Transresonant Evolution of Spherical Waves Governed by the Perturbed Wave Equation
An equation of nonlinear acoustics for radial spherical waves in a solid body has been derived. An approximate solution to this equation is presented, which takes into account nonlinear, spatial, and dissipative effects. It is found that in the transresonant frequency band nonlinear spherical...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2002
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irk-123456789-468712013-07-07T19:52:19Z Transresonant Evolution of Spherical Waves Governed by the Perturbed Wave Equation Bhattacharyya, D. Galiev, Sh.U. Panova, O.P. Научно-технический раздел An equation of nonlinear acoustics for radial spherical waves in a solid body has been derived. An approximate solution to this equation is presented, which takes into account nonlinear, spatial, and dissipative effects. It is found that in the transresonant frequency band nonlinear spherical waves may be excited, which it is difficult to classify as the well-known solitonor cnoidal- or shock- or breather-type waves. These resonant spherical waves are also quite different from the well-known sawtooth spherical waves. However, some expressions for the spherical waves resemble the solutions for surface waves. Выводится уравнение нелинейной акустики для радиальных сферических волн в твердом теле. Приближенное решение этого уравнения учитывает нелинейные, пространственные и диссипативные эффекты. Установлено, что в трансрезонансной частотной полосе могут возбуждаться нелинейные сферические волны, которые трудно классифицировать как хорошо известные солитон-, кноидал-, ударные или бриз-тип волны. Эти резонансные сферические волны также существенно отличаются от хорошо известных гладких сферических волн. Однако некоторые выражения для сферических волн напоминают известные решения для поверхностных волн. Виводиться рівняння нелінійної акустики для сферичних хвиль у твердому тілі. Наближений розв’язок цього рівняння ураховує нелінійні, просторові і дисипативні ефекти. Установлено, що у трансрезонансній смузі частот можуть збуджуватися нелінійні сферичні хвилі, які важко класифікувати як добре відомі солитон-, кноїдал-, ударні або бриз-тип хвилі. Ці резонансні сферичні хвилі також суттєво відрізняються від добре відомих гладких сферичних хвиль. Однак деякі вирази для сферичних хвиль нагадують відомі розв’язки для поверхневих хвиль. 2002 Article Transresonant Evolution of Spherical Waves Governed by the Perturbed Wave Equation / D. Bhattacharyya, Sh.U. Galiev, O.P. Panova // Проблемы прочности. — 2002. — № 4. — С. 62-74. — Бібліогр.: 17 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/46871 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Bhattacharyya, D. Galiev, Sh.U. Panova, O.P. Transresonant Evolution of Spherical Waves Governed by the Perturbed Wave Equation Проблемы прочности |
| description |
An equation of nonlinear acoustics for radial
spherical waves in a solid body has been derived.
An approximate solution to this equation
is presented, which takes into account nonlinear,
spatial, and dissipative effects. It is found
that in the transresonant frequency band nonlinear
spherical waves may be excited, which it is
difficult to classify as the well-known solitonor
cnoidal- or shock- or breather-type waves.
These resonant spherical waves are also quite
different from the well-known sawtooth spherical
waves. However, some expressions for the
spherical waves resemble the solutions for surface
waves. |
| format |
Article |
| author |
Bhattacharyya, D. Galiev, Sh.U. Panova, O.P. |
| author_facet |
Bhattacharyya, D. Galiev, Sh.U. Panova, O.P. |
| author_sort |
Bhattacharyya, D. |
| title |
Transresonant Evolution of Spherical Waves Governed by the Perturbed Wave Equation |
| title_short |
Transresonant Evolution of Spherical Waves Governed by the Perturbed Wave Equation |
| title_full |
Transresonant Evolution of Spherical Waves Governed by the Perturbed Wave Equation |
| title_fullStr |
Transresonant Evolution of Spherical Waves Governed by the Perturbed Wave Equation |
| title_full_unstemmed |
Transresonant Evolution of Spherical Waves Governed by the Perturbed Wave Equation |
| title_sort |
transresonant evolution of spherical waves governed by the perturbed wave equation |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2002 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/46871 |
| citation_txt |
Transresonant Evolution of Spherical Waves Governed by the
Perturbed Wave Equation / D. Bhattacharyya, Sh.U. Galiev, O.P. Panova // Проблемы прочности. — 2002. — № 4. — С. 62-74. — Бібліогр.: 17 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT bhattacharyyad transresonantevolutionofsphericalwavesgovernedbytheperturbedwaveequation AT galievshu transresonantevolutionofsphericalwavesgovernedbytheperturbedwaveequation AT panovaop transresonantevolutionofsphericalwavesgovernedbytheperturbedwaveequation |
| first_indexed |
2025-07-04T06:24:10Z |
| last_indexed |
2025-07-04T06:24:10Z |
| _version_ |
1836696471877976064 |
| fulltext |
UDC 539.4
Transresonant Evolution of Spherical Waves Governed by the
Perturbed Wave Equation
D. B hattacharyya,a Sh. U. Galiev,a 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
Резонансные сферические волны, описываемые возмущенным
волновым уравнением
Д. Б хаттачарияа, Ш. У. Галиева, О. П. П анова6
а Отделение машиностроения Оклендского университета, Окленд, Новая Зеландия
6 Институт проблем прочности НАН Украины, Киев, Украина
Выводится уравнение нелинейной акустики для радиальных сферических волн в твердом
теле. Приближенное решение этого уравнения учитывает нелинейные, пространственные и
диссипативные эффекты. Установлено, что в трансрезонансной частотной полосе могут
возбуждаться нелинейные сферические волны, которые трудно классифицировать как хо
рошо известные солитон-, кноидал-, ударные или бриз-тип волны. Эти резонансные сфери
ческие волны также существенно отличаются от хорошо известных гладких сферических
волн. Однако некоторые выражения для сферических волн напоминают известные решения
для поверхностных волн.
1. Introduction, a Governing Equation and Approxim ate Solutions. A
linear equation for spherically symmetric waves has a simple analytical solution.
This solution was generalised for the case of resonant weakly nonlinear waves
excited in gas [1,2]. We use here this solution to study weakly nonlinear resonant
spherical waves in solid resonators. With the help of this solution, the boundary
problem reduces to the perturbed Korteweg-de Vries equation in time. A few
solutions of this equation have been constructed. Contributions from nonlinear,
spatial, transresonant, and dissipative effects can be seen from these solutions.
According to these solutions, shock waves may be excited in an inviscid medium
due to nonlinear effects. However, the formation of shock discontinuity is
prevented due to viscosity and spatial dispersion. As a result of the competition
among the nonlinear, dissipative, and spatial effects, periodic localized oscillating
spherical excitations may be generated in resonators instead of the spherical shock
waves. The shape and amplitude of these excitations depend on the excited
frequency.
Let us consider spherically symmetric solid bodies and conical-type
resonators having an approximately circular cross section. There, for the purely
© D. BHATTACHARYYA, Sh. U. GALIEV, O. P. PANOVA, 2002
62 ISSN 0556-171X. Проблемы прочности, 2002, N 4
Transresonant Evolution o f Spherical Waves
radial nonlinear waves, the following equations of continuity and motion are
valid:
2
P t + P ru t + P u rt = - ~ P u t , (1)
2
P( u tt u tu tr ) r,r G r ~ G <p ), (2)V 1
where u is the displacement, a r = X{ur + 2r lu) + 2/xur + 3 v ( u tr - r 1 u t )
- 1 2 2 -1
and a p = 2 {u r + 2r u) + —fxu + - q ( u tr - r u t ). Here the notations are
r r 3
standard and a viscoelastic model of a solid body is introduced. Equation (1)
yields the following approximate expression: p = p 0 - p 0(u r + 2 r - u ), where
p 0 is the undisturbed density. Using the expressions for p , a r , and a p , and
neglecting small terms, which are of the third order, we can rewrite Eq. (2) so that
p 0u tt(1- u r — 2 r -1 u ) + p 0u t u rt = (2 + 2/x)[urr + 2 r - 1(u r — r - 1u)] +
2 -1 4
+ 3r n (u r - r u ) t + 3 Vutrr. (3)
Then the displacement potential $ is introduced in (3) (u = $ r). After
integrating (3), we have
p 0$ tt - p 0 f $ ttr ($ rr + 2 r ~ l$ r )dr + °-5p 0$ I =
= (2 + 2 ^ )($ rr + 2r 1$ r ) + 3 V($ rr + 2r 1$ r ) t . (4)
Let the viscous term in (4) be of the second order. For this case, a linear
wave equation follows from (4):
a o( $ rr + 2 r-1 $ r ) = $ tt,
where a 0 = (2 + 2jm) /p 0. Using the wave equation, we can simplify (4). As a
result, we have
p 0$ tt(1 - °.5a0 2$ tt) + 0-5p 0($ rt)2 = (2 + 2H )($ rr + 2 r-1 $ r ) + 4 Va 0 2$ ttt ■
Let <p = $ t . For the latter case, we have the following equation of nonlinear
acoustics for a homogeneous viscoelastic solid body:
ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 4 63
D. Bhattacharyya, Sh. U. Galiev, and O. P. Panova
a o( p rr + 2r V r ) = p tt - a 0 2 p t p tt + p r p rt - (5 a 0 2 p ttt , (5)
. 4
where o = 3 It is important that Eq. (5) differs from the equation of nonlinear
acoustics for gas and liquid only by some coefficients [1, 2, 3]. Therefore, bellow
we use the results obtained for resonant waves in spherical gas layers [1, 2, 4, 5].
We emphasize that Eq. (5) does not take into account the third order effects
and the dissipative term is of the second order. The solution of (5) may be
presented as
p = p 1 + p 2 , (6)
where p 1 and p 2 are the first- and the second-order values, respectively.
Substituting (6) into (5) 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 2 p 1tt, (7)
a 0 (p 2rr + 2r 1p 2r ) = p 2tt + p 1rp 1rt - a 0 2 p 1tp 1tt - ^ a 0 2 p 1ttt. (8)
The solution of the linear wave equation (7) is the sum of two travelling
waves: p 1 = r -1 (/ + / 2). Here and below, / = £) and / 2 = f 2 (^), where
£ = a 01 - r and ^ = a 01 + r. Now we can rewrite Eq. (8) in the form
a 02(p 2rr + 2r - 1p 2r ) - p 2tt = a 0 {r-2 ( /2 - / 1' ) ( / " - / 1" ) -
- r -3 [ ( / 2 + / 1)(/2 - / 1' ) - ( / " )2 + ( /2 )2 ] + r - 4( / 2 + /1 )( /2 + / " ) -
- r - 2 ( / 2 + / " ) ( / " + / 1') - <Sr- 1( / 2" + / 1“ )},
where the primes denote a derivative with respect to the argument. One can find
p 2 from this equation following [1, 2, 4, 5]. Finally, the approximate solution of
(5) is
p = r -1 CA + / 2 + ^ 1 2) + 0 .25a-1r -2 [ ( / 1 + / 2 )2 ]' -
-0.25a0~V-1 f f r -1 ( /" + / 2 ) ( / / ' + / 2 )dU n + 0 . 2 5 ^ ^ -1 (/ " + £/2' ). (9)
Here ^ 1 = ^ ( £) and ^ 2 = ^ 2 (^), and the Unctions ^1 and ^ 2 are of the
second order. The functions / 1, / 2 , ^ 1, and ^ 2 are unknown and must be
found from the initial and boundary conditions. However, solution (9) is
complicated by the integral. Let us simplify it to a form that is more convenient
for satisfying the boundary conditions. Near any boundary surface r = R and the
multiplier 1/r under the integral is replaced by 1/R. As a result, we have
64 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, N 4
Transresonant Evolution o f Spherical Waves
V = r -1 f + f i + V 1 + V 2) + 0 .2 5 aq V -2 [ ( f + f 2)2 ]' -
-0 .25a o_1r -1R - 1 [0 .5 ](f 1)2 + 0 .5£(f 2 )2 + f f 2 + f 2 f 1] +
+ 0.25<5a-V-1 ( f + f ). (10)
Solution (10) satisfies Eq. (5) if the expression ^ a 0 r - 2 [( f ' + f 2 )2 ]' ( 1 - rR - 1 )
is of the third order. Thus, (10) is valid near the surface r = R, where
|1 - rR - 1 | < < 1.
In this article, we examine only periodical oscillations. In this case, the
velocity must not contain secular terms. The secular terms will be eliminated if
we assume in (10)
V i = + 0 .125a-1R - 1 [ £ ( f ') 2 - 2 f J l ] - 0.25<5a- X̂ f ”,
where i = 1 or 2; f 1, f 2 , W1 = W1(£), and W2 = W2(] ) are periodic functions.
As a result, near the surface r = R, we have for steady-state oscillations
V = r -1 ( f 1 + f 2 + W1 + W 2) + 0 .2 5 a-1r - 2 (1 - 0.5rR-1 ) [ ( f 1 + f 2) 2 ] -
-0 .2 5 a - 1R ~ l [ ( f { ) 2 - ( f 2)2 ]+ 0.5<5a-1( f " - f 2 ). (11)
Both expression (9) and (11) are used below to solve a boundary problem.
2. A B oundary Problem and Basic Equation. Let us consider waves
excited by an oscillating velocity at the surface r = R. Therefore, we have
V r = -(o B sin rnt ( r = R ), (12)
4 ^ r 2 v r = 0 ( r ^ 0). (13)
We have written (13) according to [6, p. 491]. When r ^ 0, the stresses
increase and the mechanical properties of the material can change strongly at the
origin. As a result, Eq. (5) and solution (9) are not valid if r = 0. Therefore, it is
possible that Eq. (13) is a rough approximation of the reality at r = 0. Let us
assume that the influence of the origin is very local and does not change the wave
pattern qualitatively. Using (9), we can rewrite condition (13) so that
r ( f 2 - f i + ^ 2 - ^ ) - f 1 - f 2 - ^ 1 - ^ 2 +
+ 0.25a o” V 2{r- 2 [ ( f 1 + f 2)2 ] }r + 0.125da - l r 2[ r - 1( ] f i ' + & 2 )]r +
+ 0 . 2 5 a - 7 / r -1 f + f 2 ) ( f ' + f 2 ) d £ d ] -
ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 4 65
D. Bhattacharyya, Sh. U. Galiev, and O. P. Panova
— 0.25a—V [ / / r ~ \ f i + / 2 ) ( / " + f 2 = 0, (14)
where r ^ 0. Equation (14) is satisfied if
f i ( a o t — r) = f ( a o t — r ) , f 2( a o t + r) = —f ( a o t + r ) ,
W1( a 01 — r ) = W( a 01 — r ), W2( a 01 + r ) = W( a 01 + r ). (15)
In (14)
/ / 1 ( f i + f 2 )^-/’x'+ f 2 )d U n = 2 / / ( f ') r / " ( I ) — f * ( 0
because I ^ ^ . Condition (12) is written now with the help of (11):
R ( f 2 — f 1 + ^ 2 — ^ i ) — f 1 — f 2 —^1 — ^ 2 +
+ 0.75a— R ~ l ( f 1 + f 2) / 1 + f 2) +
+ 0 .2 5 a ^ [ ( f 2 )2 - ( f 1)2 + ( f i + f 2 )(_ f 1 + f 2)] +
+ 0 . 5 a - R f f 1'+ f 2 f 2') - 0 .5 6 a - ^ 2(f1"+ f2") = -w B R 2 sinwt. (16)
Here we must take into account (15). As a first approximation, it follows from
(15) and (16) that
f (£) - f (n ) + R f '(£) + R f '(^) = wBR 2 sinwt. (17)
Following [1, 2, 4, 5], we assume f (£) and f (^) in the form
f (£) = - R -1 / F ( £ + R ) d £ - [a0w-1 R -1 (sinwRa-1 -
- w R a -1 cos w R a -1) - 1]F (£ + R ),
f (^) = - R 1 / F (^ - R d - [a0w-1 R -1 (sin w R a -1 -
- w R a -1 cos w R a -1) + 1]F(^ - R ).
Then Eq. (17) describes travelling waves
F [a 0t ± ( r - R )] =
= 0.5wBR2(sin w a - 1R - w a - 1R cos w a - 1R )-1 cos w a - 1[a01 ± ( r - R )]. (19)
66 ISSN 0556-171X. Проблемы прочности, 2002, N 4
(18)
Transresonant Evolution o f Spherical Waves
From (19), we obtain resonant frequencies as
Q y = n y a 0 R - 1 , (20)
where y = 1.4303, 2.4590, 3.4709, ... [6, p. 506]. Linear solutions (19), (18), and
(15) are not valid near the frequencies m = Q r . Therefore, Eq. (16) will be
considered taking into account nonlinear terms. Considering the nonlinear terms,
we assume that near a fast varying solution
| a 0R - 1 f F ( a 0l)d l| < < | F ( a 01)|. (21)
Taking into account (21), (18), and (15), we rewrite Eq. (16) in the following
way:
m 1F " + R -1 (W 2 - W") - R - 2 (W 1 + W2) - d a ^ F " ' -
- 3a- 1R -3 F F " + a - 1R -1F F " = - m B sin m l, (22)
where
m1 = 2a0m-1 R - 2 (sinm R a-1 - m R a-1 cosm R a-1).
Equation (22) is complex for the integration. This equation can be simplified if
we assume the following expressions in (11):
W = W1( £) = W ( £ ) = 0.25a - 1[F "(£ + R )]2 ,
1 2 (23)
W 2 = W 2(f) = -W ( f ) = - 0 .2 5 a - 1[F "(f - R )]2.
As a result, we obtain the following basic equation from (22), which is valid
if (21) takes place:
m 1F " - d a - lF"' - 3 a - 1 R -3 F F " = - m B sin ml. (24)
Equation (24) resembles Eq. (3.17) from [7]. After integrating, we take a
tant of i
rewritten as
_2 2
constant of integration in the form c = a 0B(1 - 8^ R ). Then Eq. (24) may be
(F - 2 G 4 e n 1)2 + q0£ 0'5F"" = £ cos2 r. (25)
Here
r = m l/2, G = ^ m ^ 0£ 05R 3/6, q 0 = - d m 2£ 05a 0 2R 3/6 , £ = 4a2B R 3/3.
This equation has a nonlinear term that tends to produce a ‘discontinuity’
solution. The second term, which is generated due to viscosity of the medium, is
responsible for the dispersion of the waves.
ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 4 67
D. Bhattacharyya, Sh. U. Galiev, and O. P. Panova
3. Resonant Waves. InviscidM edium . For this case, Eq. (25) does not have a
smooth periodic solution with the same period as forced oscillations. Therefore,
following [7], we construct a discontinuous 2n/m-periodic solution of (25):
F = cos r , (26)
where 0 < r < n . This solution is valid for exact resonance (G = 0). Now with the
help of (18), we can calculate the functions f (a o t ± r ). Then, taking into account
(15) and (11), the expression for p may be written. We recall again that (11)
does not take into account correctly the second-order values far from the
boundaries. Therefore, we must consider only the first-order terms in this
expression so that
/ - - i f 1 -1 p = V£r jc o s ^ m t — ma0 ( r — R)] +
1 —1 —1 —1 1 —1 + cos^[m t + ma0 (r — R )] — 2a0m R sin^[m t — ma0 (r — R )] +
+2a 0 m—1R — 1 sin^[m t + ma—1( r — R )]j, (27)
where 0 < mt ± ma—1(r — R ) < 2n. Within subsequent intervals of length n , we
1 —1can find <p by the periodic continuation of cos^[m t ± mao (r — R )] and
1 —1sin^[m t ± mao (r — R )] in (27). As a result, the solution is obtained with the
same period as the forced oscillations. The function p is discontinuous along
straight lines: mt ± ma— (r — R ) = 2nn (n = 0, 1, 2, 3, ...). Generally speaking,
solutions (26) and (27) take place near the shock jumps, where condition (21) is
valid.
Thus, according to the inviscid model of the material, resonant shock waves
may be excited in a sphere. These waves are the sum of the spherical
saw-tooth-like travelling waves. However, this result changes dramatically if we
take into account the spatial dispersion [the second term in Eq. (25)].
Effect o f Spatial Dispersion. Transresonant Process. I f the combined effects
of the nonlinearity and the spatial dispersion compensate each other, then
soliton-like waves may be excited in the system. Following [8], we seek solutions
of (25) for this case in the form F = V« [2Gn —1 + 0 ( r ) c o s r ], where 0 ( r ) is an
unknown function. As a result, we obtain the following equation:
O" — 2 0 ' tan r — O = ^ —1(1— O 2 )cos r. (28)
We assume that q 0 < < 1. We seek a localized fast varying solution of (28).
Let
68 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, N2 4
Transresonant Evolution o f Spherical Waves
0 = {Asech2[y(sinM 1 r - G )]+ C }cosr, (29)
where A, y , and C are constant values, and M = 1 ,2 ,3 ,.. . . Solution (29) is
localized near the points, where sin M -1 r ^ G. Solution (29) approximately
satisfies Eq. (28) if A = 6q0y 2M - 2 , y 2 = 0.5M 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 -1 M 2 ,and A = 3. For the
latter case, the expression for F written for the travelling waves is as follows:
F (2 a 0rn 1 p ± ) = 2^fe G n 1 +
+ V^{3sech2[M (sinM -1 p ± - G ) / \]2q0 ] - 1}cos2 p ± , (30)
1 1 _1
where p ± = ^ wt ± ^ wa (r _ R ). Strictly speaking, solution (30) is valid if
G ~ 0. Then condition (21) takes place. According to (30), when sin M 1 p ± « G,
a peak of the function F (2 a o w 1 p ± ) is generated and then a crater occurs. This
excitation resembles the so-called ‘oscillon’ [9]. By contrast to oscillons,
expressions (30), (18), (15), and (11) describe travelling spherical oscillons.
Generally speaking, solution (30) defines a spectrum of subharmonic localized
waves if M = 2 ,3 , 4 , . . . . I f M = 1, oscillations are possible with a forced
frequency wt. Thus, near and at resonance, periodic resonant localized waves are
predicted by (30).
Now, using (18) and (15), it is possible to find f 1(£) and f 2(7]). Then we
can write expressions for <p, stresses, and velocity. However, we emphasise that
(11) does not take into account the second-order values far from the boundaries.
Therefore, we must only consider the first-order terms in the expressions. For
example, for the velocity we have
<P r = _ r _2 [rf '( £ ) + f ( £ )]_ r _2[rf '( ] ) _ f ( ])] , (31)
where the functions f (a 01 ± r ) are found approximately according to (30) and
(18):
f (a 01 ± r ) = _(±)V 3{2G ^_1 + 3sech2 [(sin p ± _ G ) / - j 2q0 ]cos2 p ± _
_ cos2 p ± }+ a 0w _ 1R ~ l 4 ë { p ± _ 4 G ^ _ 1 p ± + 0.5sin2p ± _
_ 2^j2q 0 tanh[(sinp ± _ G )/-yJ2q 0 ]}. (32)
We assume here M = 1. At the same time, according to (18), we have
f ' ( a 0 1 ± r ) = - R ~ l F ( 2 a 0 w _1 p ± ) , ( 3 3 )
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D. Bhattacharyya, Sh. U. Galiev, and O. P. Panova
where expression (30) is valid. One can see that the shape of the wave (31) may
be complex. At the same time, the most important contributions in (31) are
defined by expressions 3sech2[(sinp ± — G )/y/2q0 ]cos2 p ± . Thus, near and at
resonance, the localized oscillating spherical waves are excited, which are quite
different from the well-known saw-tooth spherical waves.
Free Oscillations or Trapped Waves. Oscillations may be generated due to a
change in the velocity at r = R at the moment t = 0. We assume that, after a
sufficiently long time, transient oscillations are damped, and we are interested in
free nonlinear steady-state oscillations. Oscillations with a frequency of are
considered. For this case, it follows from (24) that
ô a —1F " + 1.5a—1R —3 F 2 = c , (34)
where c is some constant of integration. We assume that c is defined by the
1
In the latter expression, the second term localizes near the points ^ t =
= (K - 1)n (K = 1, 2, 3, ...). Then one can approximately find from (34):
E 1 = - ( - 0.66ca0R 3)°'5 , E 2 = 0.1875(5-1 a f a - 2(-0 .6 6 ca0R - 3 )05 , and A =
= - 3E 1 . For the travelling waves,
F (2a0a>~1 p ± ) = E 1[1 - 3sech2(E sinQ y a>- 1 p ± )cos2 Q y a>- 1 p ± ]. (35)
Solution (35) is valid near the lines, where sinQ y m - 1 p ± ~ 0. Thus, if the
coefficient d ensures condition | E | > > 1, then nonlinear localized free spherical
waves may be generated in resonators. These waves are defined by expressions
(35), (18), (15), and (11).
4. Discussion. Thus, strongly localized spherical waves can travel within
resonators according to the above analysis. Now we can calculate the stresses and
velocity in the medium. We recall again that we must only consider the first-order
terms for them. For example, we have expression (31) for the velocity. Pictures of
the variation of the velocity p r£ -0 '5 are presented in Figs. 1-3. The
dimensionless time x and radius r /R are used, and (2q0)-0 '5 = 3 and a 0 =
= 340 m/s. There is strong amplification of the waves near r = 0. Figures 1 and 2
display particularities of the transresonant process for the case of spherical waves.
It is known that passage through resonance is a classic problem. However, usually
one- or several-degree-of-freedom models are used. From Figures one can see
that sometimes the properties of nonlinear waves may be very important. The
waves depend on the excited frequency. They are localized and strongly amplified
at resonance (G = 0). The fast varying waves transform into harmonic waves
when |G| increases. If |G| ~ 1, two-peak localized waves with small amplitude are
excited (Figs. 1 and 2). The process, which resembles a transresonant process,
takes place at resonance if the dissipative effect changes (Fig. 3).
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Transresonant Evolution o f Spherical Waves
Fig. 1. Transresonant evolution of spherical waves (y = 1.4303, q0 = 0.02).
We have considered resonant localization of waves in spherical resonators.
On the one hand, these waves are strictly different from the waves in elongated
natural resonators [8, 10-12] and tubes [7, 12]. On the other hand, solutions (30)
and (35) resemble the expressions for travelling localized plane surface waves
(see solutions (35) and (39) from [8]). The localization of surface waves has been
considered recently [9, 13, 14]. These localized waves are usually observed in
parametrically excited dispersive systems [9-11, 13, 14].
Thus, solutions (30) and (35) of the perturbed wave equation describe a
variety of wave processes in dispersive systems. One can see from (25) that in
spherical systems dispersive effects are defined by viscous properties of the
material. Periodic localized oscillating spherical waves are generated because
spatial dispersive and nonlinear effects balance each other within the sphere.
Thus, smooth localized waves rather than shock waves are formed in the system.
This result agrees qualitatively with the data of numerical calculation [15].
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D. Bhattacharyya, Sh. U. Galiev, and O. P. Panova
031̂ 2 03
Fig. 2. Transresonant evolution of spherical waves (y = 2.4590, q0 = 0.02).
Localization also takes place because of the wave focusing [1, 14]. The order of
the amplitude O (£ 0 5) of resonant spherical waves is the same as for plane waves
in elongated resonators with fixed boundaries [7, 8, 10-12].
Resonant spherical nonlinear waves, in contrast to plane resonant waves,
practically were not studied. At the same time, a spherical model for the
simulation of different physical objects is very popular. Indeed, on the one hand,
the model o f a pulsating sphere is widely used in astrophysics [16]. On the other
hand, this model is used to study sonoluminescence in liquids where the period of
oscillation and space distances are very small [15]. However, the competition
between nonlinear and spatial dispersive effects in resonant spherical systems has
not been studied. Due to this competition, the distortion of harmonic waves into
oscillating localized resonant waves can take place. Our study has been strictly
limited by the aspect o f nonlinear acoustics. However, the results presented may
be interesting for various media and circumstances [10, 11, 17].
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Transresonant Evolution o f Spherical Waves
Fig. 3. Localization of waves due to dissipative and spatial dispersive effects.
Р е з ю м е
Виводиться рівняння нелінійної акустики для сферичних хвиль у твердому
тілі. Наближений розв’язок цього рівняння ураховує нелінійні, просторові і
дисипативні ефекти. Установлено, що у трансрезонансній смузі частот мо
жуть збуджуватися нелінійні сферичні хвилі, які важко класифікувати як
добре відомі солитон-, кноїдал-, ударні або бриз-тип хвилі. Ці резонансні
сферичні хвилі також суттєво відрізняються від добре відомих гладких
сферичних хвиль. Однак деякі вирази для сферичних хвиль нагадують
відомі розв’язки для поверхневих хвиль.
1. Sh. U. Galiev, “Nonlinear one-dimensional oscillations o f a viscous
diathermic gas in a spherical layer” in: Proc. Seminar on Shell Theory, 2,
Kazan, Kazan Physico-Techical Institute (1971), pp 240-253.
2. Sh. U. Galiev, “Passing through resonance of spherical waves,” Phys. Let. A,
260, 225-233 (2001).
3. V. P. Kuznetsov, “Equations o f nonlinear acoustics,” Sov. Phys. Acoust., 16,
467-470 (1970).
4. Sh. U. Galiev, Nonlinear Waves in Bounded Continua [in Russian], Naukova
Dumka, Kiev (1988).
5. Sh. U. Galiev and O. P. Panova, “Periodic shock waves in spherical
resonators (survey),” Probl. Prochn., No. 10, 49-73 (1995).
ISSN 0556-171X. Проблеми прочности, 2002, № 4 73
D. Bhattacharyya, Sh. U. Galiev, and O. P. Panova
6. H. Lamb, Hydrodynamics, 6th edition, Dover Publications, New York,
(1932).
7. W. Chester, “Resonant oscillations in a closed tube,” J. F luid M ech., 18,
44-64 (1964).
8. Sh. U. Galiev and T. Sh. Galiev, “Resonant travelling surface waves,” Phys.
Let. A , 246, 299-305 (1998).
9. P. B. Umbanhowar, F. Melo, and H. L. Swimmey, “Localized excitations in
a vertically vibrated granular layer,” Nature, 382, 793-796 (1996).
10. Sh. U. Galiev, “Topographic effect in a Faraday experiment,” J. Phys. A:
Math. Gen., 32, 6963-7000 (1999).
11. Sh. U. Galiev, “Unfamiliar vertically excited surface water waves,” Phys.
Let. A , 256, 41-52 (2000).
12. M. A. Ilgamov, R. G. Zaripov, R. G. Galiullin, and V. B. Repin, “Nonlinear
oscillations of gas in a tube,” Appl. Mech. Rev., 49, 137-154 (1996).
13. L. Jiang, M. Perlin, and W. W. Schultz, “Period tripling and energy
dissipation of breaking standing waves,” J. F luid M ech., 369, 273-299
(1998).
14. B. W. Zeff, B. Kleber, J. Fineberg, and D. L. Lathrop, “Singularity dynamics
in curvature collapse and jet eruption on a fluid surface,” Nature, 403,
401-404 (2000).
15. H. Y. Cheng, M. C. Chu, P. T. Leung, and L. Yuan, “How important are
shock waves to single-bubble sonoluminescence?” Phys. Rev. E, 58, R2705-
R2708 (1998).
16. A. Gautschy and H. Saito, “Stellar pulsation across the HR diagram. Part 1,”
Ann. Rev. Astron. Astrophys., 33, 75-113 (1995).
17. Sh. U. Galiev and T. Sh. Galiev, “Nonlinear trans-resonant waves, vortices
and patterns: From microresonators to the early universe,” Chaos, 11,
686-704 (2001).
Received 20. 04. 2001
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