The effects of gravity in second-sound

Physica B 1658~166 North-Holland

THE

EFFECTS

(1990)

565-566

OF GRAVITY IN SECOND-SOUND

T. C. P. CHUI and D. R. SWANSON Physics Department, Stanford University, Stanford, California 94305 U.S.A.

We present a modified wave equation for second-sound propagation in an inhomogeneous medium. For extreme inhomogeneity we show that a new mode exists at low frequency, in which normal wavelike propagation is not possible. The nature of this mode is discussed.

1. INTRODUCTION The gravitational inhomogeneity near TX can be easily understood. The pressure in a fluid varies with height due to gravity, while TX is depressed at higher pressure. Thus it can be shown (1) that TX(Z) = TX(O) + Az, where A = 1.27~10~~ K/cm and z is the vertical height. Near TX the superfluid density, p,(z,T) = 2.4 p E(z,T#‘~~, where p is the density and E (z,T) = 1 - T/Ix(z).

Combining the

expressions for ps and E gives: p,(z,T) = 2.4 p [ Aznh(O) + &(O,T)l*674. (1) Recent developments of high resolution thermometers (2) and second-sound detectors (3) enable the measurements of second-sound (4) to E -lo-*. At these temperatures, ps is highly inhomogeneous. Using two-fluid hydrodynamic theory, we rederived the wave equation for second-sound propagation taking into account the effect of inhomogeneity. For small amplitude excitations, we obtain:

which can be approximated to within 10% by T=(V@)VT+U2V%,

(2)

where the local second-sound velocity is 0921-4526/90/$03.50

@ 1990 - El sevier Science

U(z,T) = [S2 ps(z,T) T / pn C(Z,T)I~‘~,

(3)

C and S are the heat capacity and the entropy. 2. THIN CELL APPROXIMATION Equation 2 is highly non-linear, and can only be solved numericaIly. However, it is intuitive to make certain assumptions to simplify this equation to a point that an exact solution can be found. We assume that the experimental cell is thin, so that U does not vary much inside, and VU can be approximated by a constant. We seek a solution of the type T(z,t) - e i(kz-ot). The solution is given by: k=-aifp (4) where a=VU/U (5) p = (o/v) JiZ5+ and (6) Depending on (VU /co), p can be real or imaginary. Therefore, two distinct types of solutions exist. The usual propagative type corresponds to the case of real /3. The case of imaginary p is a new mode in which wave-like propagation is not possible. 3. WAVE-LIKE MODE For real p, the solution is a traveling wave with an amplitude that diverges in the -z direction and decays in the +z direction. In a resonator, the resonant frequency can be found by solving the Eigenvalue problem with the boundary condition VT = 0 (insulating walls). We obtain: w=m (7) where o. is the resonant angular frequency in the

Publishers

B.V. (North-Holland)

T.C.P. Chui, D.R. Swanson

566

absence of the VU term. Thus the effect of the VU term is to increase the resonant frequency.

quantity A = TX&/A is the distance from the normal-

4. NON-WAVE-LIKE MODE

This table, shows that the effects caused by VU should be observable. For example, if a generator is located at the bottom of a cell and a detector is placed 1.7 cm above, then o, - 5 rad./sec., when the bottom is near TX.

When p is imaginary, T(z,t) - e -z/8 - iot where, 6+,_ =yp

[l**JI-(o/w)21

(8) S+ and 6_ are decay lengths. In this case k is putely imaginary, and there is no phase shift between a sound generator and a detector at different locations. This situation would appear to violate causality, since a finite time is required for a signal to travel. To understand this, we consider the case of a sinewave second-sound generator being activated at t = 0. The driving signal can be written as the product of a step function at t = 0 and a sine wave. The step function contains Fourier components of high 6.1,therefore the signal would propagate in the wave-like region until a steady state is reached in which no phase shift exists between the generator and the detector. Since this is only a steady state phenomenon, causality is not violated. Figure 1 shows the expected steady state phase difference as a function of o. Below the critical frequency of oc = VU, no phase shift exists. Also shown is the usual propagative mode in which the phase shift approaches zero as o -> 0.

e ..4

non-wave-like

G; # a”

Figure 1. Phase shift between a second-sound generator and a detector. 5. ORDER OF MAGNITUDE ESTIMATION We estimate the order of magnitude of the various quantities discussed above using U - 46.28 E .387(rn/sec) (5). Table 1 shows the quantities VU (or ccc), and the frequency shift of a .3 cm resonator caused by the VU term (eq. 7), as a function of E. Assuming an infmite column of helium, the

superfluid interface when a fluid element is at

E.

Table 1 Estimation of the order of magnitude of several relevant quantities. E

vu-WC (

lE-3 lE-4 lE-5 lE-6 lE-7 lE-8 lE-9 lE-10 lE-11

WJbYo,

A

set-1)

(.3cm cell)

(cm)

7.21E-2 2.96E-1 1.21E+O 4.98E+O 2.04Ecl 8.38E+l 3.44E+2 1.41E+3 5.78E+3

2.32E-10 2.32E-08 2.32E-06 2.32E-04 2.30E-02 1.38E+OO 2.06E+Ol 2.15E+02 2.16E+O3

1.71E+03 1.71E+02 1.71E+Ol 1.71E+OO 1.71E-01 1.71E-02 1.71E-03 1.71E-04 1.71E-05

ACKNOWLEDGMENTS We thank Q. Li, J. A. Lipa and M. J. Adriaans for helpful comments, and NASA for its support with contract JPL 957448. REFERENCES (1) G. Ahlers, Phys. Rev. 171 (1968) 275 . (2) T. C. P. Chui and J. A. Lipa, Proc. of LT-17, eds. U. Eckern, A. S&mid, W. Weber, and H. Wurhl, (North-Holland, Amsterdam, 1984) P.93 1, (3) T. C. P. Chui and D. Marek, J. Low Temp. Phys. 73 (1988) 161. D. Marek, J. A. Lipa and D. Philips, Phys. (4) Rev. B 38 (1988) 4465 . (5) G. Ahlers, The Physics of Liquid and Solid