Propagation of longitudinal waves in a random binary rod

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1 Downloaded By: [University of North Carolina, Charlotte At: 7:3 2 May 28 Waves in Random and Complex Media Vol. 6, No. 4, November 26, Propagation of longitudinal waves in a random binary rod YURI A. GODIN Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA (Received 2 August 25; in final form 3 December 25) Propagation of waves in a composite elastic rod consisting of rods with alternating properties of random length is considered. We calculate exactly the Lyapunov exponent and find its short and long wave asymptotics. Finally, we discuss conditions for propagation and localization of waves in a binary random medium.. Introduction The one-dimensional wave equation describes many phenomena in physics [. In a periodic medium, the solution of the wave equation has a Floquet Bloch structure. Very long waves propagate without attenuation and the medium can be viewed as homogeneous. However, for higher frequencies there are certain intervals (so-called gaps) where waves cannot propagate even though the system is perfectly conservative. Due to multiple scattering, destructive interference occurs, so that the wave amplitude decreases exponentially in the medium. The rate of decay of the solution is called the Lyapunov exponent (the inverse of the localization length). The Lyapunov exponent is strictly positive inside the gaps and equals zero in the spectral bands. The Floquet Bloch structure of the solution is completely destroyed if the medium is not perfectly periodic because of random independent variations of geometric or material parameters [2. Now as the wave propagates, its amplitude decreases exponentially with probability one and the medium can be considered homogeneous for long waves only on a finite time interval. This phenomenon is known as localization. Calculation of the Lyapunov exponent for disordered systems usually employs the transfer matrix formalism, asymptotic analysis, and numerical modelling [3 6. Our approach is based on the phase-amplitude representation of the solution and stochastic methods. This allows us to find exactly the joint probability density distribution for the phase of the solution that leads to an exact formula for the Lyapunov exponent. ygodin@uncc.edu Waves in Random and Complex Media ISSN: (print), (online) c 26 Taylor & Francis DOI:.8/

2 4 Yuri A. Godin Downloaded By: [University of North Carolina, Charlotte At: 7:3 2 May 28 x ρ s ρ s ρ s ρ s ρ s ρ s ρ s ρ s Figure. An elastic binary rod composed of two alternating materials with parameters ϱ, s and ϱ, s, where ϱ is the density and s is the elastic compliance, respectively. 2. Formulation of the problem We consider longitudinal vibrations in an infinite elastic rod composed of alternating rods made of two types of materials M and M. The properties of successive rods alternate and take on only two values: either ϱ, s in M or ϱ, s in M (figure ), where ϱ is the density of the rod s material and s is its elastic compliance (the inverse of the Young modulus). We also suppose that the random length l i of the i-th component is distributed exponentially with parameters λ i P{l i > x} =e λi x, i =,. () Assuming harmonic time dependence of the displacement u(x, t) = X(x)e iωt in the x- direction, X(x) must satisfy the wave equation ( ) d dx + ω 2 X = (2) ϱ(x) dx s(x) dx with { { ϱ, x M ; s, x M ; ϱ(x) = and s(x) = ϱ, x M, s, x M. To obtain analytical results for the asymptotic behaviour of the solution, we introduce the phase ϑ(x) and the amplitude r(x) using the Prüfer transform [7 X(x) = r(x) sin ϑ(x), ϑ [,π) = S, (3) X (x) = s(x)r(x) cos ϑ(x). (4) Then r(x) and ϑ(x) obey the system dϑ dx = s(x) cos2 ϑ + ω 2 ϱ(x) sin 2 ϑ, (5) dr dx = 2 r sin 2ϑ[s(x) ω2 ϱ. (6) Following [8, 9, it is constructive to view the rod as a Markov chain ξ x with two states ξ ={, } depending on whether x M or x M. From () we derive transition intensities for ξ x P{ξ x+dx = ξ x = } =λ dx, P{ξ x+dx = ξ x = } =λ dx, so (ξ x,ϑ(x)) is an ergodic Markov process on the product space ξ S and { p (ϑ), ξ = ; p(ξ,ϑ) = p (ϑ), ξ =, is the invariant probability measure for (ξ x,ϑ). For example, the probability that the random phase belongs to the interval (ϑ, ϑ + dϑ) while x M is p (ϑ)dϑ. (7) (8)

3 Propagation of longitudinal waves 4 Downloaded By: [University of North Carolina, Charlotte At: 7:3 2 May 28 The infinitesimal operator L of the Markov process (ξ x,ϑ(x)) acts on the vector functions f (ξ,ϑ) = [ f (ϑ), f (ϑ) and is given by [9 where [ λ λ L f = f + λ λ a (ϑ) df dx a (ϑ) df dx, (9) a i (ϑ) = s i cos 2 ϑ + ω 2 ϱ i sin 2 ϑ, i =,. () One can show [9 that the probability density p(ξ,ϑ) = [p (ϑ), p (ϑ) satisfies the adjoint equation L p = orthe linear system with periodic and normalization conditions (a (ϑ)p ) = λ p + λ p, (a (ϑ)p ) = λ p λ p () p i () = p i (π); p i (ϑ) dϑ = λ i λ + λ, i =,. (2) The fact that the system () has first integral allows us to find an explicit solution of () a (ϑ)p (ϑ) + a (ϑ)p (ϑ) = c (3) p i (ϑ) = cλ { [ i K i exp η(ϑ ) dϑ dϑ [ } + a i (ϑ) a i (ϑ ) exp η(ϑ ) dϑ, ϑ (4) where η(ϑ) = λ a (ϑ) + λ a (ϑ), (5) K i = c = dϑ a i (ϑ ) exp [ exp λ i i= + η(ϑ ) dϑ ϑ, i =,, (6) η(ϑ ) dϑ [ [ K i exp dϑ a i (ϑ ) exp [ η(ϑ ) dϑ [ dϑ η(ϑ ) dϑ ϑ a i (ϑ). (7)

4 42 Yuri A. Godin Downloaded By: [University of North Carolina, Charlotte At: 7:3 2 May Calculation of the Lyapunov exponent The Lyapunov exponent γ (the inverse of the localization length l)isanon-random quantity which determines the asymptotic growth or decay of the solution. If we denote by L the total length of the rod, then by definition ln r(l) γ = lim = lim L L L L L 2 (s(ξ u) ω 2 ϱ(ξ u )) sin 2ϑ du = 2 (s(ξ) ω2 ϱ(ξ)) sin 2ϑ = 2 (s ω 2 ϱ ) p (ϑ) sin 2ϑ dϑ + 2 (s ω 2 ϱ ) p (ϑ) sin 2ϑ dϑ, (8) where the limit holds with probability one. Using (4), one can calculate the Lyapunov exponent exactly. However, for the sake of analytical tractability we will find asymptotic values of γ for long and short waves. To this end, we analyse the dependence of p (ϑ) and p (ϑ) involved in (8) on ω. Figure 2 shows the dependence of the probability density function p (ϑ)onω for small and large values of the frequency. It has a maximum at the points where the phase ϑ (5) has slowest rate of change. Therefore, when ω then p (ϑ)isconcentrated around π/2 where cos ϑ in (5) is small, while if ω then p (ϑ) islocated near zero or π to minimize sin ϑ in (5). We will use these results below to find the asymptotics of γ. 3. Long wave asymptotics of the Lyapunov exponent The dependence of the probability density functions in figure 2 suggests that its asymptotic behaviour for small values of ω can be found from the Laplace method. Using this and the formulas in the appendix, we obtain K i s i λ s + λ s, i =,, ω. (9) ω =. ω =.3 ω =.6 ω = 5. ω = 2 p (θ) θ/π Figure 2. Dependence of the probability density function p (ϑ) (4) on the excitation frequency ω for ϱ = ϱ = and 3s = s = 3, 2λ = λ = 2.

5 Propagation of longitudinal waves 43 Downloaded By: [University of North Carolina, Charlotte At: 7:3 2 May 28 where dϑ a i (ϑ) [ dϑ ϑ a i (ϑ) exp η(ϑ ) dϑ s i, i =,, ω. (2) λ s + λ s dϑ [ a i (ϑ ) exp η(ϑ ) dϑ ϑ π ω(λ + λ ), s ϱ i =,, (2) s = s λ + s λ, ϱ = ϱ λ + ϱ λ (22) λ + λ λ + λ are the average values of the elastic compliance and density, respectively. Finally, from (8) we obtain the asymptotic value of the Lyapunov exponent γ (ω) ω Short wave asymptotics of the Lyapunov exponent λ λ (s ϱ s ϱ ) 2, ω. (23) (λ + λ ) 2 ϱ s Asymptotics of integrals in (4) for large values of ω can easily be found using integration by parts and the integrals in the appendix. Thus, si ϱ i K i, i =,, ω (24) λ s ϱ + λ s ϱ and c ω λ s ϱ + λ s ϱ, π λ + λ ω. (25) Evaluation of the integrals in (8) gives γ (ω) λ [( ) λ s ϱ /8 ( ) s ϱ /8 2, λ + λ s ϱ s ϱ ω. (26) Figure 3 shows typical dependence of the Lyapunov exponent γ on the frequency ω. It does not exhibit oscillating behaviour due to the infinite number of alternating rods (cf. [3). Unlike the random Schrödinger equation, waves in a random rod become more localized for high frequencies γ(ω) Figure 3. Dependence of the Lyapunov exponent for ϱ =.5,ϱ =, s =, s =.5,λ =.5,λ =. The dotted line denotes the asymptotic value (26) for ω, while the dashed parabola corresponds to (23) for ω. ω

6 44 Yuri A. Godin Downloaded By: [University of North Carolina, Charlotte At: 7:3 2 May Reflectionless propagation It is useful to rewrite formulas (23) and (26) using the impedance notation. By definition, the impedance Z i, i =,, of each rod is equal to Z i = ϱ i, i =,. (27) s i With this notation, asymptotic formulas for the Lyapunov exponent take the form γ (ω) ω2 4 λ λ (λ + λ ) 2 s 2 s2 ϱ s (Z Z ) 2, ω (28) and γ (ω) λ [( ) /8 ( ) /8 2 λ Z Z, ω. (29) λ + λ Z Z Then when the impedances of the two rods are equal, Z = Z, the Lyapunov exponent equals zero. In this case waves in the composite rod propagate without reflection. Indeed, consider an infinite elastic rod consisting of only two semi-infinite homogeneous elastic rods with parameters ϱ, s and ϱ, s (figure 4). Propagation of waves in both rods is described by the wave equation 2 u i t = 2 u i 2 v2 i x, v 2 i =, i =,, (3) ϱi s i for x > and x < with the the following boundary and initial conditions at x = u (, t) = u (, t), u (x, ) = f ( xv ), u (, t) s x u (x, ) t = u (, t) ; s x ) ( xv, = f t where u (x, t) = f (t x/v )isthe displacement in the left rod for x < and t. The solution of equations (3) for t > with conditions (3) have the form (3) u (x, t) = f (t x/v ) + Z Z f (t + x/v ), x <, Z + Z u (x, t) = 2Z f (t x/v ), x >. Z + Z Thus, waves in a coupled rod propagate without reflection in the case of matching impedances Z = Z. This result remains true for multi-coupled rods as well. Finally, we note that for the long waves the rod can be considered as homogeneous with effective parameters (22) if the propagation distance is significantly shorter than the localization length l. Equation (2) in this case can be approximated by the wave equation with effective (32) x= x ρ s ρ s Figure 4. An elastic rod composed of two semi-infinite perfectly connected rods with parameters ϱ, s and ϱ, s, where ϱ is the density and s is the elastic compliance, respectively.

7 Propagation of longitudinal waves 45 Downloaded By: [University of North Carolina, Charlotte At: 7:3 2 May 28 coefficients [ d 2 X ϱ s dx + 2 ω2 X =. (33) From here we derive the effective velocity v = (34) ϱ s which is related to the average wave velocity as follows v = vλ + vλ λ + λ (35) v =v + λ λ ( (λ + λ ) s s 2 Z ) 2v Z 2. (36) Therefore, the effective velocity is always less than the average one and can be equal in the case of matched impedances. 5. Conclusions We have considered propagation of longitudinal waves in an elastic composite rod consisting of rods with alternating properties and random length. The use of the phase-amplitude representation of the solution allows us to reduce the problem to a system of two linear equation with respect to the joint probability distribution function whose solution is found exactly. Then we calculate exactly the Lyapunov exponent (the inverse of the localization length) and explicitly find its short and long wave asymptotics. Analysis of the asymptotic formulas shows that longitudinal waves in a composite rod propagate without reflection in the case of matched impedances of the adjacent rods. The methods used in the paper also allow us to analyse the variance of fluctuations of the Lyapunov exponent. Acknowledgment The author has the pleasant duty to thank Prof. S.A. Molchanov for encouragement and valuable discussions. Appendix Below we give the values of some integrals used in the paper. η(ϑ) dϑ = π ( λ + λ ), ω s ϱ s ϱ where λ η(ϑ) = s cos 2 ϑ + ω 2 ϱ sin 2 ϑ + λ [ λ π = ω cot ϑ s arctan ϱ s 2 ω ϱ ϑ η(ϑ ) dϑ s cos 2 ϑ + ω 2 ϱ sin 2 ϑ. + λ ω ϱ s [ π 2 η(ϑ ) dϑ arctan cot ϑ ω s ϱ

8 46 Yuri A. Godin Downloaded By: [University of North Carolina, Charlotte At: 7:3 2 May 28 [ λ π = ω cot ϑ + arctan ϱ s 2 ω η(ϑ ) dϑ λ = [arctan ϑ ω cot ϑ ϱ s ω References + λ ω ϱ s [arctan cot ϑ ω s ϱ + λ ω ϱ s s arctan cot ϑ ϱ ω s [ π 2 ϱ s arctan cot ϑ s ϱ ω ϱ cot ϑ s + arctan ω ϱ [ Brillouin, L. and Parodi, M., 953, Wave Propagation in Periodic Structures (New York: Dover). [2 Lifshits, I.M., Gredescul, S.A., Pastur, L.A., 988, Introduction to the Theory of Disordered Systems (New York: John Wiley). [3 Sheng, P. et al., 99, Wave localization and multiple scattering in randomly-layered media, In: P. Sheng (Ed.) Scattering and Localization of Classical Waves in Random Media (Singapore: World Scientific). [4 Castanier, M.P. and Pierre, C., 995, Lyapunov exponents and localization phenomena in multi-coupled nearly periodic systems. Journal of Sound and Vibration, 83, [5 Bouzit, D. and Pierre, C., 2, Wave localization and conversion phenomena in multi-coupled multi-span beams. Chaos, Solitons Fractals,, [6 Kissel, G.J., 99, Localization factor for multichannel disordered systems. Physical Review A, 44, 8 4. [7 Hartman, P., 964, Ordinary Differential Equations (New York: John Wiley). [8 Benderskij, M.M. and Pastur, L.A., 97, On the spectrum of the one-dimensional Schrödinger equation with a random potential. Mathematics of the USSR: Sbornik,, [9 Pastur, L. and Figotin, A., 992, Spectra of Random and Almost Periodic Operators (New York: Springer- Verlag). [ Molchanov, S.A., 99, Ideas in the theory of random media, Acta Applicandae Mathematicae, 22,