Mutual interactions of oscillating quartz tuning forks in superfluid ⁴He
The quartz tuning fork has recently become a popular experimental tool for investigations of both classical and quantum turbulence in cryogenic helium. Its increased use in low-temperature experiments and a number of puzzling results obtained in the past have led to many questions concerning the int...
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Sheshin, G. Gritsenko, I. 2017-05-31T10:08:15Z 2017-05-31T10:08:15Z 2013 Mutual interactions of oscillating quartz tuning forks in superfluid ⁴He / G. Sheshin, I. Gritsenko // Физика низких температур. — 2013. — Т. 39, № 10. — С. 1062–1067. — Бібліогр.: 21 назв. — англ. 0132-6414 PACS: 67.25.dg, 67.25.dt https://nasplib.isofts.kiev.ua/handle/123456789/118817 The quartz tuning fork has recently become a popular experimental tool for investigations of both classical and quantum turbulence in cryogenic helium. Its increased use in low-temperature experiments and a number of puzzling results obtained in the past have led to many questions concerning the interaction of multiple tuning forks or the interaction of tuning forks with other oscillators. We report measurements performed in He II at low temperatures around 360 mK, on the mutual interaction of tuning forks placed in the same volume of fluid, and examine the responsible mechanisms in an effort to discriminate between acoustic coupling and interaction via quantized vortices. To this end, the interaction of two tuning forks is investigated by analyzing their recorded resonance curves, looking for any nonelectrical crosstalk. Further, the forcevelocity characteristics of a detector tuning fork are measured for different operating velocities of a generator tuning fork. As a complementary measurement, the intensity of sound waves is recorded using a set of miniature receivers. We confirm the current knowledge on acoustic emission by tuning forks in He II and verify properties of their radiation patterns. We conclude that in our experiment the interaction is almost entirely mediated by sound waves. This research is supported by the programs of bilateral cooperation between the National Academy of Sciences of Ukraine and the Academy of Sciences of the Czech Republic; by the fund of the National Academy of Sciences of Ukraine and the Russian Academy of Sciences 10-2012. DS and LS acknowledge the support of the programme UNCE 2040 of Charles University in Prague. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Квантованные вихри и турбулентность Mutual interactions of oscillating quartz tuning forks in superfluid ⁴He Article published earlier |
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Mutual interactions of oscillating quartz tuning forks in superfluid ⁴He |
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Mutual interactions of oscillating quartz tuning forks in superfluid ⁴He Sheshin, G. Gritsenko, I. Квантованные вихри и турбулентность |
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Mutual interactions of oscillating quartz tuning forks in superfluid ⁴He |
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Mutual interactions of oscillating quartz tuning forks in superfluid ⁴He |
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Mutual interactions of oscillating quartz tuning forks in superfluid ⁴He |
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Mutual interactions of oscillating quartz tuning forks in superfluid ⁴He |
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mutual interactions of oscillating quartz tuning forks in superfluid ⁴he |
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Sheshin, G. Gritsenko, I. |
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Sheshin, G. Gritsenko, I. |
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Квантованные вихри и турбулентность |
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Квантованные вихри и турбулентность |
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2013 |
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English |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Article |
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The quartz tuning fork has recently become a popular experimental tool for investigations of both classical and quantum turbulence in cryogenic helium. Its increased use in low-temperature experiments and a number of puzzling results obtained in the past have led to many questions concerning the interaction of multiple tuning forks or the interaction of tuning forks with other oscillators. We report measurements performed in He II at low temperatures around 360 mK, on the mutual interaction of tuning forks placed in the same volume of fluid, and examine the responsible mechanisms in an effort to discriminate between acoustic coupling and interaction via quantized vortices. To this end, the interaction of two tuning forks is investigated by analyzing their recorded resonance curves, looking for any nonelectrical crosstalk. Further, the forcevelocity characteristics of a detector tuning fork are measured for different operating velocities of a generator tuning fork. As a complementary measurement, the intensity of sound waves is recorded using a set of miniature receivers. We confirm the current knowledge on acoustic emission by tuning forks in He II and verify properties of their radiation patterns. We conclude that in our experiment the interaction is almost entirely mediated by sound waves.
|
| issn |
0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/118817 |
| citation_txt |
Mutual interactions of oscillating quartz tuning forks in superfluid ⁴He / G. Sheshin, I. Gritsenko // Физика низких температур. — 2013. — Т. 39, № 10. — С. 1062–1067. — Бібліогр.: 21 назв. — англ. |
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AT sheshing mutualinteractionsofoscillatingquartztuningforksinsuperfluid4he AT gritsenkoi mutualinteractionsofoscillatingquartztuningforksinsuperfluid4he |
| first_indexed |
2025-11-26T09:52:53Z |
| last_indexed |
2025-11-26T09:52:53Z |
| _version_ |
1850619900738928640 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10, pp. 1062–1067
Mutual interactions of oscillating quartz tuning forks
in superfluid 4He
G. Sheshin and I. Gritsenko
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: sheshin@ilt.kharkov.ua
D. Schmoranzer and L. Skrbek
Faculty of Mathematics and Physics, Charles University, V. Holešovičkách 2, Prague 8 18000, Czech Republic
Received March 21, 2013
The quartz tuning fork has recently become a popular experimental tool for investigations of both classical
and quantum turbulence in cryogenic helium. Its increased use in low-temperature experiments and a number of
puzzling results obtained in the past have led to many questions concerning the interaction of multiple tuning
forks or the interaction of tuning forks with other oscillators. We report measurements performed in He II at low
temperatures around 360 mK, on the mutual interaction of tuning forks placed in the same volume of fluid, and
examine the responsible mechanisms in an effort to discriminate between acoustic coupling and interaction via
quantized vortices. To this end, the interaction of two tuning forks is investigated by analyzing their recorded
resonance curves, looking for any nonelectrical crosstalk. Further, the force-velocity characteristics of a detector
tuning fork are measured for different operating velocities of a generator tuning fork. As a complementary meas-
urement, the intensity of sound waves is recorded using a set of miniature receivers. We confirm the current
knowledge on acoustic emission by tuning forks in He II and verify properties of their radiation patterns. We
conclude that in our experiment the interaction is almost entirely mediated by sound waves.
PACS: 67.25.dg Transport, hydrodynamics, and superflow;
67.25.dt Sound and excitations.
Keywords: superfluid helium, interacting oscillators, acoustic emission.
1. Introduction
Since the discovery of superfluidity of 4He by Kapitza
[1], Allan and Misener [2] in 1938, experiments on various
oscillatory systems in He II yielded many contributions to
the current understanding of this interesting quantum fluid,
not least of which was the pioneering work of Androni-
kashvili [3]. Today, many oscillating systems are still used
to study various aspects of superfluidity and quantum tur-
bulence in both isotopes of helium, ranging from oscillat-
ing wires, grids, spheres to a relatively more recent addi-
tion — quartz tuning forks.
The tuning forks are well-tested detectors of both clas-
sical and quantum turbulence in all the helium fluids —
cold He gas, normal liquid 3He and 4He, their superfluid
phases and even mixtures. They were also successfully
employed for studies of cavitation in normal and superfluid
helium [4], for measurements of viscosity [5,6], and more
recently, their mutual interactions as observed in helium
became of importance [7–9], and are seen, as we shall
show, related to their capability to emit and absorb sound
waves [10–12]. Depending on the parameters of the tuning
fork, sound emission may even represent the dominant
process of energy dissipation.
To study the mutual interaction of the tuning forks, we
have constructed a dedicated cell optimized to allow meas-
uring not only the influence of one fork on another, but
also to measure the intensity of the sound waves emitted
by the oscillating tuning forks and propagating in the heli-
um liquid.
2. Experimental setup
The measurements were carried out in three experi-
mental runs using a dilution refrigerator that allows reach-
© G. Sheshin, I. Gritsenko, D. Schmoranzer, and L. Skrbek, 2013
Mutual interactions of oscillating quartz tuning forks in superfluid 4He
ing temperatures as low as 12 mK. The copper experi-
mental cell was fitted with a silver heat exchanger and
thermally connected to the mixing chamber. Either one or
two tuning forks were placed in the cell in each experi-
mental run (denoted F1, F2 in the first two runs, F3 and F4
together in the last run), in such a way that the ends of their
prongs were located on the axis of symmetry of the cylin-
drical cell. Schematic arrangement of forks in the cell in all
runs is shown in Figs. 1 (a)–(c). The experiments in He II
were performed at temperatures (360 ± 10) mK (forks F1
and F2), and (352 ± 2) mK (forks F3 and F4). The parame-
ters of the tuning forks are summarized in Table 1. The cell
also contains two sound receivers S1, S2 positioned across
the cylindrical cell. These are piezoelectric sound sensors
(type of material: PSI-5A-S4-ENH) with a resonant fre-
quency of 4.7 MHz, and allow measurement up to 10 MHz.
They were used to measure the intensity of sound waves in
helium, as in our case the tuning forks can be expected to
produce measurable acoustic signals [11,12].
The tuning forks were driven by ac voltage from two
waveform generators and their signals were detected by
two EG&G Lock-in Analyzers 5208 using a common ref-
erence. The electronics used for the readout of the tuning
forks are depicted in Fig. 1 (d). The voltage induced in one
of the sound receivers was also optionally read by one of
the lock-in amplifiers.
Based on the cell dimensions and the typical sound ve-
locities of He II around 240 m/s (in fact, in the experiments
it would be close to 238 m/s), the fundamental longitudinal
and transverse acoustic modes of the cell can be expected
at frequencies around 5,5 kHz and 15 kHz, respectively.
The fundamental frequency corresponding to the resonance
between the sound receivers ( ≈ 10 mm apart) is 12 kHz.
Neglecting the complicated geometry of the tuning forks
for purposes of the following estimation, the relevant har-
Table 1. Resonant frequencies, dimensions of the tuning forks
used and their calibration constants. L is the length of the prongs,
W stands for their width in the direction perpendicular to oscilla-
tion, T denotes their thickness and D is the separation between the
prongs, both in the direction of oscillation. The dimensions are
given as measured by optical microscopy. The tuning fork cali-
bration constant, a, relates its mechanical properties to its electri-
cal characteristics [5] and was determined using the self-
calibration method (measurement in vacuum).
Tuning fork vacf ,
Hz
L,
mm
W,
mm
T,
mm
D,
mm
610a ⋅ ,
C·m–1
F1 32710.7 3.79 0.30 0.60 0.31 17
F2 32704.7 3.14 0.34 0.38 0.20 3.5
F3 32720.3 3.14 0.34 0.38 0.20 2.69
F4 32711.7 3.81 0.34 0.60 0.36 4.5
Fig. 1. Schematic depiction of the cylindrical cell of diameter 8 mm and height 22 mm thermally connected to the mixing chamber of
a dilution refrigerator. The cell contains two sound receivers S1 and S2 separated by ≈ 10 mm and different tuning forks (F1, F2; F3,
F4) placed in different positions in three subsequent experimental runs as shown. Note that F1 and F2 have different orientations with
respect to the sound receivers S1, S2 (a)–(c). Schematics of the readout electronics shown for the case (c) in the left panel. Both lock-in
amplifiers use the same reference signal in order to measure the influence of one tuning fork on another. The dashed line represents the
boundary of the cryogenic parts of the setup (d).
(a)
F1S1
S2
22 mm
S1
S2
F4 F3S1
S24.1 mm
(d)
R
ef
er
en
ce
si
gn
al
F3
1 kΩ1 kΩ
Generator 1 Generator 2
Lock-in 1
Lock-in 2
(c)
8
m
m
F4
(b)
F2
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10 1063
G. Sheshin, I. Gritsenko, D. Schmoranzer, and L. Skrbek
monics close to 32 kHz that might potentially be excited
by the tuning forks are therefore the first transverse har-
monic around 30 kHz, and the fifth longitudinal harmonic
expected close to 33 kHz. Conversely, the wavelength is
7.5 mm≈ at 32.0 kHz (estimate of tuning fork resonance
frequency when immersed in superfluid He II).
While we cannot rule out the excitation of acoustic reso-
nances of the cell altogether, it seems that they are rather
sparsely spaced in the frequency domain (even with some
reserve for smaller satellite peaks near the main resonances
due to the complex 3D geometry) and, given that they are
usually quite sharp in superfluid helium at low temperature
[8,11] and that the resonances of the tuning forks are ex-
tremely sharp themselves, we should be able to keep away
from the acoustic standing modes during our measurements.
3. Results and discussion
First, the tuning forks were characterized at low tem-
perature in vacuum and their calibration constants [5] were
determined. Together with their measured resonant fre-
quencies and their dimensions (denoted as in Ref. 5), they
are listed in Table 1. The mutual interaction of F3 and F4
was also tested in vacuum to eliminate possibilities of
purely electrical crosstalk or interaction via vibrations
propagating through the solid walls of the cell. This was
done in such a way that fork F3 was driven near its reso-
nance and the signal of F4 was analysed both at the same
frequency F3 was oscillating at, and at its own resonance
frequency only ≈ 10 Hz below that of F3. Let us stress that
in vacuum no measurable crosstalk was detected between
the tuning forks — if it was present, it was definitely well
below noise levels, meaning that any crosstalk measured in
helium is indeed due to the presence of the fluid.
A similar test was also carried out for the signal from
sound receivers, and here we found that oscillations of
forks F3 and F4 resulted in comparable signals from the
receiver S1 in vacuum and in helium, and hence that this
signal is likely due to inductive or capacitive coupling. In
the experiments with the tuning forks F1 and F2, the situa-
tion was much better and the sound receiver signals from
these experiments will be discussed in due course.
After the vacuum measurements were completed in the
arrangement with forks F3 and F4, He II was admitted into
the cell, and at the temperature of 356 mK and pressure
0.16 bar the resonance curves of F3 and F4 were recorded,
giving resonant frequencies 31687 Hz (linewidth 10 Hz)
and 32005 Hz (linewidth 23 Hz), respectively. It is im-
portant that there is virtually no overlap between the reso-
nances and that neither distortions of the resonance curves
nor extra peaks due to coupling to acoustic resonances of
the cell were observed.
In the almost pure superfluid near 350 mK*, there are
two main ways in which different tuning forks could influ-
ence each other's behavior. One type of interaction can be
mediated by first sound waves and another by quantized
vortices. While second sound was found to be potentially
important [9,13,14] in the temperature range above 1 K, its
role is greatly diminished at these low temperatures. To
discriminate between the two mechanisms proposed, we
have performed several types of measurements.
In one set of measurements, we have been driving the
fork F3 (“generator”) near its resonance and monitoring
the signal of the fork F4 (“detector”) at the same frequency
(using the common reference of the lock-in amplifiers)
while driving it at its own resonance with a low voltage.
This measurement should mainly be sensitive to acoustic
waves propagating through the superfluid. The results are
summarized in Fig. 2.
From the data, it is evident that the tuning forks do in-
teract with each other, as a small but measurable signal
near the resonance of F3 was read from F4 as well, clearly
indicating a transfer of oscillatory motion at a specific fre-
quency from the generator fork to the detector fork. Fur-
ther, when the drive of F3 is shifted in frequency towards
the resonance of F4, the signal of F4 — instead of repro-
ducing a smooth resonance curve — starts to show beats
resulting from the fine mismatch between the (low) drive
* The experiments were carried out at this relatively high temperature, because we use natural helium, containing also trace amounts
of the isotope 3He. Its presence might have influenced results at temperatures below about 300 mK, as it constitutes an impurity
and contributes to the drag forces.
31640 31660 31680 31700 31720 31740
79
80
81
D
et
ec
to
r F
4
el
ec
tri
c
cu
rr
en
t,
nA
Generator F3 drive frequency, Hz
80
140
120
100
60
321003200031900318003170031600
Fig. 2. The signal of the detector tuning fork F4 as measured by
the lock-in amplifier referenced to the frequency of the drive of
the generator tuning fork F3. A pair of smaller peaks is detected
at the resonance of the generator fork. Oscillations of the genera-
tor fork F3 are partly transmitted via sound waves to the detector
fork F4. A series of beats is also measured near the resonance
frequency of F4, resulting from the small frequency differences
between the driving voltage of F4 (in resonance) and the refer-
ence signal from the driving waveform generator connected
to F3.
1064 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10
Mutual interactions of oscillating quartz tuning forks in superfluid 4He
of F4 and the reference signal from the drive of F3, which
are connected to the same lock-in amplifier.
Another type of measurement, more sensitive to the
presence of quantized vortices (an analogical experiment
with vibrating wires is described in Refs. 15, 16), relied on
probing the transition to turbulence with each tuning fork,
while the other was operated at its resonance with a fixed
velocity amplitude. Among other, the results obtained with
forks F3 and F4 are presented in Fig. 3 in terms of the
force-velocity dependence and as drag coefficients. In our
case, it is seen that while the linear drag may be somewhat
shifted by operating the second tuning fork, the critical
velocity and the turbulent drag are largely unaffected.
Note in passing that the drag coefficient in the limit of
large velocities behaves differently for different tuning
forks. If, for the fork F2 the limiting value is about one as
in classical flows or as observed in Ref. 18, for F3, alt-
hough the transition from linear to nonlinear drag regime is
also displayed clearly, the limiting value of the drag coef-
ficient is considerably lower than one [7].
We would like to point out that the tuning forks F1 and
F4 (data in Fig. 3) exhibit much larger linear drag than F2
or F3 and did not reliably detect the transition to turbu-
lence in these experiments. F1 and F4 are notably larger in
size than their counterparts, and in accord with Refs. 11, 12
we attribute this large increase in the linear drag, roughly
by one order of magnitude, to significant losses of energy
via emitted sound waves. This is supported by the fact that
for a constant velocity amplitude, the acoustic emission
power [11,12] scales with the fork dimensions as
2 2 2 2( )L W T T D+ , giving a factor of 10 between the emis-
sion powers of F3 and F4 and a factor of 7 between F3 and
F1 (F2 differs from F3 only slightly, they have the same
nominal dimensions). This is in good agreement with the
data shown in Fig. 3.
The inability of forks F1 and F4 (due to strong acoustic
emission) to detect the nonlinear turbulent drag does not
mean that they are unable to generate quantum turbulence
at the examined range of velocities, merely that the turbu-
lent drag is not yet sufficiently large to become dominant.
In fact, based on other measurements with similar tuning
forks [19–21], it is extremely unlikely that at velocities
around 80 or 90 mm/s, quantum turbulence would not be
generated (also compare with the critical velocity of F2
and F3 in Fig. 3 between 20 and 30 mm/s).
Therefore, based on the experiment performed with the
forks F3 and F4, we can say with moderate confidence that
while these forks interact via sound waves and weakly af-
fect their respective linear drag forces (which include drag
due to acoustic emission), no significant influence on the
critical velocities for the transition to turbulence or the
turbulent drag was detected.
Fig. 3. (Color online) Velocity amplitude v vs driving force F (a) and the drag coefficient 2= 2 /DC F Aρ v (b), where A stands for the
cross-sectional area of a prong of the tuning fork and ρ for the fluid density. The forces and velocities were obtained using the calibra-
tion constants listed in Table 1 and the procedure detailed in Refs. 5, 17. The data are shown for all the tuning forks used in the three
runs, and for the case of F3 and F4, their mutual interaction via quantized vortices was also studied by changing the fixed velocity am-
plitude of the other fork (as indicated) between successive measurements of the force-velocity curve. Note that no significant influence
is observed on the magnitude of the drag force or the critical velocity. The forks F2 and F3 have very similar behavior, and both detect
the transition to turbulence around 20 to 30 mm/s (red arrow). F1 and F4 experience significantly larger drag at the same velocities
(differing by at least one order of magnitude), which is related to energy losses due to acoustic emission (see text). Any drag force aris-
ing from (quantum-) turbulent flows around these two forks is screened by this acoustic drag and therefore no clear transition to turbu-
lence is detected. The solid lines are linear dependences, the dashed line represents a quadratic law, characteristic of the turbulent drag
regime.
(a)
v,
m
m
/s
v, mm/sForce amplitude, N
F1
F2
F3 with F4 at 96 mm/s
F3 with F4 at 96 mm/s
F3 with F4 at 9.6 mm/s
F3 with F4 at 0.96 mm/s
F3 with F4 at 0.096 mm/s
F3 with F4 at 0.01 mm/s
F1
F2
F4 with F3 at 82 mm/s
F4 with F3 at 0.3 mm/s
10
–9
10
–8
10
–7
10
–6
10
–4
10
3
10
3
10
3
10
4
10
5
10
2
10
2
10
2
10
1
10
1
10
1
10
0
10
0
10
0
10
–1
10
–110
–110
–2
10
–2
(b)
C
D
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10 1065
G. Sheshin, I. Gritsenko, D. Schmoranzer, and L. Skrbek
Further we will discuss the signals from the sound re-
ceiver S1. As was mentioned earlier, for tuning forks F3
and F4, the signals of S1 in helium and in vacuum were
comparable, and likely related to capacitive or inductive
coupling. The results for tuning forks F1 and F2 are shown
in Fig. 4, where receiver signal is plotted both versus the
applied driving voltage and versus tuning fork (tip) veloci-
ty amplitude. In this case a clear difference between vacu-
um and helium is observed, implying that sound waves
propagating through He II are indeed recorded, albeit to-
gether with a background due to electrical crosstalk. This
statement is further supported by the fact that the signal
voltage seems to be proportional to the velocity rather than
the driving voltage.
Considering the relative magnitudes of acoustic emis-
sion by the forks F1 and F2 as discussed above (F1 roughly
7 times stronger), it may seem surprising that the signals
from the receivers show almost equal levels at the same
velocity amplitude. We attribute this to the different orien-
tations of the tuning forks (see Fig. 1) — fork F2 oscillates
in the direction of the axis joining the two sound receivers.
This means that the spatial radiation pattern is rotated by
90 degrees with respect to that of the fork F1 and different
intensities might be expected.
Upon consulting the radiation diagrams for tuning forks
as shown in Fig. 4 of Ref. 10, we see that this could easily
explain the apparent discrepancies in the recorded intensity
of sound radiation, as higher sound levels are expected in
the plane of oscillations than in the normal plane in dis-
tances = / 8R λ as well as = 3 / 4R λ , where λ is the
wavelength. In our case the receivers are in the distance
2 / 3R λ , i.e., in between the values taken from Ref. 10.
Thus, similar behavior can be expected, and if we substi-
tute = 2 / 3R λ into Eq. (11) of Ref. 10, we indeed find that
the ratio of the pressure amplitudes in the two principal
directions is around 6.3, i.e., it almost exactly balances the
ratio between the geometric factors in the emission powers
of the two tuning forks* and explains why similar sound
levels for a given velocity amplitude are recorded by the
receiver in both runs.
4. Conclusions
Our investigation of mutual interaction of quartz tuning
forks in He II at low temperature confirms that such an
interaction exists even in the linear drag regime and is me-
diated by sound waves emitted by either tuning fork. We
also demonstrate the increase in the linear drag force due
to this acoustic emission and its adverse effect on the ca-
pability of the tuning forks to detect quantum turbulence.
While in our experiments we have not observed any no-
table interaction via quantized vortices resulting in changes
of the critical velocity, such an interaction cannot be ruled
out, as several other experiments using different oscillating
Fig. 4. (Color online) Voltage signal of the sound receiver S1 versus tuning fork driving voltage (a) and tip velocity amplitude (b).
While electrical crosstalk in vacuum is observed for both tuning forks, it is significantly lower (20 times for F1 and 100 times for F2)
than the signal measured in He II. This represents a direct observation of acoustic emission by the tuning forks, and confirms well to the
fact that the measured signal seems to be directly proportional to the velocity amplitude and not the driving voltage. The solid lines are
linear dependences.
Tuning fork driving voltage, Vrms
F1 in He II
F2 in He II
F1 in vacuum
F2 in vacuum
So
un
d
re
ce
iv
er
v
ol
ta
ge
, V
rm
s
So
un
d
re
ce
iv
er
v
ol
ta
ge
, V
rm
s
Tuning fork velocity amplitude, mm/s
F1 in He II
F2 in He II
(a) (b)
10
3
10
2
10
1
10
1
10
0
10
0
10
–1
10
–1
10
–2
10
–3
10
–3
10
–3
10
–7
10
–7
10
–6
10
–6
10
–4
10
–4
* Note that the same geometric factor also appears in the expression for the pressure amplitude. If the velocity amplitude is used
explicitly in the formula, the emission power will be expressed as linear in the pressure amplitude because the energy flux is given
by 1 / 2 Re *( )pv , where p and v stand for the oscillating parts of pressure and velocity, respectively, and the asterisk represents
complex conjugation. See also Ref. 11.
1066 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10
Mutual interactions of oscillating quartz tuning forks in superfluid 4He
bodies have shown clear indications to the contrary
[15,16]. The most likely reason why we have not seen this
type of interaction is the large acoustic drag acting on our
tuning forks, especially F1 and F4, effectively screening
any lower drag forces that might arise from the generation
of quantized vortices.
Finally and most importantly, we have for the first time
observed the sound waves radiated by quartz tuning forks
in cryogenic conditions directly and measured their ampli-
tude by sensitive miniature sound receivers. While our
experimental technique does not allow an exhaustive quan-
titative analysis of the radiation patterns, the obtained data
are in reasonable agreement with the models presented in
the literature.
Acknowledgements
This research is supported by the programs of bilateral
cooperation between the National Academy of Sciences of
Ukraine and the Academy of Sciences of the Czech Repub-
lic; by the fund of the National Academy of Sciences of
Ukraine and the Russian Academy of Sciences 10-2012.
DS and LS acknowledge the support of the programme
UNCE 2040 of Charles University in Prague.
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