Acoustic characterization of Hofstadter butterfly with resonant scatterers
aa r X i v : . [ phy s i c s . c l a ss - ph ] J u l Acoustic characterization of Hofstadter butterflywith resonant scatterers
O. Richoux ∗ V. Pagneux † Abstract
We are interested in the experimental characterization of the Hofstadter butter-fly by means of acoustical waves. The transmission of an acoustic pulse through anarray of 60 variable and resonant scatterers periodically distribued along a waveg-uide is studied. An arbitrary scattering arrangement is realized by using the vari-able length of each resonator cavity. For a periodic modulation, the structures offorbidden bands of the transmission reproduce the Hofstadter butterfly. We com-pare experimental, analytical, and computational realizations of the Hofstadter but-terfly and we show the influence of the resonances of the scatterers on the structureof the butterfly.
PACS. 43.20.Fn-Scattering of acoustic wavesPACS. 43.20.Mv-Waveguide, wave propagation in tubes and ductPACS. 43.58.Gn-Acoustic impulse analyzers and measurements
The Hofstadter butterfly is well known since the work of Hofstadter on the electrontransmission through a two-dimensional ordered lattice perturbed by a perpendicularlyapplied uniform magnetic field [1]. Few experimental attempts have been made toobserve signatures of this phenomenon in two-dimensional electron systems [2, 3, 4].Recently, an experimental electromagnetic realization of the Hofstadter butterfly wasdone by studying the transmission of microwaves through an array of 100 scatterersinserted into a waveguide when the modulation length of this unidimensional periodiclattice changed [5]. These authors used cylindrical scatterers introduced into a rect-angular microwave waveguide. In this study, the stopbands were produced by Braggsreflection on the transmission spectra and the illustration of the Hofstadter butterflyexhibited self-similar structures. ∗ Laboratoire d’Acoustique de l’Universit´e du Maine UMR CNRS 6613 - Avenue O. Messiaen, 72085 LeMans Cedex 9, France. E-mail : [email protected] † Laboratoire d’Acoustique de l’Universit´e du Maine UMR CNRS 6613 - Avenue O. Messiaen, 72085 LeMans Cedex 9, France. E-mail : [email protected]
1n the present paper, we study the transmission of an acoustic pulse through a lat-tice composed of an assembly of variable Helmholtz resonators that are the scatterers[6]. A particularity of this work is the use of scatterers that can exhibit resonant scatter-ing at wavelength much larger then the scatterers (well known examples of Helmholtzresonators are the bottle of wine that have resonances for frequencies correspondingto wavelengths of the order of the meter). The resonators are uniformly distributedalong a cylindrical waveguide. Each Helmholtz resonator is constituted by two cylin-drical tubes which play the role of neck and cavity of the resonator. The length of theHelmholtz cavity is a variable parameter and the lattice spacing constant d betweeneach resonator is fixed.The propagation of sound wave into a tunnel with a periodic array of Helmholtzresonators has been already studied [7] and different kinds of stopband appear in thetransmission coefficient. In addition to Bragg stopbands caused by the spatial periodic-ity of the lattice, resonances of the Helmholtz resonator create other ”stopping bands”called scatterer stopbands which are non-existent in [5]. We show that a realization ofthe Hofstadter butterfly is obtained experimentally with acoustic waves. The effects ofthe scatterer stopbands are examined. Experimental work is completed by an analyticaland a numerical analysis, and all show self-similar structure. We study a monochromatic pressure wave ˜ p ( x , t ) with the form ˜ p ( x , t ) = p ( x ) e j w t where w is the wave pulsation. A cylindrical waveguide with Helmholtz resonatorsconnected axially constitutes the one-dimensional lattice. In this lattice, the pressure p ( x ) is solution of the following equation [8], d p ( x ) dx + k p ( x ) = (cid:229) n d ( x − x n ) s n p ( x ) (1)where k = w / c , s n = − j wr s i S Y n with s i and S the derivation and the main tube areasrespectively (fig. 1), r is the density of air and c is the celerity of the wave. Y n is theadmittance of the n th resonator placed at x = x n , defined by Y n = v ( x n ) / p ( x n ) = v n / p n where p ( x n ) and v ( x n ) are respectively the pressure and the acoustic velocity at x = x n .The solutions of eq. (1) are found by the transfer matrix method. In the region x n ≤ x ≤ x n + , which determines the n th cell, the solutions of eq. (1) are separated into twoplane waves with opposite propagation direction associated with the amplitudes A n and B n : p ( x ) = A n e jk ( x − x n ) + B n e − jk ( x − x n ) . The propagation through one cell is described by (cid:18) A n + B n + (cid:19) = M n (cid:18) A n B n (cid:19) = ( − s n jk ) e − jkd n s n jk e jkd n − s n jk e jkd n ( + s n jk ) e − jkd n !(cid:18) A n B n (cid:19) (2)with d n = x n + − x n . For a finite lattice which is composed of N resonators, the wavepropagation is described by ( A n B n ) t = (cid:213) Nn = M n ( A B ) t , where ( A B ) t are the ampli-tudes of the inlet of the lattice. 2 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) cavityneck liSs siF (end)E (entran e) l ri Figure 1: Description of the Helmholtz resonator.The resolution of the eigenvalue problem for the matrix M n allows us to find thestructure of the transmission (allowed and forbidden bands). By setting l ± = e ± jqd forthe eigenvalues, the dispersion relation takes the formcos ( qd ) = cos ( kd ) + s n k sin ( kd ) , (3)where q is called the Bloch wave number. The transmission of the lattice is deter-mined by the value of cos ( qd ) : if cos ( qd ) belongs to [ − , ] , the wave propagatesinto the system; in contrast, when cos ( qd ) belongs to ] − ¥ , − ] or [ , + ¥ [ , the wave isexponentially damped and the lattice is completely opaque to the wave. Y To determine the impedance of a Helmholtz resonator, we need to a relation pressure p and velocity v between the entrance (E) and the end (F) of the resonator (fig. 1).Applying classical lossless plane wave technique it is found that (cid:18) p E v E (cid:19) = cos ( kl i ) jZ ic sin ( kl i ) jZ ic sin ( kl i ) cos ( kl i ) ! (cid:18) jZ ic k D l (cid:19) cos ( kl c ) jZ cc sin ( kl c ) jZ cc sin ( kl c ) cos ( kl c ) ! (cid:18) p F v F (cid:19) (4)where Z ic and Z cc are respectively the characteristic impedance of neck and cavity, l i and l c are respectively the length of the neck and the length of the volume cavity, andsubscripts E and F refer to entrance and end of the resonator. D l represents the addedlength to the neck due to the discontinuity of the sections. This length takes generallythe form [9] D l = D l i + D l c = r i ( p + p + o ( s i S )) where r i is the radius of the neck.Because the Helmholtz resonator is closed, the boundary condition at the end of theresonator is v F =
0. Eliminating p F in the eq. (4), the acoustic impedance Z = p E / v E seen from the entry (at the point E) is given by the following relation : Z = Y = cos ( kl i ) cos ( kl c ) − Z ic Z cc k ( D l ) cos ( kl i ) sin ( kl c ) − Z ic Z cc sin ( kl i ) sin ( kl c ) Z ic cos ( kl c ) sin ( kl i ) − Z cc sin ( kl i ) sin ( kl c ) + Z cc cos ( kl i ) sin ( kl c ) . (5)Because of resonances, it is very important to take into account the losses. Thetheory with losses [10] is used in the eq. (5) with k = w c ( + b s ( + ( g − ) c ) and Z c = r c ( + b s ( − ( g − ) c ) by setting s = r / d where d is the viscous boundary layer, c = √ P r with P r the Prandtl number, b = ( − j ) / √ r is the radius of thetube considered. In the experiment, a cylindrical waveguide with an inner radius r = . − m wasused. 60 Helmholtz resonators were periodically located along this waveguide andthe lattice spacing between each resonator was d = . S i = , . − m and a length l i = . − m) and a variable length cavity (cylindrical tube with an inner area S c = , . − m and a maximum length l c = . . − m) as described in the fig. 1. Thesound source was a loudspeaker with special design placed at one end of the main tube. Two microphones (BK 4136) measured the pressure at each end of the cylindricalwaveguide. We measured in the frequency range where only the first mode can prop-agate, starting from 0 Hz up to the first cutoff frequency f c = ,
061 kHz, where thepropagation of the first higher mode becomes possible.For the experimental realization of the butterfly a periodic modulation of the lengthsof the cavity resonators was applied with the period length as a parameter. The modu-lation of this lattice is given by the variation of the cavity length of the n th resonator l n = l c cos ( p n a − a ) (6)where a = pm , with p , m ∈ N . We replace this cosinusoidal variation by a rectangularone (in the same manner as [5]): l n = (cid:26) ( p n a − a ) ≤ l c for cos ( p n a − a ) > . (7)4n the experiment we choose l c = . − m and a =
0. It is important for theexperiment to use a rectangular setup and not a sinusoidal otherwise the stopbandswould destroy the butterfly.
To investigate the experimental results, we used a time-frequency method. It maps thetime domain signal (the square of the amplitude) in the time-frequency plane. In thisway, it is possible to know the arrival time of each frequency present in the input signaland to determine a ”transfer function” of the acoustical system. The Wigner-Ville [11]distribution defined as W z ( t , f ) = Z + ¥ − ¥ z x ( t + t ) z ∗ x ( t − t ) e j p f t d t where z x ( t ) is the analytical signal [12] (associated to the real signal z ( t ) ) is certainlythe most widely studied time-frequency methods. To reduce the cross terms (interfer-ences) between different components of the signal, we used the Pseudo Wigner-Villedistribution PW given by : PW z ( t , f ) = Z + ¥ − ¥ | h ( t ) | z x ( t + t ) z ∗ x ( t − t ) e j p f t d t where h ( t ) is the temporal rectangular window with a length t . This method gives theacoustical response of the lattice in the time-frequency plane by using a source pulse.We determined the transmission spectra by detecting the maximum of energy for eachfrequency upstream and downstream the lattice. The transmission was deduced fromthe ”transfer function” (in energy) defined as the fraction of transmitted and incidentenergy.Using a pulse source, this method avoids seeing, in the transmission spectra, os-cillations phenomena caused by the finite number of cells [13]. The pulse source pre-vents this phenomena, which would destroy the Hofstadter butterfly structure, when thepropagation time along a cell is greater than the pulse duration [14]. The fig. 2 showsa comparison between these two sorts of processing method : a classical way (Fouriertransform) using a shirp source and the previous one using the pseudo distribution ofWigner-Ville and a pulse source. It illustrates the oscillations in the transmission coef-ficient and shows the influence of the source on the transmission spectra. The periodic rectangular modulation implies only two values for the cavity length ofeach Helmholtz resonator. When this length vanishes, the resonator effects on thepropagation are completely invisible in the frequency range of interest. When a iswritten as a = / m , the lattice is periodic with a period length d ′ = d / a = md . In thiscase, analytical points of the Hofstadter butterfly can be found in the ( f , a ) plane.5
200 400 600 800 1000 1200 1400 1600 1800 200000.20.40.60.81
Frequency (Hz) T r an s m i ss i on T r an s m i ss i on c oe f. (a) (b) Figure 2: Comparison between two different signals processing for a = . d ′ is used in the dispersion relation (eq. (3)) which becomescos ( kd ′ ) + s k sin ( kd ′ ) = ± − ( q ) = − s k , thedispersion relation is written as cos ( k d a + q ) = ± cos ( q ) and the solutions are k d a = n p − q and k d a = n p with n ∈ N . (8)The effects of the scatterers are present in the term q and the resonance frequenciesimply its divergence. These divergences perturb branches structure of the butterflyaround these resonance frequencies and ”break” the butterfly structure as presented inthe fig. 3 and 4 for f =
283 Hz, f = f = f = ( f , a ) plane for a ’s values corresponding to p = m = a ranging from 0 to 1 with 50 values between 0 and 0 .
5. The upper part isobtained by reflection. (b) Zoom of the experimental transmission spectra for a periodicarrangement of scatterers with a ranging from 0 to 1 between [ ] Hz. The whitepoints show the analytic results from the eq. (8).
The transfer function between the end and the beginning of the lattice, in the frequencyrange [ , ] Hz which is corresponding to two p -Bragg bands, was measured. Weused 50 different values for the parameter a between 0 and 0 . a = p -Bragg stopband) at 1700 Hz.The periodic modulation of the lattice involves the arrival of forbidden subbands inthe transmission spectra (fig. 5b, c and d) which split the transmission band (allowedband) into subbands (marked with a arrow in the figure). The number of these subbandsincreases with m . The new period of the lattice is d ′ = md so the frequency location ofthe subbands are now f b = c / ( md ) :1 subband appears for p = m = p = m =
10 (fig. 5d) every 170 Hz (thisargument is not satisfactory for p = p =
13 and m = a are plotted together on a plane building withthe frequency (abscise axe) and a belonging to [ , ] (ordinate axe). The spectra areconverted to a grey scale (fig. 3 and 4) and the part between [ . , ] is obtained byreflection.In fig. 3a two p -Bragg bands are seen and two Hofstadter butterflies are complete.The structure of the butterflies is broken by the resonances of the resonators and some7igure 4: Numerical result for the transmission spectra for a periodic arrangement ofscatterers with a ranging from 0 to 1 with 100 values between 0 and 0 .
5. The upperpart is obtained by reflection.”oscillations” due to the signal processing are observed. The fig. 3b presents a zoomof the fig. 3a for a [ ] Hz frequency range. In this figure, the structure of theHofstadter butterfly are identified and can be compared to the simulation (fig. 4) ob-tained with 100 values of a . Self similar structures are observed and a lot of subbandsare perceptible. The presence of scatterers stopbands seems to stop the effects of thelattice modulation in the transmission : for example, the main subband is just shiftedduring the both stopbands caused by Helmholtz resonators and the butterfly structureis affected by this phenomenon. The white points on the fig. 3b present the analyticalresults calculated with the eq. (8). They are in agreement with the experimental resultsand they predict sufficiently the effect of the scatterer resonances.The self-similar structure of the butterfly is also observed. The simulated butterfly(fig. 4), building with 100 different arrangements of the scatterers, exhibits more self-similar structures than the experimental results. The depth of the self-similar structuresis depending on the number of modulation used to map the Hofstadter butterfly. Thesimulation allows to observe with a high definition some fine details of the Hofstadterbutterfly and to analyze more precisely the effects of the scatterer resonances. It seemsthat the presence of scatterer stopbands (the first and the second stopbands) breaks the8
200 400 600 800 1000 1200 1400 1600 1800−30−20−100 (a) (b) T r an s m i ss i on ( d B ) (c) Frequency (Hz)(d)
Figure 5: Transmission function for four different values of modulation lengths. (a) a =
0. (b) a = .
5. (c) a = . a = .
1. Some subbands due to the modulationsare marked by arrows.modulation influence by diverting the different branches of the butterfly. It implies thatthe structure is self-similar only in the region between the scatterer stopbands. Thisphenomena is also observed with the experimental results for the main branch of thebutterfly (fig. 3b).
We are grateful to the referees for helpful suggestions.
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