Emergence of acoustic and optical bands in elastic systems A. Dıaz-de-Anda and J. Flores Instituto de Fısica, Universidad Nacional Aut onoma de M exico, P.O. Box 20-364, 01000 M exico, Distrito Federal, Mexico
rrez and R. A. Me ndez-Sa ncheza) L. Gutie Instituto de Ciencias Fısicas, Universidad Nacional Aut onoma de M exico, P.O. Box 48-3, 62251 Cuernavaca, Mor., Mexico
G. Monsivais Instituto de Fısica, Universidad Nacional Aut onoma de M exico, P.O. Box 20-364, 01000 M exico, Distrito Federal, Mexico
A. Morales Instituto de Ciencias Fısicas, Universidad Nacional Aut onoma de M exico, P.O. Box 48-3, 62251 Cuernavaca, Mor., Mexico
(Received 7 May 2013; revised 16 August 2013; accepted 15 October 2013) Two elastic systems are considered in this work: A special linear chain of harmonic oscillators and a quasi one-dimensional vibrating rod. Starting in both cases with a locally periodic system formed by unit cells with a single element, these cells are converted into binary cells. The acoustic and optical bands then appear. For the vibrating rod experimental values are compared with theoretical results; in particular, the normal-mode amplitudes are obtained and the agreement is excellent. C 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4828822] V PACS number(s): 43.40.At, 43.20.Ks, 43.40.Cw, 43.20.Ye [ANN]
I. INTRODUCTION
One-dimensional (1-D) periodic systems are of interest in several areas of acoustics and vibrations due to their multiple applications. Probably the most important one is that they form a cascade of filters.1 Other applications as wave guiding,2 transduction,3 matching,4 auditory filters,5 and transmission6–9 can be found in the literature. Also, periodic systems can be used to create an acoustic time reversal mirror.10,11 To understand the properties of 1-D periodic systems, it is assumed that identical unit cells oscillate in a translationally invariant system. Classical wave systems, as chains of springs and masses, are frequently dealt with for this question12 and other condensed-matter problems.13 In order to study several phenomena in these kinds of cascade wave systems, the composite rod shown in Fig. 1(a), formed by N elastic rods separated by notches of equal length e, has been considered for different sets of lengths fdi g, i ¼ 1; …; N. For the locally periodic rod, in which case di ¼ d, torsional and compressional waves were analyzed by Morales et al.,14 and flexural vibrations by Dıaz-de-Anda et al.15 It was shown that, as N grows, allowed and forbidden (stop) bands appear. On the other hand, if di ¼ d=ð1 þ icÞ the elastic analog of the Wannier-Stark ladders is obtained.16,17 These ladders have two important properties: (1) the frequency spectrum is equispaced and (2) the wave amplitudes are localized. These properties can be used to localize eigenmodes, equivalent to “hot spots,”18 of different frequencies in a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
J. Acoust. Soc. Am. 134 (6), December 2013
Pages: 4393–4400
different parts of the elastic system and vice versa. Another two interesting cases of the system shown in Fig. 1(a) are the following: The case N ¼ 2 with d1 d2 shows the elastic doorway state phenomenon,19 in which one part of the rod acts as a “doorway” to excite the waves in the system. Another case of interest is obtained when di is a set of independent random numbers and the Anderson localization effect for closed systems appears, since all wave amplitudes show a localization length.20 It should be mentioned that in all the cases of elastic vibrations considered up to now,14–16,19,20 the normal-mode frequencies as well as the amplitudes have been measured and the experimental values agree very well with the numerical calculations. In this paper, the torsional waves of a rod formed by unit cells consisting of binary elements is discussed, that is, di ¼ d h for i odd and di ¼ d þ h for i even. This is a problem of recent interest in several areas like molecular physics,21,22 granular crystals,23,24 phononic crystals,25,26 nonlinear systems,27,28 among others. The analysis is performed both numerically and from the experimental point of view. The application of this system in acoustic and mechanical filter theory is straightforward. To start with the locally periodic rod with N identical unit cells, that is, with h ¼ 0, as depicted in Fig. 1(b), is considered. Here local periodicity means that the system has only a relatively small number N of repeating elements. The two rods at the extreme left-hand side of the system are then changed and the rod in Fig. 1(c) is obtained. When four rods are modified, the system shown in Fig. 1(d) is gotten. Finally, the locally periodic system with a binary unit cell of Fig. 1(e) is formed. As the number of binary cells grows, and a locally invariant system is recovered, the allowed bands, except for the lowest band,
0001-4966/2013/134(6)/4393/8/$30.00
C 2013 Acoustical Society of America V
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In Fig. 2(a) a locally periodic system formed by n blocks, each consisting of N1 hard springs of constant k1 and each block separated by q softer springs of constant k is shown. When N1 1 this system behaves identically to the rod of Fig. 1(b) for compressional waves of small amplitudes. This was verified by comparing the normal-mode amplitudes and frequencies for the rod without notches. In Fig. 2(b) two blocks are modified in such a way that one block is formed by N1 l and the other by N1 þ l springs. This process is repeated until the local periodicity is recovered with a binary unit cell as shown in Fig. 2(d). For these systems of harmonic oscillators, normal modes are discussed numerically, while for the system formed by rod torsional waves are also studied experimentally. The spectra and the wave amplitudes are obtained by means of the experimental setup described in Sec. III B. II. A BINARY CHAIN OF HARMONIC OSCILLATORS FIG. 1. The rod in (a) consists of N unit cells, each of length di and radius R, and separated by a notch of length e and radius r < R. (b) The locally periodic rod with N identical unit cells, that is, with h ¼ 0, is shown; (c) and (d) two and four bodies, respectively, are modified. (e) The local periodicity is recovered but with a binary unit cell.
break into two different classes. The lower frequency states form the analog of what is called in solid-state physics the acoustic band, and the states corresponding to higher frequencies form the optical spectrum. Although the theoretical description of compressional and torsional waves is very similar, the experimental method used here is simpler when measuring torsional vibrations with free ends. The emergence of the acoustic and optical bands is also considered for another system, the one-dimensional harmonic oscillator chain, shown in Fig. 2 and discussed in Sec. II. This classical mechanics model for a diatomic chain has been discussed in many papers for several decades.29–32
Before analyzing the properties of the systems shown in Fig. 2, the construction of the chain of Fig. 2(a) from the homogeneous chain, in which all masses and springs are equal to each other, will be discussed first. When the N particles have the same mass m and are coupled by a set of springs with constants ki , the normal modes can be obtained by diagonalizing the tridiagonal matrix H with elements Hi; j ¼ ðki =mÞ di; j1 þ ððki þ kiþ1 Þ=mÞ di; j ðkiþ1 =mÞ di; jþ1 ;
(1)
where di; j is the Kronecker delta. Since large values of N will be used in this work, numerical instabilities can occur, in particular for small values of the non-diagonal matrix elements. Therefore, it is appropriate to have a case whose solution is known in closed form, so that the numerical calculations can be verified.
FIG. 2. (Color online) (a) A locally periodic system formed by n blocks, each consisting of N1 stiff springs of constant k1 , and separated by a set of q softer springs of constant k, is shown. (b) and (c) Two and four blocks, respectively, are converted into blocks formed by N1 l and N1 þ l springs. (d) The local periodicity is recovered but with a binary unit cell. 4394
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Dıaz-de-Anda et al.: Emergence of acoustic and optical bands
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Indeed, the locally periodic chain in which ki ¼ k provides this case since it was solved by Lagrange more than two centuries ago, as can be read in his fundamental book Mecanique Analytique published in Paris in 1788.33 As is well known, the normal-mode frequencies xJ , J ¼ 1; …; N, must be such that the N N determinant DN of the matrix ðH x2 IÞ be zero; here I is the N N unit matrix. For the homogeneous chain, DN obeys the recurrence relation DN ¼ cDN1 DN2 ;
(2)
where c ¼ 2 ð4x2 =x2c Þ. Equation (2) defines the well known Chebyshev polynomial of the second kind:34 DN ¼
½N=2 X
ð1Þ
i¼0
¼
i
Ni i
ð2xÞN2i
sinððN þ 1Þ arccosðc=2ÞÞ : sinðarccosðc=2ÞÞ
(3)
(4)
Here ab is the binomial coefficient and ½a is the largest integer not greater than a. In the Appendix the solution, derived by the authors, to a more general recurrence relation than Eq. (2) is presented in Eq. (A2), which reduces in the case considered here to Eq. (3). Setting DN ¼ 0, the values of xJ can be obtained from Eq. (4) in closed form: xJ ¼ 2x0 sin
Jp : 2ðN þ 1Þ
(5)
As it is very well known and can be seen from this formula, the homogeneous chain is a high frequency filter, that is, all normal mode frequencies are less than xC . All normal modes are extended in the sense that in general all particles move from their equilibrium position. This is so since the eigenvectors XiJ of the matrix H are different from zero for all values of i ¼ 1; …; N. When a single defect is introduced, one normal-mode frequency exceeds xC and the corresponding eigenvector is localized.35 By the same token the chain with N1 defects has N1 localized states with frequencies xJ > xC . As a first step to constructing the binary chain of harmonic oscillators shown in Fig. 2(d), one set of N1 springs with a constant k1 is introduced in the homogeneous system. Another set of N1 defects are then introduced, causing further N1 states to be localized and have a frequency larger than the critical one. This procedure is continued until a locally periodic system of n blocks each with N1 defects, is constructed as shown in Fig. 2(a). All states are no longer localized and a band spectrum emerges. Since, as has been mentioned before, when k1 k each block behaves as a continuous rod in compressional oscillations, the above results have been established in Ref. 14. What has been done here is to destroy, by including defects, the original invariance with respect to translations that the locally periodic chain has and recover the periodicity again, but now the system is formed by n identical blocks coupled by softer springs. The systems of Fig. 2 will now be analyzed numerically. All the results of this section were obtained for n ¼ 24 J. Acoust. Soc. Am., Vol. 134, No. 6, December 2013
FIG. 3. (Color online) The two blocks at the left-hand extreme have been modified as indicated in Fig. 2(b). The states n þ 1 and 2n have frequencies in the forbidden gap and have localized amplitudes as indicated in the insets. Here XiJ are the amplitudes of the ith mass in the Jth normal-mode.
blocks, N1 ¼ 50, and N ¼ 1249 masses. The values k1 ¼ 4000 kg/s2 much larger than k ¼ 100 kg/s2 and m ¼ 0:01 kg were used. The construction of the binary chain, shown in Fig. 2(d), will now be considered. In Fig. 2(b) the two sets at the extreme left of the locally periodic chain of Fig. 2(a) are changed, so that they now have N1 l and N1 þ l springs, respectively. Using perturbation theory it is easy to see that all the normal-mode frequencies of the lowest band, shown in Fig. 3, are only slightly altered. On the second band, however, the states marked n þ 1 and 2n jump into the forbidden band. As given in the two insets of this figure, the corresponding normal-mode amplitudes are localized. When the spring system of Fig. 2(c) is analyzed, four states are localized. Finally, when all bodies are modified, the local periodicity of Fig. 2(d) is recovered but with a binary unit cell. As is shown in Fig. 4, the second band is broken into two bands: the acoustic and the optical bands. As exemplified in the insets of this figure, all states are now extended.
FIG. 4. (Color online) When the local periodicity is recovered the acoustic and optical bands appear and all states are extended. Dıaz-de-Anda et al.: Emergence of acoustic and optical bands
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Until now, what could be called Variations on a theme by Lagrange has been solved numerically. The “theme by Lagrange” is the homogeneous chain of harmonic oscillators. Measuring the time displacements of many particles subject to different springs is, however, a tough problem, which can become very cumbersome. Therefore, to perform experiments, the oscillator chain will be replaced by the elastic rod with notches shown in Fig. 1(a).
the wave amplitudes u at cylinder j, 1 < j < N. The torque and the wave amplitude can be written as uðxÞ A j ¼ W ðxÞ ; j Cj u0 ðxÞ Bj where Wj ðxÞ ¼
III. THEORETICAL AND EXPERIMENTAL METHODS
In this section the methods used to compute and measure normal-mode frequencies and amplitudes of the vibrating rod of Fig. 1(a) will be presented. A. Theoretical considerations
The Poincare method36 used to solve compressional and torsional vibrations in a rod with notches is summarized, only for torsional waves, as follows. A rod formed by N coaxial cylinders shown in Fig. 1(a) is considered. The solution of the torsional wave equation for each cylinder is given by uðxj1 x xj Þ ¼ Aj eijx þ Bj eijx ;
0 B Mðj þ 1Þ ¼ @
cosðjðxjþ1 xj ÞÞ jCjþ1 sinðjðxjþ1 xj ÞÞ
eijx iCj eijx
eijx : iCj eijx
(8)
Here Cj ¼ pa4j Gj =2 is the torsional rigidity with Gj and aj the shear modulus and area of cylinder j, respectively. The continuity conditions of the amplitude and torque can be written as Aj Ajþ1 Wj ðx ¼ xj Þ ¼ Wjþ1 ðx ¼ xj Þ : (9) Bj Bjþ1 Evaluating the last equation in x ¼ xj , and multiplying by Mjþ1 ¼ Wjþ1 ðx ¼ xjþ1 Þ W1 jþ1 ðx ¼ xj Þ, one gets
(6)
where j is the wave number. The Poincare map method consists of finding a recurrence relation, i.e., a map, that yields
(7)
uðx ¼ x Þ uðx ¼ xjþ1 Þ j ¼ Mjþ1 : 0 Cjþ1 u ðx ¼ xjþ1 Þ Cj u0 ðx ¼ xj Þ
(10)
The matrix Mjþ1 for the rod of Fig. 1(a) is given by
1 sinðjðxjþ1 xj ÞÞ C A: jCjþ1 cosðjðxjþ1 xj ÞÞ
(11)
From the first line of the matrix (10) with j þ 1 ! j it is obtained M11 ðjÞ uðxj Þ uðxj1 Þ; Cj1 u ðxj1 Þ ¼ M12 ðjÞ M12 ðjÞ 1
0
(12)
M22 ðNÞ uðxN Þ uðxN1 Þ ¼ 0: M12 ðNÞ
M22 ðjÞ M21 ðjÞ M12 ðjÞ uðxj Þ þ uðxj1 Þ M12 ðjÞ M12 ðjÞ
M11 ðjÞ M22 ðjÞ uðxj1 Þ; M12 ðjÞ
(13)
and since detðMÞ ¼ 1, Cj u0 ðxj Þ ¼
M22 ðjÞ 1 uðxj Þ þ uðxj1 Þ: M12 ðjÞ M12 ðjÞ
J. Acoust. Soc. Am., Vol. 134, No. 6, December 2013
(15)
(16)
Since this equation depends on the wave number j, the normal-mode values are those that satisfy Eq. (16). Once they are obtained, the uðxj Þ are calculated again so the coefficients Aj and Bj can be evaluated. The problem is therefore solved.
(14)
Finally, the Poincare map, which relates uðxjþ1 Þ to uðxj Þ and uðxj1 Þ, can be obtained substituting the last equation in the first line of Eq. (10): 4396
M12 ðj þ 1Þ uðxj1 Þ: M12 ðjÞ
The boundary conditions are now imposed. For a rod with free ends CN u0 ðxN Þ ¼ 0, which implies
and using the second line of Eq. (10), it follows Cj u0 ðxj Þ ¼
M12 ðj þ 1Þ M22 ðjÞ uðxjþ1 Þ ¼ M11 ðj þ 1Þ þ uðxj Þ M12 ðjÞ
B. Experimental method
To perform the measurements, the electromagneticacoustic transducer (EMAT) developed in Refs. 14 and 15 was used. The EMAT consists of a permanent magnet and a Dıaz-de-Anda et al.: Emergence of acoustic and optical bands
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coil, and can be used either to detect or excite the oscillations. The transducer operates through the interaction of eddy currents in the metallic rod with a permanent magnetic field. According to the relative position of the magnet and the coil, the EMAT can either excite or detect selectively compressional, torsional, or flexural vibrations. Used as a detector, the EMAT measures acceleration. This transducer has the advantage of operating without mechanical contact with the rod, which is crucial to avoid perturbing the shape of the localized wave amplitudes. Both the detector and exciter are moved automatically along the rod axis by a computerized control system and then the wave amplitudes can be measured easily (see Fig. 5). Whenever the wave amplitude is measured, it is necessary to keep the system at resonance, so the wave generator must be as stable as possible. In order to achieve this, a Stanford Research Systems DS 345 wave generator with an ovenized time base with stability > s1 < = ð‘ þ 1Þ i‘ C YB im C B N i1 @ A @ A ‘¼2 > > : ‘¼3 m¼2 ; i1 i‘ Ps1 s N ðmþ1Þ im Y i m¼1 p‘‘1 ; (A2) p1 0
‘¼2
where 2
3 j1 X 6N ð‘ þ 1Þ i‘ 7 6 7 4 5 ‘¼1 N ; ij ¼ jþ1
j ¼ 1; …; s 1;
(A3)
which define what could be called generalized Chebyshev polynomials. Here ½a indicates the largest integer not greater than a and mn is a binomial coefficient. By convention, all sums and products in which the upper index is less than the lower index are zero and one, respectively. For s ¼ 2, the s 1 summations in Eq. (A2) reduce to a single sum over the index i1 , which runs from 0 to ½N=2. All binomial coefficients, except for the second one, become unity. As a consequence, for s ¼ 2, Eq. (A2) leads immediately to Eq. (3) with p1 ¼ 2x and p2 ¼ 1. Another particular case of Eq. (A2) is obtained when p1 ¼ p2 ¼ 1: the Fibonacci sequence arises. And not only this, for pi ¼ 1 and s ¼ n > 2, the Fibonacci sequence of order n is obtained. 1
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Dıaz-de-Anda et al.: Emergence of acoustic and optical bands
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rrez and R. A. Me ndez-Sa ncheza) L. Gutie Instituto de Ciencias Fısicas, Universidad Nacional Aut onoma de M exico, P.O. Box 48-3, 62251 Cuernavaca, Mor., Mexico
G. Monsivais Instituto de Fısica, Universidad Nacional Aut onoma de M exico, P.O. Box 20-364, 01000 M exico, Distrito Federal, Mexico
A. Morales Instituto de Ciencias Fısicas, Universidad Nacional Aut onoma de M exico, P.O. Box 48-3, 62251 Cuernavaca, Mor., Mexico
(Received 7 May 2013; revised 16 August 2013; accepted 15 October 2013) Two elastic systems are considered in this work: A special linear chain of harmonic oscillators and a quasi one-dimensional vibrating rod. Starting in both cases with a locally periodic system formed by unit cells with a single element, these cells are converted into binary cells. The acoustic and optical bands then appear. For the vibrating rod experimental values are compared with theoretical results; in particular, the normal-mode amplitudes are obtained and the agreement is excellent. C 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4828822] V PACS number(s): 43.40.At, 43.20.Ks, 43.40.Cw, 43.20.Ye [ANN]
I. INTRODUCTION
One-dimensional (1-D) periodic systems are of interest in several areas of acoustics and vibrations due to their multiple applications. Probably the most important one is that they form a cascade of filters.1 Other applications as wave guiding,2 transduction,3 matching,4 auditory filters,5 and transmission6–9 can be found in the literature. Also, periodic systems can be used to create an acoustic time reversal mirror.10,11 To understand the properties of 1-D periodic systems, it is assumed that identical unit cells oscillate in a translationally invariant system. Classical wave systems, as chains of springs and masses, are frequently dealt with for this question12 and other condensed-matter problems.13 In order to study several phenomena in these kinds of cascade wave systems, the composite rod shown in Fig. 1(a), formed by N elastic rods separated by notches of equal length e, has been considered for different sets of lengths fdi g, i ¼ 1; …; N. For the locally periodic rod, in which case di ¼ d, torsional and compressional waves were analyzed by Morales et al.,14 and flexural vibrations by Dıaz-de-Anda et al.15 It was shown that, as N grows, allowed and forbidden (stop) bands appear. On the other hand, if di ¼ d=ð1 þ icÞ the elastic analog of the Wannier-Stark ladders is obtained.16,17 These ladders have two important properties: (1) the frequency spectrum is equispaced and (2) the wave amplitudes are localized. These properties can be used to localize eigenmodes, equivalent to “hot spots,”18 of different frequencies in a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
J. Acoust. Soc. Am. 134 (6), December 2013
Pages: 4393–4400
different parts of the elastic system and vice versa. Another two interesting cases of the system shown in Fig. 1(a) are the following: The case N ¼ 2 with d1 d2 shows the elastic doorway state phenomenon,19 in which one part of the rod acts as a “doorway” to excite the waves in the system. Another case of interest is obtained when di is a set of independent random numbers and the Anderson localization effect for closed systems appears, since all wave amplitudes show a localization length.20 It should be mentioned that in all the cases of elastic vibrations considered up to now,14–16,19,20 the normal-mode frequencies as well as the amplitudes have been measured and the experimental values agree very well with the numerical calculations. In this paper, the torsional waves of a rod formed by unit cells consisting of binary elements is discussed, that is, di ¼ d h for i odd and di ¼ d þ h for i even. This is a problem of recent interest in several areas like molecular physics,21,22 granular crystals,23,24 phononic crystals,25,26 nonlinear systems,27,28 among others. The analysis is performed both numerically and from the experimental point of view. The application of this system in acoustic and mechanical filter theory is straightforward. To start with the locally periodic rod with N identical unit cells, that is, with h ¼ 0, as depicted in Fig. 1(b), is considered. Here local periodicity means that the system has only a relatively small number N of repeating elements. The two rods at the extreme left-hand side of the system are then changed and the rod in Fig. 1(c) is obtained. When four rods are modified, the system shown in Fig. 1(d) is gotten. Finally, the locally periodic system with a binary unit cell of Fig. 1(e) is formed. As the number of binary cells grows, and a locally invariant system is recovered, the allowed bands, except for the lowest band,
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C 2013 Acoustical Society of America V
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In Fig. 2(a) a locally periodic system formed by n blocks, each consisting of N1 hard springs of constant k1 and each block separated by q softer springs of constant k is shown. When N1 1 this system behaves identically to the rod of Fig. 1(b) for compressional waves of small amplitudes. This was verified by comparing the normal-mode amplitudes and frequencies for the rod without notches. In Fig. 2(b) two blocks are modified in such a way that one block is formed by N1 l and the other by N1 þ l springs. This process is repeated until the local periodicity is recovered with a binary unit cell as shown in Fig. 2(d). For these systems of harmonic oscillators, normal modes are discussed numerically, while for the system formed by rod torsional waves are also studied experimentally. The spectra and the wave amplitudes are obtained by means of the experimental setup described in Sec. III B. II. A BINARY CHAIN OF HARMONIC OSCILLATORS FIG. 1. The rod in (a) consists of N unit cells, each of length di and radius R, and separated by a notch of length e and radius r < R. (b) The locally periodic rod with N identical unit cells, that is, with h ¼ 0, is shown; (c) and (d) two and four bodies, respectively, are modified. (e) The local periodicity is recovered but with a binary unit cell.
break into two different classes. The lower frequency states form the analog of what is called in solid-state physics the acoustic band, and the states corresponding to higher frequencies form the optical spectrum. Although the theoretical description of compressional and torsional waves is very similar, the experimental method used here is simpler when measuring torsional vibrations with free ends. The emergence of the acoustic and optical bands is also considered for another system, the one-dimensional harmonic oscillator chain, shown in Fig. 2 and discussed in Sec. II. This classical mechanics model for a diatomic chain has been discussed in many papers for several decades.29–32
Before analyzing the properties of the systems shown in Fig. 2, the construction of the chain of Fig. 2(a) from the homogeneous chain, in which all masses and springs are equal to each other, will be discussed first. When the N particles have the same mass m and are coupled by a set of springs with constants ki , the normal modes can be obtained by diagonalizing the tridiagonal matrix H with elements Hi; j ¼ ðki =mÞ di; j1 þ ððki þ kiþ1 Þ=mÞ di; j ðkiþ1 =mÞ di; jþ1 ;
(1)
where di; j is the Kronecker delta. Since large values of N will be used in this work, numerical instabilities can occur, in particular for small values of the non-diagonal matrix elements. Therefore, it is appropriate to have a case whose solution is known in closed form, so that the numerical calculations can be verified.
FIG. 2. (Color online) (a) A locally periodic system formed by n blocks, each consisting of N1 stiff springs of constant k1 , and separated by a set of q softer springs of constant k, is shown. (b) and (c) Two and four blocks, respectively, are converted into blocks formed by N1 l and N1 þ l springs. (d) The local periodicity is recovered but with a binary unit cell. 4394
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Indeed, the locally periodic chain in which ki ¼ k provides this case since it was solved by Lagrange more than two centuries ago, as can be read in his fundamental book Mecanique Analytique published in Paris in 1788.33 As is well known, the normal-mode frequencies xJ , J ¼ 1; …; N, must be such that the N N determinant DN of the matrix ðH x2 IÞ be zero; here I is the N N unit matrix. For the homogeneous chain, DN obeys the recurrence relation DN ¼ cDN1 DN2 ;
(2)
where c ¼ 2 ð4x2 =x2c Þ. Equation (2) defines the well known Chebyshev polynomial of the second kind:34 DN ¼
½N=2 X
ð1Þ
i¼0
¼
i
Ni i
ð2xÞN2i
sinððN þ 1Þ arccosðc=2ÞÞ : sinðarccosðc=2ÞÞ
(3)
(4)
Here ab is the binomial coefficient and ½a is the largest integer not greater than a. In the Appendix the solution, derived by the authors, to a more general recurrence relation than Eq. (2) is presented in Eq. (A2), which reduces in the case considered here to Eq. (3). Setting DN ¼ 0, the values of xJ can be obtained from Eq. (4) in closed form: xJ ¼ 2x0 sin
Jp : 2ðN þ 1Þ
(5)
As it is very well known and can be seen from this formula, the homogeneous chain is a high frequency filter, that is, all normal mode frequencies are less than xC . All normal modes are extended in the sense that in general all particles move from their equilibrium position. This is so since the eigenvectors XiJ of the matrix H are different from zero for all values of i ¼ 1; …; N. When a single defect is introduced, one normal-mode frequency exceeds xC and the corresponding eigenvector is localized.35 By the same token the chain with N1 defects has N1 localized states with frequencies xJ > xC . As a first step to constructing the binary chain of harmonic oscillators shown in Fig. 2(d), one set of N1 springs with a constant k1 is introduced in the homogeneous system. Another set of N1 defects are then introduced, causing further N1 states to be localized and have a frequency larger than the critical one. This procedure is continued until a locally periodic system of n blocks each with N1 defects, is constructed as shown in Fig. 2(a). All states are no longer localized and a band spectrum emerges. Since, as has been mentioned before, when k1 k each block behaves as a continuous rod in compressional oscillations, the above results have been established in Ref. 14. What has been done here is to destroy, by including defects, the original invariance with respect to translations that the locally periodic chain has and recover the periodicity again, but now the system is formed by n identical blocks coupled by softer springs. The systems of Fig. 2 will now be analyzed numerically. All the results of this section were obtained for n ¼ 24 J. Acoust. Soc. Am., Vol. 134, No. 6, December 2013
FIG. 3. (Color online) The two blocks at the left-hand extreme have been modified as indicated in Fig. 2(b). The states n þ 1 and 2n have frequencies in the forbidden gap and have localized amplitudes as indicated in the insets. Here XiJ are the amplitudes of the ith mass in the Jth normal-mode.
blocks, N1 ¼ 50, and N ¼ 1249 masses. The values k1 ¼ 4000 kg/s2 much larger than k ¼ 100 kg/s2 and m ¼ 0:01 kg were used. The construction of the binary chain, shown in Fig. 2(d), will now be considered. In Fig. 2(b) the two sets at the extreme left of the locally periodic chain of Fig. 2(a) are changed, so that they now have N1 l and N1 þ l springs, respectively. Using perturbation theory it is easy to see that all the normal-mode frequencies of the lowest band, shown in Fig. 3, are only slightly altered. On the second band, however, the states marked n þ 1 and 2n jump into the forbidden band. As given in the two insets of this figure, the corresponding normal-mode amplitudes are localized. When the spring system of Fig. 2(c) is analyzed, four states are localized. Finally, when all bodies are modified, the local periodicity of Fig. 2(d) is recovered but with a binary unit cell. As is shown in Fig. 4, the second band is broken into two bands: the acoustic and the optical bands. As exemplified in the insets of this figure, all states are now extended.
FIG. 4. (Color online) When the local periodicity is recovered the acoustic and optical bands appear and all states are extended. Dıaz-de-Anda et al.: Emergence of acoustic and optical bands
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Until now, what could be called Variations on a theme by Lagrange has been solved numerically. The “theme by Lagrange” is the homogeneous chain of harmonic oscillators. Measuring the time displacements of many particles subject to different springs is, however, a tough problem, which can become very cumbersome. Therefore, to perform experiments, the oscillator chain will be replaced by the elastic rod with notches shown in Fig. 1(a).
the wave amplitudes u at cylinder j, 1 < j < N. The torque and the wave amplitude can be written as uðxÞ A j ¼ W ðxÞ ; j Cj u0 ðxÞ Bj where Wj ðxÞ ¼
III. THEORETICAL AND EXPERIMENTAL METHODS
In this section the methods used to compute and measure normal-mode frequencies and amplitudes of the vibrating rod of Fig. 1(a) will be presented. A. Theoretical considerations
The Poincare method36 used to solve compressional and torsional vibrations in a rod with notches is summarized, only for torsional waves, as follows. A rod formed by N coaxial cylinders shown in Fig. 1(a) is considered. The solution of the torsional wave equation for each cylinder is given by uðxj1 x xj Þ ¼ Aj eijx þ Bj eijx ;
0 B Mðj þ 1Þ ¼ @
cosðjðxjþ1 xj ÞÞ jCjþ1 sinðjðxjþ1 xj ÞÞ
eijx iCj eijx
eijx : iCj eijx
(8)
Here Cj ¼ pa4j Gj =2 is the torsional rigidity with Gj and aj the shear modulus and area of cylinder j, respectively. The continuity conditions of the amplitude and torque can be written as Aj Ajþ1 Wj ðx ¼ xj Þ ¼ Wjþ1 ðx ¼ xj Þ : (9) Bj Bjþ1 Evaluating the last equation in x ¼ xj , and multiplying by Mjþ1 ¼ Wjþ1 ðx ¼ xjþ1 Þ W1 jþ1 ðx ¼ xj Þ, one gets
(6)
where j is the wave number. The Poincare map method consists of finding a recurrence relation, i.e., a map, that yields
(7)
uðx ¼ x Þ uðx ¼ xjþ1 Þ j ¼ Mjþ1 : 0 Cjþ1 u ðx ¼ xjþ1 Þ Cj u0 ðx ¼ xj Þ
(10)
The matrix Mjþ1 for the rod of Fig. 1(a) is given by
1 sinðjðxjþ1 xj ÞÞ C A: jCjþ1 cosðjðxjþ1 xj ÞÞ
(11)
From the first line of the matrix (10) with j þ 1 ! j it is obtained M11 ðjÞ uðxj Þ uðxj1 Þ; Cj1 u ðxj1 Þ ¼ M12 ðjÞ M12 ðjÞ 1
0
(12)
M22 ðNÞ uðxN Þ uðxN1 Þ ¼ 0: M12 ðNÞ
M22 ðjÞ M21 ðjÞ M12 ðjÞ uðxj Þ þ uðxj1 Þ M12 ðjÞ M12 ðjÞ
M11 ðjÞ M22 ðjÞ uðxj1 Þ; M12 ðjÞ
(13)
and since detðMÞ ¼ 1, Cj u0 ðxj Þ ¼
M22 ðjÞ 1 uðxj Þ þ uðxj1 Þ: M12 ðjÞ M12 ðjÞ
J. Acoust. Soc. Am., Vol. 134, No. 6, December 2013
(15)
(16)
Since this equation depends on the wave number j, the normal-mode values are those that satisfy Eq. (16). Once they are obtained, the uðxj Þ are calculated again so the coefficients Aj and Bj can be evaluated. The problem is therefore solved.
(14)
Finally, the Poincare map, which relates uðxjþ1 Þ to uðxj Þ and uðxj1 Þ, can be obtained substituting the last equation in the first line of Eq. (10): 4396
M12 ðj þ 1Þ uðxj1 Þ: M12 ðjÞ
The boundary conditions are now imposed. For a rod with free ends CN u0 ðxN Þ ¼ 0, which implies
and using the second line of Eq. (10), it follows Cj u0 ðxj Þ ¼
M12 ðj þ 1Þ M22 ðjÞ uðxjþ1 Þ ¼ M11 ðj þ 1Þ þ uðxj Þ M12 ðjÞ
B. Experimental method
To perform the measurements, the electromagneticacoustic transducer (EMAT) developed in Refs. 14 and 15 was used. The EMAT consists of a permanent magnet and a Dıaz-de-Anda et al.: Emergence of acoustic and optical bands
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coil, and can be used either to detect or excite the oscillations. The transducer operates through the interaction of eddy currents in the metallic rod with a permanent magnetic field. According to the relative position of the magnet and the coil, the EMAT can either excite or detect selectively compressional, torsional, or flexural vibrations. Used as a detector, the EMAT measures acceleration. This transducer has the advantage of operating without mechanical contact with the rod, which is crucial to avoid perturbing the shape of the localized wave amplitudes. Both the detector and exciter are moved automatically along the rod axis by a computerized control system and then the wave amplitudes can be measured easily (see Fig. 5). Whenever the wave amplitude is measured, it is necessary to keep the system at resonance, so the wave generator must be as stable as possible. In order to achieve this, a Stanford Research Systems DS 345 wave generator with an ovenized time base with stability > s1 < = ð‘ þ 1Þ i‘ C YB im C B N i1 @ A @ A ‘¼2 > > : ‘¼3 m¼2 ; i1 i‘ Ps1 s N ðmþ1Þ im Y i m¼1 p‘‘1 ; (A2) p1 0
‘¼2
where 2
3 j1 X 6N ð‘ þ 1Þ i‘ 7 6 7 4 5 ‘¼1 N ; ij ¼ jþ1
j ¼ 1; …; s 1;
(A3)
which define what could be called generalized Chebyshev polynomials. Here ½a indicates the largest integer not greater than a and mn is a binomial coefficient. By convention, all sums and products in which the upper index is less than the lower index are zero and one, respectively. For s ¼ 2, the s 1 summations in Eq. (A2) reduce to a single sum over the index i1 , which runs from 0 to ½N=2. All binomial coefficients, except for the second one, become unity. As a consequence, for s ¼ 2, Eq. (A2) leads immediately to Eq. (3) with p1 ¼ 2x and p2 ¼ 1. Another particular case of Eq. (A2) is obtained when p1 ¼ p2 ¼ 1: the Fibonacci sequence arises. And not only this, for pi ¼ 1 and s ¼ n > 2, the Fibonacci sequence of order n is obtained. 1
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