relation [2.77] and substituting the potentials [2.80] into the expressions [2.79] of *ψ*_{θ} and *ψ*_{ϕ} leads to:

and:

The first term on the right-hand side of these two equations is eliminated by multiplying the first equation by and the second one by and then by integrating over *θ*, from 0 to *π*. Indeed, it appears the integral:

which is zero, since the Legendre functions are either even or odd. Adding each side of the transformed equations [2.85] and [2.86] leads to:

Using the spherical harmonic expansion [A1.48] of the plane wave and noting that the term between square brackets is equal to the integral of the left-hand side is calculated by parts:

considering the normalization [A1.41] of the Legendre functions Given relation [A1.43], the equality of each term of the sum in equation [2.88] leads to:

The incident wave is a transverse plane wave propagating in the *z* direction with the vector potential __Ψ__ _{i} = (*ia _{T}*

where the scattering coefficients are to be determined. The components of the mechanical displacement on the sphere surface are expressed by:

where the coefficients with *p* = *i, d* and M = L, T, are given by expressions [2.49] and the coefficients by:

The use of the orthogonality relations [A1.41], [A1.43] and [A1.44] allows us to decouple the system resulting from the cancellation of the three displacement components, given by relations [2.92], into one system for each harmonic *n*:

Since the third equation is independent of the first two, the transverse wave S is decoupled from the L and T waves. The scattering coefficients are expressed by:

In the far field, considering expressions [2.43] and [2.44], the components of the displacement and stresses for the scattered field are expressed as:

with

Expression [2.69] for the scattered intensity takes the form:

with:

The scattered power is the average of *I _{d}*(

with:

Given the orthogonality relations of the Legendre functions [A1.41] to [A1.44] and the normalization by *πR*^{2} and by the intensity *I _{inc}* of the incident plane wave, the scattering cross-sections are expressed by Einspruch

Figure 2.12 illustrates the scattering of a transverse plane wave by an elastic sphere buried in an epoxy resin. For the steel sphere (Figure 2.12(a)), the scattering cross-sections for the longitudinal and transverse T waves exhibit a maximum for the dimensionless wave number _{T} ≈ 0.83, corresponding to the dipolar resonance frequency *f*_{1} of the translational motion of the sphere in epoxy. The scattering cross-section of the transverse S wave has a maximum for the dimensionless wave number _{T} ≈ 1.32. This peak is associated with the rotational dipolar resonance of the particle. The same resonances are visible for the lead sphere (Figure 2.12(b)), but with larger amplitudes, since lead is denser than steel and therefore the quality factor [2.66] associated with its resonance is larger. In addition, it appears as a series of peaks, almost regularly spaced apart by a value Δ_{T} = 0.71 for higher orders. These resonances are due to the existence of a Stoneley wave propagating at the interface between the lead and the epoxy with the velocity *V _{Sto}* = 651 m/s when this interface is plane. This value, that is, the solution to the characteristic equation

Given the transverse velocity in epoxy *V _{T}* = 953 m/s, the gap between two peaks, corresponding to Δ(

The characteristic equation *d _{Sto}* = 0 has no solution for the steel-epoxy couple. There is no Stoneley wave in this case, which explains the absence of resonances due to circumferential waves in Figure 2.12(a).

A wave propagating in a heterogeneous medium is scattered by all obstacles it encounters. If the number of scatterers present on its path is small, the interactions between the fields scattered by each obstacle may not be taken into account. This *simple scattering* regime is used in certain imaging techniques based on the measurement of the paths of direct waves. If the wave encounters several obstacles, the multiple interactions between all the scattered fields lead to the *multiple scattering* regime. In a dilute medium, that is, with a low concentration of heterogeneities, coherent waves can be observed. These waves, which are the part of the field that is resistant to averaging over the disorder, are attenuated and their velocity is close to that of the host medium; they propagate as in a homogeneous medium, both dissipative and dispersive, called an *effective medium*.

Let us study the propagation, in a heterogeneous medium, of a wave represented by the scalar potential Φ_{inc}, for example, a longitudinal wave in a perfect fluid. By hypothesis, the distribution is monodisperse in size and shape: the obstacles are all identical and randomly distributed in the volume *V*. The interaction of the incident wave with the *N* obstacles positioned at * x_{m}* (with

This potential takes into account the primary interactions between the incident wave and the obstacles but also all the secondary interactions between the fields scattered by each obstacle (Figure 2.13).

The incident field on the *p*^{th} scatterer can be written as the sum of the incident field Φ_{inc} and the fields scattered by all the other obstacles:

The key for solving this problem is to determine the transition operator *T* (* x_{p}*) that connects the incident field arriving at the

Using this relation, without knowing, at this stage, the expression for *T* (* x_{p}*), equations [2.104] and [2.105] are written in the form:

and:

Several strategies can be considered for solving the system of equations [2.107] and [2.108]. A first method consists of solving the problem by using the addition theorem to write the incident fields on each scatterer in its own reference frame (Rohfritsch *et al.* 2019). This procedure requires the inversion of large linear systems, which can generally only be done numerically. An example of the result is shown in Figure 2.14(a), which illustrates the field scattered by a plane wave propagating in the direction of increasing *x* and interacting with a random distribution of 400 steel cylinders immersed in water, between the two planes represented by the dashed vertical lines.

The field averaged over 20 different distributions of the positions of the cylinders is presented in Figure 2.14(b). This shows the creation of a coherent wave front, inside the layer, propagating in the direction of increasing *x*. This result illustrates another way to approach this problem, which consists of searching for the effective properties of coherent waves. For this purpose, the fields are averaged over all possible positions of the scatterers. In this perspective, let us consider a random distribution of *N* scatterers, characterized by a probability density function *p*(__x___{1}, __x___{2},...,__x___{N}). The configurational average is defined by the relation:

Physically, the average field is the statistical mean of the field *f* (* x*) over all possible positions of the scatterers in the volume

where *p*(__x___{1}*, ..., x*

Applying the configurational average [2.109] to relation [2.107] over all possible positions of the scatterers leads to the equation:

Using expressions [2.110] and [2.111], we get:

with:

The distribution being monodisperse, the probability density *p*(* x_{m}*) is related to the number of scatterers per unit volume

translates that the incident field on the fixed scatterer *m*, averaged over all possible positions of the other scatterers, is approximately equal to the averaged field. According to Twersky, this approximation leads us to neglect all the scattering loops between each pair of scatterers (Ishimaru 1978). Considering the Foldy–Lax approximation [2.115], relation [2.113] takes the form of the Foldy–Twersky equation (Foldy 1945):

Assuming a low concentration of localized scatterers, uniformly distributed, the number of scatterers per unit volume corresponds approximately to the average number of scatterers per unit volume *n*_{0} : *n*(* x_{m}*) ≈

where *f* (__e___{i}*, e_{j}*) is the far-field scattering function, calculated for a single scatterer. The vector

In the following, the Foldy–Twersky equation is solved in the case of perfectly spherical scatterers of radius *R*, contained within two parallel planes located at *z* = 0 and *z* = *d* (Figure 2.15). The distribution is random and uniform, and the particles do not interpenetrate each other.

The incident wave is a plane wave of unit amplitude, propagating in the *z* direction:

Considering that the coherent wave also propagates in the *z* direction, the substitution of expression [2.117] for the transition operator into the Foldy–Twersky equation [2.116] leads to the relation:

The double integral between square brackets is calculated by applying the stationary phase method, discussed in section 1.2.2.1, first to the variable *x _{m}* and then to the variable

hence:

The vector * e_{j}* is collinear to

The first two terms of the right-hand side represent the propagation of progressive waves, while the third term represents the propagation of regressive waves. It then follows that the functions *f* (__e___{z}*, e_{z}*) and

Upon multiplying this equation by *e*^{−ikz} and by taking the derivative of the result with respect to *z*, the integral equation is reduced to a first-order differential equation:

whose solution is a simple plane wave 〈Φ(*z*)〉 = *e*^{ik}_{eff} ^{z}, with an effective wave number equal to:

In the case of spherical scatterers, and given formula [2.68] with *P _{n}*(1) = 1 and

where *T _{n}* is the scattering coefficient of order

In the case of cylindrical scatterers, the same reasoning leads to the relation:

The two previous expressions are approximations of the formulae:

which are often attributed to Foldy, although this author considered point-like scatterers with isotropic radiation. They remain the simplest formulae giving the effective wave number. If backward scattering is not negligible, a reasoning similar to that just described leads to the effective wave number of Waterman and Truell (Waterman and Truell 1961). Moreover, formula [2.128] are still valid for longitudinal waves propagating in solids using the scattering coefficients instead of *T _{n}*. For shear coherent waves, the formula for cylinders remains valid by using the scattering coefficients instead of

equation [2.113] was obtained by fixing a particle and by averaging over all possible positions of the other particles. This mathematical operation can be repeated for several particles, leading to a hierarchy of equations. However, if the number of particles *N* is large, it is preferable to break up this hierarchy by introducing an approximation. Previously, this hierarchy has been restricted to the first-order by using the *independent scattering approximation* (ISA). Lax (1952) introduced a second-order approximation, by performing the calculation for two fixed particles. This *quasi-crystalline approximation* (QCA) gives rise to several different expressions for the effective wave numbers. In particular, Fikioris and Waterman (1964) take into account the fact that two scatterers cannot interpenetrate by introducing the *hole correction*, which imposes an exclusion distance in the conditional probability density.

Following these works, many formulae, more or less complex, have been proposed to describe the propagation in a medium containing scatterers (Linton and Martin 2005; Conoir and Norris 2010). The advantage of approaches based on the multiple scattering theory lies in the simplicity of the expressions for the effective wave numbers because these only require the determination of scattering coefficients inherent to the problem being studied. However, these formulae are valid as long as the concentration of scatterers and the scattering cross-section remain small.

The properties of the coherent wave that can be measured are its phase velocity and its attenuation, defined by:

We will study these properties in two cases: a forest of immersed cylindrical rods and an elastic material containing a distribution of resonant spherical particles.

Coherent waves propagating in a forest of cylindrical steel rods immersed in water were measured using the phase difference method introduced in section 2.4.2.4.2 of Volume 1. The rods, of diameter 1.5 mm, are distributed in a layer of thickness *d* = 15 mm and 20 cm wide. The sample is laterally displaced over thirty positions and transmission measurements are carried out for each position using circular piezoelectric transducers of diameter 13 mm and center frequency 2.25 MHz. The acquired signals are averaged in order to reduce the incoherent field. A reference signal *s _{ref}* (

In order to observe the influence of averaging on the signal, the average signals *s _{avg}* and the first recorded signal

Since there is no physical interface, the phase velocity and attenuation of the coherent wave are given by relations [2.120] in Volume 1, with *h* = *d*, *ρ* = *ρ _{f}*, and

These quantities are plotted in Figure 2.17 as a function of the frequency for a plate containing a random distribution of steel cylinders, of concentration *φ* = 15%. The experimental results are compared to expression [2.128], as well as to the Fikioris and Waterman model (1964), which is a second-order approximation with respect to the concentration. Its first-order term is the same as that in relation [2.128]. This comparison shows, for the velocity and attenuation, that the second-order term in concentration must be taken into account in the modeling (Derode *et al.* 2006). These results must be compared with the scattering cross-section of a single steel cylinder immersed in water, as depicted in Figure 2.5. The attenuation and phase velocity are strongly impacted by the propagation of circumferential waves (see section 2.1.3).

The velocity and attenuation of the coherent longitudinal wave in a composite material, made up of an epoxy resin containing a distribution of tungsten carbide beads, are plotted in Figure 2.18 *versus* frequency. The distribution is disordered and monodisperse and the concentration of particles is *φ*_{0} = 2% (Duranteau *et al.* 2016). The experimental results obtained by a transmission and immersion method (see section 2.4.2.4.2, Volume 1) are compared to the Foldy effective wave number (formula [2.128]), as well as to the LCN multiple scattering model (Luppé *et al.* 2012), which is the generalization of the Fikioris and Waterman model to the elastic case. The translational dipolar resonance of the tungsten carbide beads alters the propagation of the coherent waves: the phase velocity becomes strongly dispersive and an attenuation peak appears in the vicinity of the resonance, whose approximate frequency (*f*_{1} = 450 kHz) is given by equation [2.66]. The LCN model is in better agreement with experimental results because it takes into account the conversion into transverse waves, which is particularly large in the vicinity of the resonance, as shown in Figure 2.10(b) for a lead bead.

- 1 For example, the scattering by a rigid cylinder at high frequencies.

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