On the effective interaction of waves in inhomogeneous, nonstationary media
The description of new conditions of effective interaction of waves in periodically inhomogeneous and periodically nonstationary media is given. New conditions, as a special case, contain the known conditions of interaction of waves (synchronism conditions). Examples of the interaction of waves whic...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2017
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| Цитувати: | On the effective interaction of waves in inhomogeneous, nonstationary media / V.A. Buts, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2017. — № 6. — С. 71-75. — Бібліогр.: 6 назв. — англ. |
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Buts, V.A. Tolstoluzhsky, A.P. 2018-06-16T06:58:12Z 2018-06-16T06:58:12Z 2017 On the effective interaction of waves in inhomogeneous, nonstationary media / V.A. Buts, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2017. — № 6. — С. 71-75. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 41.20.Jb; 02.30.Jr; 02.60.Cl https://nasplib.isofts.kiev.ua/handle/123456789/136193 The description of new conditions of effective interaction of waves in periodically inhomogeneous and periodically nonstationary media is given. New conditions, as a special case, contain the known conditions of interaction of waves (synchronism conditions). Examples of the interaction of waves which characteristics satisfy the new interaction conditions are considered. These examples allow to detect new conditions in an experiment. Даний опис нових умов ефективної взаємодії хвиль у періодично неоднорідних та нестаціонарних середовищах. Нові умови як окремий випадок містять відомі умови взаємодії хвиль (умови синхронізму). Розглянуто приклади взаємодії хвиль, характеристики яких задовольняють новим умовам взаємодії. Розглянуті приклади дозволяють виявити нові умови в експерименті. Дано описание новых условий эффективного взаимодействия волн в периодически неоднородных и нестационарных средах. Новые условия, как частный случай, содержат известные условия взаимодействия волн (условия синхронизма). Рассмотрены примеры взаимодействия волн, характеристики которых удовлетворяют новым условиям взаимодействия. Рассмотренные примеры позволяют обнаружить новые условия в эксперименте. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Динамика пучков On the effective interaction of waves in inhomogeneous, nonstationary media Про ефективну взаємодію хвиль у неоднорідних, нестаціонарних середовищах Об эффективном взаимодействии волн в неоднородных, нестационарных средах Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
On the effective interaction of waves in inhomogeneous, nonstationary media |
| spellingShingle |
On the effective interaction of waves in inhomogeneous, nonstationary media Buts, V.A. Tolstoluzhsky, A.P. Динамика пучков |
| title_short |
On the effective interaction of waves in inhomogeneous, nonstationary media |
| title_full |
On the effective interaction of waves in inhomogeneous, nonstationary media |
| title_fullStr |
On the effective interaction of waves in inhomogeneous, nonstationary media |
| title_full_unstemmed |
On the effective interaction of waves in inhomogeneous, nonstationary media |
| title_sort |
on the effective interaction of waves in inhomogeneous, nonstationary media |
| author |
Buts, V.A. Tolstoluzhsky, A.P. |
| author_facet |
Buts, V.A. Tolstoluzhsky, A.P. |
| topic |
Динамика пучков |
| topic_facet |
Динамика пучков |
| publishDate |
2017 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Про ефективну взаємодію хвиль у неоднорідних, нестаціонарних середовищах Об эффективном взаимодействии волн в неоднородных, нестационарных средах |
| description |
The description of new conditions of effective interaction of waves in periodically inhomogeneous and periodically nonstationary media is given. New conditions, as a special case, contain the known conditions of interaction of waves (synchronism conditions). Examples of the interaction of waves which characteristics satisfy the new interaction conditions are considered. These examples allow to detect new conditions in an experiment.
Даний опис нових умов ефективної взаємодії хвиль у періодично неоднорідних та нестаціонарних середовищах. Нові умови як окремий випадок містять відомі умови взаємодії хвиль (умови синхронізму). Розглянуто приклади взаємодії хвиль, характеристики яких задовольняють новим умовам взаємодії. Розглянуті приклади дозволяють виявити нові умови в експерименті.
Дано описание новых условий эффективного взаимодействия волн в периодически неоднородных и нестационарных средах. Новые условия, как частный случай, содержат известные условия взаимодействия волн (условия синхронизма). Рассмотрены примеры взаимодействия волн, характеристики которых удовлетворяют новым условиям взаимодействия. Рассмотренные примеры позволяют обнаружить новые условия в эксперименте.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/136193 |
| citation_txt |
On the effective interaction of waves in inhomogeneous, nonstationary media / V.A. Buts, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2017. — № 6. — С. 71-75. — Бібліогр.: 6 назв. — англ. |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2017. №6(112) 71
ON THE EFFECTIVE INTERACTION OF WAVES
IN INHOMOGENEOUS, NONSTATIONARY MEDIA
V.A. Buts
1,2,3
, A.P. Tolstoluzhsky
1
1
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
2
V.N. Karazin Kharkiv National University, Kharkov, Ukraine;
3
Institute of Radio Astronomy of the National Academy of Sciences of Ukraine (IRA NASU),
Kharkov, Ukraine
The description of new conditions of effective interaction of waves in periodically inhomogeneous and periodi-
cally nonstationary media is given. New conditions, as a special case, contain the known conditions of interaction of
waves (synchronism conditions). Examples of the interaction of waves which characteristics satisfy the new interac-
tion conditions are considered. These examples allow to detect new conditions in an experiment.
PACS: 41.20.Jb; 02.30.Jr; 02.60.Cl
INTRODUCTION
It is known that the effective interaction of waves in
weakly inhomogeneous, nonstationary and nonlinear
media occurs when the following conditions are ful-
filled: k 0i
i
k , 0i
i
. These condi-
tions mean that the detuning of the frequencies and
wave vectors of the interacting waves should be mini-
mal (see, for example, [1 - 3]). This also means that the
synchronism conditions between the interacting waves
must be satisfied along each of the four axes of the four-
dimensional space-time space. We note that often these
four conditions are called the laws of conservation of
energy and momentum in the interaction of waves. In-
deed, if each of these conditions is multiplied by the
Planck constant, then these are the laws of conservation
of energy and momenta in the interaction of individual
photons with each other. In our previous works [4, 5] it
was shown that in the general case, in some distributed
systems, some other relationships for the frequencies
and wave vectors of the interacting waves can be per-
formed for effective wave interaction. This possibility is
due to the fact that detuning along one of the directions
of the four-dimensional space can be compensated by
detunings along other directions. As a result, certain
lines (characteristic lines) can be identified in space
along which an effective exchange of energy is possible.
Effective exchange occurs, in spite of the fact that the
known conditions of interaction between waves (see
above) are not fulfilled. In this paper we consider the
simplest examples of the realization of such a wave in-
teraction. It is shown that in the interaction of two
waves in an inhomogeneous nonstationary medium, can
arise the waves, whose frequencies do not satisfy the
known conditions given above.
1. PROBLEM STATEMENT.
BASIC EQUATIONS
Let's consider a medium whose permittivity can be
represented as two terms. The first term is a constant.
The second term is assumed small, but is a periodic
function of space and time. As an example, we can con-
sider the following expression for such a permittivity:
0 , cos( ), 1.q r t q (1)
Let two electromagnetic waves propagate in such a
medium, the wave frequencies of which are different.
We will be interested in the conditions for the effective
interaction of these waves in such a medium. The equa-
tions for each of these waves are the Maxwell equation.
From the Maxwell equations it is easy to find the equa-
tions for the electric field vectors of each of these elec-
tromagnetic waves:
2
2 2
1 ( ) 1
.
E
E E
c t
(2)
By assumption, we have two waves, so we will seek
the solution of (2) as the sum of two terms:
0 0 0 1 1 1( , )exp( ) ( , )exp( ),E A r t i t ik r A r t i t ik r (3)
here
2 2 2 2 2 2
0 0 0 1 1 0/ , /k c k c .
Let us consider the simplest case, which shows the
most important characteristics of the new interaction
conditions and which, apparently, is most easily realized
in the experiment. We will assume that the interaction
occurs between transverse waves, that the medium is
periodically non-uniform in only one direction (in the z
direction). In this case, the time detuning can be com-
pensated only by a detuning along the z axis. Moreover,
we will assume that the waves are located on one dis-
persion linear branch. In this case, the phase velocities
of the waves and their group velocities coincide. As will
be seen below, all of these restrictions are non-essential
(they are imposed only to simplify the formulas) and, if
necessary, can be easily removed. We will also assume
that the waves propagate only in one direction in the
direction of the axis. In this case, substituting (3) in (2),
we can obtain the following equations for finding the
amplitudes
iA :
2 2
0 0 0 0 0 0 0
02 2 2 2
2
1
12
2 2
0 0 11 1 1 1
12 2 2 2
2
0
02
2
exp ( , ) ,
2
2
exp ( , ) ,
2
A A A A
i k
z tz c t c
q
A i z t
c
A A A A
i k
z tz c t c
q
A i z t
c
(4)
where ( , )r t k r t , 1 0k k k ,
1 0 .
ISSN 1562-6016. ВАНТ. 2017. №6(112) 72
This system is regorous. It is supposed that the de-
tuning though is arbitrary however it is chosen in
such a way that only these two waves can interact.
Characteristics of the equations (4) without the second
derivatives (i.e. subcharacteristics) are parallel to
straight lines:
( , )r t k z t const .
It means that derivatives along these subcharacteris-
tics are equal to zero: 0
, here z ,
0/t c .
The interaction of waves is due to the small inhomo-
geneity ( 1q ) of the dielectric constant. It is natural
to expect that the wave amplitudes will change slowly.
Therefore, in the system of equations (4), we can omit
the second derivatives.
It should be noted that this assumption always re-
quires additional analysis. In particular, as a minimum,
the obtaining solutions should be tested to satisfy this
assumption. We note that taking into account the second
derivatives in the system of equations (4), of course,
opens the possibility of the appearance of new solutions,
which may be interesting in their own way.
However, the questions arises: "How the presence of
second derivatives can change solutions that are ob-
tained without taking into account these derivatives."
Will the solutions obtained (within the framework of
accounting only the first derivatives) be stable with re-
spect to accounting for second derivatives? "This ques-
tion can be quite easily studied. Indeed, following [6],
we consider one equation from system (4), in which we
omit the right-hand side:
2 2
2 2
0
A A A A
z z
. (5)
In equation (5) and are arbitrary constants.
Add the following new variable: z . Equation (5)
in the new variables has the form:
2
4
A A A
. (6)
We will consider the dynamics of the jumps along
the sub-characteristics: 0z const and
0z const . For example, the amplitude jump
at propagation along the subcharacteristic
0 const has form:
0 0( , ) ( , )
A A
s
. (7)
Substituting into Eq. (6) one by one 0 and
0 , also combining obtained the equations, we find
the following equation, which describes the dynamics of
jump:
4
s
s
. (8)
From this equation it follows that the dynamics will
be stable if the following conditions are fullfield:
Re( ) 0 . Similarly, we can find the stability con-
dition for the solution as the jump propagates along the
second subcharacteristic. Finally, the stability condi-
tions for the solutions obtained by neglecting the second
derivatives will look:
Re( ) 0 , Re( ) 0 . (9)
These conditions are quite general. They are suitable
for the stability analysis in many applied problems. For
example, in the case of the propagation of wave beams
in inhomogeneous, nonstationary and nonlinear media.
In our case, it is easy to see that the coefficients and
purely imaginary. This means that in our case the
second derivatives are unable to radically change the
dynamics of the solutions obtained considering only the
first derivatives.
We drop the second derivatives on the left-hand side
of system (4). Then the left-hand side of these equations
can be regarded as a derivative along the characteristic
lines: z C const . Moreover, these directions
for interacting waves coincide. We also pay attention to
the fact that the right-hand sides of the system of equa-
tions (4) contain complex conjugate factors
exp ( , )i z t . In this case, taking into account that
0
or 0/k c (10)
in the system of equations (4) differentiating the left and
right sides of the equations with respect to the new vari-
able, we obtain the following equations for determining
the amplitudes
0A and
1A :
2
2
2
0 0,1i
i
A
K A i
, (11)
where 2 2
0 1 / 64K q k k .
Solutions of equations (11) can be, for example,
functions:
0 cos( )A a ,
1 sin( )A b .
This choice of solutions, in particular, can mean that
there is a periodic transfer of energy from one wave to
the second wave in the process of their interaction.
The interaction of waves considered above occurs
when they propagate in the same direction, in the direc-
tion of the positive z axis (
0 10, 0k k ). It is easy to
show that if they propagate in the opposite direction
(
0 10, 0k k ), then the interaction will occur under
the same laws of interaction. However, this interaction
will occur along another subcharacteristic. Namely,
along z . In this case, the system of equations
(11) will be transformed into a system of equations:
2
2
2
0,i
i
A
K A
(12)
where z , 0,1i .
2. INTERACTION IN THE LAYER
In this paper we study new conditions for the effec-
tive interaction of waves in inhomogeneous media. In
addition to general theoretical considerations, it is of
interest to consider some simple case in which the basic
elements of the new interaction conditions under con-
sideration were contained and that would be as simple
as possible so that they could be realized in the experi-
ISSN 1562-6016. ВАНТ. 2017. №6(112) 73
ment. In this section such case will be considered. It is a
layer of thickness L of an inhomogeneous medium.
The boundaries of the layer are ideally reflective. Inside
the layer, two waves propagate perpendicular to its
boundaries. Each of these waves represents the sum of
two waves, one of which propagates along the z axis,
and the other is a wave reflected from the boundary
(Fig. 1).
Fig. 1. Interacting waves in layer
Dynamics of interaction of each of these waves has
been described above. Expression for the field compo-
nents in such a system can be represented in such form:
0 0 0
1 1 1
0 0 0
1 1 1
A exp( ) exp( ) exp( )
Cexp( ) exp( ) exp( ),
A exp( ) exp( ) exp( )
Cexp( ) exp( ) exp( ).
x
y
E ik z B ik z i t
ik z D ik z i t
H ik z B ik z i t
ik z D ik z i t
(13)
In formula (13) is homogeneous part (constant)
dielectric permeability of the layer 0 , 1z L , but
, , ,A B C D are slow functions of time and coordinate.
The analytical type of these functions is determined by
the solutions of equations (11) and (12). To determine
the constants of these functions, it is necessary to use
the boundary conditions. In this case, they are simple:
0 :z 0; 0A B C D .
:z L
0 0
1 1
( ) exp( ) ( )exp( ) 0;
exp( ) exp( ) 0.
A L ik L B L ik L
C ik L D ik L
(14)
In the system equations (14) functions , , ,A B C D
can be presented in the form:
exp(i )A a ; exp( i )B b ;
exp(i )C c ; exp( i )D d . (15)
In the expressions (15) a,b, ,dc constants. Taking
these expressions into account, the algebraic system of
equations (14) can be rewritten as follows:
0 :z a b ; c d .
:z L
0 0exp ( ) exp ( )a i k L i b i k L i . (16)
We note that for the effective interaction of waves in
the considered dielectric layer, it is necessary that the
wave numbers satisfy the following relation:
0k L n . (17)
Taking into account the relations (16), the expres-
sion for the electric component of the total field in the
layer can be expressed by the following formula:
0 0
0 0 1
1 12
1
sin exp
2 e
sin exp
4
i
k z i
E ia k k k
k z i
k
(18)
or
0 0
0 0 1
1 12
1
Re sin sin
sin sin .
4
E a k z
k k k
a k z
k
. (19)
3. INVESTIGATION OF THE DYNAMICS
OF INTERACTION OF WAVES
BY NUMERICAL METHODS
Conditions of the effective interaction of waves in
inhomogeneous media have been verified by numerical
methods. For this purpose introducing new dimension-
less variables
0 1 0 0 1/ ;k k t c k k z , and
also introducing new dimensionless amplitudes
0 0
3 34
0 1
1
( , ) ( , )a A
k k
, 1 1
3 34
0 1
1
( , ) ( , )a A
k k
from (4) (without taking into account the second deriva-
tives) we will obtain the system of equations for the first
derivatives in the form:
0 0
1
1 1
0
exp ( , ) ,
4
exp ( , ) .
4
a a q
a i
i
a a q
a i
i
(20)
Here ( , ) k and introduce dimen-
sionless detunes 01
0 1 0 1
kk
k
k k k k
,
001
0 1 0 1
kk
k k c k k
.
The initial and boundary conditions are chosen in
accordance with the analytical solutions. Amplitude
value of the field
0 1a . The parameter 0.8, 0.1q .
The results of the analytical and numerical analysis
of the system of first-order equations obtained from (4)
are presented in Figs. 2-4. In Fig. 2 at 0.8q presented
dynamics of the interaction of waves for the case of ful-
fillment of known synchronism conditions
( 0k ), also for the case of identical detunings
( 0k ). As can be seen from this plot, the dy-
namics of the interaction of waves is practically the same.
If there is a detuning along one of the directions (for ex-
ample 1.095k ), there is no effective interaction be-
tween the waves (Fig. 3). Analytic and numerical solu-
tions coincide with a good degree of accuracy.
Fig. 2. Analytic and numerical values of the field ampli-
tudes along the line for detuning 0k ,
1.095k , 0.707k , 0.8q ,
0Re( )a red line,
1Im( )a blue line (dotted line)
Fig. 3 shows that the interaction of waves is practi-
cally absent. Indeed, the amplitude of the wave oscilla-
tions, which is caused by interaction with other waves in
1
0
0 L
z
ISSN 1562-6016. ВАНТ. 2017. №6(112) 74
absolute magnitude, is negligible. Besides, it is seen that
the frequency of these oscillations of the amplitude cor-
responds to the value of detuning.
Fig. 3. Value of amplitude
0Re( )a along the line
for detuning 1.095k , 0 ; 0.1q
Fig. 4. Plots of amplitude
0Re( )a
a – 0k , b – 0.707k ; 0.8q
Fig. 4 shows that both in the case of the known syn-
chronism conditions and in the case of equal detunings,
the amplitude of the initial wave varies periodically
along the characteristic line . Notice that under
the known conditions of synchronism the real part of the
amplitude of the initial wave is strictly transferred into
the imaginary part of the second wave and back. When
the new conditions are fulfilled, the energy exchange
process is more complicated, which is represent in
Fig. 4,b. It should also be noted that, unlike the known
synchronism conditions, in which the energy exchange
between waves can be observed in the ordinary space,
the energy exchange between the waves in the presence
of detunings in the general case is effectively observed
along the selected subcharacteristic. This feature of the
interaction of waves in the presence of detuning is clear-
ly visible from Fig. 2 at 1.095k .
CONCLUSIONS
Thus, the results obtained above show that, in addi-
tion to the known conditions for the effective interaction
of waves ( 0k ), there are more general condi-
tions for energy exchange between the waves.
In the case considered above, these conditions have
a simple form k
c
. Under these conditions,
detunings k and are almost arbitrary values. As
can be seen, these new conditions contain, as a particu-
lar case, known synchronism conditions 0k .
To visually see the difference in old and new condi-
tions, in the Fig. 5 shows an example of waves which
can effectively interact is presented. If to use the known
conditions of interaction, then for most of these waves
they can't be fulfilled. We use the new conditions of
synchronicity. It is easy to see that any triplets of waves
that are represented in the figure satisfy the new syn-
chronism conditions /k c . They can effi-
ciently exchange energy. There are infinitely many such
triples.
An important example of the conditions under con-
sideration is an example of the interaction of waves in a
plane layer, which was considered above. Indeed, as can
be seen from the formula (19), the expression for the
real component of the electric field can be easily meas-
ured in real experiments. It can be seen that the appear-
ance of waves with new frequencies (
1 ) can easily be
observed if at the initial point of time only one wave
with a frequency (
1 ), in a layer there was only one
wave with a frequency (
0 ).
Fig. 5. Dispersion effectively interaction of waves
at 0k
If we consider an unbounded region of interaction,
then the new synchronism conditions are not very con-
venient to observe. They can effectively manifest them-
selves, for example, in systems with electron beams. In
other cases, it is rather difficult to observe these condi-
tions. However, if the wave interaction region is bound-
ed, for example, as was done in Section 3 above, then it
follows from formulas (18), (19) that it is easy to observe
these conditions. Indeed, it is enough to fix the probe at
an arbitrary point in the layer. And in time, the probe will
appear oscillations both at the frequency of the original
wave and at frequencies that might not appear when the
known synchronism conditions are fulfilled.
REFERENCES
1. G.M. Zaslavsky, R.Z. Sagdeev. Introduction to Non-
linear Physics: From the Pendulum to Turbulence
and Chaos. M.: “Nauka”, 1988, 368 p. (in Russian).
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Article received 11.10.2017
ISSN 1562-6016. ВАНТ. 2017. №6(112) 75
ОБ ЭФФЕКТИВНОМ ВЗАИМОДЕЙСТВИИ ВОЛН В НЕОДНОРОДНЫХ,
НЕСТАЦИОНАРНЫХ СРЕДАХ
В.А. Буц, А.П. Толстолужский
Дано описание новых условий эффективного взаимодействия волн в периодически неоднородных и не-
стационарных средах. Новые условия, как частный случай, содержат известные условия взаимодействия
волн (условия синхронизма). Рассмотрены примеры взаимодействия волн, характеристики которых удовле-
творяют новым условиям взаимодействия. Рассмотренные примеры позволяют обнаружить новые условия в
эксперименте.
ПРО ЕФЕКТИВНУ ВЗАЄМОДІЮ ХВИЛЬ У НЕОДНОРІДНИХ,
НЕСТАЦІОНАРНИХ СЕРЕДОВИЩАХ
В.О. Буц, О.П. Толстолужський
Даний опис нових умов ефективної взаємодії хвиль у періодично неоднорідних та нестаціонарних сере-
довищах. Нові умови як окремий випадок містять відомі умови взаємодії хвиль (умови синхронізму). Розг-
лянуто приклади взаємодії хвиль, характеристики яких задовольняють новим умовам взаємодії. Розглянуті
приклади дозволяють виявити нові умови в експерименті.
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