Transcript Nonlinear waves in superfluid helium

Experimental Study of NonLinear Second Sound
Waves in He-II
Victor Efimov
Institute of Solid State Physics RAS, Chernogolovka,
Russia;
Lancaster University, Lancaster, UK
I.Borisenko, O. Griffiths, P.Hendry,
G.Kolmakov, A.Kuliev, E.Lebedeva, P.E.V.
McClintock, L.Mezhov-Deglin
Nonlinear second sound
waves in superfluid helium
9 December 2005
Warwick
2
Hydrodynamic Equations of
Ideal Fluids



For any an ideal fluid (incompressible, without energy
dissipation) it may be written the equation of
continuity (mass conservation)





 div j  0
where j   v mass flux density
t
the equation of motion

j
Newton’s second law
 P  0
t
the equation of adiabatic

S
motion of an ideal fluid
 divS v  0
t
the “equation of entropy continuity”
where
“entropy flux”
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3
Two Fluid Hydrodynamic Equations
In
superfluid
state
helium
behaves as if it were a mixture of
two different fluids. Superfluid
moves with zero viscosity and the
other is a normal viscosity fluid.
Then density is =n+s, and
mass flux is

In superfluid state only normal fluid
has entropy. The hydrodynamic
equations transform into
mass conservation


t

j
t
S
t
0 ,1 6
0 ,1 2
3
0 ,1 0
0 ,0 8
0 ,0 6
 div j  0
 P  0
Euler’s equation

 divS v n  0
isentropic motion
and add equation describing the
forces in superfluid state
 S (T )
0 ,1 4
 , g/cm

0 ,0 4
0 ,0 2

vs
 N (T )
1  2
  (  vs )  0
t
2
0 ,0 0
-0 ,0 2
0 ,0
0 ,5
1 ,0
1 ,5
2 ,0
T, K
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4
Sounds in Superfluid Helium
The
differentiation
of
two
equations with respect to time
and substituting
into another
relationships we obtain for small
disturbance (linear waves)
t
2
 S
2
 P,
t
2

sS
n

v1
180
T
v 1 / 3
160
F irst a n d S eco n d so u n d
140
4
120
v elo sity in H e a t S V P
100
80
v2
60
40
20
0
0
1
2
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3
4
T, K
(P /  ) S
waves of temperature or entropy
(countermovement motion normal
and
superfluid
components,
=n+s=const
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200
2
Solutions are
 wave
of density (co-moving
motion normal and superfluid
component)
u1 
220
v, m /s
 
2
240
TS  s
2
u2 
C n
5
Experimental
device
8
2 .5 m A
6
dU /dT , V /K

The experimental technique used
for study second sound is shown on
sketch. For heater and sensitive
thermometer we used the low
inertial evaporated metal film
resistors. MoO or Cu-Sn films were
used as a heater.
We used superconducting Re or
Cu-Sn
film
bolometers
with
resistance in normal state some
kOm. The wide of superconducting
transition was less 0.1 K. Tc was
possible to move by applied
magnetic field.
3 mA
2 mA
4
2
0
1 ,6
1 ,8
2 ,0
T, K
2 ,0
JH=3 m A
J H = 2.5 m A
1 ,5
U, V

JH=2 m A
1 ,0
0 ,5
1 ,6
1 ,8
2 ,0
T, K.
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



9 December 2005
The time resolution of heat pulses by
heater and bolometer was better 0.3 s.
The sensitivity of registration pulse by
temperature was less 10-6 K and defined
by noise levels.
The high pressure chamber allowed us to
study the processes of generation of
second and first sound waves at different
pressures
as
well
as
pressure
dependence of nonlinear effects .
We used cylinder wave guide diameter
15-30 mm for experiments with second
sound wave resonator, L=15-70 mm.
Glass tube with diameter 3 mm with
moving heater insert inside the tube was
used in experiments with shock waves
propagation. Small sizes of the heater
allowed us to reach the extremely high
intensity of heat pulses. The power
density in these experiments was up to
250 W/cm2
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
Linear waves
v
t

t

x
0
where u0 is sound velocity
Nonlinear waves
v

 u0
v
 ( u 0   ( P , T ,...) * v )
v
x
0
 may be wave of density (x,t) first sound n(x,t) / s(x,t) =const
 may be wave of temperature
T(x,t) – second sound
n(x,t) + s(x,t) = const
where
u  u   
1
10
1
u 2  u 20   2  T
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1 
where
2 
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u1
4
2V
ST
CV
3
  2V

 P 2




S
  3  S 
 u2


T 
 T  V
8
How will transfer nonlinear waves?

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When we have multiflux
wave (noninteract particles,
surface waves et al.), at
t the wave overturns
and it is appearing the
familiar picture…
9
One flux wave

If we have nonlinear wave of interacting
particles (wave of density, temperature),
motion of knap doesn’t surpass the
movement of lower part of wave profile.
The “gradient catastrophe” leads to
appear wave breakdown or shock wave
Transformation of rectangular
pulse 
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Formation of stationary waves


Competition of nonlinearity
and dissipation formats a
stationary wave. The shape
of wave front conserves.
Mutual balance of
nonlinearity and dissipation
gives equation
u2/2 ~νux
and it is possible to write
relationship for wave
amplitude a
a*Δ/2ν=const
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a
Δ
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Second sound shock waves in
coefficient
superfluid helium  Nonlinear
for first sound
Temperature dependence of second
sound nonlinear coefficient. Right
hump is connected with roton
waves, left one – with phonon’s
temperature excitation

Nonlinear coefficient may be
positive as well as negative for
second sound. It means, the
breakdown appears on front
(2>0) or on backside (for 2<0)
of wave depending from T.
12
1
2
3
4
5
6
7
8
P = 3atm ., U = 7.32V ,  = 10  s.
8
10
5
8
A rb. units.

is positive
7
4
6
6
4
3
2
2
T = 2.103K
T = 2.071K
T = 2.044K
T = 1.999K
T = 1.968K
T = 1.81K
T = 1.783K
T = 1.774K
1
0
-2
-4
0
20
40
60
80
100
T im e,  s.
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Transformation of second sound waves
with its intensity and propagation
Temperature is lower T, nonlinear
coefficient is positive, breakdown
appears on a wave front, which
comes quicker for more high wave
intensity.
50
T = 1 .5 K , L = 2 .5 cm
= 10  s
T im e of pulse,  s
40
L = 2 .5 cm , T = 1 .5 0 K ,  = 1 0  s
30
20
10
7
0
6
a - Q = 2.4 W /cm
5
A m plitude, m K

2
b - Q = 5.2 W /cm
c - Q = 9.3 W /cm
4
0
e - Q = 20.2 W /cm
e
d
c

0
-1
-1 0
0
10
20
30
40
50
60
T im e,  s
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4
5
6
7
2
a
b
3
2
2
1
2
2
d - Q = 14.2 W /cm
3
1
A m p lid u d e, m K
2
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As a result, the recording pulse
duration depends from wave
amplitude. Back side and its
slope of the waves stay the
same.
13
Transformation of second sound waves
with its intensity and propagation (II)
Temperature is higher T, nonlinear
coefficient is negative, breakdown
appears on a wave back side. The
hump moves slower the wave
pedestal. Amplitude and pulse
duration
change
with
wave
propagation while wave’s area
(pulse energy) stays constant.
T = 2 .0 2 K ,  = 1 0  s, Q = 2 0 W /cm
2
3
a - L = 0 .7 cm
b - L = 2 .5 cm
c - L = 4 .4 cm
d - L = 8 .5 cm
a
b
A m plitude, m K
2
c
d
1
0
0
10
20
30
40
50
60
2 ,5
2 ,0
1 ,5
1 ,0
0 ,5
0 ,0
0
2
4
6
8
10
50
40
40
30
E nergy in pulse, m K *  s
3 ,0
D u ration of trian gu lar in  s
T im e,  s
A m plitude, m K

30
20
20
10
10
0
0
2
4
6
8
D ista n ce, cm
D ista n ce, cm
10
0
0
2
4
6
8
10
D istan ce, cm
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3-D geometry pulse
propagation
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Wave propagation in 3-dimensional
 The first term is outgoing wave,
propagated from the origin. The
geometry


Spherical or 3-Dimensional waves
have dependence density, velocity,
temperature, pressure only on the
distance from some point.
The general solution of the wave
equation will be determined by
equation for velocity potential: Δφ(1/c2)  2φ/ t 2=0. In spherical coordinates the Laplacian transforms 
the equation into  
1  
 
2
t
2
c
2
r
2
r
r 
2

r 
We seek a solution in the form
φ=f(r,t)/r. Substitution gives us the 
ordinary
one-dimensional,
and
general solution of the spherical
equation is
f ( ct  r )
f ( ct  r )
 

1
r
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2
r
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second term is a wave coming to the
centre. A spherical wave has an
amplitude
which
decreases
inversely as a distance from the
centre. The intensity in the wave
falls as the square of the distance.
And the total energy flux in the wave
is distributed over a surface whose
area increases as r2.
A
monochromatic
stationary
spherical wave is of the form
where k=/c
 i  t sin kr
  Ae
r
An
outgoing
monochromatic
spherical wave is given by
i ( kr   t )
and φ=0 before and
e
  A
after passage of wave.
r
16
Second Sound Wave 

in 3-D geometry
The solution of this system is
(r , t )  
a
 (' t )
u 20 C
r
 u 't 
u  
 (' t )  exp   20   d  q ( ) exp  20 
a 

 a 
't

The linearized equation of motion for
the second sound can be derived
from general equations of two-fluids
hydrodynamics. We can introduce a
wave potential (r,t) by the relation
p/S=  , where p is a momentum of
a unit mass of a helium. The physical
quantities can be expressed via the
potential by the following way
T  

t
v s    
vn 
s
n

  Tdt


t
0

 u 20    0
2
and boundary condition as q=TSvn, or

 n
where q is function of
|r  a 
q
,
2
r
TS
time and a – radius of
heater.
9 December 2005
The variation of pressure in the
first sound wave P=-/t gives
the similar result 
  Pdt
2
2
0

Then wave equation is rewritten as
 
where the ‘retarded’ time ‘t is
introduced ‘t=t-(r-a)/u20
The variation of temperature in the
wave T=-/t and if we integrate
T over all time at any r, the result

is zero

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This means that, the both waves
of condensation and rarefaction
must be observed in spherical
geometry.
17

In
accordance
with
last
relationships the waves of
compressions and rarefaction,
waves of heating and cooling
must appear in 3-dimensional
geometry at distance more
longer the size of heater. The
nonlinear wave velocity forms
the breakdown in front or back
side of pulses in dependence of
the sign of nonlinear coefficient.
9 December 2005
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A m p litu d e , m V
Experimental study of
cooling second sound
waves
T = 1.50 K , L = 5.8 cm
 = 10  s
10
d q /d t= 16 W /cm
2
8
6
0
150
100
50
11
200
T = 2.02 K , L = 5.8 cm
 = 10  s
10
d q /d t= 23 W /cm
2
9
8
7
6
0
20
40
60
80
100
T im e ,  s
18
Propagation of long pulses in 3-D
geometry
One can see, that the emission of
a heat in superfluid helium in three
dimensional geometry leads to the
propagation of wave of heating
followed by the wave of cooling
(bipolar pulse). The sharp front of a
heating pulse relates to the
differentiation of the step function
q(t) in accordance with equation
T  

The amplitude of the wave during
the propagation decreases as 1/t.
The broadening of breakdown b of
the temperature pulse is b  a/c2.
This time b coincides with the time
during which the second sound
travels the radius of the heater a.
T = 1 .5 0 K , L = 4 .7 cm
d q /d t= 4 W /cm
2
22
= 50  s

20
t
18
Am p litu d e , m V

16
= 30  s
14
12
10
= 16 s
8
6
0
50
100
150
T im e ,  s
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Propagation of pulses in mix geometry:
3-D 1-D
Most curious situation appears in
case when we launch spherical
wave in long thin capillary. At
temperature near T, where
nonlinear coefficient is negative,
the two sign wave with
breakdown in a middle formats at
distance between heater and
capillary
tip.
This
wave
propagates along the capillary
without changing of it’s longitude.
It is the possibility to format heat
wave
finite
longitude
at
temperature near T.
9 December 2005
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0 ,3
3 -D ---> C a p illa ry
T = 2 .0 2 K ,  = 1 0  s
c
0 ,2
Am plitude, m K

b
0 ,1
a
0 ,0
-0 ,1
a - Q = 2 .4 W /cm
b - Q = 9 .2 W /cm
-0 ,2
c - Q = 2 0 W /cm
20
30
40
50
60
70
80
2
2
2
90
T im e,  s
20
Stationary nonlinear second
sound waves.
Energy transformation in Superfluid
Helium-4
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Pulse wave in resonator

We applied short rectangular pulse to heater and observed
reflected signals
 Time of signal recording is defined as τ=[2*(N-1)+1]*L/c0
Launched heat pulse
3
4
5
6
7
8
9
2
1
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1 ms
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δT
x
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δT
δT
x
x
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δT
δT
x
x
δT
x
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δT
δT
x
x
δT
δT
x
x
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δT
δT
x
x
δT
?
x
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δT
δT
x
x
δT
?
x
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Sine wave in resonator
B
10
500
L=70 mm,
T=2.09 K,
2
Q/S=1 mW/cm
400
300
5
200
Amplitude

We applied harmonic
signal to heater
U~sin(*t),
P~sin2(*t)=1+sin(2*t),
Resonant frequency
fG=c0/4L
f, Hz

100
0
0
2
4
6
8
10
0
12
N Resonance
ω~k
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29
Energy transformation into high
frequency modes.
?
FNT, 1988




High quality of resonator Q~100-1000;
Experimental observed energy transformation into
higher modes;
Model system, you know Q(f), nonlinear coefficient
change drastically from +1 to - 
Frequency dependence of energy transformation A(f)?
 Influence of perturbation processes (increasing of
vortex density by pulse heating) on energy
9 December 2005 transformation?
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Experimental studies of one dimensional
nonlinear second sound waves in the
cylindrical cell
12
0,010
Applied signal A=A0*sin(ωt)
6
Temperature in second
sound wave
A, arb. units
0,005
0
0,000
-6
-0,005
-12
0,001
0,002
t, ms
Fast Fourier
Transformation
AH~f
0,01
-3.2
or Pn~
1E-3
Amplitude
Spectrum of temperature
oscillations of the nonlinear
second sound waves in a
resonator
1E-4
1E-5
100
9 December 2005
-1.62
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1000
10000
Frequency (Hz)
31
Multiple harmonics appear
with
increasing
the
amplitude of second sound
waves.
Evolution of shape of the stable wave in the
resonator with increasing the power of generator
T=1.767 K, 11 Resonance, f~739 Hz
0,1
1000
0,01
1E-3
1E-5
800
UG=1.63 V
100
1000
UB, mV
1E-4
10000
0,1
0,01
1E-5
600
400
200
1E-3
1E-4
T=1.767 K, 11 Resonance, f~739 Hz
A2  b * U G
UG=1.32 V
100
1000
0
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
2*UG, V
10000
0,1
0,01
1E-3
1E-4
UG=0.54 V
30
1E-5
1000
10000
20
1/2
UB
Y Axis Title
0,1
1E-3
2
25
100
0,01
Q~6 mW/cm
UG=0.17 V
15
10
1E-4
5
1E-5
0
0,0
100
1000
10000
0,5
1,0
1,5
2,0
2,5
3,0
3,5
2*UG, V
X Axis Title
9 December 2005
T=1.767 K, 11 Resonance, f~739 Hz
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32
Formation of harmonics at different
conditions.
UG=0.93V*sin(t)
0,1
0,01
f=758.5 Hz
11 Resonance
T= + 1.8 mK
 Stable waves in resonator at
temperature T near 1.88 K, where
the nonlinear coefficient changes its
sign.
1E-3
1E-4
1000
0,1
10000
T= + 0.6 mK
0,01
1E-3
6
1E-4
1000
4
10000
0,1
-1
T0=1.88 K
 ,K
0,01
2
1E-3
1 5 atm
1000
10000
0,1
A, mV
2 5 atm
-2
S .V .P .
-4
1E-4
0,01
0
1 0 atm
-6
r
5 atm
1 ,0
T= - 0.6 mK
1 ,2
1 ,4
1 ,6
1 ,8
2 ,0
2 ,2
T, K
1E-3
1E-4
1000
10000
f, Hz
9 December 2005
Pressure dependence of the nonlinear
coefficient of the second sound
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33
Energy transfer in k-space over scales
Kolmogorov, Obukhov,
1941,
Log Ek
INERTIAL
RANGE
In the inertial range
~km
Ek
Ek~km
For the turbulence of
incompressible fluid
Log K
Energy flux
DAMPING
PUMPING
m= -5/3
Nonlinear dynamics of 1-D second sound
waves: Results of numerical simulations
The effective scaling exponent
s(n)=d log P(n) / d log n
T=2.096 K
AH~f
UG=4.5 V
UG=3 V
AH~f
-2.0
AH~f
-3.3
0,01
1E-3
Amplitude
Evolution of the calculated steady-state
spectrum P(n) of 2-nd sound oscillations in
a resonator with increasing the amplitude
of the pumping signal. The driving
frequency is equal to the lowest resonant
frequency of the resonator. Effective
amplitude of the driving force r=0.001, 0.1,
and 1.
-1.8
1E-4
UG=2 V
1E-5
1E-6
100
1000
10000
Frequency, Hz
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35
CONCLUSIONS



Experiments with the waves in the bulk of superfluid helium
present a unique possibility to study nonlinear and turbulent
behaviour in a system of interacting waves.
At high amplitudes of the driving force (or high quality of
resonator) a sine second sound wave transforms into periodic
one with many multiple harmonics. At high frequencies the
power-like spectrum is violated due to transition from the
nonlinear wave regime of the energy transfer to the viscous
damping.
Experimental studies of turbulence of the 2-nd sound waves
showed the dependence of power-like spectrum from the value
of the nonlinear coefficient and amplitude of second sound
waves.
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36
Nonstationary energy
transformation in resonator.
Formation and decay spectrum.
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37
Formation of
Kolmogorov
spectrum
FFT spectrum
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38
A=t
A=t
0.77
Harmonics
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1/2
Amplitude, a.u.
0,01
Change of regime, t=0.17 s
1E-3
1E-4
Switch on of Sine
Change of regime
0,01
Amplitude, a.u.
1E-5
N of harmonics
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1E-3
1E-4
0,1
1
Time, s
1E-5
-4,2
-4,0
-3,8
-3,6
-3,4
-3,2
Time, s
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Self-Similar Formation of Second
Sound Turbulence
Dependence of difference of times
(t0 - tf) on the frequency wf plotted for the
driving amplitudes f0=0.01 (squares)
and 1.0 (circles),
in logarithmic scale. The straight lines
correspond to the self-similar
dependence with the exponent s=5.
f (t)= (t0 - t)-s
where f (t) is the boundary frequency, at time t=tf.
Computer simulation
Decay of nonlinear energy spectrum
Amplitude
0,01
1E-3
1E-4
1E-5
b)
100
1000
10000
Frequency (Hz)
9 December 2005
0,01
Amplitude
 We applied harmonic (~sin(ωt))
signal from generator to heater in
cylinder resonator.
 After formation the nonlinear
wave spectrum (a) we switched off
the pumping signal and have
observed transformation of the
harmonics with time (b).
a)
1E-3
1E-4
1E-5
100
1000
10000
Frequency (Hz)
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9 December 2005
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We found in Fourier Transformation
of recording signal change in
behavior of harmonic amplitude:
B1
C2
D3
E4
F5
G6
H7
I8
J9
K10
L11
M12
N13
O14
Change of regime
Amplitude, a.u.
0,01
A=a0*exp(t/5.2)
1E-3
1E-4
 Stop of energy pumping in the
resonator drastically reduces multiple
harmonics. The amplitudes of harmonics
chaotically derange. Typically time of
chaotic energy random walk is order of
hundreds wave periods.
 After 2 seconds the amplitude of the
main (first) harmonic reduces with slower
rate and can be described by an exponent
dependence which correspond to the
quality of resonator about 5000.
Chaos in energy spectrum
1E-3
Amplitude, a.u.
Chaos in harmonics
after switch off the
pump
1E-4
1E-5
-8
0
8
Time, s
0
1
Time, s
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Conclusions

The equation of nonlinear second sound wave in
resonator
utt- (с0 +α u )2uxx=-ν uxxx at 0<x<L
utt- (с0 +α u )2uxx=-ν uxxx + F(t)
at x=0,
where F(t)=A*sin (w*t) or =0
ux=0 at x=0, x=L
 Formation of kolmogorov spectrum – energy
transformation from low to high frequencies.
 Decay of nonlinear harmonic spectrum can be
explained by chaotic energy transformation.
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Disturbance in superfluid helium.
L = 5 .8 c m ,
T = 1 .8 8 K ,  = 5 0  s
d q /d t= 6 .4 W /c m
2

60
55
d q /d t= 1 4 W /c m

2

50
d q /d t= 2 0 W /c m
45
2
A m p litu d e
40
35
d q /d t= 2 6 W /c m
2
30
25
d q /d t= 3 2 W /c m
2
20
15
d q /d t= 4 0 W /c m
10
2
5
0
20
40
60
80
100
120

Measurements in 3-D geometry;
Long pulse;
Bipolar pulse (pulse of heating,
undisturbed medium – normal
and superfluid components
counter-flow, pulse of cooling)
forms in helium at distance >>
size of heater;
Intensive heat pulse counter-flow
increases the vortex tangle and
relaxation in medium, second
pulse significant decreases
T im e,  s
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Conclusions
 The
superfluid helium is perfect model
medium for study nonlinear effects, energy
transformation and interaction of waves of
different nature – wave of first and second
sounds.
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Amplitude of FFT
Set 1, 0
UG=10V, fG=194.2 Hz
1E-4
first resonance
T=2.053 K
1E-5
Signal of Bolometer, mV
Properties of resonator
120
100
80
60
40
20
0
518,8
T=2.08 K, 11 Resonance
519,0
519,2
519,4
519,6
fG, Hz
1E-6
1E-7
100
0,0010
1000
10000
f, Hz
Temperature in SS wave
0,0005

0,0000
-0,0005
-0,0010
-0,0015
-0,010
Y Axis Title
6
-0,005
0,000
0,005
0,010
Signal of Generator

4
2
0

-2
-4
-6
-0,010
-0,005
0,000
0,005
Time, s
9 December 2005
0,010
We applied sine signal for heater.
FFT indicates absence of any
higher harmonics in it.
We recorded the signal of
bolometer
The recording signal has higher
frequency, it reduces electric
influence of applied signal.
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One direction waves
Wave equations
Linear wave
ut+c0ux=0
nonlinearity
dissipation
ut+c0ux= νuxx
ut+(c0+αu)ux=0
In moving system x’=x-c0t
ut=0
ut+uux=0
ut= νuxx
Two direction waves
9 December 2005
ut+(c0+αu)ux=νuxx
ut+ uux = νuxx
utt- с02uxx=-νuxxx
utt- (с0 -α u )2uxx=0
utt- с20 uxx - (u ux)x=0
Utt-c02uxx=0
Nonlinearity
and dissipation
Nonlinearity
and dissipation
utt- (с0 -α u )2uxx=-ν uxxx
utt- с20 uxx - (u ux)x =-νuxxx
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Change of sine signal
shape
α<0
Temperature in SS wave
T=2.039 K, 2<0,
UG=0.9 V*sin, f=565.2 Hz
0,04
0,03
2
AB, mV
0,02
0,01
0,00
-0,01
0
-0,02
-0,03
-0,04
-0,0020
-0,0015
-0,0010
-0,0005
0,0000
α<0
Time, s
Signal of Generator

We apply harmonic signal to heater  We got breakdown on
back side of sine wave.
U~Asin(*t),
P~A2sin2(*t)=A2[1+sin(2*t)],
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