Applied Mathematical Sciences, Vol. 8, 2014, no. 118, 5875 - 5886HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2014.46473

Peculiarities of Elastic Wave Refraction from

the Layer with Fractal Distribution of Density

A.V. Anufrieva

Department of Applied MathematicsInstitute of Computer Mathematics and Information Technologies

Kazan Federal University, Universiteskaya, 18Kazan, Russia, 420008

K.B. Igudesman

Department of GeometryInstitute of Mathematics and Mechanics

Kazan Federal University, Universiteskaya, 18Kazan, Russia, 420008

D.N. Tumakov

Department of Applied Mathematics,Institute of Computer Mathematics and Information Technologies

Kazan Federal University, Universiteskaya, 18Kazan, Russia, 420008

Copyright c© 2014 A.V. Anufrieva, K.B. Igudesman and D.N. Tumakov. This is an openaccess article distributed under the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original work

is properly cited.

Abstract

In this study, we underline the peculiarities of the refraction prob-lem of elastic waves from a layer with a fractal density distribution.The refraction problem is reduced to the system of ordinary differentialequations with linear coefficients. Analytic solutions for each of its equa-tions are found. The case for the layer with fractal density distributionis investigated numerically. Characteristic maxima of the reflected wave

5876 A.V. Anufrieva, K.B. Igudesman, D.N. Tumakov

energy are outlined. Graphs illustrate the dependence of the reflectedenergy from self-similar properties of the fractal curve.

Mathematics Subject Classification: 78A45, 28A80

Keywords: refraction, elastic wave, fractal interpolation

1 Statement of the problem

Let harmonic elastic wave of the form u0(x) exp{iωt}, in which

u0(x) = A0e−ik1x, k1 =

ω

v1,

fall from domain {x < 0} onto the layer of thickness L(medium denoted as2 {0 < x < L}, with density ρ2(x) and velocity v2). As a consequence ofdiffraction, a portion of u1(x) of the incident wave is reflected, part of theu3(x) passes to medium 3 {x > L}. We need to find the completed diffractedfield.

For harmonic oscillations of a homogeneous and isotropic elastic medium,displacement satisfies the wave equation

u′′(x) + k2u(x) = 0, (1)

where k = ω/v is wave number of the medium. The solution of Eq. (1) hasthe form

u(x) = Ae−ikx +Beikx,

with stressσ(x) = iωρv(−Ae−ikx +Beikx).

In the general case [25], oscillations of elastic waves are described by equa-tion

ρ(x)∂2u

∂t2=∂σ

∂x, (2)

where stress associated with displacement via the following relation:

σ(x) = ρ(x)v2u′(x).

Here ρ(x)v2 is the modulus G(x). Then Eq. (2) for harmonic waves takes theform

∂

∂x(ρ(x)v2

∂u

∂x) + ρ(x)ω2u = 0.

Reflected and transmitted waves are solutions of Eq. (1). They will besought in the form u1(x) = B1e

ik1x and u3(x) = A3e−ik3(x−L).

Peculiarities of elastic wave refraction 5877

At the interface, displacement and stress must be continuous. Then forx = 0 we have

A0 +B1 = u2(0), (3)

iωρ1v1(−A0 +B1) = ρ2(0)v22u′2(0), (4)

for x = L:u2(L) = A3, (5)

ρ2(L)v22u′2(L) = −iωρ3v3A3. (6)

Thus, by eliminating the unknowns B1 and A3 of Eqs. (3)–(6), we reduce thesolution of the diffraction problem to the determination of displacement u2(x),which satisfies the equation

(ρ2(x)v22u′2(x))

′ + ρ2(x)ω2u2(x) = 0 (7)

with the boundary conditions of the third type

ρ2(0)v22u′2(0)− iωρ1v1u2(0) = −i2ωρ1v1A0,

ρ2(L)v22u′2(L) + iωρ3v3u2(L) = 0.

(8)

We consider the case where v1 = v2 = v3 = v, ρ1 = ρ3 = ρ, and ρ2(x) isthe fractal interpolation function.

2 Solution of the problem

We consider the set of N adjacent sectors located close to each other insteadof the layer (0 < x < L). In each of the layer, the density distribution isgiven by ρn(x) = knx + bn, and v is velocity, which is constant. Moreover,ρn(x)v

2 is a continuous function all ever. Then the wave equation for eachlayer (xn−1 < x < xn) takes the form

((knx+ bn)v2u′n(x))

′ + (knx+ bn)ω2un(x) = 0. (9)

We make the substitution

x =v

ωz − bn

kn

and consider the following function

yn(z) = un(v

ωz − bn

kn).

Equation (2) becomes the Bessel equation

zy′′n(z) + y′n(z) + zyn(z) = 0.

5878 A.V. Anufrieva, K.B. Igudesman, D.N. Tumakov

The sign of z for each interval (xn−1 < x < xn) in the obtained equationremains constant and depends on the sign of kn. After combining the solutionsfor positive and negative z, the general solution can be written as

yn(z) = AnJ0(|z|) +BnY0(|z|).

Thus, the solution of Eq. (2) will be of the following form:

un(x) = AnJ0

(∣∣∣∣ωv x+ ωbnvkn∣∣∣∣)+BnY0(∣∣∣∣ωv x+ ωbnvkn

∣∣∣∣) . (10)The following conditions must be satisfied at the junction of the layers

un(xn) = un+1(xn), u′n(xn) = u

′n+1(xn), n = 1, . . . , N − 1,

which leads to the following equations:

AnJ0(|ξn|) +BnY0(|ξn|) = An+1J0(|ζn|) +Bn+1Y0(|ζn|),

AnJ1(|ξn|) +BnY1(|ξn|) =signζnsignξn

(An+1J1(|ζn|) +Bn+1Y1(|ζn|)),

where

ξn =xnkn + bn

vknω, ζn−1 =

xn−1kn + bnvkn

ω, n = 1, . . . , N.

Conditions at boundaries of layers with half-planes are

B1vu′(0)− iωρu(0) = −i2ωρA0,

(kNL+ bN)vu′(L) + iωρu(L) = 0.

The conditions define relationships between coefficients

A1[−b1 signζ0J1(|ζ0|)− iρJ0(|ζ0|)]+

+B1[−b1 signζ0Y1(|ζ0|)− iρY0(|ζ0|)] = −i2ρA0,

AN [−(kNL+ bN)signξNJ1(|ξN |) + iρJ0(|ξN |)]+

+BN [−(kNL+ bN)signξNY1(|ξN |) + iρY0(|ξN |)] = 0.

The above equations for coefficients An and Bn can be written in the ma-trix form via Ω · c = f , where c = {A1, B1,A2, B2, . . . , AN , BN} and f ={−i2A0ρ/b1, 0, . . . , 0}.

The matrix Ω is a five-diagonal matrix

Peculiarities of elastic wave refraction 5879

Ω11 Ω12 0 0J0(|ξ1|) Y0(|ξ1|) −J0(|ζ1|) −Y0(|ζ1|)J1(|ξ1|) Y1(|ξ1|) s1J1(|ζ1|) s1Y1(|ζ1|)

0 0 J0(|ξ2|) Y0(|ξ2|)0 0 J1(|ξ2|) Y1(|ξ2|)

. . . . . .

0 J0(|ξN−1|)0 J1(|ξN−1|)

0

→

→

00

−J0(|ζ2|) −Y0(|ζ2|)s2J1(|ζ2|) s2Y1(|ζ2|)

. . . . . . 0Y0(|ξN−1|) −J0(|ζN−1|) −Y0(|ζN−1|)Y1(|ξN−1|) sN−1J1(|ζN−1|) SN−1Y1(|ζN−1|)

0 ΩNN−1 ΩNN

,

where sn = −signζn/signξn,

Ω11 = −signζ0J1(|ζ0|)− iρ

b1J0(|ζ0|),

Ω12 = −signζ0Y1(|ζ0|)− iρ

b1Y0(|ζ0|),

ΩNN−1 = −signξNJ1(|ξN |) + iρ

kNxN + bNJ0(|ξN |),

ΩNN = −signξNY1(|ξN |) + iρ

kNxN + bNY0(|ξN |).

The resulting linear algebraic equation is solved using the Thomas algo-rithm [26].

3 Fractal interpolation functions

There exist two ways of constructing fractal interpolation functions. Suchfunctions are presented in [1] in the form of attractors of iterated functionsystems of a special form. In this study, we adopt a more general approachdeveloped by P. Massopust [27].

5880 A.V. Anufrieva, K.B. Igudesman, D.N. Tumakov

Let [a, b] ⊂ R be a non-empty interval, 1 < N ∈ N and {(xi, yi) ∈ [a, b]×R |a = x0 < x1 < · · · < xN−1 < xN = b} be points of interpolation. We considerthe affine transformation of the plane for each i = 1, N

Ai : R2 → R2, Ai(xy

):=

(ai 0ci λi

)(xy

)+

(αiβi

).

We require that for all i = 1, N , the following two conditions be

Ai(x0, y0) = (xi−1, yi−1), Ai(xN , yN) = (xi, yi).

In that case,

ai =xi − xi−1b− a

, ci =yi − yi−1 − λi(yN − y0)

b− a,

αi =bxi−1 − axi

b− a, βi =

byi−1 − ayi − λi(by0 − ayN)b− a

,

and λi, i = 1, N is considered as a family of parameters. Note that under thisdefinition, the operators Ai transform a given line segment connecting points(x0, y0) and (xN , yN) to a zigzag-shaped line, by sequentially connecting thepoints of interpolation to each other.

For each i = 1, N , we denote

ui : [a, b]→ [xi−1, xi], ui(x) := aix+ αi,

pi : [a, b]→ R, pi(x) := cix+ βi,

p(x) :=N∑i=1

(pi ◦ u−1i )(x)χ[xi−1,xi](x),

where χS is the characteristic function of S. In [27], it is shown that thefunctional operator T , which make use of the following rule,

(Tg)(x) := p(x) +N∑i=1

λi(g ◦ u−1i )(x)χ[xi−1,xi](x),

maps continuous functions into some other continuous functions. Moreover, if|λi| < 1 for all i = 1, N , then T is a contraction operator on the Banach space(C[a, b], ‖ ‖∞) with a contraction ratio λ := max{|λi| | i = 1, N}.

According to the fixed point theorem, there exists a unique contractionmapping function f ∈ C[a, b] such that T f = f. Moreover, for any f ∈ C[a, b],we have

limn→∞

‖T n(f)− f‖∞ = 0.

Peculiarities of elastic wave refraction 5881

The function f is called the fractal interpolation function. It is easy to see thatif f ∈ C[a, b], f(x0) = y0 and f(xN) = yN , then T (f) passes through the pointsof interpolation. In this case, the function T n(f) is called an interpolation pre-fractal function of order n.

In Fig. 1, the fractal interpolation function constructed from the interpola-tion points (0, 1000), (50, 1500) and (100, 1000) is shown for values λ1 = λ2 =0.5.

Figure 1: Graphs of pre-fractals, left one is for pre-fractal of the first level,right one is for pre-fractal of the ninth level.

4 Numerical results

We consider the differential Eq. (7) with boundary conditions (8). We seekdependence of solutions u2(0) (reflected energy) on density ρ(x) and frequencyω. In other words, if ρ : [0, L] → R+ is density of the layer, then we seek thefunction

Eρ : R+ → [0, 1], Eρ(ω) = u2(0).We investigate the behavior of the function Eρ when ρ is a pre-fractal

interpolation function. Let ρ be a fractal interpolation function constructedfrom interpolation points (0, 1000), (50, 1500) and (100, 1000) with parametersλ1 = λ2 = 0.7. We denote ρ

0 ≡ 1000; ρn is a pre-fractal interpolation functionof order n. Note that ρn is a continuous piecewise linear function consistingof 2n line segments, passing through points (xni , y

ni ), i = 0, 2

n. From theconstruction, it follows that xni = 100 · i · 2−n. In addition, the set of breakpoints An := {(xni , yni )}2

n

i=0 is a subset of An+1.

The following conclusions are inferred through comparing the graphs forEρ8 and Eρ9 (Fig. 2). First, the graphs are nearly identical except for theobserved Eρ9 maximum at frequencies close to 9700. This effect is due to thefact that the graph of ρ9 is obtained from ρ8 by replacing each line segment

5882 A.V. Anufrieva, K.B. Igudesman, D.N. Tumakov

Figure 2: Graphs of functions: left one is for Eρ8, right one is for Eρ9, where ρn

is pre-fractal interpolation curve constructed by interpolating points (0, 1000),(50, 1500) and (100, 1000) with parameters having values λ1 = λ2 = 0.7.

with a zigzag line comprising of two parts. Second, the function Eρn inheritsself-similar properties of the function ρn in the sense that the peaks of thefunction Eρn are the points C · 2k, where C is a constant and k = 1, n. Thisis due to the fact that the ratio of lengths of (xni , x

ni+1) and (x

n+1j , x

n+1j+1 ) equals

2. Third, the presence of distinct peaks in the function Eρn is due to the factthat projections onto the axis X of all line segments that make up the graphρn have the same length equal to 100·2−n. In other words, xni+1−xni = 100·2−nfor any i = 0, 2n − 1.

Figure 3: The graph of Eρ9, where ρ9 is a pre-fractal built by using interpola-tion points (0, 1000), (51, 1500) and (100, 1000) with parameters having valuesλ1 = λ2 = 0.7.

We displace one point of interpolation by one unit to the right on axis Xto verify the last conclusions. Now let ρ be a fractal interpolation functionconstructed from the interpolation points (0, 1000), (51, 1500) and (100, 1000)for the parameters λ1 = λ2 = 0.7. As in the previous case, pre-fractal ρ

n isa continuous piecewise linear function and consist of 2n line segments passingthrough the point (xni , y

ni ), i = 0, 2

n. However, xni 6= 100 · i · 2−n. It is easy toverify that the number of segments of length xni+1−xni is distributed accordingto the binomial law, i.e. for any k = 0, n has exactly Ckn intervals of length100 · 0.51k · 0.49n−k. Comparison of Figs. 3 and 2 shows that the presence

Peculiarities of elastic wave refraction 5883

of distinct peaks of the function Eρn depends on the partitioning of interval[0, 100] by points xni .

Figure 4: Graphs of functions Eρ8 and Eρ9, where ρn is a pre-fractal, con-structed by interpolating points (0, 1000) (61.8, 1500) and (100, 1000) withparameters having values λ1 = λ2 = 0.7.

Some more interesting results are also illustrated in Fig. 4. Selection of thesecond point of interpolation (61.8, 1500) is not accidental. Let α ≈ 0.618 bethe golden ratio, i.e. the smallest root of the equation x2 = 1 − x. Amongthe intervals xni+1 − xni is the Ckn intervals of length 100αk(1 − α)n−k. Sinceα2 = 1−α, we find that among the intervals of the (n+1)-th level of xn+1i+1 −xn+1iare intervals of lengths exactly equal to the lengths of intervals of the n-th level.Thus, the peaks corresponding to different pre-fractals are superimposed ontoeach other.

5 Conclusions

Through exploring behavior of the reflected energy Eρn for the case, when ρn

is a pre-fractal, the following conclusions can be made.Fine bursts of reflected energy show up only in those fractals whose line

segments in graph ρ(x) have projections onto axis X of the same length. More-over, increase in the number of segments of equal length increases the value ofreflected energy. These self-similar structures include pre-fractals derived froma pre-fractal of a smaller level by means of dividing the segments into equalparts, or dividing the segments using the golden ratio.

If the fractal is built via the principle of dividing into equal parts, theincrease in order does not alter pre-fractal peaks of the reflected energy, butonly adds new peaks at higher frequencies.

In case, when the fractal is constructed by dividing the segments in thegolden ratio, increase in the order increases pre-fractal peaks of the reflectedenergy as well as adds new ones at higher frequencies.

Acknowledgements. The work was performed according to the RussianGovernment Program of Competitive Growth of Kazan Federal University as

5884 A.V. Anufrieva, K.B. Igudesman, D.N. Tumakov

part of the OpenLab Crown.

References

[1] M.F. Barnsley, Fractals everywhere, Academic Press Inc., Boston, 1988.

[2] M.A. Slawinski, Seismic Waves and Rays in Elastic Media, Pergamon,Amsterdam, 2003.

[3] D.L. Turcotte, Fractals and chaos in geology and geophysics, CambridgeUniversity Press, New York, 1997.

[4] S.A.Shapiro, I.S.Faizullin, Fractal properties of fault systems by scatteringof body seismic waves, Tectonophysics, 202 (1992), 177 – 181.

[5] V.A. Moukhamedov, Fractal properties of high-frequency seismic noiseand mechanisms of its generation, Izvestiya Ross. AN. Ser. Fizika Zemli,3 (1992), 39 – 49, (in Russian).

[6] S.S. Krylov, N.Yu. Bobrov, Fractals in geophysics, Izd-vo SPb. univer-siteta, SPb., 2004, (in Russian).

[7] P.M. Gorjainov, G. Yu. Ivanjuk, Self-organization of mineral systems.Synergetic principles of geological researches, GEOS, Moscow, 2001, (inRussian).

[8] A.V. Dyskin, Continuum fractal mechanics of the Earht’s crust, Pure appl.geophys, 161 (2004), 1979 – 1989.

[9] A. Carpinteri, B. Chiaia, S. Invernizzi, Three dimensional fractal analysisof concrete fracture at the meso-level, Theoretical and Applied FractureMechanics, 31 (1999), 163 – 172.

[10] V.G. Bykov, , Dal’nauka, Vladivostok, 2000, (in Russian).

[11] V.N. Odincev, I.Zh. Bunin, Fractal model of a hierarchical structure ofmassifs of rocks, Nelineinyi mir 3 (2004), (in Russian).

[12] P.D. Panagiotopoulos, O.K. Panagouli, E.S. Mistakidis, in Fractals andFractional Calculus in continuum mechanics, Springer, Wien, CISM Se-ries, 378 (1997), 109 – 171.

[13] W. Wen, P. Sheng, Two-and three-dimensional ordered structures formedby electro-magnetorheological colloids, Physica B, 338 (2003), 343 – 346.http://dx.doi.org/10.1016/j.physb.2003.08.017

Peculiarities of elastic wave refraction 5885

[14] G.W. Milton, M. Briane, J.R. Willis, On cloaking for elasticity and physi-cal equations with a transformation invariant form, New J. Phys., 8 (2006)248 – 260. http://dx.doi.org/10.1088/1367-2630/8/10/248

[15] H. Chen, C.T.Chan, Acoustic cloaking in three dimensionsusing acoustic metamaterials, Appl. Phys. Lett., 91 (2007).http://dx.doi.org/10.1063/1.2803315

[16] S.J. Lee, S.E. Moon, H.C. Ryu, M.H. Kwak, Y.T. Kim, S.K. Han, Mi-crowave properties of compositionally graded (Ba,Sr)TiO3 thin films ac-cording to the direction of the composition gradient for tunable microwaveapplications, Appl. Phys. Lett., 82, 13 (2003), 2133 – 2135.

[17] A.J. Chakraborty, Prediction of negative dispersion by a nonlo-cal poroelastic theory, Acoust. Soc. Am., 123 (2008), 56 – 67.http://dx.doi.org/10.1121/1.2816576

[18] L.M. Brekhovskikh, O.A. Godin, Acoustics of layered media, Nauka,Moscow, 1989, (in Russian).

[19] A.B. Shvartsburg, N.S. Erokhin, Acoustic gradient barriers (ex-actly solvable models), PHYS-USP, 54(6) (2011), 605 – 623.http://dx.doi.org/10.3367/UFNe.0181.201106c.0627

[20] A.V. Anufrieva, D.N. Tumakov, V.L. Kipot, Elastic wavepropagation through a layer with graded-index distributionof density, Conference Days on Diffraction, (2012), 21 – 26.http://dx.doi.org/10.1109/DD.2012.6402745

[21] A.V.Anufrieva, D.N.Tumakov, V.L.Kipot, Peculiarities of propaga-tion of a plane elastic wave through a gradient layer, Days onDiffraction (DD), May 27 2013 – May 31 2013, (2013), 11-16.http://dx.doi.org/10.1109/DD.2013.6712795

[22] A.V.Anufrieva, D.N.Tumakov, Diffraction of a plane elastic wave by agradient transversely isotropic layer, Advances in Acoustics and Vibration,262067 (2013), 8 pages. http://dx.doi.org/10.1155/2013/262067

[23] D.N. Tumakov, V.L. Kipot, A.V. Anufrieva, computer code (Certifi-cate on the state registration of the computer program 2012615820 from26.06.2012).

[24] D.N. Tumakov, A.V. Anufrieva, computer code (Certificate on the stateregistration of the computer program 2012661046 from 06.12.2012).

5886 A.V. Anufrieva, K.B. Igudesman, D.N. Tumakov

[25] K. Aki, P.G. Richards, Quantitative Seismology, University Science Books,2002.

[26] V.P.Il’in, V.V.Karpov, A.MMaslennikov, Numerical methods for solvingproblems in structural mechanics, Vyshaya shkola, Minsk, 1990, (in Rus-sian).

[27] P.Massopust, Interpolation and approximation with splines and fractals,Oxford University Press, Oxford, 2010.

Received: June 1, 2014