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Paper text:
gumen _ et al _ @xcite have considered recently the consequences of introducing a topological defect , that is , a defect that breaks the long range order in an otherwise 1d periodic structure .
they analyze the case of two semi - infinite equal lattices that match at a point which is not at the center of the unit cell . as an example , they consider the potential @xmath4 where @xmath5 is related to the strength of the defect .
the authors then solve the stationary schrdinger equation , which is mathieu s equation at both sides of the defect , and find that for special values of the defect strength @xmath5 , two levels appear in the forbidden band .
the situation is different from what occurs with a point defect , where typically only one level lies in the energy gaps . in this paper
we shall demonstrate this effect for the normal - mode frequencies of torsional waves in rods with notches , both numerically and experimentally .
, @xmath6  cm , @xmath7  cm , and @xmath8  cm . in the case of the fourth band
only four frequencies are shown . ]        in this note we deal with torsional vibrations of the elastic rod shown in fig .  [ rod ] .
the rods consist of @xmath9 unit cells plus a defective cell , which is the topological defect . with the exception of the defect ,
each cell is formed by three cylinders , one of length @xmath10 with cross section area @xmath1 and two cylinders of length @xmath11 with cross section area @xmath2 . in the defect
the central cylinder has length @xmath12 .
the wavelength is much larger than the radius of the rod , so the system is indeed one dimensional .    using the electromagnetic acoustic transducer ( emat ) for low frequencies that we have recently developed @xcite , we can excite and measure normal - mode frequencies and wave amplitudes for torsional waves in metallic rods .
the experimental apparatus has been described in detail elsewhere @xcite . for locally periodic systems a band structure emerges . as shown in ref .
@xcite , these normal - mode properties can also be computed using the transfer matrix method .
the theoretical results agree very well with our experimental measurements ; we should emphasize that this is a parameter free fit .    in fig .
[ espectro ] we present the band spectrum obtained for @xmath13 , both from the theoretical and experimental points of view .
we see that in the gap between the first and second bands only one frequency appears , but in the second forbidden gap two levels lie . in some of the higher gaps
two frequencies are also found .
this is an experimental realization of the theoretical findings of gumen et al @xcite . in fig .
[ waveamplitudes ] we show the wave amplitudes of these two states ; they are localized around the topological defect with an exponential decay .
the theoretical values , both for the frequencies and wave amplitudes coincide well with our measurements , as these two figures show .
the theoretical results are easily extended for wider ranges of @xmath14 .
the band spectrum as a function of @xmath14 is given in fig .
[ chi ] . since the frequencies of the localized states are proportional to @xmath15 , where @xmath16 is an integer number , more than one level can lie in the forbidden band . for example , as will be seen in fig .
[ chi ] , for @xmath17  cm two levels appear , whereas for higher values of @xmath18 three states are located in the second forbidden band .    .
for @xmath17  cm _ two _ levels lie between the second and third bands . for larger values of @xmath18
_ three _ levels are in the forbidden band . ]
we have measured and calculated numerically normal - mode frequencies and amplitudes of torsional waves in a rod with a topological defect .
the theoretical prediction of gumen _ et al _ @xcite that for certain topological defects two levels instead of one become localized is shown to be true , and an experimental example is provided .
we also give , via a numerical calculation , a generalization of the results found in ref .
@xcite , since for other special values of the topological defect more than two frequencies can lie within the forbidden bands .      1 l. gumen , e. feldman , v. yurchenko , a. krokhin , and p. pereyra , physica e _ in press .
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a. morales , j. flores , l. gutirrez , and r. mndez - snchez , _ j. acoust .
* 112 * _ , ( 2002 ) 1961 .
it has been recently shown theoretically that a topological defect in a 1d periodic potential may give rise to _ two _ localized states within the energy gaps . in this work we present an experimental realization of this effect for the case of torsional waves in elastic rods . 
 we also show numerically that _ three , _ or even more , localized states can be present if the parameters characterizing the topological defect are suitably varied .     
 and @xmath0 is the width of the notch . 
 here @xmath1 and @xmath2 are the cross section areas of the rod and notches , respectively . 
 the defective cell has length @xmath3 . ]