Optical observations of second sound resonances in helium II

Volume 46A, number 5

PHYSICS LETTERS

OPTICAL OBSERVATIONS

14 January 1974

OF SECOND SOUND RESONANCES

IN HELIUM II

A.C. M O T A * , J.L. O L S E N , R.S. SORBELLO and V.S. TOMAR

EidgenOssische Technische Hochschule Zfirich, Laboratorium far FestkfSrperphysik, Zfirich, Switzerland

Received 13 November 1973 The resonance frequencies of second sound excited in a cylindrical cavity by axial heating are observed optically. The frequency distribution is compared with that resulting from Rayleigh's solution for sound waves in a similar geometry.

The wave equation for second sound is well known. A set of standing wave solutions for the temperature T in the cylindrical case is given by

Patterns of standing waves of second sound may be seen by using a Schlieren method to show the deformations of a free helium surface caused by the relative motion of the normal and superfluid components of the liquid [1, 2]. We have used this method to find resonance frequencies of second sound in a cylindrical cavity. The observed frequency distribution is similar to that calculated by Lord Rayleigh [3] for aerial vibrations confined in a rigid cylindrical envelope. In our experiment the axis of the cavity was vertical and its outer wall was made of plexiglass. The cavity was closed above and below by glass windows, and could be filled partially with liquid helium. Thermal waves were generated by a.c. electrical heating of a wire along the axis. Two small holes in the side of the cavity allowed normal fluid to escape and be replaced by superfluid.

T = T O + T 1 exp (iwt) Jn(z) cos

Here T O is the mean temperature, T 1 a constant determining the amplitude of the thermal wave, 60 its angular frequency, z = cor/c with r the distance from the axis and c the velocity, n is an integer and 0 is the azimuthal angle. Jn is the Bessel function of the first kind of order n. For our case there is almost zero thermal transport across the boundary and the boundary conditions are easily seen to be

aT/Or= 0

with

z=

wR/c.

Some solutions of this equation were tabulated by

1

.o

10

10 2

!

w

I

[

I

I

I I

II I

Z

I I I II IIIIIIIIII I I I IIIIIIIIIIII I I I I i llllllllll

I

I

I

n,O

'

r =R,

a [Jn(z)]/az = 0

J~(z)

n.1

at

where R is the radius of the cavity. For our resonance condition this yields

*On leave of absence from University of California, San Diego, La Jolla, California.

n-2

nO.

I II I i I t111 Ilillllllllilll Itl I

!

f(Hz)

10 3

observed

frequencies

Fig. 1. Observed second sound resonance frequencies and solutions of a [Jn(z)/az = 0. The position of the frequency scale cormponds to a velocity of 20.25 ms -1 . 343

Volume 46A, number 5

PHYSICS LETTERS

Rayleigh [3] and further solutions are easily available. On varying the heater frequency, standing wave patterns are observed at a range of well defined frequencies. The sharpness of the resonances corresponds to a value of Q/> 200 for the cavity. In fig. 1 we show the values of z for various n required by Rayleigh for resonance. Below this we show observed resonance frequencies. The frequency scale is placed to correspond with the second sound velocity of 20.25 ms -1 at the average temperature of measurement. It should be noted that the resonances with n = 0 and n = 1 were the strongest. We found many resonances with frequencies above 10 000 Hz, but these are not shown in the figure because they lie too close to be resolved easily on the scale used. The interpretation of the observed patterns can be based upon simple Bernouilly considerations for the surface height. More sophisticated calculations by Lukosz [4] and by Rudnick [5] will be discussed by one of us [6] in a separate note.

344

14 January 1974

We conclude that second sound resonances are being observed optically. A more detailed account of this work is in preparation. The authors are grateful to Mr. P. Caminada for his devoted help with these experiments. The work was financed in part by grants from the Schweizerischer Nationalfonds zur F6rderung der wissenschaftlichen Forschung and from the VolkartStiftung.

References [1] J.L. Olsen, Helv. Phys. Acta 46 (1973) 35. [2] J.L. Olsen, Physica 69 (1973) 136. [3] J.W.S. Rayleigh, The theory of sound, Vol. II, § 339 (e.g. Dover, New York). [4] W. Lukosz, J. Low Temp. Phys. 1 (1972) 407. [5] I. Rudnick, private communication. [6] R.S. Sorbello, to be published.