Konstantinov effect in helium II

We consider reflection of first and second sound waves by a rigid flat wall in helium II. A nontrivial dependence of the reflection coefficients on the angle of incidence is obtained. Sound conversion is predicted at slanted incidence.

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Дата:	2008
Автор:	Melnikovsky, L.A.
Формат:	Стаття
Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2008
Назва видання:	Физика низких температур
Теми:	Жидкий гелий
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Цитувати:	Konstantinov effect in helium II / L.A. Melnikovsky // Физика низких температур. — 2008. — Т. 34, № 4-5. — С. 395–399. — Бібліогр.: 11 назв. — англ.

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spelling	irk-123456789-1169042017-05-19T03:02:55Z Konstantinov effect in helium II Melnikovsky, L.A. Жидкий гелий We consider reflection of first and second sound waves by a rigid flat wall in helium II. A nontrivial dependence of the reflection coefficients on the angle of incidence is obtained. Sound conversion is predicted at slanted incidence. 2008 Article Konstantinov effect in helium II / L.A. Melnikovsky // Физика низких температур. — 2008. — Т. 34, № 4-5. — С. 395–399. — Бібліогр.: 11 назв. — англ. 0132-6414 PACS: 67.25.dg;67.25.dt;67.55.dm http://dspace.nbuv.gov.ua/handle/123456789/116904 en Физика низких температур 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	Жидкий гелий Жидкий гелий
spellingShingle	Жидкий гелий Жидкий гелий Melnikovsky, L.A. Konstantinov effect in helium II Физика низких температур
description	We consider reflection of first and second sound waves by a rigid flat wall in helium II. A nontrivial dependence of the reflection coefficients on the angle of incidence is obtained. Sound conversion is predicted at slanted incidence.
format	Article
author	Melnikovsky, L.A.
author_facet	Melnikovsky, L.A.
author_sort	Melnikovsky, L.A.
title	Konstantinov effect in helium II
title_short	Konstantinov effect in helium II
title_full	Konstantinov effect in helium II
title_fullStr	Konstantinov effect in helium II
title_full_unstemmed	Konstantinov effect in helium II
title_sort	konstantinov effect in helium ii
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2008
topic_facet	Жидкий гелий
url	http://dspace.nbuv.gov.ua/handle/123456789/116904
citation_txt	Konstantinov effect in helium II / L.A. Melnikovsky // Физика низких температур. — 2008. — Т. 34, № 4-5. — С. 395–399. — Бібліогр.: 11 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT melnikovskyla konstantinoveffectinheliumii
first_indexed	2025-07-08T11:17:35Z
last_indexed	2025-07-08T11:17:35Z
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fulltext	Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5, p. 395–399 Konstantinov effect in helium II L.A. Melnikovsky P.L. Kapitza Institute for Physical Problems, Moscow, Russia E-mail: leva@kapitza.ras.ru Received November 22, 2007 We consider reflection of first and second sound waves by a rigid flat wall in helium II. A nontrivial de- pendence of the reflection coefficients on the angle of incidence is obtained. Sound conversion is predicted at slanted incidence. PACS: 67.25.dg Transport, hydrodynamics, and superflow; 67.25.dt Sound and excitations; 67.55.dm Two-fluid model; phenomenology. Keywords: second sound, reflection coefficients, sound absorption, strong anisotropy. 1. Introduction Sound absorption in air at a plane surface may be sur- prisingly high and has strong anisotropy [1,2]. This phe- nomenon is very important in acoustics and is sometimes referred to as Konstantinov effect. Qualitative explana- tion of the effect is two fold [3]. — Air velocity in an oblique wave has a non-zero com- ponent tangential to the surface. At the surface itself, however, the velocity is clamped. Large velocity gradient in a thin boundary layer results is a large viscous dissipa- tion of energy. — Not only the velocity, but also the temperature, un- dergo periodic oscillation in a sound wave. Temperature of the wall (and adjoined gas), however, is constant. Ther- mal conductance in presence of a large temperature gradi- ent in the boundary layer again leads to high energy dissipation. Helium II supports two types of motion and its hydro- dynamics is much richer than that of a gas. The former features two independent sound modes: first sound and second sound [4]. Due to anomalously small thermal ex- pansion of helium, these modes can be viewed as purely pressure and temperature waves, respectively. To solve the problem of sound reflection in superfluid one must take both modes into account. Propagation of multiple bulk modes in helium II is a consequence of existence of additional hydrodynamic variables. In Sec. 2 we find all three nontrivial harmonic (i.e., proportional to exp( )i i tkr � � ) solutions of linear superfluid hydrodynamic equations. Some of the variables are effectively eliminated in re- stricted geometry. In particular, a steady cell wall elimi- nates the perpendicular component of the mass flux j and the normal velocity vn: the boundary condition* at the wall is j v� � �0 0, n . (1) Fourth sound [5] is one result of such elimination. This is the only sound mode in a narrow channel, both tempera- ture and pressure oscillating coherently in this wave. In general, the sound modes, independent in bulk liquid, be- gin to interact at the boundary*. The boundary condition (1) selects a two-dimensional subspace of solutions for particular frequency �. Specific solutions correspond to the reflection of first and second sound (Secs. 3 and 4). Interestingly enough, incident sound energy is not only reflected and dissipated as it happens in a normal fluid, but is also converted between first and second sound. © L.A. Melnikovsky, 2008 The heat transfer through the interface at low temperature can be neglected due to Kapitza resistance. The heat flux at the in- terface vanishes simultaneously with the perpendicular component of the normal velocity vn� . ** Sound reflection at the free helium surface and at the solid helium boundary is extensively explored [6–9]. 2. Harmonic solutions Consider the linearized equations of superfluid hydro- dynamics [10]: �� j x i i 0, (2) �j p x x v x v x v x i i k n i k n k i ik n l l � � � � � � � � � � � � � � � � � � � 2 3 � � � � � � � � � � � � x v x w xi n l l s l l � � � 2 1 , (3) v x x v x w x s k k k n l l s l l . � � � � � � � � � � � � � � � � � � 4 3 , (4) T v x T x n l l l � �� 2 , (5) where �, � �1 4� , � 2, � 3, �are dissipative coefficients,� is entropy per unit mass, p, �, T are pressure, chemical po- tential, and temperature, vs, vn, and w v v� �n s are superfluid, normal, and relative velocities, and � and j are mass and momentum densities. The velocities and the mo- mentum density are coupled by the equation j v w v w� � � �� s n n s . (6) Further simplification is facilitated by ignoring ther- mal expansion (we therefore disregard the difference be- tween specific heats c T / T� �� at constant pressure and at constant volume). Namely put p s� � �2� , (7) �� T p T s 2 , (8) T cT�� , (9) where s p/� �( ) /�� 1 2 is the first sound velocity. The prime denotes small deviation of the variables from their equi- librium values. In a harmonic perturbation, the space and time depend- ence of all deviations takes the form* exp( )i i tkr � � . (10) To find all possible harmonic excitations in bulk he- lium we substitute this exponential term in Eqs. (2)–(5) and keep linear terms only. Mass conservation (2) gives �� k ji i . (11) The momentum conservation law (3) can be trans- formed as follows: � � � � � � �� i j ip k k vn k k v k vi i k i k i n k ik l n l� � � 2 3 � � �k k w vi k s k n k( )� � �1 2 , and, using (7), (6), and (11) ( ) ( )� � � � � � �� i k / j iA s / k k j i k w /i i k k s i2 2 2 � �iBk k wi k k 0, (12) where the constants A / /� �( )� � �3 2 and B A s� �( )� �1 . From the energy conservation law (5) for harmonic de- viation we get T T T v k ik Tn l l�� 2 0. Using (6), (9), and (11) this can be reduced to T k w c i k Ts i i�� ( )2 . (13) Finally, substituting the exponential term (10) and (6) in (4) we obtain � � � � �i j w / iki n i i� � � �( ) � � � � �k w k j w k /i s k k k s k k( ( ))� � � � �3 4 Combining this with (8), (11), and (13) � � � ��j i s / k k j w Gk k wi i k k n i i k k� � � � �( )4 2 0, (14) where G T c i k is� � � � � � � � � � � � �� 2 2 4 3( ) � � � � � � � � � � � � T c i T k c s s � � � � � � � � � �� 2 2 2 2 2 4 3( ) . Equations (12) and (14) can be written together as �L j w � � � � 0, where �L is a square 6 6� matrix composed of the coeffi- cients from (12) and (14). The linear system is consistent if det �L � 0. Due to the system isotropy, the determinant cannot depend on individual components of k i . Instead it depends on k k ki i2 � only. We therefore can put k ky z� � 0 and treat �L as a 4 4� matrix: 396 Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5 L.A. Melnikovsky * It is important to mention that unbounded solutions (those with complex wave vector k) should not be overlooked in a re- stricted geometry. � ( ) ( ) L s / iA i / k i / B k i k / i k s � � � � � � � � � � � �� 2 2 2 2 2 0 0 0 0 � � � � � �� s n n / s / i k Gk� � � � � � � � � ( )2 4 2 20 0 0 0 . After factorization det �L simplifies to det � ( ) ( ) ( ) L iA i / s / k i / B k i s / k s� � � � � � � � � � � �� 2 2 2 4 2 2 � � � ��n Gk 2 � � � � � � � � � � �� i k / i k /s n 2 2 . All nontrivial solutions immediately follow: � � �� 2 1 2 2� � � � � � �k s i A i � � �� k s i k s1 2 2 2 1 2 24 3 � � � � , (15) � � �� 2 2 2 2 2 2 2 2 2 2� � � � � � � �k G i iB / k T c k ss n s n , (16) � � �� i k / n3 2 , (17) where s2 is the second sound velocity. Roots k1 (15) and k 2 (16) correspond to «longitudi- nal» solutions where j w ki i i� � 1 2, , while the root k 3 (17) corresponds to a «transverse» one j k w ki i i i 3 3 0� � . The approximation in (15) and (16) is based on an as- sumption of low bulk damping, i.e. \| \| \| \| \| \|k k k3 2 1�� . This implies complete splitting between first and second sound, namely w � 0 for (15) and j � 0 for (16). In the third solu- tion (17), the superfluid velocity vanishes v s � 0, i.e. the mass flux and the relative velocity are coupled by the rela- tion j w� �n . 3. First sound reflection To avoid complications associated with wall deforma- tion, consider a perfectly rigid flat surface. We therefore ignore numerous peculiarities of sound transmission into solids. By 1 denote the angle of incidence for the first sound wave (see Fig. 1). The subscripts I1, R1, and R2 re- fer to the incident first, reflected first, and reflected sec- ond sound waves, respectively. The x axis runs along the wall and the y axis is directed into the liquid. T h e b o u n d a r y c o n d i t i o n s ( 1 ) j y � 0, w y � 0, j wx s x� �� 0 can be written in the matrix form j j j j I I R R 1 1 1 1 1 1 1 1 0 0 0 0 sin cos sin cos � � � � � � � � � � � � � 0 0 2 2 2 2 3 w w w R R n x sin cos � �n y x y w w w 3 3 3 � � � � � �� sw w 0 0 , (18) where cos ( ) sin2 2 2 2 2 2 11 � � s /s , to satisfy the condition k kI x R x 1 2� . The last term on the left-hand side of (18) rep- resents the transverse surface wave with a wave vector k 3. The wave must decay away from the boundary, there- fore Im k y 3 0� . This requirement selects the sign in (19), which is the transversality relation w k3 3� : w w k k k k x y y x 3 3 3 3 3 1 1 � � � � �� sin � � � � � � � � � � �e ei i k k ! !/ / sin sin 4 1 1 4 2 2 (19) where � � �� n / . Substituting this in (18) we get j j j j I I R R 1 1 1 1 1 1 1 1 0 0 sin cos sin cos � � � � � � � � � � � � � �� s R R x n y y w w w w w 2 2 2 2 3 3 3 0 sin cos � � � 0 Konstantinov effect in helium II Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5 397 y x kR2 kR1 kI1 " # $ Fig. 1. First sound reflection. and 2 1 2 2 1 2 1 4 2 1 j w k I R s n i� � � � � �!sin sin cos cos cos si /e � n 2 1 0 � � � � . Second sound is slower than first sound, i.e. s s� 2, conse- quently %2 2! / and cos &2 0. One can therefore neglect the first term in parenthesis w j k k R I n i2 1 2 1 2 1 1 1 1 4 2 � � � sin cos sin tan /� �!e � . (20) Similarly, the amplitude of the reflected first sound is ob- tained from the equation j k j j kI n I i R n1 1 2 1 1 4 1 1 1 2� � � ! sin cos sin / � � � � � �e 1 1 4 1j R i� ! �e / cos � � � � � � � � � �� ! n x n i y k w w 1 1 3 4 3 0 sin / �e . From this we have j j k k R I n i n 1 1 1 1 2 1 4 1 1 2 1 � � � �� ! � � cos sin cos sin /e e�i!/ . 4 (21) Reflection and conversion efficiency must be charac- terized by appropriate coefficients R F /FR I11 1 1� and R F /FR I12 2 1� , respectively. Here F1 and F2 are the energy fluxes in the first and second sound waves. They are given by the expressions F s j1 2 2 � � \| \| , F s ws n 2 2 2 2 � � � � \| \| . Using (21) and (20) we get R k k n i n i11 1 1 2 1 4 1 1 2 1 � � � � � � � � � ! ! � � cos sin cos sin /e e /4 2 ' ' ' ' ' ' , (22) ( ( R s s k k s n n i 12 2 4 1 2 2 1 2 1 1 1 4 2 4 � � sin cos sin tan / � � � � ! �e . (23) Sample graph of these functions is illustrated on Fig. 2. The reflection coefficient R11 has a minimum of min R11 3 2 2� � (24) at finite angle of incidence. The value at the minimum is the same as for the sound reflection in usual hydrodynamics [2]. 4. Second sound reflection The same approach can be used to investigate the sec- ond sound wave incident at an angle 2 (see Fig. 3). The boundary conditions in this case are 0 0 0 0 2 2 2 2 2 2 2 2 w w w w I I R R sin cos sin cos � � � � � � � � � � � � � j j R R 1 1 1 1 0 0 sin cos � � � � � �� n x n y x y sw w w w w w 3 3 3 3 0 0 . After simplification this gives 2 2 1 4 1 2 2 2 1 2 � � � � � ! s n I R i n sw j k � � � � �e / cos sin sin sin cos cos � � � 1 2 . The last term in parenthesis is always negligible (the equation is meaningful only if sin %2 2s /s). This gives j w k k R s I n i n 1 2 2 2 2 4 1 2 1 2 2 � � � � � � �! sin cos sin sin/ �e . (25) 398 Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5 L.A. Melnikovsky 0 0.2 0.4 0.6 0.8 1 10 20 30 40 50 60 70 80 90 1, deg R11 R12 Fig. 2. Reflection and conversion coefficients R11 and R12 vs. the angle of incidence 1. y x kR2 kR1 kI2 # " # Fig. 3. Second sound reflection. The conversion coefficient R F /FR I21 1 2� is therefore given by ( ( R s s k k s n i n 21 2 2 2 4 2 4 1 2 1 2 2 4 � � � � � �! sin cos sin sin/ �e . (26) Its maximum max R s s s n 21 24� � � (27) is reached at the critical angle sin / �2 2s s. Amplitude of the reflected second sound wave is found from the relation w w k R I i n s 2 2 4 2 2 1 2 �� !e / sin tan tan � � � � � � � � � � � � �� !e i n s k / sin tan tan 4 2 2 1 2 , (28) where tan sin sin � � 1 2 2 2 2 2 2s / s s and Im (tan )1 0) (selected by the requirement Im k y 1 0* ). The reflection coefficient is therefore R k ki n s i22 4 2 2 1 2 2 2 4 � � � � � � � � � ! ! � � e e / / sin tan sin tan n sk k2 2 1 2 2 2 2 sin tan sin tan � ' ' ' ' ' ' � . (29) These functions for sample parameters are plotted on Fig. 4. 5. Discussion In helium II sound reflection at slanted incidence by a plane impervious wall is suppressed for both first and se- cond sound. This phenomenon is similar to the Kon- stantinov effect in ordinary gases. Coincident with the reflection suppression, a sound conversion takes place. The effect has strong angle de- pendence and should allow direct experimental verifica- tion. Moreover, there exist heat pulse propagation mea- surements [11] where the pulse transit time was often much shorter than that expected for second sound. This phenomenon is usually explained by an anomalously long phonon free path at low temperatures or by sound conver- sion in bulk (due to nonlinear effects) or at liquid-vapour interface. It seems probable that fast propagation is in fact a manifestation of the sound conversion described in this paper, so that the heat pulse is transformed at some wall into the pressure pulse and is later transformed back near the receiver. The signal therefore travels (some part of) the path with the velocity of first sound. I thank A.F. Andreev and V.I. Marchenko for fruitful discussions. The work was supported in parts by RFBR grants 06-02-17369, 06-02-17281 and RF president pro- gram 7018.2006.2. 1. K.F. Herzfeld, Phys. Rev. 53 , 899 (1938). 2. B.P. Konstantinov, Zh. Tekh. Fiz. 9, 226 (1939). 3. L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Perga- mon Press, Oxford (1987). 4. L.D. Landau, J. Phys. USSR 5, 71 (1941). 5. K.R. Atkins, Phys. Rev. 113, 962 (1959). 6. J.R. Pellam, Phys. Rev. 73, 608 (1948). 7. R.B. Dingle, Proc. Phys. Soc. A61 , 9 (1948). 8. D.M. Chernikova, Sov. Phys. JETP 20 , 358 (1965). 9. M.Yu. Kagan and Yu.A. Kosevich, Fiz. Nizk. Temp. 14, 787 (1988) [Sov. J. Low Temp. Phys. 14 , 433 (1988)]. 10. I.M. Khalatnikov, An Introduction to the Theory of Super- fluidity, W.A.Benjamin, New York-Amsterdam (1965). 11. J.R. Pellam, Phys. Rev. 75, 1183 (1949). Konstantinov effect in helium II Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5 399 0 0.5 1 1.5 2 10 20 30 40 50 60 70 80 90 R21 R22 2, deg Fig. 4. Reflection and conversion coefficients R22 and R21 vs. the angle of incidence 2. Note, that the energy conservation gives R R22 2 21 1 2cos cos cos � ) , and R R22 21 1� � is not a vi- olation!

Konstantinov effect in helium II

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