Standing second sound wave in many-layer systems
The variation of thermodynamic parameters of superfluid helium in standing second sound wave is studied. Such double-layer systems as heater–helium and helium–detector are considered. Heat transfer in the heater and the detector was described with usual thermal conductivity equation. Temperature...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2012
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irk-123456789-1167382017-05-15T03:02:54Z Standing second sound wave in many-layer systems Nemchenko, K. Rogova, S. Квантовые жидкости и квантовые кpисталлы The variation of thermodynamic parameters of superfluid helium in standing second sound wave is studied. Such double-layer systems as heater–helium and helium–detector are considered. Heat transfer in the heater and the detector was described with usual thermal conductivity equation. Temperature and heat transfer in superfluid ⁴He was described with the system of equations for superfluid with taking into accounts both dissipative thermal conductivity mode and second sound mode. Resonance frequency and amplitude dependence on dissipation at the heater and the detector was studied. The unusual resonance was found in the double-layer system helium– detector. 2012 Article Standing second sound wave in many-layer systems / K. Nemchenko, S. Rogova // Физика низких температур. — 2012. — Т. 38, № 1. — C. 3-7. — Бібліогр.: 8 назв. — англ. 0132-6414 PACS: 67.60.–g; 44.05.+e; 43.20.Ks http://dspace.nbuv.gov.ua/handle/123456789/116738 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы Nemchenko, K. Rogova, S. Standing second sound wave in many-layer systems Физика низких температур |
| description |
The variation of thermodynamic parameters of superfluid helium in standing second sound wave is studied.
Such double-layer systems as heater–helium and helium–detector are considered. Heat transfer in the heater and
the detector was described with usual thermal conductivity equation. Temperature and heat transfer in superfluid
⁴He was described with the system of equations for superfluid with taking into accounts both dissipative thermal
conductivity mode and second sound mode. Resonance frequency and amplitude dependence on dissipation at
the heater and the detector was studied. The unusual resonance was found in the double-layer system helium–
detector. |
| format |
Article |
| author |
Nemchenko, K. Rogova, S. |
| author_facet |
Nemchenko, K. Rogova, S. |
| author_sort |
Nemchenko, K. |
| title |
Standing second sound wave in many-layer systems |
| title_short |
Standing second sound wave in many-layer systems |
| title_full |
Standing second sound wave in many-layer systems |
| title_fullStr |
Standing second sound wave in many-layer systems |
| title_full_unstemmed |
Standing second sound wave in many-layer systems |
| title_sort |
standing second sound wave in many-layer systems |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2012 |
| topic_facet |
Квантовые жидкости и квантовые кpисталлы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/116738 |
| citation_txt |
Standing second sound wave in many-layer systems / K. Nemchenko, S. Rogova // Физика низких температур. — 2012. — Т. 38, № 1. — C. 3-7. — Бібліогр.: 8 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT nemchenkok standingsecondsoundwaveinmanylayersystems AT rogovas standingsecondsoundwaveinmanylayersystems |
| first_indexed |
2025-07-08T10:58:21Z |
| last_indexed |
2025-07-08T10:58:21Z |
| _version_ |
1837076108405637120 |
| fulltext |
© K. Nemchenko and S. Rogova, 2012
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 1, pp. 3–7
Standing second sound wave in many-layer systems
K. Nemchenko and S. Rogova
Department of Physics and Energy, V.N. Karazin Kharkov National University
4 Svobody Sq., Kharkov 61022, Ukraine
E-mail: nemchenko@bk.ru
Received April 19, 2011
The variation of thermodynamic parameters of superfluid helium in standing second sound wave is studied.
Such double-layer systems as heater–helium and helium–detector are considered. Heat transfer in the heater and
the detector was described with usual thermal conductivity equation. Temperature and heat transfer in superfluid
4He was described with the system of equations for superfluid with taking into accounts both dissipative thermal
conductivity mode and second sound mode. Resonance frequency and amplitude dependence on dissipation at
the heater and the detector was studied. The unusual resonance was found in the double-layer system helium–
detector.
PACS: 67.60.–g Mixtures of 3He and 4He;
44.05.+e Analytical and numerical techniques;
43.20.Ks Standing waves, resonance, normal modes.
Keywords: superfluid helium, second sound, resonances, standing wave.
Introduction
Superfluid 4He possesses a number of unique features.
One of them is the existence of second sound [1]. Unlike
usual sound in the second sound mode not full momentum
(and pressure) fluctuates but temperature (and entropy)
fluctuations take place. Second sound provides heat trans-
fer and temperature relaxation in He II.
One of the experimental methodic to study second
sound is the generation of the standing wave of second
sound. This methodic was used in the first experiments
dealt with second sound [2,3], as well as in the latest ex-
periments [4], in which the possible electric properties of
second sound were studied. The latter experiments resulted
on theoretical consideration presented in this paper.
In this work the heat transfer model in the double-layer
systems [5,6] is presented. Such double-layer systems as
heater–helium and helium–detector are considered. The
nature of heat transfer in these layers is very different. Heat
transfer in the heater and the detector is determined by
dissipative process of thermal conductivity, so, it is de-
scribed by usual thermal conductivity equation. Heat trans-
fer in helium is determined by second sound and it is de-
scribed by the system of equations for superfluid with
taking into accounts both dissipative thermal conductivity
mode and second sound mode.
2. Double-layer system heater–helium
At first we consider the double-layer system consisting
of a heater and helium (see Fig. 1,a). The first layer is a
usual substance of finite length 1l and serves as a heater.
The second layer is superfluid helium of finite length 2l .
The left end of the first layer is thermally isolated. Inside
the first layer heat sources are uniformly distributed.
To describe heat transfer in the first layer usual thermal
conductivity equation with external uniformly distributed
heat sources q(t) is used
( )1 1 1 0( ), ( ) cos ,T T q t q t Q tχ ω= + =′′ (1)
where χ1 is the temperature conductivity of the heater, 1T
is the temperature deviation of the heater from the equilib-
rium value T0, Q0 is the dimensionless (divided to heat
capacity) external heat sources amplitude, ω is the frequen-
cy of external heat source.
Fig. 1. The geometry of considered experiments: heater–helium
(a); helium–detector (b) systems.
4
He
4
He
H
ea
te
r
H
ea
te
r
l2 l2l1
a
D
et
ec
to
r
h
b
K. Nemchenko and S. Rogova
4 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 1
At the left end the boundary condition for the heat flow
in the heater has the form
(0, ) 0Q t = (2)
and at the right end of the heater we seek the heat flow as
( )1 1( , ) cos ,Q l t Q tω φ= + (3)
where l1 is the length of the first layer, Q1 and φ are pa-
rameters that are defined by equating temperatures and
heat flows at the layers boundary.
Heat transfer in the second layer is described with the
system of hydrodynamic equations for superfluids that
takes into account both second sound and dissipation [2]
2
2
2
0,
.
n
n
v u T
T v Tχ
⎧ + =′⎪
⎨
+ =′ ′′⎪⎩
(4)
Here 2u is the second sound velocity, vn is the normal
fluid velocity, χ2 is the temperature conductivity of the
helium, T is the temperature deviation from the equilibrium
value T0. The boundary condition at the right end of heli-
um has the form
2( , ) 0,T l t = (5)
where l2 is the length of the second layer. The heat flow in
helium is
0 ,nq ST v Tκ= − ′ (6)
where S is the entropy of helium and 2VCκ χ= is the
thermal conductivity coefficient of helium. Here we should
note that the second term in Eq. (6) almost at all tempera-
tures can be neglected.
3. Result of calculation
To solve the problem Laplace transform was used in
order to solve the problem about switching on [7,8] and off
the heat flow in future. Amplitude and phase of the heat
flow at the boundary of the layers were obtained by equat-
ing temperatures and heat flows at the boundary.
For temperature of the heater the following expression
was obtained
( ) ( ) ( )
( ) ( )
( )
2 2
1 1 1 1 1
1 2 2
1 1 1 1 1 1 1
0
sinh cos sin
( , )
cosh cos
sin .
V
Q l l t
T x t
l l C
Q
t
λ λ ω α
λ λ λ χ
ω
ω
+ −
= +
−
+ (7)
Here 1 1/ 2λ ω χ= is the inverse thermal wavelength in
the heater, 0 0 1 1VQ Q C l= is the normalized heat flow am-
plitude, ( )1 1/ 4 arctan ( ) ( )B x A xα φ π= − + is the phase
shift, and 1VC is the thermal capacity of the heater. Values
A1(x) and B1(x) are determined by expressions
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1
( ) cos cosh cos sinh
sin sinh sin cosh ,
( ) sin sinh cos sinh
cos cosh sin cosh .
A x x x l l
x x l l
B x x x l l
x x l l
λ λ λ λ
λ λ λ λ
λ λ λ λ
λ λ λ λ
= +
+
= −
−
(8)
Phase shift φ is given by
2 1
1 2
2
arctan ,
2
V
V
k C
k C
γ ω ς
φ
γ ω ς
−
+
⎛ ⎞+
= − ⎜ ⎟+⎝ ⎠
(9)
and heat flow amplitude Q1 takes the form
0
1
2 1 1 2
.
( cos sin ) ( cos sin ) 2
V
V
Q C
Q
C
ξμ
γ φ γ φ γ φ γ φ μ ω
=
− + −
(10)
In Eqs. (7)–(10) the following expressions were used:
( ) ( )1 2 1 2 1 2 2cosh 2 cos 2k k l k k lγ ξ ⎡ ⎤= +⎣ ⎦ ,
( ) ( )2 1 1 2 2 2 2sinh 2 sin 2k k l k k lγ ξ ⎡ ⎤= −⎣ ⎦ ,
( ) ( )1
2 2
1 1 1 1 1 1cosh cosVC l lξ λ χ λ λ⎡ ⎤= −⎣ ⎦ ,
( ) ( )1 1 1 1sinh 2 sin 2l lς μ λ λ± ⎡ ⎤= ±⎣ ⎦ ,
( ) ( )1 2 2 2cosh 2 cos 2k l k lμ = + .
Here we introduced the values k1 and k2, which respect to
typical inverse lengths of the problem: k1 corresponds to
dissipative thermal heat wave length:
2
2
1 2
1 1
2
u
k
U U
ω ⎛ ⎞
= −⎜ ⎟
⎝ ⎠
, (11)
and k2 defines the wavelength of second sound:
2
2
2 2
1 1
2
u
k
U U
ω ⎛ ⎞
= +⎜ ⎟
⎝ ⎠
. (12)
The value U determines the velocity of heat propagation:
4 2 24
2U u ω χ= + . (13)
For the second layer, which is superfluid helium, the
dependence on coordinate was obtained
( ) ( )1 2 2
2 2 2
2 2 1 2
( ) sin ( ) cos
( , ) .
cosh ( ) cos ( )V
Q A x t B x t
T x t
UC k l k l
ω ω⎡ + ⎤⎣ ⎦= −
⎡ ⎤−⎣ ⎦
(14)
The functions A2(x) and B2(x) are
( ) [ ]
( ) [ ]
2 1 2 1 3 2 2 4 2
2 1 1 4 2 2 3 2
( ) cos sin ( ) ( ) ( ) ( )
cos sin ( ) ( ) ( ) ( ) ,
UA x k k s x s l s x s l
k k s x s l s x s l
φ φ
ω
φ φ
= − + +
+ + −
(15)
and
Standing second sound wave in many-layer systems
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 1 5
( ) [ ]
( ) [ ]
2 1 2 2 3 2 1 4 2
2 1 1 3 2 2 4 2
( ) cos sin ( ) ( ) ( ) ( )
cos sin ( ) ( ) ( ) ( ) .
UB x k k s x s l s x s l
k k s x s l s x s l
φ φ
ω
φ φ
= − − +
+ + +
(16)
In Eqs. (15) and (16) we used the following relations:
( ) ( )1 2 2 1 2( ) cos ( ) sinh ( ) ,s x k l x k l x= − −
( )( ) ( )( )2 2 2 1 2( ) sin cosh ,s x k l x k l x= − −
( ) ( )3 2 1( ) cos cosh ,s x k x k x=
( ) ( )4 2 1( ) sin sinh .s x k x k x=
Obtained expression (14) for temperature allows de-
scribing of temperature in helium at any coordinate point
inside the experimental device.
For the heat flow in helium layer we obtain the follow-
ing expression:
( ) ( )1 2 2
2 2 2
2 2 1 2
( ) sin ( ) cos
( , )
cosh ( ) cos ( )
Q QQ A x t B x t
Q x t
k l k l
ω ω⎡ ⎤+⎣ ⎦=
−
, (17)
where functions A2Q(x) and B2Q(x) have the form
[ ]
[ ]
2 3 2 3 2 4 2 4 2
4 2 3 2 3 2 4 2
( ) ( ) ( ) ( ) ( ) sin
( ) ( ) ( ) ( ) cos
QA x s l x s l s l x s l
s l x s l s l x s l
φ
φ
= − − + − −
− − − −
(18)
and
[ ]
[ ]
2 3 2 3 2 4 4 2
4 2 3 2 3 2 4 2
( ) ( ) ( ) ( ) ( ) cos
( ) ( ) ( ) ( ) sin .
QB x s l x s l s x s l
s l x s l s l x s l
φ
φ
= − + −
− − − − (19)
These results allow determining of the heat flow both
inside the helium and at the right boundary of the vessel
with helium, i.e. in the point, where the bolometer is situ-
ated in experiments [4].
4. The problem of thermometer measurement
When carrying out experiments [4] temperature is
measured with the help of thermometer, which is situated
in the right side of the vessel with helium. Let’s show that
situation when thermometer measures not temperature but
heat flow is possible.
Using Eq. (17) heat flow at right side of helium at
2x l= can be written in the form
2 2( , ) sin ( ),Q l t Q tω β= + (20)
where phase shift is
3 4
4 3
cos sin
arctan ,
cos sin
s s
s s
φ φ
β
φ φ
⎛ ⎞−
= ⎜ ⎟−⎝ ⎠
(21)
and amplitude is equal to
2 2
1 3 4 3 4
2
2 2 1 2
2 sin (2 )
.
cos (2 ) cosh (2 )
Q s s s s
Q
k l k l
φ+ −
=
+
(22)
Let the thermometer does not change heat flow in heli-
um. Then thermal conductivity equation for the thermome-
ter t B xxT Tχ= can be written with the boundary conditions
2( 0) sin ( )BQ x Q tω β= = + , ( ) 0B BQ x l= = (because the
thermometer is heat insulated with non-conductor), where
Bl is the thermometer length, Bχ is the thermometer tem-
perature conductivity.
In this case the solution of the thermal conductivity
equation inside the thermometer has the form
( ) ( )
( )2
cos / ( )
( , ) sin
sin /
B B
B
B B B
l x
T x t Q t
l
ω χ
ω β
χ ω ω χ
−
= + . (23)
From this expression we obtain the average temperature in
the thermometer:
2
0
( , )1( ) ( , )
Bl
B B
B B
Q l t
T t T x t dx
l l ω
= =∫ . (24)
So, under definite conditions, when one side of a ther-
mometer is thermally isolated, the thermometer will show
the heat flow at its other side. In our case such thermome-
ter will show heat flow in helium at 2x l= .
5. Heat flow resonanses
It is known that under boundary conditions (3) in super-
fluid helium temperature and heat flow resonances appear.
This phenomenon was described in [7,8] where experi-
ments with given heat flow at both ends were carried out.
In such case temperature and heat flow resonances are ob-
served in all harmonics (Fig. 2,a). In this work it is shown
that such problem formulation when only odd resonances
Fig. 2. Heat flow amplitude dependence on frequency excitation
when at the right end is constant heat flow [7] (а) or constant
temperature (Eqs. (17)–(19)) (b).
a
b
1
1
2
2
3
3
4
4
5
5
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
Q
,
ar
b
.
u
n
it
s
Q
,
ar
b
.
u
n
it
s
ω0, arb. units
ω0, arb. units
K. Nemchenko and S. Rogova
6 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 1
appear is possible (Fig. 2,b). It is related with the diffe-
rence in boundary conditions. In classical case right end is
heat insulated ( 2( , ) 0Q l t = ) and in our case right end is at
constant temperature ( 2( , ) 0T l t = ).
Resonance frequency dependence on length of the first
layer was studied as well. It was found out that the longer
the first layer is the less is resonance frequency. In Fig. 3
an example of such dependence for the first resonance is
given.
6. Double-layer system helium–detector
Let’s study the influence of the right wall of the vessel
on resonances characteristics. So, we consider the right
wall as a layer of h width (see Fig. 1,b). Heat transfer in it
we describe with the usual thermal conductivity equation.
Heat transfer in helium we describe with the system (4).
For simplicity we consider 2 0χ = and 1 0l = . We have
obtained the double-layer system (the first layer is super-
fluid helium and the second layer is the right wall of the
vessel) with the following boundary conditions:
( )0( 0, ) cos , ( , ) 0.Q x t Q t T h tω= = = (25)
Making temperatures and heat flows equal at the
boundary of the layers we obtain the following expression
for temperature in superfluid helium:
____________________________________________________
( ) ( )
( )
0 1 2
2 2 2 2 2
2 2 1 2 2 2 2 2 3
( ) sin ( ) cos
( , ) ,
4 cos ( ) 4 sin ( ) 2 sin 2V V
Q C x t C x t
T x t
u C kl b u kl b C u kl b
ω ω
α ωχ α ωχ α
⎡ + ⎤⎣ ⎦=
⎡ ⎤+ +⎣ ⎦
(26)
where functions C1(x) and C2(x) are
( ) ( ) ( ) ( )
( )
2 2 2
1 2 2 1 2 2 2 2 2 2 3
2 2 4
( ) 4 cos sin ( ( )) 4 sin cos ( ) 2 cos (2 ) ,
( ) 2 cos .
V V
V
C x C kl k l x b u kl k l x b C u k l x b
C x C u kx b
ωχ α ωχ α
ωχ α
= + − + − +
= −
(27)
The expression for the heat flow in helium is
( ) ( )
( )
0 3 4
2 2 2 2 2
2 1 2 2 2 2 2 3
( ) sin ( ) cos
( , ) ,
4 cos ( ) 4 sin ( ) 2 sin 2V V
Q C x t C x t
Q x t
C kl b u kl b C u kl b
ω ω
ωχ α ωχ α
⎡ + ⎤⎣ ⎦=
+ +
(28)
with functions C3(x) and C4(x):
( ) ( ) ( ) ( ) ( )
( )
2 2 2
3 2 2 1 2 2 2 2 2 2 3
4 2 4
( ) 4 cos cos ( ) 4 sin sin ( ) 2 sin (2 ) ,
( ) 2 sin .
V V
V
C x C kl k l x b u kl k l x b C u k l x b
C x C u kx b
ωχ α ωχ α
ωχ α
= + + + + +
=
(29)
_______________________________________________
Hereinabove we used the following notations:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2 2 2 2
1 2
3 4
0 2
cos sinh , sin sinh ,
sinh 2 sin , sin 2 sinh 2 ,2
.V
b h h b h h
b h b h hh
T C
λ λ λ λ
λ λ λλ
α
= + = +
= − = +
=
(30)
Here / 2λ ω χ= is the inverse thermal wavelength in the
right wall of the vessel, χ is the temperature conductivity
of it, 2k uω= defines the inverse wavelength of second
sound.
Odd harmonics resonances of the heat flow appear and
characteristics of these resonances depend on h. When h is
much less then the length of heat wave in the right wall of
the vessel we obtain the expression for resonance of the
heat flow:
( )
0 2
2 2
2 0
( ) .
Q u
A
l
ω
ω ω γ
=
− +
(31)
Here 0ω is a resonance frequency and γ is the width of
the resonance peak:
2
2
2
.
V
u h
C l
γ
χ
= (32)
Temperature odd harmonic resonances of unusual form
appear (Fig. 4). When h is much less then the length of
heat wave in the right wall of the vessel we obtain the ex-
pression for resonance of the temperature:
Fig. 3. First resonance dependence on the heater width.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
f
f/
1
10
–3
10
–2
10
–1
l1 1λ
Standing second sound wave in many-layer systems
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 1 7
( )
( )
2 2
00
2 22 0
4
( )
Q
A
u
ω ω γ
ω
ω ω γ
− +
=
− +
. (33)
Such a behavior near a resonance frequency is much
unexpected. The nature of this phenomenon is connected
with the unusual type of the considered system. It consists
of two layers which have different type of heat transfer —
acoustic and dissipative. This frequency dependence could
be observed in future experiments. All the conditions under
which such resonances can be observed are presented in
this paper.
6. Conclusions
In conclusion we should note that the double-layer
model of heat emission and propagation in helium is built.
We have considered two double-layer systems: heater–he-
lium and helium–detector. The model takes into account
both second sound mode and thermal conductivity in su-
perfluid helium and dissipative heat transfer in heater and
the detector.
Resonance frequency and amplitude dependence on
width of the heater and the detector was studied. The ex-
plicit expressions for temporal and spatial dependencies of
temperature (14) and heat flow (17) were derived. In par-
ticular, these equations allow demonstrating the depend-
ences of resonance frequency on the heater width (Fig. 3).
Conditions, under which thermometer measures not tem-
perature but heat flow, are determined and the expression
(24) for the temperature in this case was obtained.
Unusual heat flow resonances behavior (33) is found
out and experimental conditions under which such phe-
nomenon may be observed are defined. Resonance fre-
quency dependence on the heater length is found out. In-
fluence of the right wall of the vessel on resonance
characteristics is studied. Particularly, it was found, that
the resonance width in helium can be determined not by
dissipative properties of helium, but the dissipation in de-
tector (32).
1. I.M. Khalatnikov, An Introduction to the Theory of Super-
fluidity, Redwood City, Addison-Wesley (1989).
2. V.P. Peshkov, J. Exp. Theor. Phys. 8, 16 (1946).
3. V.P. Peshkov, J. Exp. Theor. Phys. 10, 18 (1948).
4. A.S. Rybalko, Fiz. Nizk. Temp. 30, 1321 (2004) [Low Temp.
Phys. 30, 994 (2004)].
5. K. Nemchenko and S. Rogova, QFS, 1–7 August, Grenoble
(2010), p. 37.
6. K. Nemchenko and S. Rogova, International Conference for
Young Scientists “Low Temperature Physics”, 7–11 June,
Kharkiv (2010), p. 105.
7. K. Nemchenko and S. Rogova, J. Low Temp. Phys. 150, 187
(2008).
8. K. Nemchenko and S. Rogova, J. Mol. Liq. 151, 9 (2010).
Fig. 4. An example of temperature resonance of unusual form
with γ = 0.03ω0 (a), γ = 0.13ω0 (b).
00
00
0.50.5
1.01.0
1.01.0
0.80.8
0.60.6
0.40.4
0.20.2
1.51.5
2.02.0
2.02.0
1.81.8
1.61.6
1.41.4
1.21.2
0.50.5
0.50.5
1.01.0
1.01.0
1.51.5
1.51.5
2.02.0
2.02.0
aa
bb
ωω/ω/ω00
ωω/ω/ω00
TT
TT//
((ωω
00
))
TT
TT//
((ωω
00
))
|