HAL Id: jpa-00217547
https://hal.archives-ouvertes.fr/jpa-00217547
Submitted on 1 Jan 1978
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
SURFACE DEFORMATION AND SURFACE
MOTION DUE TO STANDING SECOND SOUND
WAVES IN HELIUM I I
S. Eckstein, Y. Eckstein, J. Olsen, H. Sigg
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C6, suppl6ment au no 8,
Tome 39, aolit 1978, page
C6-3
16
S U R F A C E DEFORMATION A N D S U R F A C E MOTION D U E T O STANDING S E C O N D S O U N D WAVES
IN HELIUM
I 1
S.G. Eckstein*, Y. Eckstein*, J . L . Olsen and H. Sigg
Laboratoriwn
ftirFestk8rperphysik, Swiss Federal I n s t i t u t e
ofTechnology,
8093
Ziirich, ~ z J i t i e r
land
R6sum6.- On a observg l a dgformation d'une s u r f a c e l i b r e d'hdlium l i q u i d e en prgsence d'ondes s t a t i o n n a i r e s d e second son. L'amplitude mesurge e s t 10 P 100 f o i s i n f g r i e u r e B c e l l e correspondant 5 l a p r e s s i o n a s s o c i d e au second son l o i n de l a s u r f a c e . La db- pendance t e m p o r e l l e d e l a s u r f a c e e s t n g g l i g e a b l e .
A b s t r a c t . - The amplitude of t h e deformation of t h e f r e e s u r f a c e of l i q u i d helium i n t h e presence of s t a n d i n g waves of second sound h a s been observed. The wave h e i g h t i s 10- 100 times s m a l l e r t h a n t h a t corresponding t o t h e second sound p r e s s u r e away from t h e s u r f a c e . The time dependence of t h e s u r f a c e l e v e l i s v a n i s h i n g l y s m a l l .
I t h a s been p o i n t e d o u t elsewhere t h a t s t a n d i n g waves of second sound i n He I1 produce de- f o r m a t i o n s of t h e f r e e s u r f a c e , and t h a t t h e s e can b e observed u s i n g o p t i c a l t e c h n i q u e s / 1 , 2 / .
The second sound f i e l d c o n t r i b u t e s t o t h e p r e s s u r e w i t h i n t h e l i q u i d , and theamount of t h i s p r e s s u r e h a s been c a l c u l a t e d by v a r i o u s a u t h o r s / 3 , 4 , 5 / . There i s l i t t l e doubt about t h e amplitude of t h i s p r e s s u r e v a r i a t i o n f o r a given v a l u e of t h e second sound f i e l d . There remain, however, theo- r e t i c a l and experimental u n c e r t a i n t i e s about both t h e time average and t h e time dependence of t h e r e s u l t i n g s u r f a c e deformation.
The problems a r i s i n g a r e most e a s i l y seen i f we c o n s i d e r s t a n d i n g waves of second sound g i v e n by $ = Qo s i n kx. c o s w t , (1) w h e r e $ :
- Vv-n i s t h e v e l o c i t y p o t e n t i a l of t h e
normal f l u i d . The p r e s s u r e c o n t r i b u t i o n p2 r e s u l - t i n g can be found u s i n g an e x p r e s s i o n g i v e n by S o r b e l l o /4/ which a c c o r d i n g t o P u t t e m a n and G a r r e t t / 5 / i s v a l i d i f c o m p r e s s i b i l i t y and t h e r - n a l expansion o f t h e l i q u i d a r e n e g l e c t e d . Ppn P2 = - k2 $02 c o s 2kx(l- c o s 2wt)
(2) 2psIt i s tempting t o conclude t h a t t h e sur- f a c e w i l l be d i s p l a c e d from i t s e q u i l i b r i u m h e i g h t byAz
= pplpg. (3) It t u r n s o u t , however, t h a t t h e time v a r i a t i o n of z i s l i m i t e d by i n e r t i a l e f f e c t s s i n c e t h e r e w i l l b e bulk f l o w i f Az i s t o f o l l o w p2 a t t h e frequency 2w.A simple c a l c u l a t i o n shows t h a t t h e ampli-; tude A of t h e o s c i l l a t i o n o f t h e s u r f a c e h e i g h t i s reduced t o
P2 v 2
A = - g
p g (vg2+vc2-v22)
Where v and v a r e t h e v e l o c i t i e s of g r a v i t y wa- g
v e s and c a p i l l a r y waves a t frequency 20, and vt i s t h e v e l o c i t y of second sound. For wavelengths of a few m i l l i m e t e r s a s used i n our experiments, A w i l l be approximately times Az a s given by
( 3 ) .
The s u r f a c e motion i s c l e a r l y v e r y small. On t h e o t h e r hand t h e time average of p2 i s non- z e r o so t h a t we expect some kind of s u r f a c e de- f o r m a t i o n . Indeed we do o b s e r v e c l e a r l y d e f i n e d s u r f a c e deformations /1,2/. Close bo t h e s u r f a c e v and v - ~ must be p a r a l l e l t o t h e s u r f a c e and
-n
c o n t a i n v e r t i c a l components. The p o t e n t i a l
( 9 )
must t h e r e f o r e c l e a r l y b e changed c l o s e t o t h es u r f a c e . T h i s w i l l l e a d t o a c e r t a i n r e d u c t i o n i n t h e time average below t h a t expected from
< Az > = < p p >
1
p g ( 5 ) We used t h e s c h l i e r e n technique d e s c r i b e d p r e v i o u s l y / I / t o o b s e r v e s u r f a c e deformations on helium c o n t a i n e d i n a v e r t i c a l c y l i n d e r e x c i t e d t h e r m a l l y on t h e a x i s . The a c t u a l shape of t h e s e*
A t t h e Technion H a i f a , I s r a e l . T h i s work was car-ried out w h i l e
s.
and Y. Eckstein were on leave of be seen and photogra- absence a t t h e Swiss F e d e r a l I n s t i t u t e o f Techno- phed. I n f i r s t approximation t h e resonances logy.correspond to those calculated by Rayleigh for
aerial vibrations in a cylindrical symmetry. The
resonance frequencies suggest a
Q
value for the
cavity of the order of 5 x lo2. Some 80 separate
resonance frequencies could be observed and identi-
fied with Rayleigh solutions.
In our experimental arrangement the tempe-
rature excursions in the liquid could not be mea-
sured directly. They may, however, be calculated
from the heater input if the heat transfer coef-
ficients at heater surface and at the outer boun-
dary of the resonator are known. We have shown
elsewhere
161that these can be calculated from
the deviation of
theobserved resonance frequen-
cies from the Rayleigh values.
We have used such information to calculate
the second sound field in our resonators and to
calculate the value of pp away from the surface.
In the case of a cavity of
1cm radius heated at
a frequency of 3000-5000 Hz with an r.m.s power
of 60 mW per cm of heater, the value of pp would
lead to wave heights of approximately 0.5
mmat a
distance 0.5 cm from the axis if the expression
(5) is used.
New careful experimental observations of
the wave height for the conditions described above
yield
<Az> 9:The reduction
of < A >below that
to be expected from
<p2>in the bulk appears well
substantiated.
To investigate the time dependence of
zthe schlieren system was illuminated using a stro-
boscope. The surface deformation at various phases
of the heating cycle could thus be obsverved. No
change in levelas a function of phase could be
observed although changes of Az of the order of
5
%should have been easily observable,
We conclude that the amplitude of the time
dependence of the surface deformation must be at
least 20 times smaller than the variation of
<z>with position. Clearly far more sophisticated expe-
riments will be needed to test equation (4).
This work was supported financially by a
grant from the Swiss National Science Foundation.
References
/I/ Olsen, J.L. Physica
69
(1973)
803/ 2 /
