WAVEGUIDES IN PERIODICALLY STRUCTURED SOLIDS
M. Ayzenberg-Stepanenkoa, E. Sherb and G. Osharovicha
a
Ben-Gurion University of the Negev, Beer-Sheva, Israel
c
Institute of Mining, Siberian Branch of the Russian Academy of Sciences, Novosibirsk,
Russia
Abstract. Some types of localized waves are discovered and analyzed existing in
periodically structured solids. Double periodic plane problems are considered. A
structured material is modeled by two-dimensional uniform lattices possessing “a layer
defect” – a layer of another stiffness and mass parameters located inside an uniform
lattice. Depending on material parameters of the defect, frequency-wavelength domains
are built determining a waveguide-like propagation of wave energy along the defect
direction. Analytical and computer solutions are obtained for oscillating waves
propagated in this direction without attenuation, while in the direction normal to it, the
energy flux associated with the wave propagation is exponentially decays.
1. INTRODUCTION
Mathematical models and approaches to analysis of wave propagation in structured
materials have a centuries old history. Certainly, a great attention was attracted to the
periodical mass-spring lattice (MSL) due to its simplicity and physical clearness. Isaac
Newton used an 1D model of the MSL in [1] to derive a formula for the sound velocity.
Famous mathematicians and physicians (J. and D. Bernoulli, Taylor, Euler, Lagrange,
Cauchy, Kelvin) studied diverse aspects of the wave propagation in the MSL. Various 1D
models of periodic materials were developed in a pioneer work by Lord Rayleigh [2].
The next fundamental work that 70 years later is the monograph by L. Brillouin [3] who
formulized the mathematical aspects of the filter by using the Floquet theorem [4] to
analyze waves in crystal lattices and periodic electric filters. Such waves are best
understood by recalling the Floquet theorem for periodic structures (along, say, xdirection), which states that the amplitude of free response, v ( x ) , obeys the identity
v ( x, )  V ( x, )  eiqx , where q is the wave number (then the wavelength is= 2/q),
and V ( x, ) is a x-spatially dependent wave amplitude that is periodic with the same
period as the structure.
A strong practical interest in periodic models was explained by Max Born in the
introduction to [3]: "… The striking feature is the number and the variety of subjects
which are accessible to the same mathematical treatment: on one side problem of pure
physics, like scattering of X-rays by crystals, thermal vibrations of crystal lattices,
electronic motion in metals, and on the other side problems of electrical engineering,
namely, propagation of electro-magnetic waves along periodic circuits and filtering
properties of such systems …".
In structure engineering, the Brillouin concept was applied to the analysis of a set
problems of periodically layered composites, satellite solar panels, wings and fuselages
of aircraft, petroleum pipelines, railway tracks, and many others. A comprehensive
review of this direction has been made by Mead [5].
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The main phenomenon  wave localization inherent to periodic solids is that a free
wave propagation occurs only in certain discrete bands of frequencies, known also as
pass bands, which alternate with the stop bands of no propagation but the spatial decay of
the signal. This phenomenon regained a new interest with the beginning 90s (see, e.g. [68]) when artificial “crystals” were revealed as the band-gap materials allowing to control
the propagation of different waves: electronic, electromagnetic, and waves of sound and
vibration having nowadays applications in optics, microwaves, nanostructure
engineering, etc.
In the frequency spectrum for band-gap materials, there exist resonant points, which
usually (in the 1D case) demarcate the pass and stop bands. In these points the group
velocity  dq  0 ( is the frequency, and the energy flows from a source possessing
these frequencies not as a wave but, roughly speaking, as heat (more precisely, the
corresponding law depends on the order of the first nonzero derivative  n dq n at this
point [9]). Recently resonant localization phenomena in MSL were analyzed in [10],
while those appeared in 2D/3D cases at the frequency located in the interior of pass bands
were discovered in [11].
In this paper we analyze some others types of localized waves that can exist in a
special type of nonuniform lattices. As it is pointed out in work [12] not related, however,
to band-gap materials, such a lattice can be a uniform square-cell lattice containing a
straight 1D chain, which is linked to the lattice by bonds possessing the higher stiffness.
The dispersion relation of this lattice has real roots within some frequency interval, the
width of which depends on the stiffness of the “defect” layer. This fact leads to an
assumption that waves of a frequency from this interval propagate along the above
mentioned layer without spatial decay. Such a material could be defined as a lattice with
a linear “defect”.
Note, the emergence of waveguide-type propagation phenomena is caused by the
solid discreteness and has no analogues in continuous solids. In latter, where wave
localization processes are caused by free surfaces and interfaces, there are three wellknown types of localization: the Rayleigh surface waves at the halfspace boundary, the
Love surface waves in a layer “welded” upon an elastic halfspace, and the Stoneley
waves at the interface of two media. As far as to the authors knowledge, a localization
phenomena of waveguide-like wave propagation along a single direction within 2D/3D
periodic materials has not been yet discovered.
In this work we study this phenomenon on the basis of a uniform square-cell lattice.
We prove the existence within the 2D lattice of a 1D waveguide, which traps the energy
flux associated with the wave propagation along its direction, while the energy flux is
exponentially decays in the directions normal to it. We obtain, depending on “defect”
peculiarities, frequency spectra, wave lengths and corresponding group velocities of
oscillating waves propagated in the waveguide. We show how a 2D spatial problem for a
lattice is transformed into a 1D problem of a chain upon an elastic foundation. An
analytical solution is obtained for the stationary wave propagation, while unsteady-state
processes (including resonant phenomena) are numerically calculated allowing the timedependent formation of localized waves to be described.
162
2
DISPERSION PROPERTIES OF WAVEGUIDES
2.1 Dispersion Properties of a Uniform Lattice with a Layer of “Defect” Bonds
We consider transversal oscillations of lattice shown in Fig. 1. In mechanical terms, this
material represents a plane net of inertionless bonds connecting point material particles at
nodes (m, n), m, n  0, 1, 2, . Excluding bonds within “defect” layer (0, 1), the
lattice have bond stiffness, g, which together with particle mass, M, and cell size, l, are
assumed to be measurement units. The stiffness of horizontal bonds within the “defect”
layer n = 0, 1 is designed as  m , while  n is the stiffness of vertical internal bonds (in
the common case:  m   n  1 ). Below, along with the integer numbers m and n we use
continuous coordinates x and y respectively.
y
g
1
g
m
n0
M
x
n
1
m
2
2
1
m0 1
2
Fig. 1 : Lattice with a nonuniform layer (n; m)  (  1,0; m)
Equations for propagation of free waves in the lattice in terms of displacement are:
um,n  um1,n  um1,n  um ,n 1  um,n 1  4um,n
um,0   m  um1,0  um1,0   um,1   2 m  1 um,0   n  um,1  um,0 
(n  0,  1 ),
(n  0),
(1)
um,1   m  um1,1  um1,1   um,2   2 m  1 um,1   n  um,0  um,1  (n  1).
There um ,n is a transversal displacement of m,n node. Then the problem has two
internal parameters:  m and  n .
To obtain the dispersion relation for free waves of propagated along the layer
direction and exponentially decayed along the normal to it we seek the solution of Eqn.
(1) in the form
um,n  t    n exp iq  m  t  
 n  0 ,
um,n  t     (1 n ) exp iq  m  t  
 n  1 

with requirement   1 resulting to waves vanished at n   .
After substituting (2) into (1) we obtain relations linked , q,  ,  m and n:
  1
2
  Q   2 ,    mQ  1  2 n   2 , Q  4sin 2  q / 2  ,
163


From relations (3) we get two possible values   1,2 , q,  m ,  n  . First, we consider
the particular case  m  1 (single free parameter,    n , remains in the problem). Then we
obtain two decay factors,
1  
1
1
, 2  ,
2  1
1
(4)
which are independent on q and . While requirement || < 1 is proved only if  > 1, the
decay factor  corresponds to  = 1. In according with this, the dispersion relation is the
following:
2
2
q
 sin 2   .
2  1
2
(5)
It possesses the pass-band    q  0 ,   q     , lower limits of which ( = 1)
are   0  2 ,     8 . The limiting dispersion relation,   2 1  sin 2  q 2  ,
corresponding to  = 1 is depicted in Fig. 2 (a) by the dashed curve (note that in this
case a localized wave does not exist). Curves for a set of  > 1 are located upper it, they
determine the dispersion pattern of waves localized within the waveguide.
3.5
(a)

 = 1.5
=3
3.0
2.5
cg
2.0
1.5
0.2
 = 2.5
=3
=1
2.5
2.0
0
(b)
0.4
/2
q
0.0

0
/2
q

Fig. 2: Dispersional curves (q) – (a) and cg(q) – (b)
Below, in Section 4, together with relation (q), we also need the dispersion relation
for group velocity cg  q     q to compare steady-state and transient solutions for
waves propagated in the waveguide. From relation (5) we have cg  sin q   q  .
Dependences cg  q  are depicted in Fig. 2 (b) for some values of  .
One can see from Fig. 2 that the higher  , the higher frequencies within the passband, while its width,         0 , and cg are decreasing with  growth. The
maximal group velocity is located in the vicinity of q = /2 (where wavelength  ~ 4).
164
Relation (5) is exactly the same that obtained for a SML upon an elastic foundation
with the foundation stiffness related to . Notably, the equation of motion and the
dispersion relation are:
um  um1  um1   2  g  um ,   g  4sin  q 2 
(6)
where g is the foundation stiffness, and dispersion relations (5) and (6 ) coincide if the
foundation stiffness is g  4 2 2  1 . So, in this case (  m  1 ), the 2D problem can
approximately be studied on the basis of the simple 1D model.
In the common case,  m  1, we obtain the following expression for decay factor 
1
   (1   m )4 sin 2 (q 2)  2 n 1 ,
(7)

In the interface of pass- and stop bands decay factors and frequencies are the
following:
 0  
1
2 n  1
    
,
 (0)  2  Γ ,
1
;
2 n  1  4   m  1
 ( )  6  Γ
 Γ    1  .
Requirement || < 1 results in the inequality
 n  2 m  3
(8)
that determines open domain G (see Fig. 3), in which localized waves exist .
n
3
2
G :  n  2 m  3
1
1
2 m
Fig. 3: Domain G possessing localized waves
0
If 0   n  1 , then in compliance with (7), inequality   1 (required for the
existence of a waveguide along the layer) is proved not at the whole interval 0  q   ,
but only within it part q*  q    q*  0 , where q* is obtained from the limiting
relation   q*   1 . So that the condition 1    0 is proved if q  q* , while waves of
165
wavelengths   2 q*  q  q*  aren’t localized. The limiting q* obtained from (7) is:
q*  2arcsin 1   n 2( m  1)
(9)
If, for example,  n   m  2 , then q*   2 and localized waves have lengths   4 .
2.2 Wave Dispersion in a Block-Stratified Medium
We will consider a lattice approximation of a class of stratified materials consisting of
two block media linked by a layer as shown in Fig. 4. The model allows wave patterns in
the stratum as in a waveguide to be analyzed. Our aim, in particular, is to reveal
frequency and wavelength spectra, in which the layer can play role of a peculiar trap for
energy emitted by a vibration source functioning within the pass band
In Fig. 4(a), the considered system is shown, in which a straight layer (with
parameters M 0 , g 0 ) separates upper and lower media of parameters M+, g+ and M, g,
respectively, while a corresponding lattice model is depicted in 4(b). Note, that the
material described in Section 2.1 is a partial case of this medium.
( a)
(b )
2
M  , g
g
1
n
M0
M
g
g0
0
g
g
1
M
YM  , g
2
2
1
0
m 1
2
Fig. 4: Block-stratified medium, (a), and its lattice model, (b)
If M  and g  are taken as measurement units, equations of a free wave propagation in
the considered system are:

 

n  0: um ,n  um 1,n  2um ,n  um 1,n  um ,n 1  2um ,n  um ,n 1  0,

u

  g u
n  0: M 0um ,0  g 0 um 1,0  2um,0  um 1,0  (um,1  um,0 )  g  (um, 1  um,0 )  0,
n  0: M um ,n  g 
m 1, n
 2um ,n  um 1,n

m , n 1
(10)

 2um ,n  um ,n 1  0.
The four free parameters, M 0 , g 0 , M  and g  , remain in the governing system As in
the previous case, we seek a solution to this system in the form of a plane moving wave
with the magnitude disappearing at n   (decay factors  and  are introduced):
um,n  t   n eiq mt 
 n  0 ,
um,n  t    neiq mt 
166
 n  0  ;  1    0 
(11)
After substituting solution (11) into system (10) we obtain the following algebraic
system to determine   q  ,   q  and   q  :
 2  Q     1    0 Q  4sin 2  q / 2   ,
2


M 0 2  g0Q  1     g  1   1 = 0,
M  2  g Q  g     1    0.
2

    q; M , M , g , g ,
  

0

0

. (12)

    q,   ,   
 

Let us consider a set of particular cases for values of problem parameters.
(i) A“light” layer inside uniform media: M0 = 0.5, M   M   1 , g0  g   g   1 .
In this case, the single free parameter M 0 remains within the problem, and system (12)
degenerates into the following system of two equations (        is denoted below):
 2  Q    1  ,  2  Q 1  M 0    1  0,
(13)
solution of which allows desired dispersion relations to be obtained.
First of all, as it is seen from (13), required condition 1    0 is proved if M 0  1.
In Fig. 5, dependencies  (q) ,  (q ) and cg ( q ) are depicted for a set of values M 0 .
2
0.0

a
M0= 0.2
6

0.3
b
M0= 0.2
0.5
cg
0.667
0.8
0.3
0.8
-0.5
4
2/3
0.8
M0= 0.9
-1.0
0.0
0.5
2
q/ 1.0 0.0
0.5
M0= 0.2
0.3
0.5
2/3
0.5
M0 = 0.9
c
1.2
q/ 1.0
0.4
M0= 0.9
0.0
0.0
0.5
0.8
q/
1.0
Fig. 5: Decay factor,  , frequency,  , and group velocity, cg , for a set of M 0
One can observe the main features of the dispersion pattern. The lesser M 0 , the
greater the wave decay in transversal directions (i.e. the greater wave energy propagated
within the waveguide), the wider the pass band, , and the greater group velocities. If,
for example, M 0  0.22 , the maximal value of cg exceeds the wave speed in surrounding
media equal to 1.
(ii) A light layer separated two media of different rigidities (Fig. 6): M0 = 0.5,
M   M   1, g0  g  1, g is varied.
Results obtained in this case testify significantly different decays realized in rigid ()
and pliable (+) media. The greater g, the greater frequencies within the pass band and
the lesser group velocities.
167
g = 3
10
g = 1.5
10
6
6
g = 5
8
10
6
4
5cg
2
0

4

/2
0
5cg
2
10
q

0

4
5cg
2
10
10
/2
0
q

0
/2
0
q

Fig. 6: Decay factors, frequencies and group velocities in the system (ii)
(iii) A light layer with different axial stiffnesses and rigid lower medium. Dispersion
curves for this case are shown in Fig. 7 ( M   1, g0  g  1, g   2 ) and Fig. 8
( M   1, g  1, g0  0.5, g  2 ).
M0 = 0.2
10cg
8
6

4
4
0
-10
-10
2
q
0
0
/2
0

M0 = 0.65
10
-10
8
6
2
M0 = 0.5
10
10
10
8
6
10cg

4

10
2 10cg
-10
/2

q
0
/2
q*=2.08 /4
q

Fig. 7: Dispersion pattern in the case M   1, g0  g  1, g   2 and varied M.
6
M0 = 0.5
M0 = 0.2
10
10cg
4
2
0
0
10

6
10
4
10
2
/2
q

00
6

10cg
4 

2

/2

8
8

M0 = 0.52
10cg
q

0
0.0
0.2
q 0.4
q**= 0.44
Fig. 8: Dispersion pattern in the case M   1, g  1, g0  0.5, g  2 and varied M.
The main aim of results presented in Fig. 7 and 8 is to show the degeneration of
waveguide properties depending on system parameters. With increase in mass M0, the
decay factor in the rigid media () decreases tending to 1, the pass band width tends to
zero at the certain parts of the wavelength spectrum. Then inequality    1 is proved at
168
M 0  M 0* , where M 0* is a critical value depended on parameters of the system.
One can detect “non waveguide” spectra
 q* ,  
*
at M 0  0.65 and
 0, q** 
at
M 0*  0.52 in Figs. 7 and 8, respectively.
3
STEADY-STATE SOLUTION FOR LOCALIZED SINUSOIDAL WAVES
Now we consider a structure described in Subsection 2.1 (i): M  g   m  1 and vertical
bonds connecting layers n  1 and n  0 possess stiffness  n    1 .
We will discuss the following steady-state problem: to find displacements u (t , m, n)
of the infinite lattice subjected in nodes (0,1) and (0,0) to monochromatic antiplane
excitations of frequency 0 :
Q(t ,0, 1)  2sin(0t ), Q(t ,0,0)  2sin(0t )
(14)
For this case, an asymptotic solution to the steady-state problem obtained by Slepyan
in [13] is:
1  2
um,0 
(15)

U
sin

t

q
m
,
U



m  ,
st
0
0
st
steady  state
2sin q0
where q0 is the wave number corresponding to 0 in dispersion equation (5).
Then we establish a correspondence between solution (15) and the dispersion pattern,
discussed in Subsection 2.1. With this aim, a schematic example is shown in Fig. 9 of
dispersion curves  (q ) and cg ( q ) , where 0 and q0 are marked by bold circles (recall,
the wavelength is 0  2 / q0 ). For a set of  n such curves can be found in Fig.2. As it
will be shown below, the group velocity c0 , corresponding to q0 is turned out be the
velocity of moving quasi-steady state wave designed as C st .

0
c0
0
cg
q0 /2
q

Fig. 9: Frequency, 0 , wave number, q0 , and group velocity, c0
169
4
COMPUTER SIMULATIONS OF TRANSIENT PROCESSES
As in the previous Section, we consider structures with M  g   m  1,  n    1 .
Transient processes excited in this system by source (14) were numerically simulated at
a wide range of parameters 0 and  . Presented below results are related to the case
  2.5 when the decay factor is   0.25 and the distribution of displacements along
axis y is um,n   0.25  um,0
n
4.1 Waveguide Propagation of Disturbances
In Fig. 10, we present results calculated at frequencies 0  2.6 , 2.8 , and 3.1 located
within the pass band [  ,   ]  [2.50, 3.20] . Pictures in the left column are envelopes
U m ,n (their positive parts) of displacements um ,n in nodes  m; n    0, 20, 50; 0  .
Pictures in the right column are distributions of displacements um ,0 , um ,1 , um ,2 along axis
m taken at t  500 ; one can detect wave forms and the peculiarities of the transient
process propagation in correspondence with the steady-state solution. Dashed straight
lines are the steady-state amplitude, U st , taken from analytical asymptote (15). One can
see that a wavy character of calculated envelopes has a tendency to spread with time; the
stabilization speed is maximal if frequency 0 is located at the median part of the pass
band, where the velocity of a steady state wave, C st = cg, is maximal (see the right
column). The eady-state amplitudes can also be sufficiently estimated (as median lines of
envelopes) in two other cases where frequencies 0  2.6 and 3.1 are relatively closed to
pass/stop band interfaces.
The presented results and the purposeful analysis of other simulations data allowed to
make a number of conclusions regarding peculiarities of transient and steady state
waveguide processes. Among them are the following:
 numerical simulations of the transient process allow the magnitude of the steady state
wave to be estimated,
 the main part of propagating waves move with the group velocity corresponding to
the source frequency,
 the transient waveform with respect to transversal axis n is sufficiently determined by
the steady-state estimate: um, n   num,0 .
 the waveform with respect to waveguide axi m possesses a frequency equal to q0 .
 the maximal cg is obtained in small vicinity q  qmax( c ) ~  2 ,
g
 the higher cg , the less the median amplitude of the transient wave (or, which is the
same, the steady state amplitude) and the less time required for stabilization of the
transient process,
 the main attributes of the wave package moving with velocity C st is sufficiently
determined by the steady-state estimate if loading frequency is located within the pass
band.
170
1.0
Um,0
0.8
U st  0.711
0.6
1.0
um,n
Ust =0.711
0.5
0.0
m=0
0.4
m = 20
m = 50
0.2
0.0
0  0.26
 = 2.6
t
0
0.8
Um,0
100
200
300
400
-0.5
-1.0
0
500
0.50
m=0
um,n
Ust = 0.475
0.6
m = Cstt, Cst = 0.250
100
150 m 200
0 = 8.48
50
0 = 4.59
n=0
t = 500
0.25
U st (b)0.480
0.4
0= 2.8
0.28
m = 50
m = 20
0.2
0.0
t = 500
n = 0, n = 1, n = 2
100
200
300
-0.50
m = 20
m = 50
Cst = 0.350, mst = Cstt
50
100
150
100
200
300
um,n
m 200
t = 500
n = 0, 1, 2
0.625
0.25
 = 3.1
0  0.31
t
0
0
0.00
 = 3.1
0.25
0.00
500
Um,0 = 0.628
m=0
0.50
400
0.50
1.00
Um,0
0.75
U st  0.628
0.00
-0.25
t
0
n=1
n=2
400
-0.25
0.625
0 = 2.52
-0.50
0
500
50
m = Cstt, Cst= 0.230
100
150
m 200
Fig. 10: Comparison of transient solutions with the steady-state asymptote for
Fig. Error! No text of specified style in document..1 Comparison of transient
frequencies 0 = 0.26, 0.28 and 0.31 lying inside the pass band. Solid curves in the left
solutions with the steady-state asymptote for frequencies 0 =
column are envelopes U m ,0 of displacements vs. time shown in the three nodes  (0,0),
0.26, 0.28 and 0.31 lying inside the pass band. The left column: solid curves are
(20,0) and (50,0), while dashed straight lines are steady-state amplitudes U st (0 ) from
computed
U mdistributions
envelopes
vs. time in the
nodes  (0,0), (20,0)
um ,three
(15). The right
column:
of displacements
,0 of displacements
n along axis m in three layers,
and
n =(50,0),
0, 1 and 2 (great, median and small amplitudes, respectively) at t = 500; the dashed
(0,
while
straightsteady-state
dashed lines
are amplitudes
polygons are
bounded
solution
for chainUnst =
dotted ones show the same
0 ) in
solution
for chainsource
n = 1not
( Ufound..
in accordance
Error!
Reference
The
column: with decay factor  = ¼ at
m ,1  ¼U
m ,0 right
along axis m in three layers n = 0, 1
  2.5 ). distributions of displacements
and 2 taken at t
= 500; solid curves are computed results, the dashed perimeter is moving (with
velocity
) steady-state solution Error! Reference source not found., dotted lines show
171
amplitudes in chain n = 1: in accordance with the decay factor
.
4.2 Resonant Phenomena
Below we present some computer simulations of resonant processes excited in this
system by source (14) with critical frequencies     and     demarcated pass- and
stop bands. The former frequency determines a long-wave resonant process, while the
latter one  a shortwave resonant process (it was analyzed in [10] for an 1D MSL).
Together with critical frequencies the near-critical those are considered, for which the
steady state solution exists. Due to the fact that the unsteady steady-state transition
requires a relatively long time, the mentioned comparison will allow peculiarities of the
resonance development to be elucidated in detail. Results shown in Fig. 11(a) and
(c)correspond to the first, 0    , and to the second, 0    , resonances, respectively.
In Fig. 11(b) two near-resonant cases, 0  1.004  and   0.997  , can be seen.
Data in the left column correspond to development with time of displacements in
some nodes: in Fig. 11(a) displacements um ,n are depicted in nodes (0,1), (0,1) and (0,2),
while in Fig. 11(b) and (c)positive parts are depicted of envelopes U m ,0 of displacements
um ,n in nodes (0,0), (10,0) and (20,0) um ,n . Distributions of u m , 0 , u m ,1 , u m , 2 along the
waveguide axis taken in some time moments can be seen in the right column. Dashed
straight lines in Fig. 11(b) are amplitudes of steady-state asymptote (15). Recall, that a
steady-state solutions are absent in resonant cases (a) and (c).
The presented data allow spatial-temporal pattern of perturbations to be sufficiently
described. One can observe a contra-phase transverse form of resonant (and near
resonant) processes. As calculations show, see the left part of Fig. 11(a), resonant
oscillations in case 0    monotonically increase as t , that is the same growth that
was analytically estimated in [10] for a MSL.
Thus, resonant processes excited in 2D lattices have the same nature as those arising
in a MSL. The latter can be used for a preliminary analysis of wave processes in more
complicated systems. The condition needed for justification of such a procedure is a
correspondence of dispersion relations within required spectrum intervals.
Fig. 11: Resonant and near-resonant regimes in a lattice with  n  2.5
172
2
n=0
n=1
n=2
4
u0,n
um,n
t = 300
n=0
1
2
n =1
(a)
0
0
n=2
-2
-1
n = 2.5, m = 1
-4  = - = 2.5
0
20
t
40
60
80
-2
100
m
0
20
40
60
80
100
120
(b)
6
um,n
4
6
Um,0
4
m=0
m = 10
n=2
2
m = 20
t =1250
n=0
n=1
0
(c)
n = 2.5, m = 1
2
 = 3.2,
q0 = , l0 = 2
+
-2
 =  = 3.2
+
-4
0
t
0
250
500
750
1000
1250
173
-6
0
m
25
50
75
100
125
150
ACKNOWLEDGMENTS
This work was supported by the Israel Science Foundation (Grant 504/08) and the
Russian Foundation for Basic research (Grant No. 08-05-00509).
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