About conditions of effective interaction of waves in non-uniform, non-stationary and nonlinear medium
It is shown that in the distributed systems there are more general conditions for effective interaction of waves (synchronism) than the well known conditions. The well known conditions of synchronism are just the particular case of these more general conditions. It has been shown that these new cond...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | About conditions of effective interaction of waves in non-uniform, non-stationary and nonlinear medium / V.A. Buts // Вопросы атомной науки и техники. — 2010. — № 6. — С. 117-119. — Бібліогр.: 5 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860238692500635648 |
|---|---|
| author | Buts, V.A. |
| author_facet | Buts, V.A. |
| citation_txt | About conditions of effective interaction of waves in non-uniform, non-stationary and nonlinear medium / V.A. Buts // Вопросы атомной науки и техники. — 2010. — № 6. — С. 117-119. — Бібліогр.: 5 назв. — англ. |
| collection | DSpace DC |
| description | It is shown that in the distributed systems there are more general conditions for effective interaction of waves (synchronism) than the well known conditions. The well known conditions of synchronism are just the particular case of these more general conditions. It has been shown that these new conditions lead to new opportunities for effective interaction of wave, particularly at interaction of waves in plasma.
Показано, что в распределенных системах существуют более общие условия эффективного взаимодействия волн (синхронизма), чем известные условия. Из этих более общих условий, как частный случай, следуют известные условия. Показано, что новые условия могут приводить к неизвестным ранее процессам взаимодействия волн, в частности, при взаимодействии волн в плазме.
Показано, що в розподілених системах існують більш загальні умови ефективної взаємодії хвиль (синхронізму), чим відомі умови. Із цих більш загальних умов, як окремий випадок, випливають відомі умови. Показано, що нові умови можуть приводити до невідомих раніше процесам взаємодії хвиль, зокрема, при взаємодії хвиль у плазмі.
|
| first_indexed | 2025-12-07T18:27:35Z |
| format | Article |
| fulltext |
ABOUT CONDITIONS OF EFFECTIVE INTERACTION OF WAVES
IN NON-UNIFORM, NON-STATIONARY AND NONLINEAR MEDIUM
V.A. Buts
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: vbuts@kipt.kharkov.ua
It is shown that in the distributed systems there are more general conditions for effective interaction of waves (syn-
chronism) than the well known conditions. The well known conditions of synchronism are just the particular case of
these more general conditions. It has been shown that these new conditions lead to new opportunities for effective inter-
action of wave, particularly at interaction of waves in plasma.
PACS: 05.45.-a; 52.35.Mw
1. INTRODUCTION
The processes of coherent interaction of the waves in
periodic – non-uniform, periodic – non-stationary and in
nonlinear medium are well investigated. At this the ef-
fective interaction occurs only at performance of condi-
tions of synchronism. In a general form these conditions
(see, for example, [1,2]) can be written down in such
kind:
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2010. № 6. 117
Series: Plasma Physics (16), p. 117-119.
=0, 0i i
i i
kω =∑ ∑
r
, (1)
here iω - frequency of the interacting waves; ik
r
- wave
vectors of these waves.
Among possible processes of interaction the greatest
importance has the processes with participation of three
waves (three-wave processes). Below we shall be limited
by consideration only of three-wave processes. Most sim-
ple is the case, when the parameters of one of waves do
not vary during interaction. It can be the waves of dielec-
tric permeability, potential waves, initial stage of
�isintergration processes and others. Namely such proc-
esses we, first of all, will be considered.
2. INTERACTION OF WAVES IN MEDIUM
WITH PERIODICALLY VARYING
DIELECTRIC PERMEABILITY
Let plain transverse wave with frequency 0ω and
wave vector is propagating in medium which dielectric
permeability can be described by the formula:
0k
r
0 , cos( ),q r t q 1ε ε ε ε κ= + = −Ω <<
rr
% % . (2)
The presence of the second addend will result that the
part of energy of a zero wave will be transformed in a
wave with frequency 1ω and wave vector
r
. Let's name
this wave by a wave with number 1. The conditions at
which, the significant part of energy from a zero wave can
be transformed in a first wave will be interesting for us.
From Maxwell equations we can get such equation for a
vector of an electrical field for waves that are propagating
in medium with dielectric permeability (2):
1k
2
2 2
1 ( ) 1EE
c t
ε E ε
ε
∂ ⎛Δ − = −∇ ⋅∇⎜∂ ⎝ ⎠
⎞
⎟
r rr r
ω
rr r r r
2c
r
. (3)
The solution of the equation (3) we shall search as
the sum of two components:
r rr
0 0 0 1 1 1( , )exp( ) ( , )exp( ).E A r t i t ik r A r t i t ik rω= − + + − +
r
(4)
First component in (4) describes a zero wave,
second – the wave, which is born as a result of presence
of periodic heterogeneity. At this we assume, that
between wave vectors and frequencies of both waves exist
usual connection: .
Substituting the solution (4) in the equation (3) after
bulky, but simple calculations, we can receive the
following system of the differential equations in partial
derivatives of the first order for definition of dynamics in
time and space of amplitudes
2 2 2 2 2
0 0 0 1 1 0/ /k c kω ε ω ε= =
0A
r
and 1A
r
:
( ) ( ) ( ) ( )
2 2
10
1 1 12
0 0 0
1 exp
4
A q i A k A
l N c
ω
κ κ δ
ε
⎧ ⎫+Ω∂ ⋅ ⎪ ⎪= ⋅ ± ±⎨ ⎬∂ ⎪ ⎪⎩ ⎭
i
r
rr rr r
( ) ( ) ( )
( )
2 2
01
0 0 02
1 1 0
1
4
exp .
A q i A k A
l N c
i
ω
κ κ
ε
δ
⎧ ⎫+Ω∂ ⋅ ⎪ ⎪= ⋅ ×⎨ ⎬∂ ⎪ ⎪⎩ ⎭
× −
r
rr rr r
m m
(5)
Here ( , )r tδ r - function determining detuning between
wave vectors and frequencies:
( , )r t k r tδ ω≡ Δ ⋅ − Δ ⋅
r r ; ( )1 0k k k κΔ ≡ − ±
r r r
;r r
1 0ω ω ωΔ ≡ − ±Ω ; 2 2 2 2 2/i ix iy iz iN k k k cε ω= + + + ⋅ - norms
for each of waves. Derivative in system (5) are taken
along characteristic directions:
{ }0,1 0,1, 0,1, 0,1, 0 0 0,1; ; ; / /x y zl k k k c Nω ε= ⋅
r
.
r
and 0l 1l
r
- the
unit vectors along characteristic directions.
At getting (5) we assumed, that the amplitudes are
slow functions of time and coordinates. Their changes are
caused only by presence of the weak spatial - temporary
heterogeneity ( q 1<< ). For this reason components with
second derivatives in system (5) had been omitted. This
fact we should remember. The solutions which are not
satisfied to these assumptions should be rejected.
For slow change of amplitudes it is necessary, that
the right parts of the equations (5) were varied slowly. It
will be occur when ( ),r t C constδ = =
r . This equation
represents the equation of hyperplane in four dimensional
space ( ,r tr ). This detuning will be not varying during an
exchange of energy between waves, if the characteristic
straight lines will be parallel to this hyperplane.
So, the characteristic lines
mailto:vbuts@kipt.kharkov.ua
0 0;xx k xα= + 0 0;yy k yα= + 0 0;zz k zα= +
1 1 1;xx k xα= + 1 1;yy k yα= + 1 1 1;zz k zα= + (6)
( )2
0 0 0/ ;t c t
118
α ω ε= ⋅ ⋅ + ( )2
1 1 0 1/t c tα ω ε= ⋅ ⋅ + ,
should be parallel to hyperplane ( ),r t C constδ = =
r .
These conditions have such form:
( ) 2
0 0 0k k c ω ω εΔ ⋅ − Δ ⋅ ⋅ =
r r
0 ,
( ) 2
1 1 0k k c ω ω εΔ ⋅ − Δ ⋅ ⋅ =
r r
0 .
(7)
These conditions are represented on Fig.1. It is easy
to check that derivations along l
r
и
r
from detuning at
conditions (7) will be equal to zero:
0 1l
( ) ( )0 1, / , / 0r t l r t lδ δ∂ ∂ = ∂ ∂
r r
= .
Fig. 1. The circuit of a mutual arrangement of vectors
r
,
r
, and k
r
necessary for effective interaction
of waves
kΔ 0k 1
It is interesting to analyze cases which are not man-
aged by the known schema of interaction. The simplest of
them, apparently, is the case of interaction of waves in
stationary medium (Ω = ). Besides we shall assume
that
0
0ωΔ = . From conditions (7) for this case, in particu-
lar, follows, that the effective interaction of waves will
occur at performance of conditions:
0 1k k= −
r r
, ( )0 12 / cosk kκ κ=
r rr
⋅
r . (7а)
It is easy to see, that the second condition differs
from a well known Bragg condition ( 02 kκ =
r
). It is easy
to show, that at performance (7а) occur full reflection of
waves from non-uniform medium (crystal). More details
see in [3].
r
3. INTERACTION OF WAVES
IN NONLINEAR MEDIUMS
The equations, which describe interaction of three
waves ( 0 1 2, ,ω ω ω ; k k
r r
) in nonlinear mediums, are
possible to present as:
0 1 2, ,k
r
( ) ( )0 0 0 0 0 0 1 2/ ea l a V a a a ixpσ δ∂ ∂ ≡ + ∇ = − ⋅ ⋅ ⋅ − ⋅
rr
& ;
( ) ( )*
1 1 1 1 1 1 0 2/ ea l a V a a a ixpσ δ∂ ∂ ≡ + ∇ = ⋅ ⋅ ⋅ ⋅
rr
& ;
( ) (*
2 2 2 2 2 2 0 1/a l a V a a a i )expσ δ∂ ∂ ≡ + ∇ = ⋅ ⋅ ⋅ ⋅
rr
& . (8)
Here V - group velocity of waves; i
The left part of each of the equations of system (8) is
submitted as derivative along characteristic directions.
For distinctness we shall examine interaction of waves
with positive energy ( 0iσ > ), and also we shall be
guided by processes of disintegration. The conditions of
synchronism of interacting waves will be a conditions of
parallelism between characteristic lines and hyperplane
( , )r t constδ =
r i.e. such conditions:
0ik VωΔ −Δ ⋅ =
r r
. (9)
The linear stage of the disintegration process pro-
ceeds as instability. At this stage it is possible to consider
a wave with the maximal frequency as don't changed
wave ( 0a const= ). In this case the systems (8) are con-
venient to rewrite as:
( )*
1 1 1 0 2/ exa l a a ipσ δ∂ ∂ = ⋅ ⋅ ⋅ ⋅ ;
( )2 2 2 0 1/ expa l a a iσ δ∗ ∗∂ ∂ = ⋅ ⋅ ⋅ − ⋅ .
(10)
Substituting in (10) the solution in form
~ exp( )i t i rκΩ −
rr , we will get such dispersion equation:
( ) 22 2
1 2 1 2 1 2 0 0V V V V aκ κ σ σ⎡ ⎤Ω −Ω⋅ + + ⋅ ⋅ + =⎣ ⎦
r r r rr . (11)
At the solving (11) we should from all set of the pos-
sible solutions to consider only that which frequency Ω
and vector κr will be small sizes ( 0~ ~ aκΩ ). Solving
the equation (11) for Ω in this case we shall get:
0 1 2V i aκ σ σΩ = ± ⋅
rr . (12)
The imaginary part of frequency
( )0 1 2Im a σ σΩ = determines the increment of the
decay instability.
It is interesting to find those new opportunities for
disintegrations, which are not management by known
conditions of disintegrations. Let's consider simplest. Let
there is a disintegration of a transverse wave on a
transverse wave and on one of the own waves of the
magnetized plasma waveguide ( t t ). Let's
consider that all three waves are inside a linear part of
dispersion curve (Fig. 2). In this case the group velocities
of all three waves coincide (V V
r r
). Moreover,
the group velocities in this case coincide with phase
velocity. The conditions of synchronism (9) will be
carried out in this case for any three of the waves. Really,
the conditions (9) in this case get the kind of identity:
0 1 l→ +
V V= = ≡
r r
0 1 2
0 1 2
0 1 2 V
V V V
ω ω ωω ω ω ⎛− − = − −⎜
⎝ ⎠
⎞
⎟ . (13)
Fig. 2. The circuit of disintegration of a cross
wave on a cross wave and on a wave of
magnetized plasma waveguide
r
iσ - matrix elements
of nonlinear interaction.
It is significant also that the right parts in equations (5)
and (10) have detuning phases ( ( , )r tδ r ) with opposite
signs. That is why this detuning was disappeared from
dispersion equation (11).
Thus, in a considered case in process disintegration
can to take part the large number ensembles from three
waves. And, the characteristics of these waves can a little
differ from each other. In this case, as shown in work [4],
the process of disintegration becomes chaotic. It is
necessary to notice, that the considered case of coinciding
of all velocity can be solved analytically not only at a
linear stage, but also on nonlinear. For this purpose it is
enough to use a method of effective potential (see, for
example, [5]). However, because this process became
chaotic character, this solution loses interest.
REFERENCES
1. G.M. Zaslavsky, R.Z. Sagdeev. Introduction to non-
linear physics / The main editorial of the physical and
mathematical literature. Мoscow: “Science”, 1988,
p. 368
2. B.B. Kadomthev. The collective phenomena in plasma /
The main editorial of the physical and mathematical
literature. Мoscow: “Science”. 1976, p. 238.
4. CONCLUSIONS
In the conclusion we shall formulate the basic reason
of an opportunity of occurrence of new conditions of
synchronism. If in the equations (5) and (8) we carry out
the averaging, for example, on a small interval of
time ~ 1/t ωΔ Δ , the slowness of change of amplitudes of
interacting waves will be only at performance of a usual
condition 0ωΔ = . Similarly, integrating on a small spatial
interval, we shall get habitual conditions: . 0kΔ =
r
3. V.A. Buts. About conditions of synchronism at
interaction of waves in heterogeneous, non-stationary
and non-linear mediums // Progress of contemporary
radioelectronics. 2009, N5, p. 13-22.
4. V.A. Buts, A.N. Kupriyanov, O.V. Manuilenko,
A.P. Tolstoluzhsky. Instability and dynamical chaos at
weak nonlinear interaction of the waves // Izvestiya
Vuzov: «Aplied nonlinear dynamics». 1993, v.1, N 1-2,
p. 57-62 (in Russian). However process of wave interaction occurs not only
in time, but also in space. He occurs along the
characteristics. Therefore it is enough (for effective
interaction of waves) that the right parts of the equations
(5) and (8) changed slowly along these directions. This
fact is expressed by equality to zero derivatives from
detuning along characteristic directions (see (7)).
5. J. Weiland, H. Wilhelmson. Coherent non-linear
interaction of waves in plasmas. Мoscow:
“Energoatomizdat”, 1981, p. 224.
Article received 13.09.10
ОБ УСЛОВИЯХ ЭФФЕКТИВНОГО ВЗАИМОДЕЙСТВИЯ ВОЛН В НЕОДНОРОДНЫХ,
НЕСТАЦИОНАРНЫХ И НЕЛИНЕЙНЫХ СРЕДАХ
В.А. Буц
Показано, что в распределенных системах существуют более общие условия эффективного взаимодействия
волн (синхронизма), чем известные условия. Из этих более общих условий, как частный случай, следуют из-
вестные условия. Показано, что новые условия могут приводить к неизвестным ранее процессам взаимодейст-
вия волн, в частности, при взаимодействии волн в плазме.
ПРО УМОВИ ЕФЕКТИВНОЇ ВЗАЄМОДІЇ ХВИЛЬ У НЕОДНОРІДНИХ, НЕСТАЦІОНАРНИХ ТА
НЕЛІНІЙНИХ СЕРЕДОВИЩАХ
В.О. Буц
Показано, що в розподілених системах існують більш загальні умови ефективної взаємодії хвиль (синхроні-
зму), чим відомі умови. Із цих більш загальних умов, як окремий випадок, випливають відомі умови. Показано,
що нові умови можуть приводити до невідомих раніше процесам взаємодії хвиль, зокрема, при взаємодії хвиль
у плазмі.
119
|
| id | nasplib_isofts_kiev_ua-123456789-17476 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:27:35Z |
| publishDate | 2010 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Buts, V.A. 2011-02-26T21:57:57Z 2011-02-26T21:57:57Z 2010 About conditions of effective interaction of waves in non-uniform, non-stationary and nonlinear medium / V.A. Buts // Вопросы атомной науки и техники. — 2010. — № 6. — С. 117-119. — Бібліогр.: 5 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/17476 It is shown that in the distributed systems there are more general conditions for effective interaction of waves (synchronism) than the well known conditions. The well known conditions of synchronism are just the particular case of these more general conditions. It has been shown that these new conditions lead to new opportunities for effective interaction of wave, particularly at interaction of waves in plasma. Показано, что в распределенных системах существуют более общие условия эффективного взаимодействия волн (синхронизма), чем известные условия. Из этих более общих условий, как частный случай, следуют известные условия. Показано, что новые условия могут приводить к неизвестным ранее процессам взаимодействия волн, в частности, при взаимодействии волн в плазме. Показано, що в розподілених системах існують більш загальні умови ефективної взаємодії хвиль (синхронізму), чим відомі умови. Із цих більш загальних умов, як окремий випадок, випливають відомі умови. Показано, що нові умови можуть приводити до невідомих раніше процесам взаємодії хвиль, зокрема, при взаємодії хвиль у плазмі. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Плазменная электроника About conditions of effective interaction of waves in non-uniform, non-stationary and nonlinear medium Об условиях эффективного взаимодействия волн в неоднородных, нестационарных и нелинейных средах Про умови ефективної взаємодії хвиль у неоднорідних, нестаціонарних та нелінійних середовищах Article published earlier |
| spellingShingle | About conditions of effective interaction of waves in non-uniform, non-stationary and nonlinear medium Buts, V.A. Плазменная электроника |
| title | About conditions of effective interaction of waves in non-uniform, non-stationary and nonlinear medium |
| title_alt | Об условиях эффективного взаимодействия волн в неоднородных, нестационарных и нелинейных средах Про умови ефективної взаємодії хвиль у неоднорідних, нестаціонарних та нелінійних середовищах |
| title_full | About conditions of effective interaction of waves in non-uniform, non-stationary and nonlinear medium |
| title_fullStr | About conditions of effective interaction of waves in non-uniform, non-stationary and nonlinear medium |
| title_full_unstemmed | About conditions of effective interaction of waves in non-uniform, non-stationary and nonlinear medium |
| title_short | About conditions of effective interaction of waves in non-uniform, non-stationary and nonlinear medium |
| title_sort | about conditions of effective interaction of waves in non-uniform, non-stationary and nonlinear medium |
| topic | Плазменная электроника |
| topic_facet | Плазменная электроника |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/17476 |
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