Normal transmission of phonons with anomalous dispersion through the interface of two continuous media
In this work the problem is solved of normal transmission of quasiparticles through the interface of two continuous media, one of which is quantum fluid. The quantum fluid is described as a continuous medium with correlations. Within the framework of this approach the dispersion relation of the q...
Saved in:
| Date: | 2006 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
|
| Series: | Физика низких температур |
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/120140 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Normal transmission of phonons with anomalous dispersion through the interface of two continuous media / I.M. Adamenko, K.E. Nemchenko, I.V. Tanatarov // Физика низких температур. — 2006. — Т. 32, № 3. — С. 255-268. — Бібліогр.: 22 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-120140 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1201402025-02-23T19:25:42Z Normal transmission of phonons with anomalous dispersion through the interface of two continuous media Adamenko, I.M. Nemchenko, K.E. Tanatarov, I.V. Квантовые жидкости и квантовые кpисталл In this work the problem is solved of normal transmission of quasiparticles through the interface of two continuous media, one of which is quantum fluid. The quantum fluid is described as a continuous medium with correlations. Within the framework of this approach the dispersion relation of the quantum fluid Ω(k) can be arbitrary. The integral equation describing it in a half-space is solved by the Wiener–Hopf method, and its general solution is obtained. This approach is applied to the dispersion relation of the Bose–Einstein condensate. It is shown that the solutions of equations of quantum fluid in a half-space are traveling waves deformed near the border by specific surface standing waves. By means of boundary conditions the general solution in the whole space is obtained. Expressions for transmission and reflection factors of waves in both directions are derived, depending on their frequency. The results are important for describing the creation of helium II phonons on the boundary with a solid, and are of interest for classical acoustics. We would like to thank A.F.G. Wyatt for many helpful discussions. We also express our gratitude to EPSRC of the UK (grant EP/C 523199/1), and to GFFI of Ukraine (grant N02.07/000372) for support for this work. 2006 Article Normal transmission of phonons with anomalous dispersion through the interface of two continuous media / I.M. Adamenko, K.E. Nemchenko, I.V. Tanatarov // Физика низких температур. — 2006. — Т. 32, № 3. — С. 255-268. — Бібліогр.: 22 назв. — англ. 0132-6414 PACS: 67.40.Bz, 67.40.Db, 67.40.Pm https://nasplib.isofts.kiev.ua/handle/123456789/120140 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Квантовые жидкости и квантовые кpисталл Квантовые жидкости и квантовые кpисталл |
| spellingShingle |
Квантовые жидкости и квантовые кpисталл Квантовые жидкости и квантовые кpисталл Adamenko, I.M. Nemchenko, K.E. Tanatarov, I.V. Normal transmission of phonons with anomalous dispersion through the interface of two continuous media Физика низких температур |
| description |
In this work the problem is solved of normal transmission of quasiparticles through the interface
of two continuous media, one of which is quantum fluid. The quantum fluid is described as a
continuous medium with correlations. Within the framework of this approach the dispersion relation
of the quantum fluid Ω(k) can be arbitrary. The integral equation describing it in a half-space
is solved by the Wiener–Hopf method, and its general solution is obtained. This approach is applied
to the dispersion relation of the Bose–Einstein condensate. It is shown that the solutions of
equations of quantum fluid in a half-space are traveling waves deformed near the border by specific
surface standing waves. By means of boundary conditions the general solution in the whole
space is obtained. Expressions for transmission and reflection factors of waves in both directions
are derived, depending on their frequency. The results are important for describing the creation of
helium II phonons on the boundary with a solid, and are of interest for classical acoustics. |
| format |
Article |
| author |
Adamenko, I.M. Nemchenko, K.E. Tanatarov, I.V. |
| author_facet |
Adamenko, I.M. Nemchenko, K.E. Tanatarov, I.V. |
| author_sort |
Adamenko, I.M. |
| title |
Normal transmission of phonons with anomalous dispersion through the interface of two continuous media |
| title_short |
Normal transmission of phonons with anomalous dispersion through the interface of two continuous media |
| title_full |
Normal transmission of phonons with anomalous dispersion through the interface of two continuous media |
| title_fullStr |
Normal transmission of phonons with anomalous dispersion through the interface of two continuous media |
| title_full_unstemmed |
Normal transmission of phonons with anomalous dispersion through the interface of two continuous media |
| title_sort |
normal transmission of phonons with anomalous dispersion through the interface of two continuous media |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| topic_facet |
Квантовые жидкости и квантовые кpисталл |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120140 |
| citation_txt |
Normal transmission of phonons with anomalous dispersion through the interface of two continuous media / I.M. Adamenko, K.E. Nemchenko, I.V. Tanatarov // Физика низких температур. — 2006. — Т. 32, № 3. — С. 255-268. — Бібліогр.: 22 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT adamenkoim normaltransmissionofphononswithanomalousdispersionthroughtheinterfaceoftwocontinuousmedia AT nemchenkoke normaltransmissionofphononswithanomalousdispersionthroughtheinterfaceoftwocontinuousmedia AT tanataroviv normaltransmissionofphononswithanomalousdispersionthroughtheinterfaceoftwocontinuousmedia |
| first_indexed |
2025-11-24T15:47:33Z |
| last_indexed |
2025-11-24T15:47:33Z |
| _version_ |
1849687281508548608 |
| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, No. 3, p. 255–268
Normal transmission of phonons with anomalous disper-
sion through the interface of two continuous media
I.N. Adamenko1,2, K.E. Nemchenko1, and I.V. Tanatarov3
1 V.N. Karazin Kharkov National University, 4 Svobody Sq., Kharkov 61077, Ukraine
2 Electro-Physical Scientific and Technical Centre of the National Academy of Sciences of Ukraine
28 Chernyshevsky Str., Kharkov 61002, Ukraine
3 Akhiezer Institute for Theoretical Physics National Science Center
«Kharkov Institute of Physics and Technology» of the National Academy of Sciences of Ukraine
1 Academicheskaya Str., Kharkov 61108, Ukraine
E-mail: igor_tanatarov@inbox.ru
Received August 4, 2005
In this work the problem is solved of normal transmission of quasiparticles through the inter-
face of two continuous media, one of which is quantum fluid. The quantum fluid is described as a
continuous medium with correlations. Within the framework of this approach the dispersion rela-
tion of the quantum fluid �( )k can be arbitrary. The integral equation describing it in a half-space
is solved by the Wiener–Hopf method, and its general solution is obtained. This approach is ap-
plied to the dispersion relation of the Bose–Einstein condensate. It is shown that the solutions of
equations of quantum fluid in a half-space are traveling waves deformed near the border by spe-
cific surface standing waves. By means of boundary conditions the general solution in the whole
space is obtained. Expressions for transmission and reflection factors of waves in both directions
are derived, depending on their frequency. The results are important for describing the creation of
helium II phonons on the boundary with a solid, and are of interest for classical acoustics.
PACS: 67.40.Bz, 67.40.Db, 67.40.Pm
Keywords: superfluidity, quasiparticles, dispersion, interface
1. Introduction
Many physical properties of continuous media at
low temperatures can be described in terms of quasi-
particles which are observed experimentally. So, for
example, in the experiments of [1,2] the quasiparticles
of superfluid helium were created on the interface
with a heated solid. Then the propagation of quasi-
particles in superfluid helium and their reflection
from various interfaces was investigated.
When incident on an interface of two continuous
media, a quasiparticle brings to it its energy and is de-
stroyed. On both sides of the interface new quasi-
particles are created with certain probabilities, and
carry away from the interface the energy of the de-
stroyed one. We shall name the described process
briefly «transfer of quasiparticles through the inter-
face». This process underlies a series of observable
phenomena. In particular, it determines the tempera-
ture discontinuity (Kapitza’s discontinuity, see, for
example, [3]) which always exists on the boundary of
two continuous media at low temperatures.
The dispersion relation of real continuous media
�( )k is nonlinear. Direct experiments [1,2] have
shown, that in superfluid helium there are phonons
with anomalous dispersion, which is close to linear,
and rotons with dispersion essentially different from
linear. The dispersion relation of quasiparticles of Bo-
se–Einstein condensate (BEC), according to [4], at
small wave vectors k is close to linear, and with
growth of k becomes strongly nonlinear. Meanwhile
the problem of quasiparticles’ transfer through the in-
terface of two continuous media has been solved till
now only for the case of strictly linear dispersion laws
of both the adjacent media.
© I.N. Adamenko, K.E. Nemchenko, and I.V. Tanatarov, 2006
The aim of the present work is the rigorous solution
of this problem for the case when the dispersion rela-
tion of one of the media is nonlinear. Thus the descrip-
tion of Kapitza’s temperature discontinuity [3] is gen-
eralized to the case of nonlinear dependence �( )k . In
this work we consider the dispersion relation of BEC
— the quantum continuous medium, the experimental
realization of which has led to occurrence of numerous
experimental and theoretical works devoted to the re-
search of its physical properties (see, for example, the
reviews [5,6]). The dispersion relation of BEC�( )k at
small wave vectors k well describes the dispersion rela-
tion of the phonons of superfluid helium. That is why
the results obtained in this work may appear impor-
tant not only for BEC, but also for research on mecha-
nisms of creation and evolution of phonon beams in
superfluid helium. In recent years a number of experi-
mental and theoretical works devoted to these ques-
tions have been published (see, for example, [7–10]).
The results obtained appear to be important also for
classical acoustics, where the problem of wave passage
through the interface has been solved only for the case
when the dispersion relations of both the adjacent me-
dia are strictly linear (see, for example, [11]). This
problem with respect to real media, that have nonlin-
ear dispersion, was of interest in the middle of the last
century (see, for example, [12]) and is still relevant.
For example, in recent works [13,14] simulations of
wave packets transfer through the interfaces of real
crystals were carried out. In [13] the passage of
phonon wave packets through the interface of two
semiconductors was investigated by means of molecu-
lar dynamics simulations. In [14] the phonons trans-
port in superlattices made up of the same crystals was
investigated.
We describe a quantum fluid with nonlinear disper-
sion relation within the framework of the theory sug-
gested in [15], in which it is considered as a continu-
ous medium with correlations. This theory is based on
the fact that the thermal de Broglie wavelength of a
particle of a quantum fluid exceeds the average inter-
atomic separation. Then the variables of the conti-
nuous medium can be assigned values at each ma-
thematical point of space in the probabilitic sense, but
the relations between them become nonlocal. This
nonlocality allows us to describe a continuous medium
with arbitrary dispersion relation. The approach was
also used earlier in [16] to describe the hybridization
of rotons and ripplons of superfluid helium, but with-
out a substantiation for the possibility of application
of this theory to the description of a quantum fluid.
It turned out that the theory introduced in [15],
which describes a quantum fluid in the infinite space,
cannot be trivially generalized to the half-space. In
the brief work [17] a way of such generalization was
offered, and the equation describing a quantum fluid
in half-space was obtained. One of the solutions of
this equation was obtained there (here we name it the
«out-solution») that corresponds to wave packets tra-
veling from the interface. In this work the two lin-
early independent solutions of the equation (the in-
and out-solutions) are obtained. This enabled us to
solve the problem of the phonon passage not only from
a solid into the quantum liquid, but also in the oppo-
site direction. Besides, only the in-solution gives us
the phase shifts of waves at their reflection from the
interface of a quantum fluid with vacuum or with an
infinitely hard solid. We find that the probabilities
of quasiparticles passage through the solid-quantum
fluid interface in both directions are equal at the same
frequencies. This means that at equal temperatures of
the continuous media on both sides of the interface
there exists the thermodynamic equilibrium between
them — the heat flows to the right and to the left are
compensated.
In section 2 of this work we, following [17], formu-
late the problem; we derive the equation describing a
quantum fluid in a half-space and introduce the solu-
tion of this equation for arbitrary dispersion relation
of the quantum fluid obtained by Wiener–Hopf me-
thod [18]. In theoretical physics this method was used
earlier, in particular, for the description of guided
electromagnetic waves [19]. It has been applied to
the solution of the equations of quantum fluid in
half-space for the first time. This solution is of interest
by itself and is given in detail in the mathematical ap-
pendix.
In section 3, working from the general solution, we
investigate the dispersion relation �( )k of Bose–Ein-
stein condensate. At small wave vectors this disper-
sion law well describes the anomalous dispersion rela-
tion of phonons of superfluid helium. It is shown that
if the dispersion relation of the continuous medium
differs from linear, then specific standing waves ap-
pear near the boundary which correspond to the imagi-
nary roots of equation �2 2( )k � � . Neither these sur-
face waves nor the traveling waves which correspond
to the real roots of equation �2 2( )k � � are by them-
selves the solutions of the equations of a continuous
medium with correlations in the half-space. The solu-
tions of the equations of quantum fluid in the half-
space are certain linear combinations of them —
traveling waves that are distorted near the interface
by additives that decay exponentially deep into the
medium. The parameters of the solution that are deter-
mined by the properties of the continuous medium re-
ceive physical interpretation in terms of the micro-
scopic theory of BEC in subsection 3.2.
256 Fizika Nizkikh Temperatur, 2006, v. 32, No. 3
I.N. Adamenko, K.E. Nemchenko, and I.V. Tanatarov
In subsection 3.3 we obtain the solution in the
whole space with the help of boundary conditions and
derive the amplitude reflection and transmission fac-
tors of wave packets passing through the interface, de-
pending on their frequency. Using them, we obtain
the probabilities of creation of quasiparticles on the
interface. From them we get the correction to the cre-
ation probability of any quasiparticles with anoma-
lous dispersion relation at small wave vectors. In par-
ticular, this gives us the correction to the transmission
factor through the solid-superfluid helium interface at
small k.
In subsection 3.4 we take into account that far from
the boundary the energy in a quantum fluid is trans-
ported in a wave packet with the group velocity [20]
and show that in our solution the energy is conserved.
This is not a trivial fact because the standard, and it
could seem simpler methods (see, for example, [16]),
appear to lead to such solutions in which energy is not
conserved.
2. Formulation of the problem, derivation of
equations, and their solution for arbitrary disper-
sion relation
Let us consider two continuous media separated
with a sharp interface x � 0. To the left of the inter-
face, in the region x � 0, there is an ordinary continu-
ous medium with linear dispersion relation � sol ( )k .
Its equilibrium density is �sol , sound velocity is ssol .
The one-dimensional problem is solved, in which the
variables of continuous medium � s ,Vs , and Ps depend
on one coordinate x and time t. Here � s is the devia-
tion of density from the equilibrium value, Vs is the
velocity of continuous medium and Ps is the deviation
of pressure from the equilibrium value. They are re-
lated to each other through the equations of continu-
ous medium (see system (2)), in which the relation
between pressure and density is local
P x t s x ts s( , ) ( , )� sol
2 � .
The solution of these equations is well known and
gives the linear dependence of frequency of the cha-
racteristic oscillations on their wave vector �( )k
� sol sol
2 2 2( )k s k� . (1)
The region x � 0, to the right of the boundary, is
filled with a quantum fluid with a nonlinear disper-
sion relation �( )k that can be essentially nonlinear.
We describe it with the help of the nonlocal hydrody-
namics developed in [15]. According to it, in the infi-
nite space the quantum fluid obeys the ordinary
linearized equations of a continuous medium:
� ,
� ,
� �
�
� � �
� � �
�
��
0
0
1
V
V P
(2)
where �0 is the equilibrium density of the quantum
fluid, in which density depends on pressure through a
nonlocal relation
�
( ) ( ) ( )x d h x P� �
��
�
� . (3)
Here the dot indicates a derivative with respect to
time t, and the prime a derivative with respect to x.
Fourier transform of system (2) and (3) gives us a
simple relation between the Fourier transform of the
kernel of the nonlocal relation h x( )
H k dx h xikx( ) ( )�
��
��
�
� e (4)
and the dispersion relation of the medium [15] �( )k :
H k
k
k
( )
( )
�
2
2�
. (5)
The suggested model describes well the interface
between a quantum fluid (superfluid helium or BEC)
and a solid, because on the frequency scale of the dis-
persion curve of helium II (and all the more so for
BEC) the dispersion curves of most solids, such as Cu
or Au, are very close to linear and these solids can be
described as ordinary continuous media. In the prob-
lem considered, that of normal incidence on the inter-
face, the transverse waves in a solid can be neglected.
Equations (2) and (3) give an equation for pressure
�� � � � �� �
��
�
�P x t d h x P t x( , ) ( ) ��( , ), ( , ),
(6)
which is the generalization of the wave equation to
the case of a nonlinear dispersion relation and de-
scribes a continuous medium with correlations in the
infinite space.
To formulate the problem for the case when such a
medium fills a confined region, we consider the inter-
face sharp, so that the kernel of nonlocal relation (3)
stays the same in the presence of the surface. Then if
we take into account the fact that the fluid fills the re-
gion x � �( , )0 , we get the problem stated on the
half-line:
�� � � � �
�
�P x t d h x P t x( , ) ( ) ��( , ), ( , ) .
0
0 (7)
Normal transmission of phonons through interface with quantum fluid
Fizika Nizkikh Temperatur, 2006, v. 32, No. 3 257
This is an integro-differential equation with differ-
ence kernel, given on the half-line x � �( , )0 , not on
the whole real line. In this form it does not contain a
convolution product in the sense of the Fourier trans-
form because of the finite lower limit. There is no con-
volution product in the sense of either the Laplace or
one-sided Fourier transform because the function h x( )
is even [15] and cannot become zero at x � 0. In order
to solve equation (7) by the Fourier transform we
have first to define the unknown function in the re-
gion x � 0 and to widen the limits of integration to in-
finity.
However, here we cannot just search for an even
(or odd) solution, as is often done when solving differ-
ential equations. The wave equation describing an or-
dinary continuous medium in the infinite space as well
as in a confined geometry has symmetry with respect
to the substitution x x� � . This gives us reason to
search for a solution with the same symmetry. But the
integral equation that describes continuous medium
with correlations in a half-space (7) does not have
such symmetry. Therefore we cannot claim the func-
tion P x( ) to be even, and if we make this assumption,
we may encounter a contradiction. Indeed, let us de-
mand the equation (7) to be true for P x( ) on the
whole interval x � �� ��( , ). Then the left side of
Eq. (7) for any x � �� �( , ) can be obtained from the
right side, and the right side is defined on this interval
only by the values of the function P x( ) at x � ��( , )0 .
So the values of P x( ) at x � ��( , )0 are completely de-
termined by the values of P x( ) at x � ��( , )0 . Then the
function P x( ) on the whole interval x � �� ��( , ),
composed of those two functions, does not have to be
either odd or even.
This means that if we follow the authors of [16]
and try to find an even solution of the equation from
problem (7) stated on the real line, not on the
half-line as in (7), we will come to a mathematical
problem different from the initial one (7), and its so-
lution will not be the solution of problem (7). Our es-
timations with the even solution showed that energy is
not conserved in it. Exceptions are the cases of reflec-
tion from vacuum and from an infinitely hard surface,
but in those cases the solutions themselves are also dif-
ferent from ours. This means that the mathematical
problem with the demand for an even function is not
adequate to the initial physical formulation (7).
Therefore we solve this equation on x � �( , )0 with the
help of the Wiener–Hopf method. This method does
not use any additional assumptions, and as we shall
see it gives us a unique solution in which energy is
conserved.
The solution, given in the appendix, leads to the
two linearly independent solutions of problem (7)
with different asymptotes:
P k
C
k k
k k
k k
k
i
i
i
out
out�
�
�
�
� � �
�
�
�( , )
( )
( )
( )
( )
,�
�
�
�
�1
0
C
(8)
P k
C
k k
k k
k k
k
i
i
i
in
in�
�
�
�
� � �
�
�
�( , )
( )
( )
( )
( )
.�
�
�
�
�1
0
C
(9)
Here the function P k�( , )� is the Fourier transform of
a new function P x t�( , ), that is equal to P x t( , ) at
x � 0 and to zero at x � 0. The products are taken over
all the roots ki of the equation �2 2( )k � � in the up-
per half-plane C � of the complex variable k. The real
roots are assumed to be shifted from the real line in
accordance to the rule (À.14) for the out-solution,
and in the opposite direction for the in-solution. The
primes over the products designate that the «main»
root is omitted in them — it is k1( )� for the out-solu-
tion and ( ) ( )�1 1k � for the in-solution. Taking into ac-
count the roots shifting, those main roots lie in the
upper half-plane for the both solutions and are ob-
tained continuously from zero with the increase of �.
The functions Cin and Cout are determined by bound-
ary conditions.
Let us consider the extreme case of low frequencies
�� 0, when the dispersion relation is close to linear
�2 2 2( )k s k� . In this case the products in relations
(8) and (9) tend to unity, k s1 � �� , and those expres-
sions are brought to the form
P k
C
k s
P k
C
k sout
out
in
in� ��
� �
�
� �
( , )
( )
; ( , )
( )
.�
�
�
�
�
�
(10)
Then we make the inverse Fourier transform by x,
shifting the roots up, and obtain
P x iC i
s
x
P x iC
out
lin
out
in
lin
i
exp( , ) ( ) ;
( , )
� �
�
�
� �
�
�
�
�
�
� n exp( ) .�
�
��
�
�
�
�
�i
s
x
(11)
Those are the well-known solutions for the simplest
special case of linear dispersion relation, that corre-
spond to a set of plane waves traveling from the inter-
face (Pout ) or towards it (Pin). The expressions (8)
and (9) give the generalization of these solutions to
the case of nonlinear dispersion relation of the contin-
uous medium.
258 Fizika Nizkikh Temperatur, 2006, v. 32, No. 3
I.N. Adamenko, K.E. Nemchenko, and I.V. Tanatarov
3. Transfer of wave packets through
the boundary with a quantum fluid
3.1. In- and out-solutions for a quantum fluid
on the half-line
Bose–Einstein condensate is an example of a quan-
tum fluid with anomalous dispersion relation. Its ex-
perimental realization has lead to a large number of
experimental and theoretical works devoted to investi-
gation of its physical properties (see, for example, the
reviews [5,6]).
In Bogoliubov’s theory [4] the dispersion relation
�( )k of the quasiparticles of BEC in the approxima-
tion of point interaction has the form
�2 2 2
2
0
2
1
2
( )k s k
k
k
� �
�
�
�
�
�
�
�
�
. (12)
Here s n m� �� is the sound velocity; k s0 � �
� �2 m �; � is the matrix element of the binary inter-
action of the particles of BEC; m is their mass; n is
their number per unit volume.
The dispersion relation (12) is the simplest compli-
cation with regard to the linear case in the framework
of our approach to the description of continuous media
with nonlinear dispersion relation. At small k this re-
lation approximates the dispersion of any continuous
medium with anomalous dispersion relation, that is
when �( )k at small k deflects upward from the linear
law. This also applies to superfluid helium at a range
of pressures.
The equation�2 2( )k � � with�2( )k from Eq. (12)
has four roots: �kR and �kI , where
k k
k ik
R
I
( ) ,
( ) .
�
�
� �
� �
�
��
0
0
1
1
�
�
(13)
Here the symbols are introduced
� � �1 2� , where � �� �2 0k s (14)
is the dimensionless frequency.
Then the general solution (8), (9) applied to the
dispersion relation (12) gives
P k
C
k k
k k
k kR
m
I
in
out
in
out� �
�
�
�
( , )
( )
( ) ( )
,�
�
� �
(15)
where k k ikm I� � �( )� 0 20 . The upper sign corre-
sponds to the in-solution, the lower sign — to the
out-solution. On shifting kR( )� from the real line
(À.14) and taking the inverse Fourier transform by x,
we get
P x P
k k
k k
k k
k k
ik x R m
R I
ik x m I
R I
R I
out out e e( , ) ( , )� ��
�
�
�
�
�
0
�
�
�
,
(16)
P x P
k k
k k
k k
k k
ik x R m
R I
ik x I m
R I
R I
in in e e( , ) ( , )� ��
�
�
�
�
�
�0
�
�
�
.
(17)
We consider the wave packets passing through the
interface, and that means that the pressure on the in-
terface oscillates with frequencies close to the carrier
frequency of the wave packet �0. So
P t P t P t P ti t i t
out ine e( , ) ~( ) ; ( , ) �( ) ,0 00 0� �� �� � (18)
where functions ~( )P t è �( )P t are the envelopes of pack-
ets in time, that change much more slowly with time
than the oscillating parts.
Now we expand the quantities in the exponents of
Eqs. (16) è (17) into series by ( )� �� 0 and restrict
ourselves to the linear terms, the coefficients by the
exponents are taken at � �� 0. Such expansion was
used in works [21,22] in order to investigate the wave
packets of electromagnetic field and it gives the de-
scription of a packet’s propagation without deforma-
tion.
We omit the index 0 at �0 and have in mind here
and below that kI and kR are taken at the carrier fre-
quency � �� 0. Then taking the inverse Fourier trans-
form with respect to t we get
P x t
k k
k k
P t
x
u
k k
R m
R I R
ik x i t
m
R
out e( , ) ~�
�
�
�
�
�
��
�
�
�� �
�
�
� �
I
R I
k x i t
k k
P t I
�
� �~( ) ,| |e � (19)
P x t
k k
k k
P t
x
u
k k
R m
R I R
ik x i t
I
R
in e( , ) ��
�
�
�
�
�
��
�
�
�� �
�
�
� � �
m
R I
k x i t
k k
P t I
�
� ��( ) .| |e � (20)
Here u u kR R� ( ( ))�0 is group velocity of the wave
packet with carrier frequency �0.
As we know from the equations of continuous
media (2) the relation between the Fourier transforms
of pressure and velocity
V k
k
P k( , ) ( , ) ,�
� �
��
1
0
(21)
we can put down from solutions (19) and (20) the ex-
pressions for velocity in the in- and out-solutions:
Normal transmission of phonons through interface with quantum fluid
Fizika Nizkikh Temperatur, 2006, v. 32, No. 3 259
V x t
k k k
k k
P t
x
u
R R m
R I R
ik x i tR
out e( , ) ~�
�
�
�
�
�
��
�
�
��
�1
0� �
� �
�
�
�
� �1
0� �
�k k k
k k
P tI m I
R I
k x i tI~( ) ,| |e (22)
V x t
k k k
k k
P t
x
u
R R m
R I R
ik x iR
in e( , ) �� �
�
�
�
�
�
��
�
�
��
� �1
0� �
�t
I I m
R I
k x i tk k k
k k
P t I
�
�
�
�
� �1
0� �
��( ) .| |e (23)
According to Eqs. (19), (20), (22) and (23), struc-
turally the solutions consist of two parts — a travel-
ing wave packet (toward the interface in the in-solu-
tion and away from it in the out-solution) and a
standing wave that exponentially decreases with dis-
tance from the boundary. Thus the solution that in the
infinite space has the form of a traveling wave, is dis-
torted in the presence of the boundary by the additives
that decrease exponentially from it. It must be empha-
sized that neither the traveling nor standing waves are
the solutions of equation (7) by themselves.
The coefficients of all the summands in Eqs. (19),
(20), (22), and (23) are essentially complex, which
means that there are phase shifts between all the
waves. The range of penetration | |kI
�1 of the standing
waves into the medium is of the order of the wave-
length of the travelng part | |kR
�1 and therefore is
much less than the size of the wave packet. At this
scale the correction � �x uR to the arguments of �( )P t
and ~( )P t in the corresponding terms are not essential
and therefore are omitted. We can see from the expres-
sion for kI (13) that at small frequencies the penetra-
tion depth of the standing wave is approximately con-
stant and is of the order of k0
1� , and at � � � it tends
to zero as 1 � �. The physical interpretation of the
penetration depth will be discussed in subsection 3.2.
Let us consider the contributions of standing and
traveling waves to the pressure and velocity on the in-
terface. We denote the ratio of the standing waves’
contribution to P t( , )0 to the contribution of the
traveling waves as !, and the same ratio for the contri-
butions to V t( , )0 as ". The subscripts «in» and «out»
designate those ratios for the in- and out-solutions
corresponding. Then we can obtain these quantities di-
rectly from Eqs. (19), (20), (22), and (23). With the
help of relations (13) we can show that ! !in out� * ,
" "in out� * , where the star means complex conjuga-
tion, and
| | | | ,
| | | | ,
! !
" "
!
out in
out in
outarg
� � �
�
� �
� �
�
�
1
2
1
1 2
1
�
�
�
� � �
�
�
� � #
#
# #
#
#
� �
arg arcsin arcsin
arg a
in
out
!
"
�
�
1
1
k
k
R
I
,
rg arccos arccosin" �
�
�
� #
#
# #
#
#
�
�
�
�
�
�
�
�
�
�
�
�
�
1
1
k
k
R
I
.
(24)
According to Eqs. (24), at small frequencies, � �� 1,
the contributions of standing waves are small and
! �in,out � �2 8, " �in,out � � �i 4. As the frequency in-
creases, their contributions also increase and become
comparable with the contributions of the traveling
waves. At large frequencies the ratios of the contribu-
tions of standing and traveling waves to velocity and
pressure on the interface tend to unity as1 � �: at � �� 1
! �out � � � � �i i( )1 2 , " �out � � � �1 1 2( ) .i
The relative phases of velocity and pressure oscilla-
tions in all the waves also depend on frequency. For
the in-solution the picture on the phase diagram is the
following. The velocity in the traveling wave oscil-
lates in antiphase with pressure. The velocity in the
standing wave is shifted by ( )� �! 2 relative to the
pressure in it. At � �� 1 the phase difference between
the pressures of standing and traveling waves arg in!
is small and positive and it rises monotonically with
increase in frequency. At � � � the phase shift be-
tween pressures of standing and traveling waves tends
to ! � 2 and the velocities are in antiphase. For the
out-solution the scheme differs only by the sign of the
relative phase of pressures of standing and traveling
waves.
3.2. Correlation length and penetration depth
In order to describe a quantum fluid in general, and
the Bose–Einstein condensate in particular, we used a
model in which nonlinearity of the dispersion relation
is equivalent to nonlocality of the relation between
pressure and density with kernel h x( ). As we know the
relation (5) between the Fourier image of the kernel
H k( ) with the dispersion relation �( )k , we can easily
obtain this kernel. For the dispersion of BEC (12) in
the one-dimensional case it is
� �� �h x
k
s
m k xm( ) � �
2 2
e . (25)
We see that the correlation length $ is of the order of
$ ∼ ∼| |k km
� �1
0
1. (26)
260 Fizika Nizkikh Temperatur, 2006, v. 32, No. 3
I.N. Adamenko, K.E. Nemchenko, and I.V. Tanatarov
If we express k0 in terms of the microscopic theory
of BEC, we can express it in the form $ ∼ � �ms. This
quantity may be interpreted as the de Broglie wave-
length of a particle with mass m moving with the
sound velocity s. If we use the expression for the
sound velocity in BEC, we will get $ �∼ � � mn,
which is the de Broglie wavelength of a particle of
mass m with energy % �p n� , equal to the average en-
ergy of a particle’s interaction with the condensate.
The penetration depth of the standing wave into the
condensate &, in accordance with Eq. (16), is equal to
| ( )|kI � . At � �� 1 it is nearly constant and of the order
of | |km , and therefore is of the order of the correlation
length $. Thus in the limit of small frequencies there is
only one quantity with the dimensions of length that
characterizes the system, i.e., the correlation length
$ (26).
At large frequencies the penetration depth tends to
zero as & � �( ) ( )� �k0
1. On substituting k0 and �, we
get & �∼ � �� 2m . This is the de Broglie wavelength
of a free particle of mass m and energy ��. In the limit
of large frequencies this is also the de Broglie wave-
length of the condensate quasiparticle of frequency�.
3.3. Reflection and transmission factors
We will call the amplitude reflection factor r the
ratio of the envelopes of pressures in the reflected and
the incident wave packet. In the quantum fluid the en-
velopes are taken of the traveling parts of the solu-
tions, which transport energy far from the interface.
As the packets are composed of sets of plane waves
with close frequencies, evidently the plane wave with
the carrier frequency has the same reflection factor.
We obtain the reflection and transmission factors with
the help of boundary conditions that demand continu-
ity of pressure and velocity on the interface.
Let us at first consider the problem of a wave
packet incident on the interface from the solid. In this
case there are both the incident and the reflected
waves in the solid (region x � 0), and in the quantum
fluid (region x � 0) there exists only the out-solution.
As the time dependence of all the solutions is the
same, the boundary conditions can be satisfied only if
all the frequencies are equal. Then the incident and
the reflected waves have equal frequencies and wave
vectors that differ only in sign. This is enough to ex-
press the reflection factor r through pressure and ve-
locity on the interface:
r
Z
Z
�
�
�
1
1
, where Z
P t V t
s
�
�( , ) ( , )0 0
�sol sol
. (27)
In the well-known case, when the region x � 0 is
filled with an ordinary continuous medium, with den-
sity �0, a linear dispersion relation, and sound velo-
city s0, Z turns into the real impedance of the inter-
face ( ) ( )� �0 0s s� sol sol and then the expression for the
reflection factor takes the ordinary real form.
In the general case the quantities P t( , )0 andV t( , )0
can also be found from the out-solution (19) and (22)
with the help of boundary conditions. As a result the
relation (27) for Z gives
Z
Z
i
R�
�1 '
, (28)
where the notations are introduced for real quantities
Z
s
sR
R�
�
�
0
sol sol
and ' �
�| |k k
k
I m
R
. (29)
According to definitions (29), ZR is the impedance
of a boundary between a solid and an ordinary continu-
ous medium with density �0 and linear dispersion law
with sound velocity s kR R� �� , which is the phase ve-
locity of a wave packet with carrier frequency �0. On
substitution of expressions for kR I m, , into Eq. (29),
one can see that ' depends only on the dimensionless
frequency �.
After substituting relation (28) into Eq. (27),
we’ll get the amplitude reflection factor for a wave in-
cident on the interface with a quantum fluid from the
solid r�:
r
Z i
Z i
R
R
� �
� �
� �
( ) ( )
( ) ( )
� ' �
� ' �
1
1
. (30)
This quantity is complex and therefore there is a
nontrivial (i.e., different from zero or �!) phase shift
between the incident and reflected waves, equal to
arg r�.
The transmission factor D is the share of the inci-
dent wave’s (wave packet’s) energy that is transferred
through the interface into the second medium. As the
share of the reflected energy is | |r 2, the transmission
factor equals D r� �1 2| | . Then from Eq. (30) we ob-
tain
D
Z
Z
R
R
�
� �
4
1 2 2( ) '
. (31)
This quantity can be expressed in terms of only two
parameters,
D Z
Z
Z
( , )0
0
0
2 2
4
1
2
1
2
1
1 2
1
�
�
� �
�
�
�
�
�
�
�
��
�
�
�� �
� �
�
�
�
��
�
�
��
,
(32)
where
Normal transmission of phonons through interface with quantum fluid
Fizika Nizkikh Temperatur, 2006, v. 32, No. 3 261
Z
s
s0
0�
�
�sol sol
(33)
is the (ordinary) impedance of the boundary in the
limit �� 0 and does not depend on frequency, and �
is �( )� from definition (14).
At �� 0 (which is equivalent to � � 1), the ex-
pression (32) turns into the classic expression for en-
ergy transmission factor through an interface between
two continuous media with impedance Z0
D
Z
Z
0
0
0
2
4
1
�
�( )
. (34)
At small frequencies � �� 1 the nonlinearity brings in
a small correction, quadratic in frequency:
D D
Z
Z
( )
( )
�
�
� �
�
�
�
�
�
�
�
�
�
�0
2
0
2
0
2
1
16
2 1
1
. (35)
This expression gives the correction for the transmis-
sion factor through the interface with any continuous
medium, that has an anomalous dispersion relation
(i.e., it deflects upward from the linear law) at small
frequencies � �� 1.
At � �� 1 the factor D slowly tends to zero
D
Z
( )� �� �
4
2
0
. (36)
There are many interfaces, for example between
solids and liquids or solids and gases, for which the
strong inequality Z0 1�� is true. The transmission fac-
tor through such interfaces with liquids that have
anomalous dispersion relation at small frequencies is
given by Eq. (35), in which we should put Z0 0� .
In particular, the dispersion relation of superfluid
helium at small frequencies has the form
�( ) ( )k sk k� �1 2( . (37)
Therefore for the transmission factor with superfluid
helium at small k we obtain
D k D k( ) � ��
�
�
�
�
�0
21
1
2
( . (38)
Now let us consider the problem of a wave packet
incident on the interface from the quantum fluid. The
solution in the quantum fluid (region x � 0) then is a
sum of in- and out-solutions
P x t P x t P x t� � �( , ) ( , ) ( , )in out at x � 0 , (39)
and in the solid (region x � 0) there is only the trans-
mitted wave.
As we know the relation between velocity and pres-
sure in a plane wave (21), which is the only wave in
the solid in this case, we can take into account the sign
of its wave vector and obtain that P t V t( , ) ( , )0 0� �
� � �sol sols . From the other side of the boundary we
find P t( , )0 from Eq. (39), on substituting there Pout
and Pin from Eqs. (19) and (20). Similarly, we get
V t( , )0 taking Vout and Vin from Eqs. (22) and (23).
Then the quantity P t V t( , ) ( , )0 0� is expressed in terms
of the ratio ~( ) �( )P t P t� , and we find the last one using
the boundary conditions. The reflection factor is
found as the ratio of the coefficients at exp ( )�ik xR in
solutions (19) and (20):
r
Z i
Z i
k k
k k
k k
k k
R
R
R m
R m
R I
R I
� �
� �
� �
)
�
�
)
�
�
( )�
'
'
1
1
. (40)
If we use Eqs. (29), (33) and (13), the expression
(40) can be reduced to
r Z
Z i
Z i
� �
�
�
�
� �
�
�
�
�
� �
�
*
*
�
( , )0
0
0
1
1
2
1 2
1
1
1
2
1 2
1
1
�
� �
�
� �
�
� �
� �
)
� � �
� � �
i
i
i
i
2
1 2
1 1
1 1�
� �
� �
. (41)
According to Eq. (41), there is a phase shift arg r�
between the incident and the reflected waves when a
packet falls on the interface from the quantum fluid,
and it is different from arg r�.
It is important that the absolute values of reflec-
tion factors in both directions coincide | | | |r r� �� .
Then the energy transmission factors in both direc-
tions are the same and equal to D from Eq. (31). This
leads to the fact that at equal temperatures on both
sides of the interface the heat flows in opposite direc-
tions are compensated and the continuous media are in
thermodynamic equilibrium.
Let us now consider the extreme cases of reflection
of quasiparticles of a quantum fluid from the free sur-
face and the opposite extreme case of its reflection
from an infinitely hard surface. The reflection factor
at the reflection from vacuum is obtained from
Eq. (41) by passing to the limit Z0 � �. This is true if
the phonon’s energy is small enough not to cause
quantum evaporation at the free surface [2]. Reflec-
tion from an infinitely hard wall is also obtained from
Eq. (41) if we put Z0 0� . We can see that in both
cases the absolute value of the reflection factor goes to
1 (and therefore the transmission factor to zero), but
its argument remains nontrivial. So, in both cases
there are standing waves by the surface of the quan-
tum fluid, and phase shifts between the incident and
the reflected waves.
262 Fizika Nizkikh Temperatur, 2006, v. 32, No. 3
I.N. Adamenko, K.E. Nemchenko, and I.V. Tanatarov
3.4. Energy flows. Conservation laws
Working from the general solutions in the two adja-
cent continuous media, with the help of boundary con-
ditions we obtain the solution in all the interval
x � �� ��( , ). As there are two boundary conditions, we
can express all the amplitudes in terms of one of them,
for example, the amplitude of the incident wave.
The expression for the energy flow in an ordinary
liquid is commonly known. The expression for the en-
ergy flow density in a wave packet of a quantum fluid
in the infinite spaceQ x t� ( , ) was obtained in the work
[20]. It was shown in [20], that in a quantum fluid
with nonlinear dispersion relation the energy of a
wave packet is transported with the group velocity, so
that
+ , � + ,� �Q x t u x tR( , ) ( , )% . (42)
Here %� ( , )x t is the energy density in the wave
packet. It was defined in [15] as % �� � � �0
2 2V
� �� �P 2 0. The brackets in Eq. (42) indicate averag-
ing over rapid oscillations. The expression (42) was
derived in the way similar to the one used in [21] and
[22], where the propagation of wave packets and
transport of energy in them was investigated for an
electromagnetic field in a medium. Thus we can ex-
press the energy flows in the incident, reflected and
transferred wave packets in terms of one amplitude.
But then the conservation of energy is not trivial and
requires verification.
It can be shown from the general solution in the or-
dinary continuous medium that the average energy
flow through the interface when a wave packet is pass-
ing through it in any direction is equal to
+ , � + ,Q t P t V ts ( ) ( , ) ( , )0 0 . (43)
Let us consider the problem of a wave packet inci-
dent on the interface from the solid. Then we can find
P t( , )0 and V t( , )0 from the out-solutions in the quan-
tum fluid (19) and (22) if we put x � 0. When we sub-
stitute those expressions into Eq. (43), we get
+ , �Q t
P t
ss
R
( )
| ~( )|2
02�
. (44)
From the other side, far enough from the boundary,
at x kI�� �| | 1, where the influence of interface can be
neglected, the average energy flow density + ,�Q is de-
fined by Eq. (42). Then it can be brought to the same
form as in an ordinary continuous medium
�0
2+ ,V x t( , ) . When we substitute here the envelope of
the velocity in the traveling wave packet from
Eq. (22), use the expressions for kR, kI and uR, and
compare the result with Eq. (44), we find that
+ , � + � � ,�Q x t Q t x us R( , ) ( ) . (45)
This equality means that the energy brought to the
interface by the incident wave packet is taken away by
the reflected and the transmitted packets, i.e., that
energy is conserved in the process. It can be shown in a
similar manner that energy is conserved when a wave
packet passes through the interface in the opposite di-
rection as well. This result is not at all trivial, as it is
easy to check that if we search for an even solution (as
in [16], for example), the energy in the general case
will not be conserved.
4. Conclusions
In this work we have solved the problem of normal
transmission of quasiparticles through the interface of
two media, one of which is a quantum fluid. It is
solved as a problem of wave packets transmission
through a boundary of an ordinary continuous medium
and a continuous medium with correlations [15]. In
the framework of this approach the continuous me-
dium with correlations may have arbitrary dispersion
relation �( )k . The solution in the whole space is ob-
tained with the help of boundary conditions that de-
mand pressure and velocity continuity on the interface.
The equations of quantum fluid in half-space are
brought to an integro-differential equation (7), which
is solved by the Wiener–Hopf method. Its general so-
lution (8), (9) is obtained. In the extreme case of lin-
ear dispersion relation of the quantum fluid and in the
limit of small frequencies, when dispersion law is
close to linear, the solution turns into plane waves
(11) and gives the well-known reflection and trans-
mission factors for the case of linear dispersion rela-
tion (34). The general solution is investigated for the
dispersion relation of quasiparticles of Bose–Einstein
condensate (BEC) (12). At small frequencies it ap-
proximates well to the dispersion relation of phonons
of superfluid helium (37) and of quasiparticles of
other continuous media with anomalous dispersion.
It is shown that in a medium with dispersion rela-
tion �( )k that differs from the linear law, there al-
ways exist specific surface waves that correspond to
imaginary roots of the equation �2 2( )k � � . These
waves, however, are fundamentally different from the
Rayleigh surface waves because they appear at normal
incidence of traveling waves on the interface. They
differ also from thermal waves as they are not by
themselves the solutions of equations of the continu-
ous medium with correlations. Their solutions are only
definite linear combinations of ordinary traveling
waves and the standing surface waves (19), (20).
Thus, the solutions of equations of continuous medium
with correlations are plane waves that are distorted
over distances of the order of the wavelength from the
boundary by exponential terms. At small frequencies
Normal transmission of phonons through interface with quantum fluid
Fizika Nizkikh Temperatur, 2006, v. 32, No. 3 263
the penetration depth of the standing waves is of the
order of the correlation length (26) and it is the only
quantity with dimensions of length that describes the
quantum fluid. At large frequencies the penetration
depth is of the order of de Broglie wavelength of a
quasiparticle of BEC. There are phase shifts between
the standing and traveling waves that depend on their
frequency.
The amplitude reflection factors are obtained for
the wave packets incident on the interface both from
the solid (30) and from the quantum fluid (40). They
depend on the carrier frequency of the wave packet as
well as on the properties of the adjacent media. The
complexity of the reflection factors leads to the exis-
tence of nontrivial (not equal to zero or !) phase shifts
between incident and reflected waves. The extreme
cases of reflection from vacuum or from infinitely hard
surface are obtained from the general expression (41)
if we pass to the limits Z0 � � or Z0 0� , respectively.
The energy transmission factors in both directions are
derived (31) and they coincide. This ensures compen-
sation of the heat flows in both directions when tem-
peratures on both sides of the interface are equal.
Those transmission factors are the probabilities of new
quasiparticles’ creation on the interface and they char-
acterize the heat exchange between a solid and a quan-
tum fluid. A correction is obtained for the transmis-
sion factor through an interface between a solid and a
medium with anomalous dispersion relation for small
wave vectors (35). For the interface with superfluid
helium this correction takes the form (38).
It is shown, that due to the existence of standing
waves near the boundary the energy conservation,
shown by Eq. (45), follows from the boundary condi-
tions. This is not a commonplace fact because standard
methods of solution of the stated problem, that seem
to be also simpler, lead to solutions in which energy is
not conserved (see, for example, [16]). Such contra-
dictions appear, in particular, if we try to find an even
solution of equation (7), that is stated on a half-line
and does not possess that symmetry. In this work we
do not make any additional assumptions of this kind
and use the Wiener–Hopf method to solve the equa-
tion (7) on the half-line.
The results obtained in this work may be applied to
describe Kapitza’s temperature discontinuity and heat
exchange between solids and various quantum contin-
uous media (superfluid helium, BEC and others).
They are necessary for understanding the mechanism
of creation of phonon beams in superfluid helium by a
heated solid [7–10]. The results are important for clas-
sical acoustics as well, as an example of solution of the
problem of waves transmission through the interface
of two continuous media in the case when dispersion
relation of one of them is nonlinear and may be essen-
tially nonlinear. Further development of this theory
may give description of the transfer of quasiparticles
with arbitrary dispersion relation through the inter-
face at all incidence angles.
5. Mathematical appendix: solution of our
problem with the Wiener–Hopf method
5.1. Reduction of our equation to a Riemann
boundary problem
Taking the Fourier transform of Eq. (7) by t, we
get
�� � � �
�
�P x d h x P( ) ( ) ( )�
2
0
. (A.1)
This is an integro-differential equation with one
variable. We will use the Wiener–Hopf method to
solve it. It should be noted, that the function P x( ) is
yet defined only on x � �( , )0 . So in order to take the
Fourier transform of Eq. (A.1), let us introduce two
new functions P x�( ) and P x� ( ). The first one is de-
fined as
P x P x� �( ) ( ) at x � 0 , (A.2a)
and
P x� �( ) 0 at x � 0 . (A.2b)
Let us likewise demand for the second function
P x� �( ) 0 at x � 0 , (A.3)
and at x � 0 let it be defined so that the equation
( ) ( ) ( )P P d h x P� �
��
�
�� �� � � ���
2 (A.4)
should hold true on x � ��( , )0 . At x � �( , )0 the equa-
tion (À.4) is fulfilled automatically.
We take the Fourier transform of Eq. (A.4) by x
P k dx ikx P x�
��
�
�� ��( ) exp { } ( ) . (A.5)
We are looking for the solutions P x� ( ), that grow
not faster than power law at infinity. Then the defini-
tions of P k� ( ) (À.5) and demands (A.2b) and (A.3)
lead to important analytical properties of these func-
tions. Let us indicate as C � the upper half-plane of
complex variable k and as C � the lower half-plane.
Then P k� ( ) is analytical in C � , and P k�( ) is analyti-
cal in C �.
264 Fizika Nizkikh Temperatur, 2006, v. 32, No. 3
I.N. Adamenko, K.E. Nemchenko, and I.V. Tanatarov
Then on taking Fourier transform (A.5) of Eq. (A.4)
and using the convolution theorem, we get
� � � �� � �k P k P k H k P k2 2( ( ) ( )) { ( ) ( )}� . (A.6)
Here H k( ) is expressed through the dispersion rela-
tion of the quantum fluid �( )k through relation (5)
[15]. Then relation (A.6) is brought to the form
P k
k
k
P k� ��
� �( )
( )
( )
( )
�
�
2 2
2
0
�
. (A.7)
This equality defines on the real line k� �� ��( , ) a
linear relation between the limit values of functions
P� and P� on it, that are analytical correspondingly
in the upper and lower half-planes of the complex
variable k. Therefore it defines a Riemann boundary
problem for the half-plane.
5.2. Solution of Riemann boundary problem without
singularities on the contour
In its simplest form the Riemann boundary problem
is formulated as follows (see, for example, [18]).
There is a simple smooth closed contour L that di-
vides the complex plane z into the interior domain D�
and the exterior domain D�.G p( ) (the Riemann prob-
lem factor) is a function of the points of the contour
which satisfies the condition of Goelder and doesn’t
go to zero. We need to find the two functions: -� ( )z ,
analytical in the domain D� , and -�( )z , analytical in
domain D� including z � �, that satisfy on the con-
tour L the linear relation
- -� ��( ) ( ) ( ).p G p p (A.8)
A functionG p( ) is said to satisfy the Goelder condi-
tion on contour L, if for any p p L1 2, � there are such
A � 0 and $ � ( , ]01 , that
| ( ) ( )| | |G p G P A p p1 2 1 2� � � . (A.9)
Index of the function G p( ) is the increment of its ar-
gument at path-tracing the contour, divided by 2!. It
can be expressed through the logarithm
Ind arg lnG p G p
i
d G pL
L
( ) [ ( )] ( )� � �
1
2
1
2! !
. (A.10)
If G p( ) satisfies the listed conditions and remains
real on the contour, then evidently Ind G p( ) � 0. In
this case it can be shown [18], that it follows from re-
lation (A.8) that the functions -� ( )z and -�( )z do
not turn into zero in the domains of their analyticity.
Then on taking the logarithm of Eq. (A.8), we get
ln ( ) ln ( ) ( )- -� �� �p p G pln , (A.11)
where ln G p( ) satisfies the Goelder condition,
ln ( )-� p and ln ( )-� p are single-valued on L, and
ln ( )-� z and ln ( )-� z are analytical correspondingly
in D� and D�.
Then Eq. (A.11) represents the problem of finding
a piecewise analytical function from its step [18]. Its
solution is given by the formula of Sohotskij [18] and
it is defined up to an additive constant:
ln ( )
( )
- z
i
G p
p z
dp
L
�
�
��
1
2!
ln
const . (A.12)
Here ln ( )- z is a piecewise analytical function, equal
to ln ( )-� z at z D� � and to ln ( )-� z at z D� �.
This solution can be extended [18] to the case when
contour L is the real line, D�
�� C , and D�
�� C .
5.3. Complications in our problem with regards to
the simplest case. Roots shifting
The factor of the Riemann boundary problem (A.7) is
J k
k
k
( )
( )
( )
�
��
�
2 2
2
�
. (A.13)
It turns into zero in the roots of equation
�2 2( )k � � . Also we are interested in the dispersion
relation such that �2 2( )k k∼ at k � 0, so at this
point the factor J k( ) is non-bounded and therefore
does not satisfy the Goelder condition. It can be seen
from (A.13), that at k � � function J k( ) is bounded
and separated from zero. So the problem (7) contains
two complications with regard to problem (A.8).
First, the factor J k( ) turns into zero in the points k of
the real line, such that �2 2( )k � � . Second, function
J k( ) is unbounded at k � 0.
The first complication is eliminated this way: we
assume that the points k, in which�2 2( )k � � , always
have at least a small imaginary part, so that J k( ) turns
into zero not on the contour.
The sign of this addition is found from the demand
of damping of any waves in the direction of their pro-
pagation. If we search for a solution that should con-
tain wave packets running from the interface, the rule
for shifting of a root kj � �� ��( , ) from the real line
can be put down symbolically in the form
k k i uj j j� � )sign 0 , (A.14)
where u d k dkj kj
� �( ( ) )� is the group velocity in
the point k kj� . This means that the points kj are
shifted in accordance with the signs of group veloci-
ties in them. If u j � 0, the point is shifted up, if
u j � 0 — down. We will call such solutions the
«out-solutions».
In a similar way we may search for solutions that
far from the interface represent wave packets running
Normal transmission of phonons through interface with quantum fluid
Fizika Nizkikh Temperatur, 2006, v. 32, No. 3 265
to the interface. Those will be «in-solutions» and for
them the rule for roots shifting will be the opposite. If
u j � 0, the root kj is shifted down, and if u j � 0 it is
shifted up. As we take � �( ) ( )� �k k , in both cases
there are as many roots shifted up as down. So
Ind J k( ) remains equal to zero.
5.4. Solving Riemann boundary problem with
singularities on the real line
Let us now eliminate the second complication, asso-
ciated with the unboundedness of factor J k( ) at zero.
We define a new function I k( ) such that
J k
A k
k
I k( ) ( )�
�2 2
2
, (A.15)
where A � 0 is an arbitrary number. It can be seen
from (A.15) that Ind IndI k J k( ) ( )� � 0.
As I k( ) satisfies, after the roots shifting, all the
conditions laid on G k( ) in problem (A.8), the
Riemann boundary problem (A.8) on the real line
with the factor I k( ) instead of J k( ) can be solved by
the scheme given in subsection 5.2. In other words,
I k( ) can be in a unique way (up to a multiplicative
constant) presented in the form
I k
I k
I k
( )
( )
( )
�
�
�
, (A.16)
where I k� ( ) is analytical in C � , I k�( ) is analytical in
C � and in accordance with subsection 5.2 they do not
turn into zero in the domains of their analyticity.
Until now the solution did not require any restric-
tion on the form of the dispersion relation of our con-
tinuous medium with correlations �( )k . But the
asymptotical behavior of a real relation �( )k at k � �
must not have any influence on the solution of the
problem of transmission of a wave packet with a finite
carrier frequency though a boundary. From the other
side, any real dispersion relation can be approximated
on a finite interval of k by a polynomial function of ar-
bitrary power with any required precision.
Then, without reducing the generality of our con-
sideration, we may assume�2 2( )k to be a polynomial.
In this case the procedure of factorization of I k( )
(A.16) is significantly simplified with regard to the
general procedure given in subsection 5.2. The differ-
ence ( ( ) )�2 2k � � can be presented in the form
�2 2 2 2( ) ( ( ))k B k ki
i
� � ��� � , (A.17)
where the product is taken over all the roots ki
2 of
equation �2 2( )k � � relative to k2 in the complex
plane.
As we consider such dispersion relation, that
�2 2( )k k∼ at k � 0, on taking into account
Eq. (A.17), �2( )k can be presented in the form
�2 2 2 2 0( ) ( ( ))k B k k k
i
i� � � �� � . (A.18)
Here the prime over the product means that it does
not contain the factor with kj
2 0 0( )� � � , which is
taken out of the product. When we substitute
Eqs. (A.17) and (A.18) into relation (A.13) and take
into account definition (A15), we get
I k
I k
I k k A
k k
k k
i
i
i
i
( )
( )
( )
( ( ))
( ( )
� �
�
�
� � �
�
�
�
�
1
02 2
2 2
2 2
�
� )
.
(A.19)
Every term of the products k k k ki i
2 2� � � *( )
* �( )k ki contains one multiplier with its zero in C �
and one with zero in C �. If we group together all the
factors of the numerator and the denominator that
turn into zero in C � , we’ll get a function that is ana-
lytical in C � and doesn’t go to zero in this domain.
But, taking into account subsection 5.2, those are the
properties of function I k�( ), which is defined only
within a multiplicative constant. Therefore
I k
k iA
k k
k k
i
i
� �
�
�
� � �
�
�
�
�
( )
( ( ))
( ( ))
1
0
C
C
�
�
, (A.20)
where the product is taken over all the roots of equa-
tion �2 2( )k � � with regard to k in C � . In the pro-
duct with prime the factor ( ( ))k kj� �� 0 is omitted
with kj � �C such that kj ( )� � �0 at �� 0. In the
in- and out-solutions those factors are different. In-
deed, let us indicate as k1( )� that real positive root of
�2 2( )k � � (if it exists), in which the root
kj ( )� � �0 0 passes continuously when � increases
from zero. Then, as � �( ) ( )k k� � , u k( )1 0� and
u k( )� �1 0. In accordance to our rule for roots shifting
(A.14), for the out-solution the root k1 is shifted up
and ( )�1 1k down; for the in-solution ( )�1 1k is shifted
up and k1 is shifted down.
Then if we factor out the corresponding multiplier
from the product of the numerator, we get finally for
I � in the in- and out-solutions (here the upper signs
correspond to the in-solutions, and the lower signs to
the out-solutions)
I k
k k
k iA
k k
k k
k
i
i
i
in
out
�
�
�
�
�
�
� �
�
�( )
( ) ( )
( )
1
0
� �
�
C
. (A.21)
266 Fizika Nizkikh Temperatur, 2006, v. 32, No. 3
I.N. Adamenko, K.E. Nemchenko, and I.V. Tanatarov
Taking into account Eqs. (A.15) and (A.16), we
can rewrite the equality (A.7), given on k� �� �( , ), in
the form
( ) ( ) ( ) ( ) ( ).k iA P k I k
k
k iA
P k I k� � �
�
� � � �
2
(A.22)
The left side of equality (A.22) is the limit value on
k� �� �( , ) of a function analytical in C �, and the
right side is the limit value of a function analytical in
C � . Then by the analytic continuation theorem those
functions are analytical continuations of each other
over the real line and are equal to some function
.( )k , analytical in the whole complex plane, with ex-
ception possibly only in the points on the real line
and complex infinity. It is fully determined by those
poles.
We demand for the solution to be fully determined
by the properties of the continuous medium and the
physical statement of the problem. Then we rely on
the fact that the dispersion relation of a medium �( )k
defines completely its dynamical properties and postu-
late that the equality (A.22) contains all the physical
factors determining the poles of .( )k . So we search for
the functions P k� ( ), that may possibly have poles in
the complex plane only in the poles and zero points of
J k( ) (its zero points are the roots of �2 2( )k � � , and
the only pole is of the second order in zero) and of
no greater order. Then .( )k must be analytical on
k� �� �( , ) and can have poles only at zero or infinity.
At x � �� the solution must turn into the well-
known solution in the infinite medium [15], which is
the set of plane waves. The behavior of P x( ) at
x � �� is determined by the asymptote of P k�( ) at
k � 0 and then it can be seen from Eq. (A.22), that
.( )k must be bounded at zero.
For the problem stated for the function P x( ) to be
fully determined by equation (7), we should look for
P x( ) in the category of functions, that do not contain
&-function or its derivatives at zero. In the opposite
case we should specify how to understand the lower
limit of integration in Eq. (7), which is quite baffling.
Then we should demand P k O k� � �( ) ( )1 at k � �,
and so .( )k must be bounded at infinity. All the fore-
going means eventually that we are searching for the
solution P k� ( ), bounded on the contour, as it is usu-
ally done when solving mathematical problems of this
kind [18].
Then .( )k is analytical in the whole complex
plane, including infinity, and by the (generalized)
Liouville theorem (see, for example, [18]) is equal to
a constant:
( ) ( ) ( ) ( ) ( )
( ) ( ) .
k iA P k I k
k
k iA
P k I k
k C
� � �
�
�
� �
� � � �
2
. � (A.23)
The last equality is true for all k�C. It gives us the
solution of the boundary problem (A.7):
P k
C
I k k iA
�
�
�
�
( )
( )
( )( )
�
. (A.24)
The constant A is only formally present in
Eq. (A.24), as the factor ( )k iA� on factorization
(A.20) always gets into the denominator of I k�( ).
When we substitute I � from Eq. (A.21) into
Eq. (A.24), we finally get the expressions for the Fou-
rier transforms of pressures in the in- and out-solu-
tions (9) and (8).
Acknowledgements
We would like to thank A.F.G. Wyatt for many
helpful discussions. We also express our gratitude to
EPSRC of the UK (grant EP/C 523199/1), and to
GFFI of Ukraine (grant N02.07/000372) for support
for this work.
1. A.F.G. Wyatt, N.A. Lockberie, and R.A. Sherlock,
Phys. Rev. Lett. 33, 1425 (1974).
2. M.A.H. Tucker and A.F.G. Wyatt, Science 283, 1150
(1999).
3. I.M. Khalatnikov, An Introduction to the Theory of
Superfluidity Addison-Wesley, New York (1998).
4. N.N. Bogoliubov, J. Phys. USSR 11, 23 (1947); re-
printed in D. Pines, The Many Body Problem Ben-
jamin, New York (1961).
5. C.J. Pethick and H. Smith, Bose-Einstein Condensa-
tion in Dilute Gases, Cambridge University Press,
Cambridge (2002).
6. L. Pitaevskii and S. Stringari, Bose-Einstein Conden-
sation, Clarendon Press, Oxford (2003).
7. Ruslan V. Vovk, Charles D.H. Williams, and Adrian
F.G. Wyatt, Phys. Rev. B68, 134508 (2003).
8. Ruslan V. Vovk, Charles D.H. Williams, and Adrian
F.G. Wyatt, Phys. Rev. Lett. 91, 235302 (2003).
9. I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and
A.F.G. Wyatt, Fiz. Nizk. Temp. 30, 579 (2004) [Low
Temp. Phys. 30, 432 (2004)].
10. I.N. Adamenko, K.E. Nemchenko, V.A. Slipko, and
A.F.G. Wyatt, J. Phys.: Condens. Matter 17, 2859
(2005).
11. L.M. Brehovskih, Waves in Layer Media, Nauka,
Moskow (1973).
12. L.I. Mandelshtamm, Lectures on Optics, Relativity
Theory and Quantum Mechanics, Nauka, Moskow
(1972).
13. P.K. Schelling, S.R. Phillpot, and P. Keblinski, Appl.
Phys. Lett. 80, 2484 (2002).
Normal transmission of phonons through interface with quantum fluid
Fizika Nizkikh Temperatur, 2006, v. 32, No. 3 267
14. P.K. Schelling and S.R. Phillpot, J. Appl. Phys. 93,
5377 (2003).
15. I.N. Adamenko, K.E. Nemchenko, and I.V. Tanatarov,
Phys. Rev. B67, 104513 (2003).
16. L. Pitaevskii and S. Stringari, Phys. Rev. B45, 13133
(1992).
17. I.N. Adamenko, K.E. Nemchenko, and I.V. Tanatarov,
J. Low Temp. Phys. 138, 10909-005-1584-9 (2005).
18. F.D. Gahov, Boundary Problems, Gos. izdat. fiz-mat.
lit., Moskow (1958).
19. R. Mittra and S.W. Lee, Analytical Techniques in the
Theory of Guided Waves, The MacMillan Company,
New York (1971).
20. I.N. Adamenko, K.E. Nemchenko, and I.V. Tanatarov,
J. Mol. Liquids 120, iss. 1–3, p. 167 (2005).
21. S.M. Rytov, Zh. Eksp. Teor. Fiz. 17, 930 (1947).
22. Ya.L. Alpert, Izvestia Akademii Nauk SSSR, seria.
Fiz. 12, 241 (1948).
268 Fizika Nizkikh Temperatur, 2006, v. 32, No. 3
I.N. Adamenko, K.E. Nemchenko, and I.V. Tanatarov
|