INTRODUCTION
According to the universal conclusion of the scaling theory, it is well understood
that in OneDimensional (1D) disordered systems, all the elementary excitations
are localized in the Anderson sense. The destructive quantum interferences appear
to be the dominant mechanism in localizing the electronic eigenstates (Anderson,
1958; Abrahams et al., 1979; Sheng, 1979; Anderson et al., 1980;
Ya Azbel, 1983). It is clear that the disorder precludes the presence of longrange
propagation. However, relevant theoretical approaches have successfully examined
different ways in delocalizing the eigenstates indicating that disorder can
act also in creative fashion (Economou et al., 1988; Sanchez et al.,
1994). In particular a challenging scenario has been put forwards in binary
alloys to suppress localization allowing the propagation of waves: Namely correlation
in disorder. Originally introduced by Dunlap et al. (1990), the Random
Dimer Model (RDM) has been applied to various domains: Conducting polymers (Wu
et al., 1991a, b), semiconductor disordered superlattices (Diez et
al., 1995; Diez et al., 1996; Bentata et al., 2002) pointing
out the existence of truly extended states supported by experimental evidences
(Bellani et al., 1998; Kuhl et al., 2000). The key idea is that
the RDM within a short length correlation restores the tunnel effect and then
the necessary condition for delocalising the particle. However all this matter
holds only for the quantum case since the competition between destructive interference
and tunnel effect is the major cause leading to the localization or delocalisation
of the electronic eigenstates.
Although extensive interest has been devoted to the electronic Random Dimer Model (RDM) case (Huang et al., 2001; Sedrakyan, 2004; Zhang et al., 2006), very few has been done in the context of mechanical waves. Indeed such classical ones are good candidates to illustrate the modelengineering structures for a better observation of the Anderson Localization in 1D systems (He et al., 1986;
Richoux et al., 1999; Albuquerque et al., 2005; Sigalas et
al., 2005). This prompts us to examine the effect of the binary correlated
disorder via the random dimer model on the propagation of classical waves since
the 1D mechanical systems illustrate perfectly the analogy between electronwave
and classicalwave. In fact, the propagating medium is constituted by a large
string having negligible mass submitted to a uniform tension; A mass is linked
at each regular lattice point to a ground foundation by a spring forming a linear
oscillator unit cell. In this situation, the wave field consists of transverse
amplitudes along the string. It could be noticed that such system may be easily
experimented. In a perfect ordered system all the unit cells are identical while
for a disordered system the variables mass and/or spring are random. Here we
introduce the dimer effect by assuming a concentration of two successive identical
cells at random through a host lattice of identical cells. Typically it corresponds
to a random KronigPenney (KP) binary alloy with dimer. It is expected that
such system restores the existence of extended modes in the conventional sense
(Sanchez et al., 1994).
Theoretical model: A semiinfinite tight string with homogeneous density ρ is submitted to a uniform tension T_{0}. The string is formed by a large number of subsystems at each lattice discrete points x_{n} = nd, d denoting the lattice spacing. Each subsystem is a harmonic oscillator for which the mass M_{n} is connected to a grounding rigid foundation by a spring having a linear stiffness K_{n} (Fig. 1).
In a perfect analogy with the electronic KP model (Sanchez et al., 1994), namely:
we are interested by the propagation of transverse wave in the vertical plane. The transverse displacement y at the longitudinal coordinate x is solution of the equation of motion:

Fig. 1: 
The
host and dimer unit cell in the binary correlated disordered one dimensional
mechanical system 
with
k and v_{φ} denoting the wave vector and the wave (or sound) velocity through the whole system, respectively while ω is the incident frequency.
The term λ_{n} related to the vibration mode at each delta peak is given by:
where
Physically Ω_{n} defines the free frequency of the nth subsystem while the parameter λ_{n}≡ λ_{n} (ω) has the meaning of effective potential delta peak strength since it depends on both ω and the media unit cell intrinsic parameters.
In this study the mechanical randomness may be introduced in different ways: Disorder in mass and/or stiffness, referred to the cellular disorder. Thus in this description, the masses are statistically independent variables given by a common probability distribution.
In the region nd ≤ x ≤ (n +1) d, the solution of Eq. 2 is a superposition of forward and backward scattering waves:
A_{n} and B_{n} denote the amplitude coefficients in the nth region. Introducing the reflection the transmission amplitudes r_{N} and t_{N} of the system, y(x) satisfies the limit conditions:
where L = Nd is the system size. The amplitudes A_{n} and B_{n} through the initial and final amplitudes can be linearly expressed using boundary conditions giving out the total transfer matrix M(L, 0) of the system:
where S (L, 0) refers to the total scattering matrix.
This allows one to determine the transmission coefficient, which is the fundamental physical quantity of interest:
Once the transmission coefficient is known, the nature of the propagating modes may be characterized by the reduced Lyapunov coefficient given by the ratio:
where ξ represents the localization length.
RESULTS AND DISCUSSION
Here, we first deal with the ordered case as a reference with the aim to show how the deterministic principal physical magnitudes can be achieved. Disorder is then considered by means uncorrelated random binary mass distribution. Finally correlated case is examined with an interesting manner to construct the conventional dimer unit cell.
The propagating wire media is characterized with a mass density ρ = 5 kg m^{–1} submitted to a tension T_{0} = 10N with the oscillator unit cell length d = 5.10^{2}. The above parameters have been chosen according to the available data in the literature (Richoux et al., 1999). We have studied rather large systems up to N = 1000 unit cells, i.e., 50 m length, to get convincing observation of the mechanical Anderson localization. All statistical averaged procedures are taken with a satisfactory convergence limit.
Ordered case: The system is constituted by identical oscillator cells
formed by a mass M_{0} = 0.100 kg with a free frequency Ω_{0}
= 40 rd s^{–1}. The corresponding analytical KP formula a (ω)
provides the principal frequency for
which and the band edges ω_{lower} ≈ 21 rd s^{–1}
and ω_{upper} ≈ 21 rd s^{–1} by the condition a
(ω) = 1 as shown in Fig. 2. The symbol *
denotes the absolute value.
The nature of the vibration modes can also be characterized with the reduced Lyapunov coefficient as depicted in Fig. 2. The criterion indicating the existence of forbidden or allowed bands may be simply formulated through of the ratio
One has to note the existence of a divergence in the spatial extent ξ at the singular mode Ω_{0}. This feature occurs only for the first allowed band and has not quantum equivalent in the ordered case (Sanchez et al., 1994). Indeed as the frequency ω reaches the free frequency Ω_{0}, the reduced Lyapunov coefficient behaves like:
with the critical exponent v = 1.99 ±0.06.
Moreover one has to note the existence of a spectacular phenomenon in T(L) at the free frequency Ω_{0}. Indeed the transmission coefficient reaches it maximum unity value, independently from the system length since the effective potential profile vanishes at Ω_{0}.
Consequently, the wave propagates as freely as possible along an effective empty wire, i.e., λ_{n}(Ω_{0})_{n=1, N} = 0, settling down a very interesting ballistic regime.

Fig. 2:  Reduced Lyapunov coefficient and the absolute KronigPenney function for the
perfectly ordered host lattice (M_{0 }= 0.100 kg, Ω_{0} =40 rd s^{–1}, 
In order to appreciate more deeply the spatial coherence of a diffusive propagating wave, the variation of transmission coefficient T (N)versus the length of the system size L up to 2500 is also depicted in Fig. 3 for an allowed frequency, i.e., ω = 35 rd s^{–1}. Indeed, the envelope function is constituted by a periodic superposition of forward and backward waves with uniform amplitudes. In the corresponding deterministic ordered limit, this feature is a signature of the obvious Bloch extended nature of the allowed modes. Furthermore, the envelope amplitude and its period depend strongly on the frequency of the allowed vibration mode. Stationary waves occur whenever the system length becomes a multiple of its half incident wavelength:
as well known.
Binary disordered case: In this section, the effects of the binary disorder on the nature of the vibration modes are examined. Binary disorder is introduced by assuming that the host and impurity masses M_{A} and M_{B}, respectively are statistically independent random variables given by a binary alloy distribution:
c_{A} = 1c_{B} and c_{B} represent their corresponding concentrations.
As did in the previous ordered case, the impurity oscillator unit cell will be characterized with the free and principal frequencies, i.e., Ω_{B} and , respectively. With this in mind, the corresponding averaged frequency responses <T(ω)> and are investigated in Fig. 4 and 5. The uncorrelated disorder is especially considered for the comparison with the electronic case (Sanchez et al., 1994).
The uncorrelated case: The propagating properties are described within a statistical procedure. A satisfactory convergence for 10^{4} random samples has been checked up getting with an appropriate tolerance the averaged principal physical magnitude <T (ω)>. This statement is necessary to determine with enough accuracy all the other quantities of interest such the size dependence of the transmission coefficient.

Fig. 3:  Transmission coefficient T(N) versus system length for different allowed frequencies at the perfectly ordered case (ω = Ω_{0} and ω = 35.00 rd s^{–1}) 

Fig. 4:  Averaged transmission frequency response <T (ω)> versus frequency for binary correlated (black color) and uncorrelated (grey color) cases with the parameters: Host lattice: M _{A} = 0.100 kg, Ω _{A} = 40. 00 rd s ^{–1}, =43. 20 rd s ^{–1}. Impurity sub lattice: M _{B} = 0.060 kg, Ω _{B} = 10.75 rd s ^{–1}, =40.00 rd s ^{–1}, c _{B} = 0.4 
In this part, we deal with the oscillator binary disorder when the impurity and host unit cells are built with different masses M_{B}≠M_{A} and different springs k_{B}≠k_{A} leading to different free frequencies Ω_{B}≠Ω_{A}. For instance, the impurity unit cell is characterized with the Ω_{B}= 10.75 rd s^{–1} and M_{B} = 0.060 kg unit cell while the host lattice is defined with the unit cell Ω_{A} = 40.0 rd s^{–1} and M_{A} = 0.100 kg.

Fig. 5: 
Reduced Lyapunov frequency response versus frequency for binary correlated (black color) and uncorrelated (grey color) cases with the parameters: Host lattice: M _{A} = 0.100 kg, Ω _{A} = 40.00 rd s ^{–1} and =43. 20 rd s ^{–1}. Impurity sub lattice: M _{B} = 0.060 kg, Ω _{B} = 10.75 rd s ^{–1}, =40.00 rd s ^{–1}, c _{B} = 0.4 
The commute resonance ω_{c}: In contrast with the quantum electronic model (Sanchez et al., 1994), the classical uncorrelated delta peak binary disorder presents constructive effects around the frequency ω_{c} ≈ 10^{3} with the existence of a set of extremely delocalized states:
From an analytical point of view and taking the advantage of the δfunction limit, the wave propagation equation can also be achieved within the Poincaré map representation (Bellisard et al., 1982), which in turns enables one to relate the displacements at successive lattice points. Indeed defining y_{n} ≡ y (x = n^{+}d), Eq. 1 may be exactly transformed into a recursive site description:
where R_{n} (ω) is the translating matrix and
is the standard KP formula yielding the frequency spectrum corresponding to the delta peak strength λ_{n}.
This statement may be understood from an analytical consideration using the on site description given in Eq. 14 and 15. The corresponding translating matrices associated to the host and impurity unit cells are defined by:
and
In particular, at ω_{c}, the two corresponding KP formulas cross over, i.e., a_{A} (ω_{c}) = a_{B} (ω_{c}). Hence, the resulting matrix elements become identical and consequently, R_{A} (ω_{c}) and R_{B} (ω_{c}) commute. From a phenomenological point of view, the incident propagating wave does not distinguish the host unit cells from impurity ones since they present the same local translating attitudes. The propagating wire is felt as a perfect ordered lattice with constant effective delta peak strength λ_{c} = λ_{A} (ω_{c}) = λ_{B} (ω_{c}). Such a characteristic frequency (ω_{c}) referred as the commute frequency, can be determined analytically, from the conditions:
At this resonant vibrating mode ω_{c} = 61.86 rd s^{–1}, the two indiscernible unit cells, present an additive behaviour, giving rise to deterministic features. This conclusion is in agreement with the correlated electronic case reported by Hilke et al. (1997), Hakobyan et al. (2000), Bentata et al. (2001) and Bentata (2005) where the origin of such resonance is related to the commuting properties of the binary individual unit cells. We have also to notice that the existence of the set of extended states in a mini band is due to the nonabrupt character of the corresponding transition since the recursive matrix elements get still very close together i.e., a_{A} (ω_{c}) = a_{B} (ω_{c}) around the discrete resonance ω_{c}.
The binary correlated case. A shortrange correlation in such binary disorder is considered by introducing at random the impurity oscillators by pairs without any aggregates. As well known from the electronic case, the conventional dimer resonance can be realized with the total transparency of the dimer unit cells within the host allowed band. In other words, the dimer resonance happens when the principal impurity frequency j_{B} belongs to the host allowed band.
The ballistic dimer resonance Ω_{A}: The new interesting case we examine in our study concerns the conventional dimer effects when the impurity principal frequency coincides with the host free frequency:
For the above parameters, this condition is verified. So the corresponding averaged responses < T (ω) > and of such correlated case is reported in Fig. 4 and 5.
The commute resonance still survives at ω_{c} while the dimer resonance appears at the free frequency Ω_{A} with more singular and larger localization length, i.e., ξ (Ω_{A})_{cor} ≈ 5.10^{6} ξ (Ω_{A})_{uncor}. The two different resonances Ω_{0} and ω_{c} ≈ 61,86 rd s^{–1} appear simultaneously within the same allowed band providing attractive various ways to propagate incident waves. Moreover the dimer resonance Ω_{A} presents a challenging advantage since it considerably improves the magnitude of the localization length, i.e., ξ (Ω_{A}) ≈ 1000 ξ (ω_{c}).
The nature of the resonant vibration modes is studied with the description of the transmission coefficient T(N) versus the system length. As depicted in Fig. 6, the presence of periodic envelope function with uniform amplitude similar to an allowed vibration in the ordered case justifies the extended Bloch wave character at the commute resonance ω_{c}. Indeed, the incident wave does not distinguish the host unit cells from the separately impurity one within the dimer unit cells.
In the other hand, the dimer resonance at the free frequency is similar to the ballistic resonance in ordered case since T(N) gets the unity transmission value independently from the system length. In fact with the conventional dimer statements at Ω_{A}, the dimer unit cells becomes totally transparent as well as all of the host unit cells where the effective potential profile vanishes over whole of the lattice sites. Consequently the incident wave does not discern again host elements from the dimer unit cells within a deterministic transparent propagating media.

Fig. 6:  Averaged Transmission Coefficient T(N) versus system length for the correlated case at (ω = Ω_{A} and ω = ω_{c} = 61. 86 rd s^{–1}) 
In this case, the dimer resonance seems to be equivalent to the commute one suggesting the existence of Bloch extended vibration modes at the free frequency Ω_{A}. This controversy finding contradicts the general conclusion in the transport properties in one dimensional short range correlated disorder (Huang et al., 1997; Hilke et al., 1998).
CONCLUSION
With the aim to observe the phenomenological aspects of the Anderson localization. the propagation of mechanical waves in random media has been studied by using an analogy with the well known electronic disordered KronigPenney model. In this description, two particular frequencies characterize the corresponding ordered case: The principal frequency vanishes the KronigPenney analytical equation, i.e., a (j) = 0_while at the free frequency Ω_{0}, the ballistic regime settles down i.e., λ_{n} (Ω_{0}) = 0 Singular behavior happens around the free frequency Ω_{0} since the spatial extent length diverges pointing out the Bloch wave functions.
Dimer unit cells can be constructed with a conventional manner or with a new interesting way that enhances the transport properties and ensures the deterministic potential profile even in presence of short range correlated binary disorder: At the ballistic dimer resonance, the Bloch extended vibration modes can be restored in controversies with the general belief in onedimensional disordered systems.
To conclude, we have studied for the first time numerical results describing the ballistic dimer resonance in the random dimer effect within classical mechanic situation. At this stage, such model presents the main advantage to be checked experimentally within a rather simple method to conceive ballistic mechanical filters with high quality responses.
ACKNOWLEDGMENT
Authors thank A. Brezini for helpful discussions during his passage at LPTPM.