Y C Angel

C Aristégui

M Caleap

Sound propagation in a solid through a screen of cylindrical scatterers

Keywords: multiple scattering, coherent wave, random media, acoustics

The propagation of SH waves in a solid containing a screen of line-like scatterers is investigated. When the scatterers are uniformly distributed, the amplitudes of the coherent waves inside and outside the screen are evaluated in closed form. In the analysis, multiple scattering effects are taken into account within the context of a first-order approximation. A Global Closure Assumption is proposed, which yields an effective wavenumber identical to that of Waterman and Truell. The scatterers can be fibers of circular or elliptical cross-sections; they can also be two-dimensional cracks with slit-like or elliptical cross-sections. Specific analytical and Angel, Aristégui, Caleap Sound through cylinders 2 numerical results are presented for flat cracks and empty cavities of circular cross-sections. In those two cases, figures are presented to illustrate the variations of the reflection and transmission coefficients as functions of frequency and of scatterer concentration. The crack and cavity results, respectively, are compared with those of earlier works.

Introduction

Waves propagating in biological, or structural materials, containing fibers are subjected to multiple scattering when the wavelengths of the probing signals are sufficiently short relative to a characteristic cross-length of the fibers. With high fiber concentrations, diffusive effects may also be observed [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF]. Here, we are interested in ultrasound measurements in materials with low fiber concentrations, where multiple-scattering effects can be significant and must be taken into account, if correct interpretations are to be inferred from reflected or transmitted signals.

Experimental and analytical difficulties associated with the phenomenon of multiple scattering have been dealt with for many years. Yet, the task is complicated, and the phenomenon is still being investigated, both experimentally and analytically [START_REF] Challis | Ultrasound techniques for characterizing colloidal dispersions[END_REF][START_REF] Derode | Influence of correlations between scatterers on the attenuation of the coherent wave in a random medium[END_REF][START_REF] Kim | Antiplane shear wave propagation in fiber-reinforced composites[END_REF][START_REF] Kim | On the generalized self-consistent model for elastic wave propagation in composite materials[END_REF][START_REF] Bas | Reflection and transmission by randomly spaced elastic cylinders in a fluid slab-like region[END_REF][START_REF] Linton | Multiple scattering by random configurations of circular cylinders: second-order corrections for the effective wavenumber[END_REF], 2006; [START_REF] Wang | Effective Antiplane Dynamic Properties of Fiber-Reinforced Composites[END_REF][START_REF] Wilson | Phase speed and attenuation in bubbly liquids inferred from impedance measurements near the individual bubble resonance frequency[END_REF]).

Here we look at analytical aspects of the problem. We derive concise and simple expressions for reflection and transmission (back-scattering and forward-scattering). These expressions will provide a tool to test, in the laboratory, the validity of the theory.

The present work applies to the case of SH (antiplane) waves in a linearly-elastic solid.

Scatterers are solid fibers or cavities, of circular or elliptical cross-sections. They can also be two-dimensional cracks with slit-like or elliptical cross-sections. Solid fibers can be coated or layered. Propagation is perpendicular to the cylinder axes, and particle displacements are parallel to those axes. Thus, the displacements are solutions of scalar wave equations, and we can talk of "scalar waves."

Our analytical approach follows that of Foldy-Twersky and of Waterman-Truell (Ishimaru 1997). Their equations are reexamined in detail. In particular, their closure assumption concerning the field scattered by a fixed scatterer in the presence of all other scatterers is reformulated and labeled "Global Closure Assumption" in Section 6. This allows us to state, in mathematical terms, the approximation that is made to arrive at the solution of the problem. In the end, we obtain an expression for the effective wavenumber that is identical to that of [START_REF] Waterman | Multiple scattering of waves[END_REF].

Our work is an extension of theirs. Indeed, we determine closed-form expressions for the reflection and transmission coefficients -which they had not obtained. These expressions depend on scatterer concentration and size, frequency, the properties of the solid matrix, the thickness of the scattering region, as well as on the forward and backward scattering properties of a single scatterer.

Discussions available in the literature seem to agree that closure assumptions such as that of Waterman and Truell or such as our Global Closure Assumption yield predictions that are in good agreement with experimental measurements -but only in the limit of low scatterer concentration. However, to the best of our knowledge, this observation has always been made on limited sets of experimental data. There exist no extensive data collections of ultrasonic measurements for a wide variety of materials containing scatterers. This work on SH waves follows an earlier work where P waves are considered [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF]. Because of more immediate potential applications in (fluid-like) biological tissue, we started with P waves, which are also called "vector waves". In [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF], scalar displacement potentials are used in the formulation of the problem. Here, for SH waves, potentials are not needed. Thus, one cannot simply infer from [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF] the results of this paper, although there are some similarities between the two works. For this reason, we have included in Section 9 a comparative discussion of the two methods of solution.

In that section, we point out that the amplitude formulae for the SH and P problems, respectively, have been made to look identical. To arrive at those expressions, however, the respective Q factors must be defined differently (see (9.33) and (9.34)). In [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF], the theory only is developed. No numerical examples are presented.

We present in Section 10 numerical results corresponding to a distribution of flat cracks and a distribution of cylindrical cavities, respectively. The results are compared with those of earlier works, where scatterers are not replaced with equivalent line scatterers. We also show numerical predictions for the reflection from a half-space containing a distribution either of flat cracks or of cylindrical cavities.

Screen of scatterers SECTION D'ÉQUATION 2

We consider an SH wave that encounters cylindrical scatterers, as shown in Fig. 1. The scatterers, of identical geometry and elastic properties, are uniformly and randomly distributed in a layer of thickness 2h , where their number concentration per unit area in the ( ) It follows from the preceding assumptions on the distribution and geometry of the scatterers that the wave system generated by the SH wave in Fig. 1 is two-dimensional. The solid particles move in the 2 y direction, with 2 u the displacement measured from the equilibrium configuration.

The stress components in the context of small elastic deformations are given by Both R and T in (2.2) are a priori frequency dependent. They represent, respectively, averages on disorder of the amplitudes of the backscattered and transmitted waves generated by a continuous harmonic wave. These amplitudes depend also on the number concentration 0 n , the size and material properties of the scatterers, the density and speed of sound in the surrounding solid, and on the layer thickness 2h . Another parameter is the angle of incidence, but in this work we focus attention on normal incidence.

2 21 1 u y s m ¶ = ¶ , 2 23 3 u y s m ¶ = ¶ , ( 2
The scattering properties of a single scatterer are described by a set of coefficients n C .

Calling 2 u the scattered displacement in the solid, one has ( ) (2 . 4 )

( ) ( ) (1) 2 0 , c o s n n n n n u r A C H kr n q e q + ¥ = = å . ( 2
These coefficients are those of the incident displacement in the expression ( )

( ) ( ) 3 i i c o s 2 0 0 , e e cos i k k r n n n n u r u A J kr n x q q e q + ¥ = = = å , (2.5)
where n J denotes the cylindrical Bessel function.

The rule (2.3) and (2.5) says that the n th mode of amplitude n A is scattered away from the local origin 0 r = , where the scatterer is centered, with amplitude n n A C . The coefficient n C

acts as an amplification factor and is dimensionless. Further, each scattered mode has a Logr

( ) 0 n = or n r -( ) 1 n ³
square-root singularity as r approaches zero.

Observe that the origin 0 r = is inside the scatterer and the scattered field is defined outside the scatterer. Thus, there is no singularity in the region where (2.3) holds. The coefficients n C depend on frequency, as well as on the properties of the scatterer and those of the solid outside.

In the next section, we shall reduce the geometry of the scatterer to a line. To this line, we associate the scattered displacement (2.3), where the coefficients n C are evaluated by imposing appropriate continuity conditions on the boundary of an actual-size cylinder. Thus, the actual cylinder will be replaced with a new object; the geometries are different, but the scattering properties are identical. This geometrical simplification, however, causes the singularities of (2.3) to be no longer embedded inside the scatterer.

Coherent motion SECTION D'ÉQUATION (SUIVANTE)

The coherent motion in the solid of Fig. 1 is the sum of the incident motion and of the wave motions scattered by all the scatterers [START_REF] Waterman | Multiple scattering of waves[END_REF]. In terms of the displacement, one has [START_REF] Angel | Complex-valued wavenumber, reflection and transmission in an elastic solid containing a cracked slab region[END_REF] ( ) ( )

( ) 2 3 2 3 0 2 3 1 3 1 3 1 , ; , h i h u y u y n u y y d d x x x x ¥ - -¥ = + ò ò . (3.1)
The second term on the right-hand side of (3.1) represents the global contribution of all scatterers located at all possible positions ( ) 3 1 , x x in the layer of thickness 2h , under the assumption that the distribution is random and uniform. The dimension of the number density 0 n in (3.1) is an inverse area. The coordinate 1 y is integrated out on the right-hand side of (3.1) because of the translational invariance in the 1 y direction.

The term ( )

2 3 1 3 1
, ; , u y y x x in (3.1) is an average displacement corresponding to the wave motion scattered by the line at ( ) 3 1 , x x . The average on disorder is taken over all configurations keeping the line at ( ) 3 1 , x x fixed. Thus, it is a partial average.

The exciting wave motion on a fixed scatterer is caused by scattering processes of all orders (single, double, triple…) [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF]) inside the layer. Keeping the scatterer fixed and averaging the displacements corresponding to those scattering processes, one defines the average exciting displacement 2 E u . Assuming that this displacement is a bounded solution of the twodimensional Helmholtz equation in the solid with wavenumber k , one finds that the most general expression of 2 E u corresponding to a scatterer fixed at x is ( )

( ) ( ) ( ) 2 3 1 3 1 0 3 0 , ; , cos E n n n n u y y u E J k n x x e x q + ¥ = = - å y x , (3.2)
where the angle q is defined by ( ) 3 3 cos / y q x = --y x and the distancex y is defined as in (2.3). The coefficients n E in (3.2) depend only on 3 x , owing to the translational invariance in the 1 y direction.

Next we recall the scattering rule (2.3) and (2.5). Thus, we infer from (3.2) that the average scattered displacement discussed above is given by ( )

( ) ( ) ( ) ( ) 1 2 3 1 3 1 0 3 0 , ; , cos n n n n n u y y u C E H k n x x e x q + ¥ = = - å y x . (3.3)
Note that the singularity of ( ) 1 n H at = y x is not integrable over the two-dimensional domain 1

x < ¥ and 3 h x £ when y is inside the domain and when 2 n ³ . Thus, the double integral of (3.1), with 2 u given by (3.3), is evaluated as in [START_REF] Waterman | Multiple scattering of waves[END_REF], page 526. Namely, an entire infinite slab 3 3 y x e -< is omitted from the range of integration, and then the limit is taken as e approaches zero.

Without loss in generality in the present context, one can set 1 0 y = on the right-hand side of (3.3). Substituting (3.3) into (3.1), one finds that the coherent displacement in the solid has the form ( ) ( ) ( ) ( )

2 3 2 3 0 0 3 3 3 3 0 h i n n n n h n u y u y n u C E y d e x j x x + ¥ - = = + - å ò . (3.4)
The quantities n j on the right-hand side of (3.4) are defined by ( )

( ) ( ) ( ) 1 3 3 1 cos n n y H k n d j x qx + ¥ -¥ - = - ò y x , 3 y x ¹ . ( 3
( ) ( ) 3 3 i 2 3 3 2 1 e n k y n y k x j x - - = - , (3.6) ( ) ( ) ( ) 3 3 i 2 1 3 3 3 3 2 i 1 sgn e n k y n y y k x j x x - + - = - - , ( 3
( ) ( ) ( ) ( ) ( ) 3 3 i i 2 3 2 3 0 0 1 3 0 0 2 i s g n e + e h h k y k y i h h u y u y n u y d n u d x x s x x x s x x - - - - = + - ò ò . (3.8)
The functions ( )

1 s x and ( ) 2 s x in (3.8) are defined by ( ) ( ) ( ) 1 2 1 2 1 0 4 1 j j j j C E k s x x + ¥ + + = = - å , ( )
( ) ( ) 2 222 0 2 1 j j j j j C E k s x e x + ¥ = = - å . (3.9)
It follows from (3.8) that the reflected coherent motion is a plane wave of wavenumber k and amplitude 0 u R and the transmitted coherent motion is a plane wave of wavenumber k and amplitude 0 u T , as in (2.2). Expressions for the coefficients R and T are obtained by setting, respectively, 3 y h < -and 3 y h > on the right-hand side of (3.8). We find that ( ) ( ) where the angle a is defined by ( )

( ) i 0 2 1 i e h k h R n d x s x s x x - = + ò , (3.10) ( ) ( ) ( ) i 0 2 1 1 i e h k h T n d x s x s x x - - = + - ò . ( 3

Scattered motionSECTION D'ÉQUATION (SUIVANTE)

For 3 h y h -< < ,
3 3 cos / y a z = - -z y
. After some rather lengthy but straightforward calculations, one finds that (3.8), for

3 h y h -< < , can be written in the equivalent form ( ) ( ) ( ) ( ) ( ) 2 3 2 3 0 0 3 3 0 , c o s i n n n n u y u y n u G y J k n e z a + ¥ = = + - å y z . (4.2)
In this equation, the term n G can be written as the sum of a term that depends only on 3 z and another term that depends on both 3 z and 3 y . Thus, one has ( ) ( ) ( )

3 3 3 3 3 , , n n n G y F D y z z z = + , (4.3)
with 3 y h < and 3 h z < . For even and odd values of the subscript, one finds that ( ) ( ) ( )

2 3 0 3 1 n n F F z z = - , ( ) ( ) ( ) 2 1 3 1 3 1 n n F F z z + = - , (4.4) ( ) ( ) ( ) 2 3 3 0 3 3 , 1 , n n D y D y z z = - , ( ) ( ) ( ) 2 1 3 3 1 3 3 , 1 , n n D y D y z z + = - , (4.5) ( ) ( ) ( ) ( ) 3 3 i i 0 3 1 3 2 i s g n e e h h k k h h F d d x z x z z s x x z x s x x - - - - = - + ò ò , (4.6) ( ) ( ) ( ) ( ) 3 3 i i 1 3 1 2 3 e i s g n e h h k k h h F d d x z x z z s x x s x x z x - - - - = - - ò ò , (4.7) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 3 3 0 3 3 1 3 2 3 , 2i cos 2i sin y y D y k d k d z z z s x x z x s x x z x = - - - - ò ò , (4.8) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 3 3 1 3 3 1 3 2 3 , 2i sin 2i cos y y D y k d k d z z z s x x z x s x x z x = - - + - ò ò . (4.9)
The series expansion with the coefficients n G in (4.2) can be decomposed according to (4.3) into two series expansions. The series with the coefficients n F is a solution of the two- dimensional Helmholtz equation in the solid with wavenumber k . The series with the coefficients n D is not in general a solution of that equation, because n D depends on 3 y . Observe that the expansion (4.2) holds also in 3 y h > and 3 y h < -. In these intervals, the coefficients n G are independent of 3 y . Exact expressions can be obtained readily from (3.8) and (4.1).

Exciting motion SECTION D'ÉQUATION (SUIVANTE)

The exciting wave motion on a fixed scatterer at z is the sum of the incident wave and all the motions scattered by the other scatterers except that at z [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media[END_REF]. Averaging on disorder, keeping the scatterer at z fixed, one finds that ( ) ( ) ( )

2 3 1 3 1 2 3 0 2 3 1 3 1 3 1 3 1 , ; , , ; , ; , h E i h u y y u y n u y y d d z z x x z z x x ¥ - -¥ = + ò ò . (5.1)
On the right-hand side of (5.1), the integrand represents the average displacement scattered by the line at ( )

( ) ( ) ( ) 2 3 1 3 1 2 3 0 0 3 0 , ; , cos E i n n n n u y y u y n u M J k n z z e z a + ¥ = = + - å y z . (5.2)
The coefficients n M in (5.2) depend only on 3 z , owing to the translational invariance in the 1 y direction.

In (5.2), we expand 2 i u by using the series (4.1), and the average exciting Eventually, all the scatterers are fixed. This approach would in principle allow us to determine the quantities n E exactly but, unfortunately, it is too complicated and cannot be implemented in practice [START_REF] Waterman | Multiple scattering of waves[END_REF].

As an alternative to the approach described above, we propose a closure assumption that will allow us to determine the quantities n E . The most simple assumption corresponds to the situation of simple scattering, where the scatterers are sufficiently distant from each other and the interactions between them can be neglected. In this case, it is assumed that the exciting displacement on each scatterer is equal to the incident displacement. The coefficients n E corresponding to simple scattering are obtained explicitly from (5.3) by setting the second term on the right-hand side of that equation equal to zero. Substituting these expressions for n E into (3.8), one finds the corresponding coherent displacement in the solid and, substituting into (3.10) and (3.11), one finds the reflection and transmission coefficients.

When interactions between scatterers cannot be neglected, one must take into account multiple scattering effects. These effects are represented by the quantities n M in (5.3). The n M can be viewed as a representation of the integral on the right-hand side of (5.1). Likewise, the n G in (4.2) can be viewed as a representation of the integral on the right-hand side of (3.1).

Comparing the integral in (3.1) with that in (5.1), we recall that the former is not a solution of the Helmholtz equation with wavenumber k in the interval 3 y h < , and the latter is indeed a solution of the Helmholtz equation with wavenumber k for all 3 y . Both integrals represent the same global scattered displacement averaged on disorder, except that in (5.1) one of the scatterers is fixed at z and in (3.1) no scatterer is fixed.

In view of the preceding discussion, we assume that n M is equal to the part of n G that is a solution of the two-dimensional Helmholtz equation in the solid. Thus, referring to (4.3), we write ( ) ( )

3 3 n n M F z z = , (6.1) for 0,1, 2, n = K .
The part of n G that is left out in (6.1) is n D which, we observe from (4.5), (4.8) and (4.9), approaches zero as 3 y approaches 3 z . Equation (6.1) is our Global Closure Assumption.

To clarify the physical meaning of the assumption (6.1), we substitute (6.1) into (5.2), and then we recall (4.2) and (4.3). Then, we find that

( ) ( ) ( ) ( ) ( ) 2 3 1 3 1 2 3 0 0 3 3 0 , ; , , cos E n n n n u y y u y n u D y J k n z z e z a + ¥ = = - - å y z , (6.2)
where the functions n D are defined by (4.5), (4.8), and (4.9). Equation (6.2) can be viewed as an equivalent statement of the assumption (6.1); it provides an expression for the exciting displacement 2 E u on a scatterer fixed at z in terms of the coherent displacement 2 u .

The expression (6.2) is defined at all points y in the neighbourhood of z and away from that neighbourhood. We see that 2 E u approaches 2 u when y approaches z , since the coefficients n D vanish when 3 3 y z = .

Substituting (6.1) into (5.3), and using (4.4), one finds that ( ) ( ) ( ) Substituting (6.6) into (4.6) and (4.7), one can write 0 F and 1 F in terms of 0 E and 1 E . It follows that (6.4) and (6.5) represent a system of two coupled integral equations for the unknown functions 0 E and 1 E .

2 3 0 3 1 n n E E z z = - and ( ) ( ) ( ) 2 1 3 1 3 1 n n E E z z + = - , (6.3) with ( ) ( ) 3 i 0 3 0 0 3 e k E nF z z z = + , ( 6

Effective wavenumber SECTION D'ÉQUATION (SUIVANTE)

To simplify the notation, we use the variable x in place of 3 z . Then, with x h < , the system of equations for 0 E and 1 E has the form

( ) ( ) ( ) ( ) i i i 0 0 1 1 0 2 0 e i sgn e e h h k x k x kx h h E x n E x d n E d x x s x x x s x x - - - - = + - + ò ò , (7.1) ( ) ( ) ( ) ( ) i i i 1 0 1 1 0 2 0 i e e i sgn e h h k x k x kx h h E x n E d n E x d x x s x x s x x x - - - - = + - - ò ò . (7.2)
We differentiate (7.1) and (7.2) two times with respect to x , keeping the variable x in the interval ( ) , h h -. We use the properties that the derivatives of , where ( ) x d denotes the Dirac delta function. Thus, we find that ( ) ( )

2 2 0 0 2 d E x K E x dx = - and ( ) ( ) 2 2 1 1 2 d E x K E x dx = - . (7.3)
The coefficient K is independent of x and is given by ( )( )

2 0 1 0 2 2i 2i K k n k n s s = - - . (7.4)
The general solutions of the second-order ordinary differential equations (7.3) are

( ) i i 0 0 0 e e Kx K x E x V W - = + , (7.5) ( ) i i 1 1 1 e e Kx K x E x V W - = + , (7.6)
where the coefficients 0 V , 0 W , 1 V , and 1 W are independent of x . We now infer from (3.8), (6.6) , (7.1), and (7.5) that the coherent displacement in the interval 3 y h < is given by ( ) ( ) . Equation (7.7) shows that, inside the layer where the scatterers are distributed, there is one wave traveling in the forward direction and a second wave traveling in the backward direction. Both waves are governed by K , which is the effective wavenumber.

In general, the wavenumber K is complex-valued and depends on frequency. It follows that the two waves inside the layer are dispersive and undergo attenuation as they propagate.

It is customary to write the effective wavenumber K of (7.4) in terms of the far-field scattered displacement corresponding to a single scatterer subjected to the incident wave of (2.2) [START_REF] Waterman | Multiple scattering of waves[END_REF]. To arrive at the usual formula for K , we recall that the cylindrical Hankel function is given in the far-field as r tends to infinity by the expression

p q q ae ö ÷ ç + ÷ ç ÷ ç è ø = .
(7.9)

When the scatterer is located at = 0 x , the angular shape function f in (7.9) is given by

( ) ( ) 0 1 cos i n n n f C n q e q p + ¥ = = å .
(7.10)

We can see from Fig. 1 that, relative to the incident wave, the forward direction corresponds to 0 q = and the backward direction to q p = . It follows that ( ) 0 f and ( ) f p , respectively, are measures in the forward and backward directions of the displacement scattered by a single scatterer. We infer from (6.7) and (7.10) that ( )

( ) ( ) 1 i 0 f f k p s p = - and ( ) ( ) ( ) 2 i 0 f f k p s p = + .
(7.11) Equation ( 7.11) shows that the factors 1 s and 2 s depend only on the far-field amplitudes scattered by a single scatterer in the forward and backward directions, apart from the wavenumber k . Substituting (7.11) into (7.4), and rearranging terms, one finds the well-known formula [START_REF] Waterman | Multiple scattering of waves[END_REF] (7.12) ( )

( ) 2 2 2 2 2 0 0 2 2 2 2 1 0 K k n f k n f k k p p p ae ö ae ö ÷ ÷ ç ç = + - ÷ ÷ ç ç ÷ ÷ ç ç è ø è ø .

Reflection and transmission SECTION D'ÉQUATION (SUIVANTE)

We determine the coefficients 0 V , 0 W , 1 V , and 1 W in closed form by substituting (7.5) and

(7.6) into (7.1) and ( 7.2), as in [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF]. Next, we substitute these coefficients, together with (6.6), (7.5), (7.6), into (3.10) and (3.11). Integrating the exponential functions in closed form, we find that the reflection and transmission coefficients are given by ( )

2i 4i 2 4i e 1 e 1 e kh Kh Kh Q R Q - = - - , (8.1) ( ) 2 2i 2 4i 1 e 1 e K k h Kh Q T Q - - = - , (8.2)
where

1 1 Q -Q = + Q (8.3) and 0 2 0 1 2i 2i k n K K k n s s - Q = = - . (8.4)
Outside the screen, the reflected and transmitted waves have amplitudes 0 u R and 0 u T , respectively, where R and T are given by (8.1) and (8.2). Inside the screen, where 3 y h < , the forward and backward waves have amplitudes 0 u A + and 0 u A -, respectively, where A + and A - are given by i

( ) 0 2 4i 1 e 1 e K k h Kh Q V A Q - + + = = - , (8.5) ( ) i( 3 ) 0 2 4i 1 e 1 e K k h Kh Q Q W A Q - - + = = - - . (8.6)
The factor Q in (8.1), (8.2), (8.5) and (8.6) is defined by (8.3) and (8.4), together with (6.7) or (7.11).

The formulae (8.1), (8.2), (8.5) and (8.6) for the reflected, transmitted, forward, and backward waves, respectively, apply also to a homogeneous elastic plate. To see this, replace in Then, one finds, by imposing continuity of stress and displacement on the boundaries 3 y h = ± , that the amplitudes of the four waves generated by the incident wave (2.2) are still given by (8.1) , (8.2), (8.5) and (8.6), provided that K be replaced with 1 k and Q with 0 0 1 1

0 0 1 1 c c Q c c r r r r - = + . (8.7)
The factor Q is a ratio of acoustic impedances. This shows that the system of waves propagating in the presence of scatterers in Fig. 1 is formally identical to that propagating outside and inside a homogeneous elastic plate. The effective wavenumber K is replaced with 1 k ; the factor Q of (8.3) and (8.4) is replaced with (8.7).

Comparison between SH and P solutions SECTION D'ÉQUATION (SUIVANTE)

We compare the solution obtained in this paper -for SH waves with particle displacements along the 2 y direction -with that obtained in an earlier work [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF] -for P waves with particle displacements along the 3 y direction. In both cases, the geometry is that of Fig. 1 and the scatterers have cylindrical geometry with axes parallel to the 2 y direction.

In both cases, after making a closure assumption, the formulation of the scattering problem reduces to a pair of coupled integral equations for two unknown functions 0 E and 1 E . For the purpose of this discussion, we call 0 S E and 1 S E the solutions of (7.1) and ( 7.2); we call 0 P E and 1 P E the solutions of ( 56) and ( 57) in the P-wave work [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF]. Likewise, the factors 1 s and 2 s are labeled, respectively, with an S or a P superscript. Then, if 0 S E and 1 S E are solutions of (7.1) and ( 7.2), one infers that 0 0

1 i P S E E k = , 1 1 1 i P S E E k = , (9.1)
are solutions of ( 56) and ( 57) in the P-wave work [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF]. The solution of the system (7.1) and (7.2) is ( )

i i 0 0 0 e e S S K x S K x E x V W - = + , (9.2) ( ) i i 1 1 1 e e S S K x S K x E x V W - = + , (9.3)
where the wavenumber K is given by (7.4) and no superscript is attached to K for simplicity.

The coefficients in (9.2) and ( 9.3) are given by

( ) i( ) 0 1 e / S S Kk h S V Q D - = + , (9.4) ( ) i(3 ) 0 1 e / S S S K kh S W Q Q D - = - + , (9.5) 1 0 i S S S V V = Q , (9.6) 1 0 i S S S W W = -Q , ( 9.7) where ( ) 2 4i
1 e

S S K h D Q = - , (9.8) 1 1 S S S Q -Q = + Q , (9.9) 0 2 0 1 2i 2i S k n K K k n s s - Q = = - . (9.10)
In equations (9.4)-(9.10), the superscript S has been omitted on K , 1 s and 2 s . It follows from (9.1) that the solutions (9.2) and ( 9.3) -with the superscript P replacing S -can be used to determine 0 P E and 1 P E . Then, one infers from (9.1) and (9.4)-(9.10) that

( ) i( ) 0 i 1 e / P K k h P kV Q D - = + ( , (9.11)
( ) i(3 ) 0 i 1 e / P K k h P kW Q Q D - = - + ( ( , (9.12
)

1 0 i P P P V V = Q , (9.13) 1 0 i P P P W W = -Q , ( 9.14)
where 2 4i

1 e

P Kh D Q = - ( , (9.15)
1 1

P P Q -Q = + Q ( , (9.16) 0 2 0 1 2i 2i P k n K K k n s s - Q = = - .
(9.17)

In equations (9.11)-(9.17), the superscript P has been omitted on K , 1 s and 2 s . Next, as in equation ( 76) of the P-wave work [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF], we define Z and P Q by

1 P Z = Q , 1 1 P Z Q Z - = + . (9.18)
Then, it follows from (9.16) and (9.18) that

P Q Q = - ( . ( 9
( ) i( ) 0 i 1 e / P P K k h P kV Q D - = - , (9.20) ( ) i(3 ) 0 i 1 e /
P P P K kh P kW Q Q D - = - , (9.21)
( ) ( ) i 1 1 e / K k h P P P kV Q D - = + , (9.22) ( ) ( ) i 3 1 1 e / K k h P P P P kW Q Q D - = - + , (9.23) where ( ) 2 4i
1 e

P P K h D Q = - . (9.24)
Next, recall from (7.7) that the SH displacement inside the layer is proportional to 0 E . Thus, it follows from (7.7) and (9.2) that for SH waves

0 S A V + = , 0 S A W -= .
(9.25) Equation (9.25), together with (9.4), (9.5) and (9.8), is in agreement with (8.5) and (8.6).

In the case of P waves, the displacement inside the layer is proportional to 1 E , as can be seen from ( 62) in the P-wave work [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF]. Thus, it follows from ( 62) and ( 77)

in [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF], and from (9.3) with the superscript S replaced with P , that for P waves 1

P A kV + = , 1 P A kW -= - . (9.26)
Equation ( 9.26), together with (9.22), (9.23) and (9.24), is in agreement with ( 78) and ( 79)

in [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF].

We observe that the sign conventions for the amplitudes of the backward traveling waves inside the layer are opposite, respectively, for the cases of SH and P waves. This can be seen from ( 7.7) in this paper and from (77) in [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF].

From (3.10), (3.11), and (6.6), the reflection and transmission coefficients for SH waves are given by ( ) ( )

( ) i 0 2 0 1 1 i e h S S k x h R n E x E x dx s s - = + ò , (9.27) ( ) ( ) ( )
i 0 2 0 1 1 1 i e h S S k x h T n E x E x dx s s - - = + - ò . (9.28)
In (9.27) and (9.28), the superscript S has been omitted on R , T , 1 s and 2 s .

For P waves, we find expressions for R and T from equations ( 24), (25), and (53) in [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF]. The results are

( ) ( ) ( ) i 0 2 0 1 1 i i e h P P k x h R kn E x E x dx s s - = + ò , (9.29) ( ) ( ) ( ) i 0 2 0 1 1 1 i i e h P P k x h T kn E x E x dx s s - - = + - ò . (9.30)
In (9.29) and (9.30), the superscript P has been omitted on R , T , 1 s and 2 s .

Substituting (9.2)-(9.10) into (9.27) and (9.28), one finds expressions for R and T that are those of (8.1) and (8.2), with a superscript S attached to the Q factor.

For P waves, expressions for R and T are obtained by substituting (9.1) into (9.29) and (9.30), and using the preceding results for (9.27) and (9.28). Then, with the factor Q ( of (9.16), one

e 1 e /

kh Kh P R Q D - = - ( , (9.31)
( ) ( ) 2i 2 1 e / K k h P T Q D - = - ( , (9.32)
where the superscript P is omitted on R , T , and K . Now, we recall from (9.19) that

P Q Q = - ( .
Then, in terms of P Q , we find that (9.31) and (9.32) are identical to ( 74) and ( 75)

in [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF].

We observe that the sign conventions for the amplitudes of the reflected waves are opposite, respectively, for the cases of SH and P waves. This can be seen from ( 2.2) in this paper and from

(2) in [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF].

We have shown in this Section that the SH solution and the P solution are related. However, they cannot be inferred easily from each other. Indeed, the P problem requires that scalar potentials -from which displacements follow by differentiation -be considered. For the SH problem, we work directly with the displacements, which are solutions of scalar Helmholtz equations.

The scattering properties of a single cylindrical scatterer are described by coefficients n C .

These n C serve to represent the scattered displacement in the SH problem and the scattered displacement potential in the P problem. The quantities 1 s and 2 s are series of the coefficients n C , with the same formal expressions in either case -as can be seen from ( 6.7) in this paper and from (54) in [START_REF] Angel | Analysis of sound propagation in a fluid through a screen of scatterers[END_REF], respectively.

The amplitudes A -, A + , R , and T have the same forms, respectively, for SH and P waves - except for the sign changes concerning A -and R , which reflect only our choice of two different sign conventions for backward-traveling waves. This formal identity in the expressions of the two sets of amplitudes is obtained by introducing two distinct definitions for the factors S Q and P Q . These two factors are formally opposite of each other, because S Q and Z are inverses of each other. Indeed, we recall from (9.9)-(9.10) and from (9.17)-(9.18), respectively, that 1 1

S S S Q -Q = + Q , 0 2 2i S k n K s - Q = , (9.33) 1 1 P Z Q Z - = + , 0 2 2i K Z k n s = - . (9.34)
The results are presented in terms of S Q for SH waves and of P Q for P waves because they become identical in form to those obtained for homogeneous plates. Indeed, we have shown that the formulae for A -, A + , R , and T are still valid for homogeneous plates, provided that K be replaced with 1 k (the wavenumber in the plate) and Q be interpreted as a ratio of real-valued impedances. In either case ( S Q or

) .35) In equation (9.35), 0 r is the mass density of the material outside the layer and 1 r is the mass density of the homogeneous layer. For SH waves, we have two solids; for P waves, two fluids.

P Q , one has 0 0 1 1 0 0 1 1 c c Q c c r r r r - = + . ( 9
The speeds 0 c and 1 c , outside and inside the layer respectively, are either speeds of transverse waves or of longitudinal waves.

Cracks and cavities SECTION D'ÉQUATION (SUIVANTE)

We illustrate the results of Section 8 first when the scatterers are flat cracks and then when they are cylindrical cavities with circular cross-sections. We focus attention on the reflection and transmission coefficients R and T of equations (8.1) and (8.2). We shall compare the results with those obtained elsewhere by different approaches.

The cracks have width 2a and infinite length along the 2 y direction. Their faces lie in planes parallel to the 1 2 ( , ) y y plane. When cracks are represented as lines of discontinuity of negligible thickness, the solution of the one-crack problem cannot be expressed in closed form. It can be expressed in terms of a dislocation density, which is a solution of a singular integral equation [START_REF] Angel | On the reduction of elastodynamic crack problems to singular integral equations[END_REF].

For a crack with a cross-section located along the segment 3 0 y = , 1 y a < , in the 3 1 ( , ) y y plane, the dislocation density b satisfies the two equations Let ( ) , r q be the polar coordinates of an arbitrary point in Fig. 1, with the angle q measured from the 3 y axis in the clockwise direction and the origin at the intersection of the 3 y and 1 y axes. Then, in the far-field, one infers from (10.4) and the asymptotic expression (7.8) of the Hankel function that ( ) y in the interval 1 y a < . From this observation and from (10.6), we infer that ( ) ( )

( ) ( ) i 4 2 1 3 2 0 2 , , e kr u y y u r u f kr p p q q ae ö ÷ ç + ÷ ç ÷ ç è ø = = , ( ) , 0 2 r a q p £ < ? . ( 10
f f q q = - and ( ) ( ) f f p q q - = - .
Taking the limit of (10.6) as q approaches zero, and using (10.2), one finds that For cracks distributed in the layer of Fig. 1, one finds from (7.4), (7.11), and (10.9) that the effective wavenumber K is given by

( ) 2 0 0 2i a a sb s ds n K k k u - ae ö ÷ ç ÷ = - ç ÷ ç ÷ ç è ø ò . ( 10
.10) From (10.9), together with (7.11), (8.3) and (8.4), one infers that .11) where K is the complex root of (10.10) that lies in the first quadrant. Next, we determine the reflection and transmission coefficients R and T by substituting (10.10) and (10.11) into (8.1) and (8.2).

K k Q K k - = + , ( 10
In Figs. We can see that (10.10) and (10.12) yield identical expressions to within first-order terms in e .

Thus, the results of this paper and those of [START_REF] Angel | Complex-valued wavenumber, reflection and transmission in an elastic solid containing a cracked slab region[END_REF] are expected to coincide for low values of e . In fact, Figures 2 and3 show that small differences can be seen for 0.05 e = . With K , 1 s , and 2 s determined from (10.14), (6.7), and (7.4), we use (8.1)-(8.4) to evaluate the reflection and transmission coefficients R and T . In the series 1 s and 2 s , we take the first fifty terms, which yields convergence of the respective sums in the frequency range 0 10 w < £ % .

Figures 5 and6 show, respectively, the modulus of R and that of T versus w %. The cavities are distributed in a layer of thickness / 5 h a = ; their concentrations are 2 0 0.01, 0.03, n a e = = and 0.05. We find that these results are nearly identical to those of [START_REF] Aguiar | Antiplane coherent scattering from a slab containing a random distribution of cavities[END_REF] for low cavity concentration.

In the approach of [START_REF] Aguiar | Antiplane coherent scattering from a slab containing a random distribution of cavities[END_REF], boundary conditions for the average exciting field are written on each finite-size cavity. As a result, the reflection and transmission coefficients in [START_REF] Aguiar | Antiplane coherent scattering from a slab containing a random distribution of cavities[END_REF] are represented by Fourier series and the precise numerical evaluation of those series is time-consuming. In this paper, the cavities are replaced with line scatterers by using the coefficients n C of (10.14). Figure 7 shows the modulus of the reflection coefficient for a distribution of cavities occupying a half-space. In this case, equation (10.13) holds.

Conclusion

For normal incidence, we have solved analytically the problem of scattering of an antiplane (SH) time-harmonic wave by a region of finite-thickness containing a distribution of cylindrical scatterers. The scatterers are identical. In each unit of area, in a plane perpendicular to the cylinder axes, the number of scatterers is equal to a constant 0 n .

We have determined the effective wavenumber K , and the amplitudes of the four coherent SH waves inside and outside the scattering layer. These amplitudes, which are denoted by , R , T A + , and A -, are given in closed form in equations (8.1), (8.2), (8.5), and (8.6), respectively.

To arrive at these results, we have used the Twersky-Foldy (TF) and Waterman-Truell (WT) approaches. We find that the amplitudes , R , T A + , and A -are completely determined -for a given solid, a given scatterer distribution, and a given frequency -when the backscattered amplitude ( ) f p and the forward-scattered amplitude ( ) 0 f of a single scatterer are known. This is a fundamental result of TF and WT. It rests on a closure assumption, because the formulation of the multiple-scattering problem yields a system of an infinite number of equations that cannot be solved either analytically or numerically. Thus, in order to arrive at some useful analytical expressions, it is necessary to truncate -or close -the system of equations. The closure assumption of TF and WT makes use of translational addition theorems for cylindrical

Figure 1 :

 1 Fig.1and cylinder axes in the 2 y direction. We assume further that the cross sections are

  modulus of the solid. We denote by 0 r the mass density and by 0 c the speed of transverse waves in the solid.If the time dependence is of the form ( amplitude factor, R is the reflection coefficient,T is the transmission coefficient, and the factor ( ) exp i t w -has been omitted. The waves (2.2)propagate in the solid outside the layer. Inside the layer, where the scatterers are distributed, there are two waves traveling, respectively, in the forward and backward directions (as we shall see later).

  .3) This equation is written in terms of the cylindrical Hankel function( The coefficients n A characterize the displacement of the exciting wave on the scatterer. In equation (2.3), the variables ( ) , r q are local cylindrical coordinates centered on the axis of the scatterer. Let x denote the position of this axis in the coordinate system of Fig.1. Then, the distance r is defined by

  is evaluated over the entire real line. It can be evaluated by mathematical induction, which yields the results[START_REF] Aguiar | Antiplane coherent scattering from a slab containing a random distribution of cavities[END_REF][START_REF] Urick | The propagation of sound in composite media[END_REF]

  denotes the sign function. It follows from (3.4)-(3.7) that the coherent displacement in the solid is given by

  which are still unknown at this point.

z

  each of the two integrals in (3.8) is written as a sum of two integrals over the is the coordinate along the 3 y direction of an arbitrary fixed point z inside the layer. On each integration interval, we use the expansion

  kept fixed. The integral of this displacement over all positions ( ) 3 1 , x x in the layer of Fig. 1 represents the total contribution of all scattered motions knowing that the line at ( ) 3 1 , z z is fixed. The point at which the wave motion is evaluated is ( ) 3 1 , y y . The average exciting displacement on the left-hand side of (5.1) is given by (3.2), where ( ) bounded solutions of the two-dimensional Helmholtz equation in the solid with wavenumber k , where the derivatives in that equation are taken with respect to the y coordinates. It follows that the integral in (5.1) is a bounded solution of the two- dimensional Helmholtz equation in the solid with wavenumber k . Thus, we write (5.1

  the average exciting displacement on a fixed scatterer at z (averaging on the disorder of all the other scatterers). The quantities represent a global scattered displacement, where the displacement scattered by a scatterer at x is averaged on the disorder of all the other scatterers except those at z and x and, subsequently, all the positions x in the layer are taken into account by the integration in (5.1). At this stage, neither n E nor n M are known.6. Global closure assumption SECTION D'ÉQUATION (SUIVANTE)Equation (5.1) expresses the exciting displacement on a fixed z in terms of the scattered displacement corresponding to two fixed x and z . Similarly, one can write an equation for the exciting displacement corresponding to two fixed z and u in terms of the scattered displacement keeping three scatterers fixed at x , z and u . Repeating this process, one can construct a sequence of equations such that an additional scatterer is fixed at each step.

Fig. 1

 1 Fig. 1 the screen of scatterers with a homogeneous elastic plate of thickness 2h and mass density

  (10.4), sgn denotes the sign function.

  .5)In equation (10.5), f is the angular shape function of the crack. It is defined u y + is an even function of 1 y , it follows from (10.7) that ( ) 1 b y is an odd function of 1

  2 and 3, respectively, we show the modulus of the reflection coefficient and that of the transmission coefficient versus the dimensionless frequency ka w 2 and 3 are in very good agreement with Figs. 5 and 7 in[START_REF] Angel | Complex-valued wavenumber, reflection and transmission in an elastic solid containing a cracked slab region[END_REF].In the approach of Angel and Koba (1998), boundary conditions for the average exciting field are written on each finite-size crack. The cracks are not replaced with line scatterers, as is done in this paper. Using our notations, the formula of[START_REF] Angel | Complex-valued wavenumber, reflection and transmission in an elastic solid containing a cracked slab region[END_REF] for the complex-valued wavenumber can be written in the form

ForFigure 2 :Figure 3 :

 23 Figure 2 : Modulus of the reflection coefficient versus the dimensionless frequency for dimensionless thickness / 3 h a = and crack densities 0.01 e = , 0.03 , 0.05 .

Figure 4 Figure 4 :Figure 5 :Figure 6 :Figure 7 :

 44567 Figure4shows the modulus of the reflection coefficient for a distribution of cracks

wavefunctions. In this paper, we have used a different approach, which is illustrated by equations (5.3) and (6.1), and which yields the same expression for the effective wavenumber as that of WT.

This paper shows that wave amplitudes inside and outside the layer can be obtained without imposing continuity conditions on the "boundaries" of the layer. In fact, the layer has no physical boundaries, since the solid is one and the same inside and outside. It would be erroneous to impose conditions on the "boundaries" -even though numerical results in particular situations could be nearly identical with those obtained in the context of the present work.

We have shown that our formulae for R and T can be easily applied to thin cracks and cylindrical cavities. In each case, we need only recall the solution of the single-scatterer problem.

The results for cracks and cavities are in very good agreement with earlier results obtained by more labor-intensive methods.


Chinnaraji Annamalai
email: anna@iitkgp.ac.in

Computation for the Summation of Binomial Expansions and Geometric Series of Multiples of Powers of Two

Keywords: MSC Classification codes: 05A10, 40A05 (65B10) computation, combinatorics, geometric series, binomial coefficient

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

Introduction

The author of this article introduces an innovative binomial coefficient [START_REF] Annamalai | Annamalai's Binomial Identity and Theorem[END_REF][START_REF] Annamalai | My New Idea for Optimized Combinatorial Techniques[END_REF][START_REF] Annamalai | Computation for the Summation of Integers and Geometric Progression of Powers of Two[END_REF] along with a traditional binomial coefficient. The innovative binomial coefficient is shown below:

𝑉 𝑟 𝑛 = (𝑟 + 1)(𝑟 + 2)(𝑟 + 3) ⋯ (𝑟 + 𝑛) 𝑛! , (𝑛 ≥ 1, 𝑟 ≥ 0 & 𝑛, 𝑟 ∈ 𝑁 = {0,1,2,3, ⋯ }). Traditional binomial coefficient: ( 𝑛 𝑟 ) = 𝑛𝐶𝑟 = 𝑛! 𝑟! (𝑛 -𝑟)!
, where 𝑛, 𝑟 ∈ 𝑁.

Let us compare the innovative binomial coefficient 𝑉 𝑥 𝑦 with the traditional binomial coefficient as follows:

Let 𝑧 = 𝑥 + 𝑦. Then, 𝑧𝐶 𝑥 = 𝑧! 𝑥! 𝑦! . Here, 𝑉 𝑥 𝑦 = 𝑉 𝑦 𝑥 ⟹ 𝑧𝐶 𝑥 = 𝑧𝐶 𝑦 , (𝑥, 𝑦, 𝑧 ∈ 𝑁).

For example, 𝑉 3 5 = 𝑉 5 3 = (5 + 3)𝐶 3 = (5 + 3)𝐶 5 = 56.

Also, 𝑉 𝑛 0 = 𝑉 0 𝑛 = 𝑛𝐶 0 = 𝑛𝐶 𝑛 = 𝑛! 𝑛! 0! = 1 𝑎𝑛𝑑 𝑉 0 0 = 0𝐶 0 = 0! 0! = 1(∵ 0! = 1).

Computation of Binomial Expansions and Geometric Series

In this section, the author of this article provides a theorem based on binomial expansions and geometric series [START_REF] Annamalai | A novel computational technique for the geometric progression of powers of two[END_REF][START_REF] Annamalai | Extension of ACM for Computing the Geometric Progression[END_REF][START_REF] Annamalai | A Novel Approach to ACM-Geometric Progression[END_REF][START_REF] Annamalai | Computational Modelling for the Formation of Geometric Series using Annamalai Computing Method[END_REF][START_REF] Annamalai | Summations of Single Terms and Successive Terms of Geometric Series[END_REF][START_REF] Annamalai | Sum of Geometric Series with Negative Exponents[END_REF][START_REF] Annamalai | Computation of Geometric Series in Different Ways[END_REF][START_REF] Annamalai | Numerical Computational Method for Computation of Binomial Expansions and Geometric Series[END_REF][START_REF] Annamalai | Algorithmic and Numerical Techniques for Computation of Binomial and Geometric Series[END_REF][START_REF] Annamalai | A Binomial Expansion equal to Multiple of 2 with Non-Negative Exponents[END_REF].

Theorem ∶ ∑ 𝑖 × 𝑉 𝑖 1-𝑖 1 𝑖=1 + ∑ 𝑖 × 𝑉 𝑖 1-𝑖 2 𝑖=1 + ∑ 𝑖 × 𝑉 𝑖 1-𝑖 3 𝑖=1 + ⋯ + ∑ 𝑖 × 𝑉 𝑖 1-𝑖 𝑛 𝑖=1 = (𝑛 -1)2 𝑛 + 1.
Proof. Let us find the value of each binomial expansion in the binomial theorem step by step.

𝑆𝑡𝑒𝑝 1: 1 × 𝑉 1 0 = 1 ⟹ ∑ 𝑖 × 𝑉 𝑖 1-𝑖 1 𝑖=1 = 1 = 1 × 2 0 , (∵ 𝑉 𝑛 0 = 𝑉 0 𝑛 = 1 & 𝑛 = 0, 1, 2,3, ⋯ ). 𝑆𝑡𝑒𝑝 2: ∑ 𝑖 × 𝑉 𝑖 1-𝑖 2 𝑖=1 = 1 × 𝑉 1 1 + 2 × 𝑉 2 0 = 2 + 2 = 4 = 2 × 2 1 . 𝑆𝑡𝑒𝑝 3: ∑ 𝑖 × 𝑉 𝑖 1-𝑖 3 𝑖=1 = 1 × 𝑉 1 2 + 2 × 𝑉 2 1 + 3 × 𝑉 3 0 = 3 + 6 + 3 = 12 = 3 × 2 2 . 𝑆𝑡𝑒𝑝 4: ∑ 𝑖 × 𝑉 𝑖 1-𝑖 4 𝑖=1 = 1 × 𝑉 1 3 + 2 × 𝑉 2 2 + 3 × 𝑉 3 1 + 4 × 𝑉 4 0 = 4 + 12 + 12 + 4 = 4 × 2 3 .
Similarly, we can continue the expressions up to "step n " such that ∑ 𝑖 × 𝑉 𝑖

𝑛-𝑖 𝑛 𝑖=0 = 𝑛2 𝑛-1 .
Now, by adding these expressions on both sides, it appears as follows:

∑ 𝑖 × 𝑉 𝑖 1-𝑖 1 𝑖=1 + ∑ 𝑖 × 𝑉 𝑖 1-𝑖 2 𝑖=1 + ∑ 𝑖 × 𝑉 𝑖 1-𝑖 3 𝑖=1 + ∑ 𝑖 × 𝑉 𝑖 1-𝑖 4 𝑖=1 ⋯ + ∑ 𝑖 × 𝑉 𝑖 1-𝑖 𝑛 𝑖=1 = ∑ 𝑖 × 2 𝑖-1 𝑛 𝑖=1 .
where ∑ 𝑖 × 2 𝑖 𝑛 𝑖=1 = (𝑛 -1)2 𝑛 + 1 is the geometric series of multiples of powers of two [START_REF] Annamalai | Computation for the Summation of Integers and Geometric Progression of Powers of Two[END_REF].

∴ ∑ 𝑖 × 𝑉 𝑖 1-𝑖 1 𝑖=1 + ∑ 𝑖 × 𝑉 𝑖 1-𝑖 2 𝑖=1 + ∑ 𝑖 × 𝑉 𝑖 1-𝑖 3 𝑖=1 + ⋯ + ∑ 𝑖 × 𝑉 𝑖 1-𝑖 𝑛 𝑖=1 = (𝑛 -1)2 𝑛 + 1.
Hence, theorem is proved.

Conclusion

This article presented a numerical computational technique for the summation of binomial expansions and geometric series of multiples of powers of two This technique and its results can be useful for researchers who are working in science, economics, engineering, management, and medicine [START_REF] Annamalai | Application of Exponential Decay and Geometric Series in Effective Medicine[END_REF].