Nonlinear and shock waves in superfluid He II
We review studies of the generation and propagation of nonlinear and shock sound waves in He II (the superfluid phase of ⁴He), both under the saturated vapor pressure (SVP) and at elevated pressures. The evolution in shape of second and first sound waves excited by a pulsed heater has been invest...
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nasplib_isofts_kiev_ua-123456789-1208862025-02-23T20:11:33Z Nonlinear and shock waves in superfluid He II Kolmakov, G.V. Efimov, V.B. Ganshin, A.N. McClintock, P.V.E. Lebedeva, E.V. Mezhov-Deglin, L.P. Quantum Liquids and Solids We review studies of the generation and propagation of nonlinear and shock sound waves in He II (the superfluid phase of ⁴He), both under the saturated vapor pressure (SVP) and at elevated pressures. The evolution in shape of second and first sound waves excited by a pulsed heater has been investigated for increasing power W of the heat pulse. It has been found that, by increasing the pressure P from SVP up to 25 atm, the temperature Tα, at which the nonlinearity coefficient of second sound reverse its sign, is decreased from 1.88 to 1.58 K. Thus at all pressures there exists a wide temperature range below Tλ where α is negative, so that the temperature discontinuity (shock front) should be formed at the center of a propagating bipolar pulse of second sound. Numerical estimates show that, with rising pressure, the amplitude ratio of linear first and second sound waves generated by the heater at small W should increase significantly. This effect has allowed us to observe at P 133. atm a linear wave of heating (rarefaction) in first sound, and its transformation to a shock wave of cooling (compression). Measurements made at high W for pressures above and below the critical pressure in He II, Pcr 22. atm, suggest that the main reason for initiation of the first sound compression wave is strong thermal expansion of a layer of He I (the normal phase) created at the heater-He II interface when W exceeds a critical value. Experiments with nonlinear second sound waves in a high-quality resonator show that, when the driving amplitude of the second sound is sufficiently high, multiple harmonics of second sound waves are generated over a wide range of frequencies due to nonlinearity. At sufficiently high frequencies the nonlinear transfer of the wave energy to sequentially higher wave numbers is terminated by the viscous damping of the waves. The authors are grateful to A.A. Levchenko, E.A. Kuznetsov, and V.V. Lebedev for valuable discussions. The investigations were supported by the Russian Foundation for Basic Research, project Nos. 05-02-17849 and 06-02-17253, by the Presidium of the Russian Academy of Sciences in frames of the programs «Quantum Macrophysics» and «Mathematical Methods in Nonlinear Dynamics», and by the Engineering and Physical Sciences Research Council (UK). 2006 Article Nonlinear and shock waves in superfluid He II / G.V. Kolmakov, V.B. Efimov, A.N. Ganshin, P.V.E. McClintock, E.V. Lebedeva, L.P. Mezhov-Deglin // Физика низких температур. — 2006. — Т. 32, № 11. — С. 1320–1329. — Бібліогр.: 28 назв. — англ. 0132-6414 PACS: 68.03.Kn, 47.35.+i, 47.27.Gs https://nasplib.isofts.kiev.ua/handle/123456789/120886 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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| topic |
Quantum Liquids and Solids Quantum Liquids and Solids |
| spellingShingle |
Quantum Liquids and Solids Quantum Liquids and Solids Kolmakov, G.V. Efimov, V.B. Ganshin, A.N. McClintock, P.V.E. Lebedeva, E.V. Mezhov-Deglin, L.P. Nonlinear and shock waves in superfluid He II Физика низких температур |
| description |
We review studies of the generation and propagation of nonlinear and shock sound waves in
He II (the superfluid phase of ⁴He), both under the saturated vapor pressure (SVP) and at elevated
pressures. The evolution in shape of second and first sound waves excited by a pulsed heater
has been investigated for increasing power W of the heat pulse. It has been found that, by increasing
the pressure P from SVP up to 25 atm, the temperature Tα, at which the nonlinearity coefficient
of second sound reverse its sign, is decreased from 1.88 to 1.58 K. Thus at all pressures
there exists a wide temperature range below Tλ where α is negative, so that the temperature discontinuity
(shock front) should be formed at the center of a propagating bipolar pulse of second
sound. Numerical estimates show that, with rising pressure, the amplitude ratio of linear first and
second sound waves generated by the heater at small W should increase significantly. This effect
has allowed us to observe at P 133. atm a linear wave of heating (rarefaction) in first sound, and
its transformation to a shock wave of cooling (compression). Measurements made at high W for
pressures above and below the critical pressure in He II, Pcr 22. atm, suggest that the main reason
for initiation of the first sound compression wave is strong thermal expansion of a layer of He I
(the normal phase) created at the heater-He II interface when W exceeds a critical value. Experiments
with nonlinear second sound waves in a high-quality resonator show that, when the driving
amplitude of the second sound is sufficiently high, multiple harmonics of second sound waves are
generated over a wide range of frequencies due to nonlinearity. At sufficiently high frequencies the
nonlinear transfer of the wave energy to sequentially higher wave numbers is terminated by the
viscous damping of the waves. |
| format |
Article |
| author |
Kolmakov, G.V. Efimov, V.B. Ganshin, A.N. McClintock, P.V.E. Lebedeva, E.V. Mezhov-Deglin, L.P. |
| author_facet |
Kolmakov, G.V. Efimov, V.B. Ganshin, A.N. McClintock, P.V.E. Lebedeva, E.V. Mezhov-Deglin, L.P. |
| author_sort |
Kolmakov, G.V. |
| title |
Nonlinear and shock waves in superfluid He II |
| title_short |
Nonlinear and shock waves in superfluid He II |
| title_full |
Nonlinear and shock waves in superfluid He II |
| title_fullStr |
Nonlinear and shock waves in superfluid He II |
| title_full_unstemmed |
Nonlinear and shock waves in superfluid He II |
| title_sort |
nonlinear and shock waves in superfluid he ii |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| topic_facet |
Quantum Liquids and Solids |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120886 |
| citation_txt |
Nonlinear and shock waves in superfluid He II / G.V. Kolmakov, V.B. Efimov, A.N. Ganshin, P.V.E. McClintock, E.V. Lebedeva, L.P. Mezhov-Deglin // Физика низких температур. — 2006. — Т. 32, № 11. — С. 1320–1329. — Бібліогр.: 28 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT kolmakovgv nonlinearandshockwavesinsuperfluidheii AT efimovvb nonlinearandshockwavesinsuperfluidheii AT ganshinan nonlinearandshockwavesinsuperfluidheii AT mcclintockpve nonlinearandshockwavesinsuperfluidheii AT lebedevaev nonlinearandshockwavesinsuperfluidheii AT mezhovdeglinlp nonlinearandshockwavesinsuperfluidheii |
| first_indexed |
2025-11-24T23:41:34Z |
| last_indexed |
2025-11-24T23:41:34Z |
| _version_ |
1849717103825780736 |
| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, No. 11, p. 1320–1329
Nonlinear and shock waves in superfluid He II
G.V. Kolmakov1,2, V.B. Efimov1,2, A. N. Ganshin2, P.V.E. McClintock2,
E.V. Lebedeva1, and L.P. Mezhov-Deglin1
1 Institute of Solid State Physics RAS, Chernogolovka 142432, Russia
E-mail: mezhov@issp.ac.ru
2 Lancaster University, Lancaster, LA1 4YB, UK
E-mail: g.kolmakov@lancaster.ac.uk
Received May 31, 2006
We review studies of the generation and propagation of nonlinear and shock sound waves in
He II (the superfluid phase of 4He), both under the saturated vapor pressure (SVP) and at ele-
vated pressures. The evolution in shape of second and first sound waves excited by a pulsed heater
has been investigated for increasing power W of the heat pulse. It has been found that, by increas-
ing the pressure P from SVP up to 25 atm, the temperature T�, at which the nonlinearity coeffi-
cient � of second sound reverse its sign, is decreased from 1.88 to 1.58 K. Thus at all pressures
there exists a wide temperature range below T� where � is negative, so that the temperature discon-
tinuity (shock front) should be formed at the center of a propagating bipolar pulse of second
sound. Numerical estimates show that, with rising pressure, the amplitude ratio of linear first and
second sound waves generated by the heater at small W should increase significantly. This effect
has allowed us to observe at P � 133. atm a linear wave of heating (rarefaction) in first sound, and
its transformation to a shock wave of cooling (compression). Measurements made at high W for
pressures above and below the critical pressure in He II, Pcr � 22. atm, suggest that the main reason
for initiation of the first sound compression wave is strong thermal expansion of a layer of He I
(the normal phase) created at the heater-He II interface when W exceeds a critical value. Experi-
ments with nonlinear second sound waves in a high-quality resonator show that, when the driving
amplitude of the second sound is sufficiently high, multiple harmonics of second sound waves are
generated over a wide range of frequencies due to nonlinearity. At sufficiently high frequencies the
nonlinear transfer of the wave energy to sequentially higher wave numbers is terminated by the
viscous damping of the waves.
PACS: 68.03.Kn, 47.35.+i, 47.27.Gs
Keywords: first and second sound, nonlinearity, acoustic turbulence.
1. Introduction
First of all, we are much indebted to the Editorial
Board of the Low Temperature Physics journal for
their invitation to report the results of our recent stud-
ies in the special issue devoted to 100-th anniversary
of the famous scientist A.F. Prikhot’ko.
We review here experimental and theoretical inves-
tigations of the peculiarities of nonlinear evolution of
solitary second and first sound pulses propagating in
superfluid 4He, and of a second sound standing wave
in a high-Q resonator filled with He II, the superfluid
phase of 4He. Entropy waves (second sound) are a
macroscopic quantum effect that may be observed in
superfluids and perfect crystals [1–3]. The properties
of second sound in He II have been studied exten-
sively, both experimentally and theoretically. More
recently, attention has been focused on the nonlinear
acoustic properties of He II [4–8].
It is known [1–3,9–11] that second sound is cha-
racterized by rather strong nonlinear properties. These
lead to the formation of a shock wave (temperature
discontinuity) during the propagation of a finite-am-
plitude wave in He II, at short distances from the
source (heater). The velocity of a traveling second
© G.V. Kolmakov, V.B. Efimov, A. N. Ganshin, P.V.E. McClintock, E.V. Lebedeva, and L.P. Mezhov-Deglin, 2006
sound wave depends on its amplitude and, to a first
approximation, can be written:
u u T2 20 21� �( ),� � (1)
where �T2 is the wave amplitude, u20 is the velocity
of a wave of infinitely small amplitude, and � is the
nonlinearity coefficient of second sound, which is de-
termined by the relation [2]
�
�
�
� �
�
�
�
T
u
C
T
ln ,20
3 (2)
where C is the heat capacity per unit mass of liquid
helium at constant pressure, and T is the temperature.
The nonlinearity coefficient � of second sound may
be either positive and negative, depending on the tem-
perature and pressure [2,10,12]. Under the saturated
vapor pressure (SVP), in the region of roton second
sound (i.e. at T � 0 9. K) the nonlinearity coefficient is
positive (� � 0) at temperatures T T
�� 188. K (like
the nonlinearity coefficient of conventional sound
waves in ordinary media), but it is negative in the
range T T T� �
. Here T� � 2176. K is the tempera-
ture of the superfluid-to-normal (He II to He I) tran-
sition. At T T� � the nonlinearity coefficient passes
through zero.
If T T� � , the nonlinear evolution leads to the cre-
ation of a shock in the profile of a travelling second
sound pulse. During the propagation of a plane,
one-dimensional wave of heating (compression) of sec-
ond sound (�T2 0� ), the shock appears at the front of
the propagating wave at temperatures 1 K
T T� ,
and on the trailing edge of the wave for temperatures
T T T� �
(see, for example, Fig. 1). The width of
the shock front lf is defined by both the nonlinearity
coefficient � and the dissipation coefficient � of second
sound. In the hydrodynamic regime, where the shock
front width exceeds greatly the mean free path of
rotons, one has l / Tf � � �� 2 for the shock front width,
and v u T /f � �20 2 2�� for the velocity of shock prop-
agation [1,2].
At large distances L from the heater the profile of a
one-dimensional shock pulse eventually acquires a tri-
angular form. The dependence of the length of the tri-
angle (duration of the pulse �) and its height (the tem-
perature jump at the shock front �T2) on the distance
can be described by a universal power law
� � � �� � �const const( ) , ( ) ,/ /L T L1 2
2
1 2 (3)
where the constants depend on the initial shape of the
second sound pulse. It is important to note here that
the evolution of a developed shock wave (i.e., the
dependence of the parameters of the triangle on
distance) is governed only by the value of the
nonlinearity coefficient and does not depend on the
value of the dissipation coefficient �. The entropy
production rate dS/dt at the shock front due to
dissipative processes remains finite as the dissipation
coefficient � tends to zero, because the small value of
� is compensated by a large temperature gradient
dT/dx T /lf� � 2 .
2. Propagation of nonlinear second sound pulses
in He II
2.1. Evolution of the shape of second sound pulses
with reduction of the temperature
The experimental arrangements were similar to
those used in our earlier studies of nonlinear and shock
second sound pulse propagation in He II [9,11]. Fi-
gure 1,a shows the evolution in shape of planar second
sound pulses excited by a rectangular heat pulse
at temperature T � 210. K under pressure P � 3 atm
( )�
0 [12]. The time duration of the electrical pulse
applied to the resistive heater was � e � 10 �s. The
Nonlinear and shock waves in superfluid He II
Fizika Nizkikh Temperatur, 2006, v. 32, No. 11 1321
1
2
3
4
5
7
6
a
b
0.4
0.3
0.2
0.1
0
–0.1
3400 3600 3800
T
,a
rb
.u
n
its
2
T
,a
rb
. u
n
its
2
Time, s�
Time, s�
7
5
3
1
0
–1
–10 0 10 20 30 40 50 60
5 4 3 2 1
1
2
3
4
5
6
7
0.16
0.66
1.47
5.89
16.3
65.4
147
1
2
3
4
5
2.4
5.2
9.3
14.2
20.2
Fig. 1. Evolution in shape of planar second sound waves
for increasing heat flux density W from the heater, where
�T2 is the temperature change measured at the bolometer:
for a He II bath temperature of T � 210. K (� � 0), pressure
P � 3 atm (a); for T � 150. K (� � 0), pressure P PSVP�
(b). The numbers beside the curves indicate the heat flux
density W expressed in W/cm2. The time duration of the
electrical pulse to the heater was �e � 10 �s.
numbers beside the curves correspond to different heat
flux densities W from the heater. The waves were de-
tected by a superconducting film bolometer [13]. At
small W
0 7. W/cm2 the bolometer detected linear
waves (curves 1, 2) of amplitude �T2 proportional to
W. Above W � 1 W/cm2 and up to 17 W/cm2 the
pulse shape is close to triangular (linear waves trans-
form to planar shock waves, curves 3–5). The ampli-
tude of the triangular pulse �T2 and its width increase
in proportion to W 1 2/ . Above 30 W/cm2 (curves 6
and 7) the amplitude and the width of the pulse de-
pend weakly on W (the range of saturation).
Figure 1,b shows the evolution of the profile of a
second sound pulse at T � 150. K (� � 0) under the
SVP, P PSVP� , while the heat flux density increases
from W � 2 4. W/cm2 to 20.2 W/cm2, at a constant
distance L � 2 5. cm from the heater [14]. The heater
pulse duration was � e � 10�s. We see that the slope of
profile a T /� � �2 does not depend onW: in accordance
with the general relations (3), the value a u / L� 2 � .
Creation of the shock on the trailing edge of a com-
pression wave at T T� � (or at the front of a rarefac-
tion wave for �T2 0
) is a specific property of second
sound in He II [2]. At temperatures very close to T�
the nonlinearity coefficient tends to infinity according
to the power law [7] � �� �1, where � � �� ( — )T T /T
is the reduced temperature. Near the lambda transition
the nonlinearity therefore plays a crucial role even for
the smallest amplitudes �T2.
We discuss here the evolution in the shape of short
second sound pulses of the relatively small amplitude
| |�T2
210
� K, so that we can neglect possible creation
of quantum vortices at the shock front and the resul-
tant wave-vortex interactions. For small amplitudes,
the description of the nonlinear evolution of the sec-
ond sound wave can be confined to the first terms in
the expansion of the velocity u2 in �T2, as done in
Eq. (1). Under this approximation the amplitude of a
second sound pulse at small W (linear wave) should
be proportional toW, i.e., �T W U2
2� � (here U is a
voltage applied to the heater). At higher W, where
a shock wave is formed, the amplitude of the propa-
gating triangular pulse should be proportional to
�T W U2
1 2� �/ .
The experimental dependence of the amplitude of
the pulse plotted as �T U2( ) (left-side scale, open cir-
cles) and �T U2
1 2/ ( ) (right-side scale, solid circles)
presented in Fig. 2 was reconstructed from the mea-
surements shown in Fig. 1,a (T � 210. K, P � 3 atm,
and heater pulse duration � e � 10 �s).
At high W the amplitude of second sound pulses
saturates, as can be seen from Figs. 1 and 2. A similar
dependence was observed at P � 3 atm and T � 17. K,
where the nonlinearity coefficient � � 0. All the
curves �T U2( ) observed at different pressures and tem-
peratures in our studies look much the same. Similar
saturation effects at high W were reported in [15,16].
A detailed discussion of the reasons for the saturation
lies beyond the scope of the present paper, but one rea-
son could be the attenuation of the propagating sec-
ond sound wave by vortices in the bulk of the He II
[17]. However it follows from our observations (see
Sec. 2.4) that, along with the nonlinear second sound
wave, a heater should generate also a first sound shock
1322 Fizika Nizkikh Temperatur, 2006, v. 32, No. 11
G.V. Kolmakov et al.
0.6
0.4
0.2
0
0.4
0.3
0.2
0.1
0 10 20 30 40
U, V
�T2 �T2
1/2
Fig. 2. Dependence of the amplitude of second sound
wave �T2 on the pulsed voltage U to the heater, recon-
structed from Fig. 1,a (left-side scale, open circles), and
the dependence on �T2
1 2/ on U (right-side scale, solid cir-
cles). Arrows show the proper scale of ordinates.
a
b
T
, a
rb
. u
n
its
2
T
, a
rb
. u
n
its
2
10
8
6
10
8
6
0 100 200
0 50 100
Time, s�
Time, s�
Fig. 3. Shape of a travelling three-dimensional second
sound pulse generated by a point-like heater; T � 150. K
( )� � 0 (a), T � 202. K (� � 0) (b). The heater pulse dura-
tion was �e � 10 �s.
wave of compression before the �T U2( ) curves satu-
rate. We may infer, first, that a significant part of the
energy could be expended in the creation of this wave
and, secondly, that vortices can be generated initially
by the shock wave of first sound propagating ahead of
the second sound shock wave.
It is known [6] that, if a three-dimensional (spheri-
cal) second sound wave is generated in He II by a
point-like heater, then the propagating wave of heat-
ing (compression) is followed by a wave of cooling
(rarefaction), see Fig. 3. In the range of temperatures
T T T� �
, where the nonlinearity coefficient is neg-
ative, the temperature discontinuity (the shock) is
created at the centre of a running pulse. Conse-
quently, the duration of the pulse does not change
with the distance propagated [6,14] (see Fig. 3,b).
This feature is important, e.g., for easier investigation
of nonlinear and dissipative phenomena in the close
vicinity of T� in He II, especially at elevated pres-
sures.
2.2. Propagation of nonlinear second sound waves
in compressed He II
Thermodynamic characteristics of He II such as the
heat capacity C, the second sound velocity u20, and
the temperature of the phase transition T�, change
considerably upon changing the pressure. The depen-
dence of the nonlinearity coefficient on pressure (see
above) was discussed in detail in the papers [12,18].
Figure 4,a shows the dependence of the
nonlinearity coefficient � on temperature for
P PSVP� (curve 1) [2] and for elevated pressures [18]
of P � 5, 10, 15, and 25 atm (curves 2–5), respec-
tively. The dependences �( , )P T for P PSVP� were
calculated by using equation (1) and known depen-
dences [3] for the heat capacity C P T( , ) and the sec-
ond sound velocity u P T2( , ). It can be seen from
Fig. 4,a that the temperature T� decreases from 1.8 to
1.58 K as the pressure is increased up to 25 atm.
The pressure dependence of T� , the temperature at
which the nonlinearity coefficient of second sound in
He II passes through zero, is shown by the dotted
curve in Fig. 4,b. The circles correspond to our experi-
mental data. The dependence �( )T at fixed pressure
was obtained from experimental observations of the
evolution in shape of the planar shock second sound
pulse with changing bath temperature T or heat flux
density W. It is evident that the experimental data
agree well with the results of our computations. The
solid curves in Fig. 4,b were reconstructed from the
data available in the literature [3]. They describe the
temperature dependence of the solidification pressure
and the pressure dependence of the superfluid transi-
tion temperature T�.
It follows from Fig. 4,b that, at all pressures up to
that of solidification, there exists a fairly wide tem-
perature range T T T� �
below T�, in which the
nonlinearity coefficient of the second sound � is nega-
tive. According to Fig. 3,b, in this range the bipolar
pulse of second sound propagating into the bulk of
He II from a point-like heater should transform to a
shock wave of constant duration with the discontinu-
ity placed at its centre. These are important consider-
ations for studies in the vicinity of T� at elevated pres-
sures.
2.3. Nonlinear second sound in superfluid
3He— 4He mixtures
The nonlinearity coefficient of second sound in a
dilute superfluid 3He—4He solution is given by the
following expression (see Refs. 19, 20 for details)
Nonlinear and shock waves in superfluid He II
Fizika Nizkikh Temperatur, 2006, v. 32, No. 11 1323
T, K
a
5
4
3
1
2
6
4
2
0
–2
–4
–6
1.0 1.2 1.4 1.6 1.8 2.0
30
20
10
0
P
,a
tm
Solid He b
Normal
He I
1.4 1.6 1.8 2.0 2.2
T, K
Superfluid
He II
,K
–
1
� > 0 � < 0
Fig. 4. Numerical evaluations of the dependence on pres-
sure of the nonlinearity coefficient of second sound �, at
SVP (curve 1), and for different elevated pressures in the
superfluid: P (atm) = 5 (2), 10 (3), 15 (4), and 25 (5)
(a). Dependence on pressure of the temperature T� at
which � changes sign (b). On the plot (b) the dotted
curve corresponds to theory; circles describe the experi-
mental results; full lines show the temperature dependence
of the pressure of solidification and the variation of the
phase transition temperature T� with the pressure.
�
� �� �
� �
�
3
2 20
2
s
n
A / T
u
( )
, (4)
where
A u
/ T T
cn
s
s
n n
�
�
�
��
�
�
�� �
�
�
��
�
�
�� �2
2
2
2
�
�
�
�
�
�( ) c
/ T T
c
n
�
�
�
�
�
�
��
�
�
��
�
�
�
�
� �
�
�
�
�
�
�
�
�
�
1
3
3
3
2
2
2
( )
—
�
�
�
�
�
�
�
�
( )
( )
/ T c
c
c
Z
c
/ T
�
�
�
�
�
� �
�
�
��
�
�
�� �
�
1
3
3
2
2
2 �
�
�
�
�
�
�
�
�
�
2
2
2
c
c
/ T T
/ c
/ T
�
�
�
�
�
�
�
�
�
�
��
�
�
��—
( )
. (5)
We use here a conventional notation [2] such that:
c x/� � is the mass concentration of 3He impurity; �
is the entropy per unit mass; � � �� �� � c / c( ); and Z
is the thermodynamical potential for impurity par-
ticles.
Figure 5 illustrates the temperature dependence of
the nonlinearity coefficient � in superfluid 4He—3He
mixture, calculated from Eqs. (4), (5) for molar 3He
content x � 4 6. , 7, and 10%. Data from Ref. 21 were
used in the calculations. The curve drawn for x � 0
(pure superfluid 4He) corresponds to the curve 1 on
Fig. 4,a. It can be seen from Fig. 5 that the tempera-
ture T� at which the nonlinearity coefficient changes
its sign is reduced from 1.88 K for pure superfluid 4He
to 1.67 K for superfluid mixture with x � 10% of 3He
impurity. Note here that the temperature T� of the
superfluid transition is reduced from 2.17 K in pure
superfluid 4He to 2.02 K in mixture with 3He content
x � 10%. This means that for dilute solutions there
exist a rather wide interval of temperatures below T�
for which �
0.
2.4. Peculiarities of the generation of second and
first sound waves by a film heater in He II at high
heat flux density
We have studied the evolution in shape of the first
sound wave that is excited, together with the second
sound wave, by a heater in superfluid He II at ele-
vated pressures.
We recall (see above) that, for all W, a pulsed
heater in He II should excite ordinary first sound
(pressure/density waves) as well as second sound
(entropy/temperature waves). The thermal expan-
sion coefficient � of He II is negative at temperatures
above 1.2 K, � � �� � � �
( )( )1 0/ / T s (here � is the
density of bulk liquid). So, a quasi-adiabatic plane
linear wave of first sound (the wave of heating of am-
plitude �T1 0� ), excited by a heater, is a wave of rar-
efaction: �� ����
T1 0.
At low heat flux densitiesW, the energy contained
in the first sound wave is much less than that in the
second sound wave. For this reason, the amplitude �T1
of the temperature oscillations in quasi-adiabatic first
sound should be much smaller than the amplitude
�T2 of the second sound wave. Using the relations
presented in Refs. 1, 2, 4 one can estimate that
at small pressures close to SVP (P � 0 05. atm),
the amplitude ratio is very small � �T / T1 2
42 10� �• .
Therefore in experiments with linear second sound
waves amplitude �T2 1
mK, the amplitude of the
first sound wave excited by a heater should be
�T1 0 2
. �K. This value is beneath the typical resolu-
tion of a superconducting bolometer. But at high heat
flux densities, at which the amplitude of the second
sound wave was close to saturation (Fig. 2), the
bolometer had registered arrival of the first sound
wave of compression (a shock wave with �� � 0 and
�T
0). Creation of the shock wave of the first sound
was associated typically with the film boiling of
superfluid liquid at the surface of the heater at pres-
sures below the critical pressure in He II (further dis-
cussion see below).
It follows from our computations based on the
two-fluid model [2] that, in the linear approximation,
the amplitude of the first sound wave should increase
with increasing pressure in He II bath, for any given
temperature T and given heat flux densityW. Figure 6
demonstrates the dependence on temperature of the
derivative ( )� �P/ T s in a quasi-adiabatic first sound
wave in both He II and He I, calculated for different
pressures P. A jump in the dependence at a given pres-
sure arises due to the discontinuity in the temperature
dependence of the thermal expansion coefficient� that
occurs at a temperature slightly above T� (for exam-
ple, in He I at P PSVP� the� coefficient changes both
its value and sign from negative to positive at
1324 Fizika Nizkikh Temperatur, 2006, v. 32, No. 11
G.V. Kolmakov et al.
4
3
2
1
T, K
–
–
,K
–
1
Fig. 5. Dependence on temperature of the nonlinearity
coefficient of second sound � in superfluid 3He—4He so-
lutions under SVP at different 3He concentrations x, %:
0 (1), 4.6 (2), 7 (3), and 10 (4).
T T� �� 0 006. K [21]). Figure 7 shows the calculated
ratio of amplitudes of temperature oscillations
� �T / T1 2 in linear waves of first and second sound
emitted by a film heater in superfluid He II at differ-
ent pressures P. It follows from Figs. 6, 7 that the
temperature oscillations in the wave of first sound
generated by a heater immersed in the superfluid
should increase considerably with rising pressure P. It
follows from Fig. 8 that, as the pressure is increased
up to 15 atm, the ratio � �T / T1 2 at T � 19. K increases
by an order of magnitude.
Consistent with earlier work [4,22], we were un-
able to detect the linear waves of heating due to first
sound in He II within the resolution of our bolometer
(threshold about 5 10 6• � K) for pressures below
10 atm. The amplitude of a linear wave of the first
sound could not be increased significantly by increas-
ing the heat flux density W: at high W the bolometer
detected the arrival of a wave of cooling of the first
sound, i.e., a compression wave (�T1 0
, �� � 0) in-
stead of a wave of heating.
On increasing the pressure to 13.3 atm, the
bolometer detected the propagation in bulk He II of
both waves of heating — linear first sound waves of
rarefaction and second sound waves of compression,
with heater pulses of relatively small W. With in-
crease of W, first sound waves of rarefaction were
transformed into waves of compression (cooling), and
then into a shock sound wave with its discontinuity (a
stepwise jump of pressure and temperature) situated
at the front of the running wave.
The threshold for creation of a first sound compres-
sion wave was observed at pressures both below the
critical pressure (Pcr � 2 2. atm) and above it where no
liquid-vapor interface can exist. In the latter case,
film boiling near the surface of the pulsed heater is ob-
viously impossible. Creation of the wave of compres-
sion in He II can be attributed to the strong expansion
of a normal He I layer created at the heater-fluid in-
terface for W W� cr : see discussion below. Note here,
that in He I the thermal expansion coefficient � is
positive, and its value is unusually high, about
10 2 1� �K . We observed a change in sign of the shock
wave of compression when the temperature of the bath
was increased above T�. As mentioned above, at pres-
sures below the critical pressure one should also take
account of pulse film boiling of helium at the inter-
face.
Nonlinear and shock waves in superfluid He II
Fizika Nizkikh Temperatur, 2006, v. 32, No. 11 1325
T, K
d
p
/d
T
,a
tm
/K
200
100
0
–100
–200
–300
–400
15 atm
10 atm
5 atm
SVP
1.5 2.0 2.5 3.0 3.5
Fig. 6. Temperature dependence of the derivative
( )� �P/ T s in a quasi-adiabatical wave of first sound in liq-
uid helium, calculated for the pressures P shown.
T, K
SVP
15 atm
10 atm
5 atm
T
/
T
1
2
0.002
0.001
0
1.4 1.6 1.8 2.0 2.2
Fig. 7. Numerical computations of the ratio of the ampli-
tudes of temperature oscillations � �T / T1 2 of linear first
and second sound generated by a plane heater in He II for
the indicated pressures.
�T1
0.2
0.1
0 10 20 30 40
U, V
P = 5 atm, T = 2.103 K
P = 3 atm, T = 2.103 K
P = 1 atm, T = 2.103 K
P = 5 atm, T = 1.76 K
P = 3 atm, T = 1.76 K
Fig. 8. Dependence of the amplitude �T1 of the tempera-
ture oscillations in the shock first sound wave of compres-
sion (� � 0, �T1 0� ) on the voltage U, measured at pres-
sures P � 1, 3, and 5 atm and temperatures T � 21. and
1.76 K. The heater pulse duration was �e � 10 �s. The
waves of compression were registered by the bolometer at
high heat flux densities W � 10 W/cm2.
Typical dependences of the amplitude of tempera-
ture oscillations in the first sound waves �T1 (in arbi-
trary units) in He II at pressures P � 1, 3, and 5 atm
and temperatures T � 21. K and 176. K on the voltage
U of the heater pulse are shown in Fig. 8. The average
heat flux density emitted by the heater in these
experiments can be estimated from the relation
W U� 01 2. W/cm2. The heater pulse duration is con-
stant at � e � 10 �s. Open circles, squares, and dia-
monds correspond to T � 210. K and P � 5, 3, and
1 atm, respectively; solid circles and squares corre-
spond to T � 176. K and P � 5 and 3 atm.
It should be emphasized that similar voltage
dependences �T U1( ) were observed at other pressures
between 1 and 9 atm. In all cases the bolometer de-
tected initiation of the first sound shock wave of cool-
ing (compression) with �T1 0
, �� � 0, just as in pre-
vious studies [4,6]. As can be seen from Fig. 8, the
cooling shock waves were created at heat flux densi-
ties beyond some critical value Wcr , which is equal to
about 10 W/cm2 for T � 21. .K and about 30 W/cm2
for T � 176. K. These values are of the order of the heat
flux densities at which saturation of the second sound
pulse amplitude takes place, as can be seen in Fig. 1.
Usually, saturation of the �T U2( ) curve at high W
is attributed to the interaction of the running second
sound pulse with vortices [17]. It follows from our
measurements that account should also be taken of the
possible formation of vortices at the heater-fluid He
interface; they can also be generated by the first sound
shock wave propagating along a waveguide ahead of
the shock wave of second sound.
Figure 9 shows the oscillograms illustrating the
evolution of shape of the first sound pulses in com-
pressed He II at P � 13 atm and T � 1895. K with in-
creasing heat flux density. The heater pulse duration �
was 3 �s. The numbers beside the curves indicate the
voltageU applied to the heater. A detailed picture of
the evolution of the shapes of first sound pulses of dif-
ferent initial duration, from 0.3 to 10 �s, has been pre-
sented earlier [23].
The curves plotted in Fig. 9 provide a clear illustra-
tion of the transformation of the wave of heating (ra-
refaction) into the wave of cooling (compression)
with increasing heat flux density, forU above 15 V. It
can be seen that the wave of compression is accompa-
nied by a wave of heating (rarefaction), which is
formed at the moment when the heater pulse is
switched off. The wave of compression is associated
with the pulsed thermal expansion of the layer of He I
created at the heater-fluid He interface in the process
of heating. Similarly, the subsequent wave of rarefac-
tion might be attributed to a fast contraction of the
liquid layer when the heat flux was switched off.
The dependence of the first sound pulse amplitude
�T1 (in arbitrary units) on the voltage applied to the
heater, for different � is presented in Fig. 10. The heat
pulse duration is marked on the plots. The solid points
correspond to a rarefaction wave, and the open points
correspond to a compression wave. As might be ex-
pected, the straight line �T U1
1 2/ � drawn through the
solid points passes through the origin of coordinates.
We estimated the ratio of the slopes of the straight
lines �T k U1
1 2
1
/ � and �T k U2
1 2
2
/ � describing for
smallU the dependences of the amplitudes �T f U1 � ( )
and �T f U2 � ( ) of linear first and second sound waves
of heating excited in He II. The ratio of the coeffi-
cients calculated from the data shown in Fig. 7 is
equal to
k k T T2 1 2 1
1 2 23/ ( / ) ./� �� � (6)
We also evaluated the ratio of the slopes of the
straight lines drawn through the experimental points
describing the dependence of the amplitudes �T1 of
the first sound waves and the amplitudes �T2 of the
second sound waves in He II at P � 13 3. atm and
T � 1895. K at given �. The ratio of the slopes calcu-
lated for � � 1, 3, and 10 �s lies in the range
k /k2 1 23 6� � , which clearly agrees with the result
(6) obtained in the linear approximation.
1326 Fizika Nizkikh Temperatur, 2006, v. 32, No. 11
G.V. Kolmakov et al.
7.32 V
9.15 V
12.2 V
15.3 V
18.3 V
21.4 V
0.010
0.005
0
–0.005
–0.010
–0.015
90 95 100 105 110
Time, s�
T
, a
rb
. u
n
its
1
Fig. 9. Evolution of shape of the first sound wave in com-
pressed He II for P � 13 atm, T � 1895. K, heater pulse du-
ration �e � 3 �s. Numbers on the oscillograms indicate the
voltage U at the heater.
The threshold heat flux Wcr , above which a com-
pression wave is observed, can be estimated from the
abscissa intersections of the lines drawn through the
open symbols in Fig. 10. It turned out that, for a given
temperature, the product Wcr�
1 2/ remains close to a
constant as � increased from 0.3 to 10 �s in the range of
pressures of 1–13.3 atm (compare, e.g., Figs. 8 and
10), whereas the value of the product increases with
reduction of the temperature below T�. The observed
transformation from a wave of rarefaction to a wave of
compression of first sound indicates that the heat
transfer mechanism at the heater-He II interface
changes qualitatively for a heat flux density above the
critical level Wcr . As mentioned above, the appear-
ance of the first sound waves of compression in
superfluid He II at elevated pressures should be attrib-
uted to the strong thermal expansion of a layer of nor-
mal liquid He I created at the heater-He II interface at
high heat loads. At low pressures P P
cr , pulse film
boiling was previously supposed to be the principal
mechanism leading to the excitation of the first sound
shock waves, and this additional effect must also be
taken into account.
3. Nonlinear dynamics of standing second sound
waves in a high-quality resonator filled
with He II
In this Section we present the results of our recent
studies of nonlinear second sound wave interactions in
a high quality (high-Q) cylindrical resonator filled
with He II.
The inner construction of the experimental cell was
similar to that used in earlier investigations of planar
second sound pulses [5,12]. The resonator was formed
by a cylindrical quartz tube L � 7 cm in length and
D � 15. cm in diameter, whose ends were capped by a
pair of flat, parallel, quartz plates. Second sound
waves were generated by a film heater on one plate
and were detected by a superconducting film
bolometer on the opposite plate. The resistance of the
heater was R � 35 � at T � 2 K. The sensitivity of the
bolometer was varied from 1.2 to 2.6 V/K at tempera-
tures 1.79 K < T < 2.08 K.
The frequency of the second sound wave f fd g� 2
excited by such a heater is twice the frequency fg of
the generator. This frequency doubling occurs because
the heat flux from the heater is proportional to the
squared voltage applied to the heater, W t U t( ) ( )� 2 ,
where U t U f tg g( ) sin ( )� 2� is the voltage applied to
the heater. The signal from the bolometer was taken to
a preamplifier, digitized with an analog-to-digital
converter, and recorded on the hard disk of a com-
puter. Before being recorded, each signal was aver-
aged automatically over 6–8 measurements to reduce
electrical noise.
Figure 11 shows the dependence of the amplitude
of the recorded signal on the driving frequency fd , at
Nonlinear and shock waves in superfluid He II
Fizika Nizkikh Temperatur, 2006, v. 32, No. 11 1327
0.3 0.3
0.30.3
0.2 0.2
0.20.2
0.1 0.1
0.10.1
0 0
00
10 10
1010
20 20
2020
30 30
3030
40 40
4040
U, V
U, V
U, V
U, V
� �= 0.3 s
� �= 3 s � �= 10 s
� �= 1 s
T
11
/2
T
11
/2
T
11
/2
T
11
/2
Fig. 10. Dependence of the amplitude �T1 of the first
sound pulse on the heater pulse voltage U, plotted as
�T1
1 2/ vs. U. The pressure was P � 133. atm, the tempera-
ture was T � 1895. K, and the heater pulse duration was
�e � 03. , 1, 3, and 10 �s as indicated. The solid symbols
correspond to the wave of rarefaction, and open symbols
correspond to the wave of compression.
f, Hz
3000 3100 3200 3300 3400
0.12
0.10
0.08
0.06
0.04
0.02
0
T
,a
rb
.u
n
its
2
Fig. 11. Frequency dependence of the amplitude of a stan-
ding second sound wave �T2 generated by the heater in the
cylindrical resonator at T � 2075. K. The driving voltage
Ug � 188. V. The ac heat flux density W � 002. W/cm2.
T � 2 075. K. The amplitude of the driving voltage was
Ug � 188. V, so the ac heat flux density was W =
= 0.02 W/cm2. The peaks at the frequencies f � 3020,
3120, 3220, and 3320 Hz correspond to excitation of
longitudinal standing second sound waves of frequen-
cies
f pu T Lp � 2 2( ) / ,
where p � 30, 31, 32, and 33 are the resonance num-
bers. Other, smaller peaks correspond to the genera-
tion of radial modes in the cylindrical resonator. The
Q-factor of the resonator determined from the data of
Fig. 11 for longitudinal waves with p � 10 was
Q � 1000.
We have observed a marked broadening of resonant
peaks like those of Fig. 11 when the pulsed voltage ap-
plied to the heater exceeds Ug1 3 75� . V (i.e., at
W � 0 08. W/cm2). This phenomenon is believed to
be attributable to interaction of the second sound
wave with quantised vortex filaments, leading in turn
to a nonlinear broadening of the resonances. In what
follows below, we discuss only the results of measure-
ments made at smallU Ug g
1.
Steady-state spectra of the standing second sound
wave in a resonator were analysed by computing Fou-
rier transforms of the signals. As an example, the spec-
trum of a signal recorded when driving at a frequency
equal to the 11th longitudinal resonant frequency of
the cell, f fd � �11 1088 Hz at T � 2 075. K, is shown in
Fig. 12. It is clearly evident that the main spectral
peak lies at the driving frequency fd , and that
high-frequency peaks appear at harmonics,
f f nn d� � , where n � 2 3, ...
It is seen that a cascade of waves is formed over a
wide range of frequencies up to f � 40 kHz, i.e., up to
frequencies 40� higher than the frequency of the sec-
ond sound f11 generated by the heater.
Following the general ideas formulated in papers
[1,24,25] one can associate formation of the cascade
with the establishment of a steady-state directed flux
of the wave energy through the wavelength scales to-
wards higher frequencies. Formation of this cascade is
similar to the creation of the Kolmogorov distribution
of fluid velocity over frequencies observed in classical
liquids [1]. At high frequencies the amplitudes of har-
monics are decreased. In this high-frequency domain
the nonlinear mechanism for almost nondissipative
transfer of the wave energy must change to viscous
damping of the waves (cf. observations of high-fre-
quency edge of the inertial range of frequencies in the
system of nonlinear capillary waves on the surface of
liquid hydrogen [26]), and the energy flux will then
be absorbed due to viscous energy loss in this fre-
quency domain. Observation of the cascade is in quali-
tative agreement with the results of our previous nu-
merical studies [27,28] of nonlinear second sound
standing waves in He II in a high-Q resonator. In pa-
pers [27,28] it was shown that, even if the amplitude
of the periodic driving force is relatively small, a tur-
bulent cascade of second sound waves of frequencies
equal to multiples of the driving frequency should be
formed over a wide range of frequencies when the
Q-factor of the resonator is sufficiently high. Thus, we
may infer that a wave turbulent state in the system of
second sound waves has indeed been formed in the ex-
periments in the frequency domain up to 40 kHz.
4. Conclusions
Our investigations have shown that, at all pres-
sures up to that of solidification of He II, there exists
a relatively wide temperature range below T� where
the nonlinearity coefficient of second sound is nega-
tive. At these temperatures the shock front (the tem-
perature discontinuity) is formed on the trailing edge
of the plane wave, or at the center of a propagating bi-
polar second sound pulse of a finite amplitude gener-
ated by a point heater. The width of the bipolar pulse
does not change with distance, so such pulses can be of
a wide use in studies of nonlinear and dissipative phe-
nomena in the vicinity of the superfluid transition
line.
Observations of the transformation of the first
sound waves of rarefaction to compression waves at
heat flux density W higher than some critical value
imply that the heat exchange mechanism at the pulsed
heater-He II interface changes essentially above Wcr :
the main reason for excitation of the compression
waves in He II at pressures above the critical pressure
is strong thermal expansion of the layer of normal
1328 Fizika Nizkikh Temperatur, 2006, v. 32, No. 11
G.V. Kolmakov et al.
f, Hz
A
m
p
lit
u
d
e
,a
rb
.u
n
its
10
0
10
–1
10
–2
10
–3
10
–4
10
–5
1000 10000
Fig. 12. Fourier spectrum of second sound waves com-
puted from signals recorded when driving at the 11th lon-
gitudinal resonance frequency of the cell, f11 1088� Hz at
T � 2075. K. The driving amplitude was Ug � 263. V. The
ac heat flux density W � 006. W/cm2.
fluid He I arising at the interface for high W, accom-
panied by fast contraction of the layer after comple-
tion of the heat pulse. At pressures below Pcr one
should consider also the film boiling of the liquid at
the interface. Both the mechanisms explain the gener-
ation of the wave of compression (cooling) followed
by a subsequent wave of rarefaction (heating) of the
first sound in He II at high heat loads. All these pro-
cesses should be taken into account in a discussion of
the peculiarities of the behavior of first and second
sound waves generated in superfluid helium by power-
ful heat pulses.
We have observed for the first time the creation of
a cascade of standing second sound waves in He II in a
high-Q resonator, formed at frequencies higher than
that at which the system is driven, over a wide range
of frequencies. The cascade is formed due to the non-
linear interactions between second sound standing
waves in the resonator. At high frequencies the non-
linear mechanism of transfer of the wave energy is
changed to viscous damping of the waves, thus termi-
nating the cascade.
The authors are grateful to A.A. Levchenko, E.A.
Kuznetsov, and V.V. Lebedev for valuable discus-
sions. The investigations were supported by the Rus-
sian Foundation for Basic Research, project Nos.
05-02-17849 and 06-02-17253, by the Presidium of the
Russian Academy of Sciences in frames of the pro-
grams «Quantum Macrophysics» and «Mathematical
Methods in Nonlinear Dynamics», and by the Engi-
neering and Physical Sciences Research Council
(UK).
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