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Submitted on 1 Jan 1978
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AN EXPERIMENT WHICH MEASURES RESONANT
MODE CONVERSION IN He II
S. Garrett, S. Adams, S. Putterman, I. Rudnick
To cite this version:
JOURNAL DE PHYSIQUE Collogue C6, supplement au n° 8, Tome 39, aout 1978, page C6-228
AN EXPERIMENT WHICH MEASURES RESONANT MODE CONVERSION IN He II*
S. Garrett, S. Adams, S. Putterman and I. Rudnick
Department of Physios,, University of California, Los Angeles, California 90024 - U.S.A.
Résumé.- Nous décrivons les résultats d'une expérience destinée à l'observation de la conversion non-linéaire du deuxième son en premier à l'intérieur d'un guide d'onde. Le terme non-linéaire qui domine le couplage entre les deux modes provient de l'invariant Galiléen (v" -v' ) . Les mesures montrent que la conversion résonnante a lieu exactement a la fréquence prévue. Des procédés origi-naux ont été utilisés pour obtenir l'étalonnage absolu des transducteurs du premier son et, à l'ai-de l'ai-de ces étalonnages, nous trouvons que l'amplitul'ai-de du mol'ai-de converti est en accord avec sa valeur théorique.
Abstract.- The results of an experiment to observe the non-linear conversion of second sound to first sound within a waveguide are described. The non-linear term which dominates the coupling between the two modes arises from the Galilean invariant (v -v )2. Measurements show that the re-sonant mode conversion occurs precisely at the predicted frequency. Unique procedures were used to get absolute calibration of the first sound transducers and using these calibrations we find the amplitude of the mode converted first sound has its theoretically predicted value.
When wavelike solutions to the two-fluid equations for He II are sought retaining terms which are second order in displacements from equi-librium, the wave equations for the second order pressure or temperature variations in the superfluid are not homogenous but have driving terms which are proportional to quadratic combinations of the first order quantities which characterize ordinary first and second sound / ] / . In the linear approximation these modes are uncoupled. In general these driving terms do not lead to the generation of propagating (first or second) sound waves unless the phase velo-city of the driving term is equal to the speed of first or second sound.
In the region of interaction of two second sound waves, the square of the sum of the waves will have components with phase velocities other than u and for the appropriate angle of intersection can be made equal to u . For two second sound waves o fequal frequency, W, and equal amplitude, traveling in different directions indicated by their wave vec-tors, k and k', the condition that the phase velo-city, u ,, of the component at 2w be that of first sound can be expressed as u - u = 2w /|k + k| or cos (9/2) = u /u where 8 is the angle between k and k'. This resonance condition leads to the ge-neration of a propagating first sound wave at a frequency 2 Hi, whose amplitude |P | grows linearly
with x in the region of interaction where x is cho-sen as the direction which bisects 8.
In order to exercise precise control over the intersection angle, 6, of the two second sound waves and to achieve an interaction region of maxi-mum length, the experiment to detect resonant mode conversion was performed in a waveguide of rectan-gular cross-section (1.467 ± 0.0008 cm x 0.474 cm at helium temperature) which was wound into a spiral of length 150 cm as shown schematically in figure I where all dimensions are in cm at room temperature.
The bottom and two sides of the waveguide are a groove machined into a single block of aluminium. The top-plate containing the first and second sound transducers is bolted to the spiral to complete the waveguide. Thewaveguide is operated in the lowest frequency non-planewave mode with the cut-off fre-* Supported in part by Contracts ONR
N00014-76-C-0246 and NSF DMR 76-22306
quency g i v e n by
w
= nu-/R, whereII
i s t h e d e p t h COo f t h e waveguide. Second sound p r o p a g a t i n g i n t h i s mode i s e q u i v a l e n t t o two i n t e r s e c t i n g planewaves whose a n g l e of i n t e r s e c t i o n i s given by cos (8/2) =
(1
-
(wco/w')
'I2.
The frequency of t h e mode conver- t e d f i r s t sound,
wmc,
i s twice t h e frequency of t h e second sound. I f we l e t umc/2 = uco+
h, t h e nh / w c o
5 u ; / ~ u :=
0.35 %. The non-planewave mode i ss e l e c t i v e l y e x i c t e d by a shaped d i p o l e source.
F i g u r e 2 shows raw f i r s t and second sound d a t a taken a t 1.31 K. The second sound t r a n s d u c e r
fa+ Cmur2 Frequency (HI)
used e x t r a p o l a t e d v a l u e s of U,. There a r e no adjus- t a b l e parameters.
Absolute c a l i b r a t i o n of microphones was made u s i n g a new a p p l i c a t i o n of t h e r e c i p r o c i t y p r i n c i p l e t h a t
w i l l
be d e s c r i b e d elsewhere / 3 / . Absolute ma- g n i t u d e ~ have o n l y been measured f o r s i x p o i n t s between 1.26 K and 1.41 K a t t h i s time b u t t h e y a r e i n agreement with t h e two-fluid model t o w i t h i n 16 % on t h e average. The accuracy and temperature range of t h e s e a b s o l u t e measurements a r e c u r r e n t l y b e i n g improved and extended i n t h e hope t h a t a p r e c i s e d e t e r m i n a t i o n of t h e coupling c o n s t a n t f o r thismode c o n v e r s i o n p r o c e s s w i l l lead t o a c c u r a t e determina- t i o n s o f the dependence of t h e f l u i d d e n s i t y on t h e s q u a r e of t h e counter-flow v e l o c i t y .References i s 7 . 4 cm downstream from t h e d i p o l a r source. The
cut-off i s n o t i n f i n i t e l y s t e e p because the group / l / P u t t e m a n , S. and G a r r e t t ,
S., J. Low Temp. v e l o c i t y i s zero a t cut-off and Lhere a r e v i s c o u s Phys.
2
(1977) 543l o s s e s a t t h e waveguide w a l l s . The f i n i t e v a l u e of / 2 / Maynard, J., Phys. Rev.
P14
(1976) 3868t h e second sound s i g n a l a t cut-off i s due t o t h e /3/ Rudnick, I . , J. Acoust. Soc. Am.
2
(May 1978) e x p o n e n t i a l l ~ decaying t a i l which p e n e t r a t e s t h ewaveguide by i n c r e a s i n g numbers of waveguide d e p t h s a s cut-off i s approached from below. The f i r s t sound s i g n a l g e n e r a t e d by mode conversion i s super- imposed on t h e second sound d a t a . The f u l l width of t h e resonance a t t h e h a l f power p o i n t s i s 3.6
HZ
corresponding t o a
