Experimental Study of Non-Linear Second Sound Waves in He-II Nonlinear second sound waves in superfluid helium Hydrodynamic Equations of Ideal Fluids Two Fluid Hydrodynamic Equations Sounds in Superfluid Helium
Byers, Ken, Operation Director has reference to this Academic Journal, PHwiki organized this Journal Experimental Study of Non-Linear Second Sound Waves in He-II Institute of Solid State Physics RAS, Chernogolovka, Russia; Lancaster University, Lancaster, UK Victor Efimov 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 Hydrodynamic Equations of Ideal Fluids For any an ideal fluid (incompressible, without energy dissipation) it may be written the equation of continuity (mass conservation) where mass flux density the equation of motion Newtons second law the equation of adiabatic motion of an ideal fluid the equation of entropy continuity where entropy flux
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Two Fluid Hydrodynamic Equations In superfluid state helium behaves as if it were a mixture of two different fluids. Superfluid moves with zero viscosity in addition to the other is a normal viscosity fluid. Then density is =n+s, in addition to mass flux is In superfluid state only normal fluid has entropy. The hydrodynamic equations trans as long as m into mass conservation Eulers equation isentropic motion in addition to add equation describing the as long as ces in superfluid state Sounds in Superfluid Helium The differentiation of two equations with respect to time in addition to substituting into another relationships we obtain as long as small disturbance (linear waves) Solutions are wave of density (co-moving motion normal in addition to superfluid component) waves of temperature or entropy (countermovement motion normal in addition to superfluid components, =n+s=const Experimental device The experimental technique used as long as study second sound is shown on sketch. For heater in addition to 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.
The time resolution of heat pulses by heater in addition to bolometer was better 0.3 s. The sensitivity of registration pulse by temperature was less 10-6 K in addition to defined by noise levels. The high pressure chamber allowed us to study the processes of generation of second in addition to first sound waves at different pressures as well as pressure dependence of nonlinear effects . We used cylinder wave guide diameter 15-30 mm as long as 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 Nonlinear waves 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 where Linear waves where u0 is sound velocity How will transfer nonlinear waves When we have multiflux wave (noninteract particles, surface waves et al.), at t®µ the wave overturns in addition to it is appearing the familiar picture
One flux wave If we have nonlinear wave of interacting particles (wave of density, temperature), motion of knap doesnt surpass the movement of lower part of wave profile. The gradient catastrophe leads to appear wave breakdown or shock wave Trans as long as mation of rectangular pulse ð Formation of stationary waves Competition of nonlinearity in addition to dissipation as long as mats a stationary wave. The shape of wave front conserves. Mutual balance of nonlinearity in addition to dissipation gives equation u2/2 ~ux in addition to it is possible to write relationship as long as wave amplitude a a/2=const a Second sound shock waves in superfluid helium Temperature dependence of second sound nonlinear coefficient. Right hump is connected with roton waves, left one with phonons temperature excitation Nonlinear coefficient is positive as long as first sound Nonlinear coefficient may be positive as well as negative as long as second sound. It means, the breakdown appears on front (a2>0) or on backside ( as long as a2<0) of wave depending from T. Trans as long as mation of second sound waves with its intensity in addition to propagation Temperature is lower T, nonlinear coefficient is positive, breakdown appears on a wave front, which comes quicker as long as more high wave intensity. As a result, the recording pulse duration depends from wave amplitude. Back side in addition to its slope of the waves stay the same. Trans as long as mation of second sound waves with its intensity in addition to 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 in addition to pulse duration change with wave propagation while waves area (pulse energy) stays constant. 3-D geometry pulse propagation Wave propagation in 3-dimensional 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 as long as velocity potential: -(1/c2) 2/ t 2=0. In spherical co-ordinates the Laplacian trans as long as ms the equation into We seek a solution in the as long as m =f(r,t)/r. Substitution gives us the ordinary one-dimensional, in addition to general solution of the spherical equation is The first term is outgoing wave, propagated from the origin. The 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 as long as m where k=/c An outgoing monochromatic spherical wave is given by in addition to =0 be as long as e in addition to after passage of wave. Second Sound Wave in 3-D geometry The linearized equation of motion as long as 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 Then wave equation is rewritten as in addition to boundary condition as q=TSvn, or where q is function of time in addition to a radius of heater. The solution of this system is where the retarded time t is introduced t=t-(r-a)/u20 The variation of temperature in the wave T=-/t in addition to if we integrate T over all time at any r, the result is zero The variation of pressure in the first sound wave P=-/t gives the similar result This means that, the both waves of condensation in addition to rarefaction must be observed in spherical geometry. Experimental study of cooling second sound waves In accordance with last relationships the waves of compressions in addition to rarefaction, waves of heating in addition to cooling must appear in 3-dimensional geometry at distance more longer the size of heater. The nonlinear wave velocity as long as ms the breakdown in front or back side of pulses in dependence of the sign of nonlinear coefficient. 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 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. Propagation of pulses in mix geometry: 3-D ð1-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 as long as mats at distance between heater in addition to capillary tip. This wave propagates along the capillary without changing of its longitude. It is the possibility to as long as mat heat wave finite longitude at temperature near T. Stationary nonlinear second sound waves. Energy trans as long as mation in Superfluid Helium-4 Pulse wave in resonator We applied short rectangular pulse to heater in addition to observed reflected signals Time of signal recording is defined as =[2(N-1)+1]L/c0
T x T x
Sine wave in resonator We applied harmonic signal to heater U~sin(wt), P~sin2(wt)=1+sin(2wt), Resonant frequency fG=c0/4L ~ k Energy trans as long as mation into high frequency modes. High quality of resonator Q~100-1000; Experimental observed energy trans as long as mation into higher modes; Model system, you know Q(f), nonlinear coefficient change drastically from +1 to – ¥; Frequency dependence of energy trans as long as mation – A(f) Influence of perturbation processes (increasing of vortex density by pulse heating) on energy trans as long as mation FNT, 1988
Wave equations One direction waves Linear wave nonlinearity dissipation Nonlinearity in addition to dissipation Two direction waves ut+c0ux=0 ut+(c0+u)ux=0 Utt-c02uxx=0 utt- (0 – u )2uxx=0 utt- 20 uxx – (u ux)x=0 ut+ uux = uxx utt- (0 – u )2uxx=- uxxx utt- 20 uxx – (u ux)x =-uxxx ut+c0ux= uxx utt- 02uxx=-uxxx ut=0 ut+uux=0 ut+(c0+u)ux=uxx ut= uxx In moving system x=x-c0t Nonlinearity in addition to dissipation Change of sine signal shape We apply harmonic signal to heater U~Asin(wt), P~A2sin2(wt)=A2[1+sin(2wt)], Signal of Generator Temperature in SS wave <0 <0 We got breakdown on back side of sine wave.
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