Localization of Resonant Spherical Waves
This paper treats radial spherical resonant waves excited in the transresonant regime. An approximate general solution of a perturbedwave equation is presented here, which takes into account nonlinear, spatial, and dissipative effects. Then a boundary problem reduces to the perturbed compound...
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| Опубліковано в: : | Проблемы прочности |
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| Дата: | 2002 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут проблем міцності ім. Г.С. Писаренко НАН України
2002
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Localization of Resonant Spherical Waves / Sh.U. Galieva, O.P. Panova // Проблемы прочности. — 2002. — № 1. — С. 102-111. — Бібліогр.: 17 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859803040894156800 |
|---|---|
| author | Galiev, Sh.U. Panova, O.P. |
| author_facet | Galiev, Sh.U. Panova, O.P. |
| citation_txt | Localization of Resonant Spherical Waves / Sh.U. Galieva, O.P. Panova // Проблемы прочности. — 2002. — № 1. — С. 102-111. — Бібліогр.: 17 назв. — англ. |
| collection | DSpace DC |
| container_title | Проблемы прочности |
| description | This paper treats radial spherical resonant
waves excited in the transresonant regime. An
approximate general solution of a perturbedwave
equation is presented here, which takes
into account nonlinear, spatial, and dissipative
effects. Then a boundary problem reduces to
the perturbed compound Burgers-Kortewegde
Vries equation (BKdV) in time. Several solutions
to this equation are constructed. Shock
waves may be excited near resonance according
to the solutions for an inviscid medium.
However, both viscosity and spatial dispersion
begin to be important very close to resonance
and prevent the formation of shock discontinuity.
As a result, periodic localized excitations
are generated in resonators instead of shock
waves.
Рассматриваются радиальные сферические резонансные волны, возбуждаемые в трансрезонансном
режиме. Приближенное общее решение возмущенного волнового уравнения
представляется в виде, учитывающем нелинейные, пространственные и диссипативные
эффекты. Граничная задача сводится к возмущенному смешанному уравнению Бюргера-
Кортевега-де Вриза, для которого построено несколько решений. Установлено, что в
невязкой среде вблизи резонанса могут возникать ударные волны. Однако как вязкость, так
и пространственная дисперсия вблизи резонанса предотвращают формирование ударного
разрыва, в результате чего в резонаторе вместо ударных генерируются периодические
локализованные волны.
Розглядаються радіальні сферичні резонансні хвилі, що збуджуються в
трансрезонансному режимі. Наближений загальний розв’язок збуреного
хвильового рівняння записується з урахуванням нелінійних, просторових і
дисипативних ефектів. Гранична задача зводиться до збуреного змішаного
рівняння Бюргера-Кортевега-де Вріза, для якого побудовано декілька розв’
язків. Установлено, що в нев’язкому середовищі поблизу резонансу можуть
виникати ударні хвилі. Однак як в ’язкість, так і просторова дисперсія
поблизу резонансу запобігають формуванню ударного розриву, в результаті
чого в резонаторі замість ударних генеруються періодичні локалізовані
хвилі.
|
| first_indexed | 2025-12-07T15:14:22Z |
| format | Article |
| fulltext |
UDC 539.4
Localization of Resonant Spherical Waves
Sh. U. Galieva and O. P. Panovab
a Department of Mechanical Engineering, The University of Auckland, Auckland, New
Zealand
b Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev,
Ukraine
УДК 539.4
Локализация резонансных сферических волн
Ш. У. Галиева, О. П. П анова6
а Отделение машиностроения Оклендского университета, Окленд, Новая Зеландия
6 Институт проблем прочности НАН Украины, Киев, Украина
Рассматриваются радиальные сферические резонансные волны, возбуждаемые в транс
резонансном режиме. Приближенное общее решение возмущенного волнового уравнения
представляется в виде, учитывающем нелинейные, пространственные и диссипативные
эффекты. Граничная задача сводится к возмущенному смешанному уравнению Бюргера-
Кортевега-де Вриза, для которого построено несколько решений. Установлено, что в
невязкой среде вблизи резонанса могут возникать ударные волны. Однако как вязкость, так
и пространственная дисперсия вблизи резонанса предотвращают формирование ударного
разрыва, в результате чего в резонаторе вместо ударных генерируются периодические
локализованные волны.
One-side travelling nonlinear waves have been the subject of intense studies
for the last decades [1-4]. In finite physical systems both left and right travelling
waves may be excited. Near the resonant frequencies, the amplitudes of these
waves increase. As a result, the balance between nonlinear, dissipative, and
dispersive effects varies together with the excited frequency. Therefore, in the
transresonant frequency band both shock and soliton-like waves may be excited in
resonators. This dynamics was studied in [5-7] for the case of plane resonant
waves in elongated resonators. Here we consider the spatial effect on the
evolution of nonlinear waves in transresonant frequency bands. For simplicity,
spherically symmetric pressure waves excited in a gas or liquid sphere are
discussed. An oscillating monopole is located at the origin. Apparently, these
types of d riven re so n a n t-d iss ip a tive th ree-d im en sion a l systems were not
considered earlier.
In accordance with [8], we write an equation of nonlinear acoustics for
spherical waves taking into account only linear and quadratic terms, respectively,
for the velocity potential p:
a o(P rr + 2r_1p r ) = p tt + ( y - W p t p t t + 2P r p r t -<Sa o 2p ttt , (1)
© Sh. U. GALIEV, O. P. PANOVA, 2002
102 ISSN 0556-171X. Проблемы прочности, 2002, № 1
Localization o f Resonant Spherical Waves
where a o is the speed of sound in an undisturbed medium, y is the polytropic
exponent of gas (Eq. (1) is also valid for fluid [8]), 5 is the so-called “sound
diffusivity.” The subscripts t and r indicate the time and space derivatives,
respectively. We emphasize that Eq. (1) does not take into account the third order
effects and the dissipative term is of the second order [8, 9]. The solution of (1)
can be presented as
P = P 1 + P 2 , (2)
where P 1 and P 2 are the first- and the second-order values, respectively.
Substituting Eq. (2) into (1) and equating the values of the same order, we obtain
a system of differential equations for p 1 and p 2 :
p 1rr + 2r 1p 1r = a 0 2p 1tt, (3)
a 0(P 2rr + 2r _1P 2r ) = P 2tt + 2P 1rP 1rt + (y ~ 1)a0 2P 1tP 1tt - ^ a 0 2P 1ttt. (4)
The solution of (3) is the sum of diverging and converging waves:
P 1 = r - 1 ( f 1 + f 2 ). (5)
Here and hereinafter, f 1 = f 1(£) and f 2 = f 2(^), where £ = a 0 1 - r and
^ = a 0 1 + r. With allowance for (5), we rewrite Eq. (4) in the form
a 0( P 2rr + 2 r -1 P 2r ) - P 2tt = 2a 0 r - 2 ( f 2 - f 1)( f 2 '- f 1,) -
- 2 a 0 r - 3 [ ( f 2 + f 1 ) (f2 ' - f / ' ) - (f 1)2 + (f 2 )2 ] + 2a0 r - 4 ( f 2 + f 1 ) ( f2 + f 1) +
+ ( y - 1)a0r - 2 ( f 2 + f ' ) ( f 2'+ f / ' ) - 5 a 0 r -1 ( f 2''+ f / ' ) , (6)
where the primes denote a derivative with respect to the argument. The solution to
(6) is
P 2 = r - 1 ( ^1 + P 2 ) + 0 5 a 0 1 r -2 [ ( f1 + f 2 )2]' -
- 0.25( y + 1)a - r -1 / / r -1 (f1 + f 2 )(f1 '+ f 2' )d^drj +
+ 0.25 (5a 0~1r -1 ( ̂ /Y'+ §/' 2'). (7)
Here and above, the functions f 1, f 2 , p 1 = ^ 1(^ ), and ip 2 = ^ 2( ̂ ) are
unknown and must be found from the initial and boundary conditions of the
corresponding problem [9]. However, it is complicated to solve boundary
problems using (2) due to the integral in (7). To simplify solution (7), let us
replace the multiplier 1/ r under the integral by 1 /R t . As a result, near the
boundary surfaces r = R t ( i = 1, 2) and
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Sh. U. Galiev and O. P. Panova
<P = r 1(/ l + f 2 + P 1 + P 2 ) + 0 1r 2 [ ( / l + f 2 )2 ] ' -
-0 .2 5 ( y + 1)a o_1r ” 1̂ r 1[0.5 ?7( f 1) + 0.5£( f 2 ) 2 + f i f 2 + f 2 f 1] +
+ 0.256a — 1r -1 ( f " + f 2' ). (8)
Solution (8) satisfies Eq. (1) if the expression 0.5 a 0 r - 2 (y + 1)[( f 1 + f 2 ) 2 ] X
X(1 — rR—1) is a value of the third order. Thus Eq. (8) is valid near the surface
r = R j , where 11 — rR—11 < < 1.
In this paper, we examine only periodical oscillations. In this case, the
velocity and the pressure perturbation
P - P 0 = —P 0 ( <P t + 0 5 <P 2 — 0.5a 0 2 <P 2) + ( 2 + 2v )a 0 2 <P tt (9)
must not contain secular terms (formula (9) defines the pressure if 2 and v are
the shear and dilatational viscosities [8 ]). The secular terms will be eliminated if
we assume in (8) that
P 1 = ^ 1 + 0.125 a —1R —1( y + 1)[£( f 1) 2 — 2 f f 1 ]-0 .2 5 6 a — ̂ 1 ' - c£ 2 ,
p 2 = ^ 2 + 0.125 a —1R —1(y + 1)fo(f 2 ) 2 - 2 f 2f 2 ] - 0.256 a — V 2'+ «7 2 ,
where c is an arbitrary constant, f 1, f 2 , W1 = W1(£), and W2 = W2(^) are
periodic functions. As a result, near the surface r = R t , the velocity potential for
steady-state oscillations is given by the expression
<p = r -1 ( f 1 + f 2 + W1 + W2 + 4ca 0 r t) +
+ 0.5a —1r - 2 [1 - 0.25rR —\ y + 1)][f ̂ + f 2 )2 ]' -
-0 .2 5 (y + 1)a—1R—1[(f1 ) 2 - (f 2 )2 ] + 0 .5 6 a 0- 1( f 1" - f 2 '). (10)
This expression will be used below to solve a boundary problem. We
consider the waves excited by a simple-harmonic source of pressure, which has
radius R 1 and is placed in the center of a sphere. It is assumed that the pressure
source region is very small relative to the excited wavelength. The other boundary
of the sphere is free. Therefore, we have
P - P 0 = —B cosm t ( r = R1), (11)
104
P - Po = 0 (r = R 2 ). (12)
ISSN 0556-171X. Проблемы прочности, 2002, № 1
Localization o f Resonant Spherical Waves
First, resonant frequencies are determined. Then, resonant oscillations are
analyzed on the basis of nonlinear relations. Using (9) and (10), we can rewrite
condition (11) as
a 0R i ( f 1 + / 2 + + 4cRi) + [1 - 0.25(y + 1)][(f 1 + / 2 )2 +
+ (/1 + /2 ) ( /1 '+ / 2 ') ] - 0.5R1( y + 1)(f i f i - /2 /2 ') + 0.5<5r2(f i " - / 2 ') +
+ 0.5[R1-1 (/1 + / 2 ) + / 1 - / 2 ]2 - 0.5(/1 + / 2 )2 -
- 0.5(/1 + / 2 )2 - p - ^ ( 2 + 2v )(/1 '+ /2 ') = p - 1BR2 coswt, (13)
where / 1 = f 1(a 0 1 - R 1), / 2 = / 2 (a 0 1 + R 1), ^ = ^ ( a 0 1 - R 1), and ^ =
= ^ 2 ( a 0 t + R1).
Condition (12) may be presented as (13) if in (13) we substitute R 2 for R 1
and assume B = 0. Then from (12) we can find
f 1( a 0t - r ) = f ( a 0 1 - r + R 2 X f 2( a 0 1 + r ) = - f ( a 0 1 + r - R 2 )
W' = - a - 1R 2-1 ( /1 ) 2 = 0.5(5a-1R 2/ " ' - 2cR 2 , ^ 2 = W'.
Let us consider Eq. (13) taking into account (14). As the first approximation,
it follows from (13) that
f '(a 0 1 - R 1 + R 2 ) - f '(a 0 1 + R 1 - R 2 ) = BR1 a -1 p - 1 cos rnt
and
f '(a 0 t - r + R 2 ) =
= 0.5BR1a - 1 p - 1 sin w a - 1( a 0 1 - r + R 2 )/sin w a - l (R 2 - R 1). (15)
From (15) we obtain resonant frequencies: Q N = n N a 0(R 2 - R1) -1
( N = 1,2, 3,...). The linear solution (15) is not valid near the frequencies m =
= Q n + ®1, where rn1 is a small value. We assume that rn1 = a 0(R 2 - R1)-1 X
Xsinrna- 1(R 2 - R 1).
Let us consider resonant frequencies. First, the function f '(a 0 1 + R 1 - R 2)
is expanded in Taylor’s series at r = R 1:
f '( a 0 1 + R 1 - R 2 ) = f ' - 2®-1 ®1( R 2 - R O f '' + 2®-2 ®2( R 2 - R 1)2 f ' ' ' - ....
(16)
It was suggested that
f '(a 0 1 - R 1 + R 2 - 2N n a 0 / w) = f '(a 0 1 - R 1 + R 2 ) = f '.
ISSN 0556-171X. npo6n.eubi npounocmu, 2002, N2 1 105
Sh. U. Galiev and O. P. Panova
Then using expansions (16) and (13), we obtain the following basic equation:
a 0R iR 2w-1 ® / '' - R iR 2 [a o®-2 ®2(R 2 - R i ) + 0 .5 ( 5 1 /+ ( f ') 2 =
= p - 1 B R 2R 12(R 2 - R1) -1 (cos2 1/2® t - 0.5) + 2a0R 1R2c. (17)
Equation (17) is the perturbed compound Burgers-Korteweg-de Vries
equation written for a travelling wave. This equation has a nonlinear term that
tends to produce “discontinuity” in this wave. The term f ' dissipates through the
viscous-like effect. This term disappears at resonance. The second term, which is
generated due to the viscosity of the medium, disperses the wave. Due to this
term, solitary waves may be excited. We write the solution of (17) for the case
c = 0 .2 5 a -1 p - 1BR1(R 2 - R1) -1 as f ' = V£ O (r)co sr. Here O ( r ) is an
_1 2 _1
unknown function and £ = B p0 R 2R1 (R 2 - R 1) , and r = ®t / 2. As a result,
Eq. (17) becomes
0.5w1R1R2(O ' - O ta n r ) - 0 .2 5 a -1®2R1R2(R 2 - R1 + 0.55®2a0 ® -2) X
Here O ' = d O /d r .
T ran sreson an t p ro c e ss . Far from the resonance, when the first term in (17) is
dominant, the acoustic solution (15) follows from (17). Near resonance, this term
reduces together with ®1. At the same time, the influence of the nonlinear and
second terms in (17) increases. To simplify the problem, let us assume that the
nonlinear term begins to distort the acoustic solution, while the dispersive effect is
still small. In this case, we seek an approximate solution of (18) as the sum
O = O 0 + 0 1, where O 0 > > 0 1. The quantity O 0 takes into account the
nonlinear and first terms in (18), while 0 1 corrects O 0 . We seek a solution,
which is valid near the points where | sin r |< < 1 . By equating the terms of the
same order in (18), we obtain two differential equations:
x (O '' - 2O ' ta n r - O ) = V£(1 - O 2)cosr. (18)
O0 = 2 j q (1 - O 2)c o s r , (19)
4^/q O 00 1 c o s r , (2 0 )
where -Jq = 4 e (®1R1R 2 ) 1. Equation (19) is locally satisfied if O 0 =
= tanh(2^/q s in r ) [8]. The approximate solution of (20) is
106 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, № 1
Localization o f Resonant Spherical Waves
O 1 = q 1 sec h 2 (2 ^[q sin r)cos r,
where
_115 1 2 _1 _2
qi = 8^ 1«o q ' (R 2 _ R 1 + ^ o® a 0 » 1 ).
For the travelling waves
f '[a 0 1 ± (R 2 _ r )] =
= V£ [tanh(^/q sin p )cos p + q 1 sec h 2 ( 2 ^[q sin p )co s2 p ] , (2 1 )
where p = » t /2 ± [® a_1(R 2 _ r ) _ n N ]/2 . This solution indicates that the
finite-amplitude travelling waves become steeper when the excitation frequency
approaches the resonant frequency. According to (21), shock waves may be
excited near resonance in an inviscid medium. For the latter case, if » 1 = 0, we
have the solution with discontinuities [7]. However, both the viscosity and spatial
dispersion begin to be important near resonance and can prevent the formation of
a shock wave [5-13]. It follows from (21) that a soliton-like wave can generate
near resonance. The amplitude of the soliton-like wave increases when » 1 ^ 0
because Eq. (20) and solution (21) are not valid very close to resonance.
Near resonance, the influence of the first term in (18) decreases.
Accordingly, the influence of the second term increases. At resonance, Eq. (18)
transforms to the Korteweg-de Vries type equation
O '' _ 2 0 ' tan r — O = q_ 1(1_ O 2 )c o s r , (22)
where q 0 = _ - 0 » 2 £ _ 05 a _ 2R 1R 2. Let 0 = [A sec h 2(y sin M _ 1r ) + C ]cos r ,
8
where A, y , and C are constant values. We have written the solution localized
near the points where |s in M _ r | < < 1 ( M = 1,2,3 ...). This solution satisfies
approximately Eq. (22) if A = 6q 0y 2M _2 , y 2 = 0 .5 M 2 (1_ q_ 1C ), and C ± =
= 4 (q 0 ± Vq 0 + 3 / 4 ) / 3 . I f |q 0 | < < 1, then C ± ~ 1, y 2 ~ _ 0 5 q _ 1M 2 , and
A ^ _ 3 . For the latter case,
f ' = V£{1_ 3sech 2[M (sinM _ 1 r _ R ) / ^ _ 2 q 0 ]}cos2 r.
For the travelling waves,
f '[a 0 1 ± (R 2 _ r )]= V£{1_ 3sech 2 [M (sinM _ 1 p ) / y l _ 2 q 0 ]}cos2 p. (23)
If M = 1, expression (23) defines oscillations at r _ R 1 at the frequency »t.
Solution (23) also describes subharmonic oscillations if M = 2, 3 ... . Since this
solution must satisfy expansion (16), subharmonic waves corresponding to M >1
may be excited only near the frequencies » = M Q n ■ For example, the case
ISSN 0556-171X. npo6n.eubi npounocmu, 2002, N2 1 107
Sh. U. Galiev and O. P. Panova
M = 2 may be realized only for even resonance. Thus, solution (23) defines the
spectrum ( M = 2, 3 ...) of subharmonic localized waves.
The case M = 1 corresponds to solution (22). For this case, near resonance
we assume 0 = 0 0 + 0 1 , where O 0 > > 0 1. The quantity O 0 takes into
account the nonlinear and the second terms in (18), while 0 1 corrects O o. Then
for travelling waves one can find
f '(a o t ± r ) = V£{[3sec h 2 (sin p / y ] 2 q 0 ) - 1]cos p +
+ Q tanh(sin p J q )}cos p. (24)
Here
Q = 3w1(2q0 )- 0 5 {4^ 1̂ 2£ 05 + 0.25<5w2a - 2 q - 1 [0.5(5 + a 0w- 2w2(R 2 - R1)]}-
Solution (24) is localized near the lines where sin p < < 1. Thus, according to (23)
and (24), periodic spherical solitons may be excited in viscous media at the exact
resonance. These waves contrast with the periodic spherical shock waves, which
are predicted by (2 1 ) for inviscid media.
Linear (15) and nonlinear (21) and (24) solutions describe some scenarios of
transresonant evolution of the waves in weakly dissipative media. Far from
resonance, we have harmonic waves. These waves are distorted due to the
nonlinear effect when the value of « 1 decreases. If 0, these waves transform
into the shock-like waves. However, 6 ^ 0 and discontinuities do not form in the
system. Very close to resonance W1 ~ 0 and spatial dispersion (the second term in
(17)) begins to distort the waves. As a result, the waves may be generated which
have some features of both shock and soliton-like waves. However, at the exact
resonance, the first term in (17) equals zero and soliton-like waves are generated.
These waves may be much localized if 0.
Now we can find pressure and velocity in the medium. However, we
emphasize again that expression ( 10) does not take into account correctly the
second-order values far from the boundaries. Therefore, we must only consider
the first-order terms in expressions for velocity and pressure. For example, instead
of (9) we have
P - P 0 = r -1 p 0a 0 [ f '(n - R 1) - f '(£ + R 1 )].
Thus, according to the above analysis, strongly localized waves travel inside
the sphere (spherical layer). Pictures of the variation of dimensionless pressure
(P - P0 ) / p 0a 0 £ 0 5 are presented in Figs. 1, 2, and 3. There the dimensionless
time r and radius (r / R 2 ) are used. We calculated pressure using (21) (Fig. 1)
and (23) (Figs. 2 and 3), and assuming R 1 = 0.01R2. There is strong amplification
of the waves near r = R 1.
In contrast to plane resonant shock waves [7, 14], resonant spherical
nonlinear waves practically have not been studied [9, 13]. At the same time, the
spherical model for the simulation of different physical objects is very popular.
Indeed, on the one hand, the model of a pulsating sphere is widely used in
astrophysics [15, 16]. On the other hand, this model is used for studying
108 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, N2 1
Localization o f Resonant Spherical Waves
sonoluminescence in liquids when the period of oscillation and the space
distances are very small [17]. The competition of nonlinear, dissipative, and
dispersive effects may be important for these systems. We considered this
competition in the transresonant regime. The distortion of harmonic waves into
shock-like and then soliton-like waves was shown. Our considerations have been
strictly limited to the aspect of nonlinear acoustics; however, the results presented
may be interesting for various media and circumstances.
Fig. 1. Forced pressure waves inside the sphere (q = 10, q1 = 0.1, and N = 2).
Fig. 2. Forced pressure waves inside the sphere (qo = 0.001, N = 2, and M = 1).
Fig. 3. Forced pressure waves inside the sphere q = 0.01, N = 4, and M = 4).
ISSN 0556-171X. npoôëeubi npounocmu, 2002, N 1 109
Sh. U. Galiev and O. P. Panova
Р е з ю м е
Розглядаються радіальні сферичні резонансні хвилі, що збуджуються в
трансрезонансному режимі. Наближений загальний розв’язок збуреного
хвильового рівняння записується з урахуванням нелінійних, просторових і
дисипативних ефектів. Гранична задача зводиться до збуреного змішаного
рівняння Бюргера-Кортевега-де Вріза, для якого побудовано декілька роз
в’язків. Установлено, що в нев’язкому середовищі поблизу резонансу мо
жуть виникати ударні хвилі. Однак як в ’язкість, так і просторова дисперсія
поблизу резонансу запобігають формуванню ударного розриву, в результаті
чого в резонаторі замість ударних генеруються періодичні локалізовані
хвилі.
1. T. I. Belova and A. E. Kudryavtsev, “Solitons and their interactions in
classical field theory,” P h ysic s-U sp ek h i, 40, 359-386 (1997).
2. M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” N atu re,
387, 880-883 (1997).
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110 ISSN 0556-171X. Проблеми прочности, 2002, № 1
Localization o f Resonant Spherical Waves
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Received 20. 04. 2001
ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 1 111
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| id | nasplib_isofts_kiev_ua-123456789-46733 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T15:14:22Z |
| publishDate | 2002 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Galiev, Sh.U. Panova, O.P. 2013-07-06T13:58:04Z 2013-07-06T13:58:04Z 2002 Localization of Resonant Spherical Waves / Sh.U. Galieva, O.P. Panova // Проблемы прочности. — 2002. — № 1. — С. 102-111. — Бібліогр.: 17 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/46733 539.4 This paper treats radial spherical resonant waves excited in the transresonant regime. An approximate general solution of a perturbedwave equation is presented here, which takes into account nonlinear, spatial, and dissipative effects. Then a boundary problem reduces to the perturbed compound Burgers-Kortewegde Vries equation (BKdV) in time. Several solutions to this equation are constructed. Shock waves may be excited near resonance according to the solutions for an inviscid medium. However, both viscosity and spatial dispersion begin to be important very close to resonance and prevent the formation of shock discontinuity. As a result, periodic localized excitations are generated in resonators instead of shock waves. Рассматриваются радиальные сферические резонансные волны, возбуждаемые в трансрезонансном режиме. Приближенное общее решение возмущенного волнового уравнения представляется в виде, учитывающем нелинейные, пространственные и диссипативные эффекты. Граничная задача сводится к возмущенному смешанному уравнению Бюргера- Кортевега-де Вриза, для которого построено несколько решений. Установлено, что в невязкой среде вблизи резонанса могут возникать ударные волны. Однако как вязкость, так и пространственная дисперсия вблизи резонанса предотвращают формирование ударного разрыва, в результате чего в резонаторе вместо ударных генерируются периодические локализованные волны. Розглядаються радіальні сферичні резонансні хвилі, що збуджуються в трансрезонансному режимі. Наближений загальний розв’язок збуреного хвильового рівняння записується з урахуванням нелінійних, просторових і дисипативних ефектів. Гранична задача зводиться до збуреного змішаного рівняння Бюргера-Кортевега-де Вріза, для якого побудовано декілька розв’ язків. Установлено, що в нев’язкому середовищі поблизу резонансу можуть виникати ударні хвилі. Однак як в ’язкість, так і просторова дисперсія поблизу резонансу запобігають формуванню ударного розриву, в результаті чого в резонаторі замість ударних генеруються періодичні локалізовані хвилі. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Localization of Resonant Spherical Waves Локализация резонансных сферических волн Article published earlier |
| spellingShingle | Localization of Resonant Spherical Waves Galiev, Sh.U. Panova, O.P. Научно-технический раздел |
| title | Localization of Resonant Spherical Waves |
| title_alt | Локализация резонансных сферических волн |
| title_full | Localization of Resonant Spherical Waves |
| title_fullStr | Localization of Resonant Spherical Waves |
| title_full_unstemmed | Localization of Resonant Spherical Waves |
| title_short | Localization of Resonant Spherical Waves |
| title_sort | localization of resonant spherical waves |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/46733 |
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