Page 18 - 2017Winter

Isadore Rudnick
Pg 14, column 1, equation 1: c1 = (∂p / ∂ρ )1⁄2 ≅ 220 m/s
Pg 14, column 1, equation 2:
show that the speed of second sound (c ) is a function of the
22 superfluid’s density,[ρs; c2 = (ρs / ρn )s (∂s / ∂T )P ]..
First Sound
With two fluid components, one may have a sound mode trhat is rthe same as ordinPargy1s6ou, ncodliunmnnor1m, aelqfluuaitdiso.nTh1e: mo-
Pvgid1e4d, bcoylupmren1⁄2ss1u,req(upa)t.ioThn 1is: sound mode, called first sound, is
2
Pg 13, column 1:
vn and vs ,
tion of both the normal-fluid and superfluid components
The speed of second sound (c2) is a strong function of tem- perature, vanishing above T and reaching a maximum
are equal and in phase,v! = v! ,with the restoring force pro- ns
Pg 14, coluλmn 2, equation 1:
speed of 20.4 m/s at 1.65 K under saturated vapor pressure.
shown schematically in Figure 2a. The speed of first sound,
c1 = (∂p / ∂ρ ) ≅ 220 m/s , is nearly independent of tempera-
ture both below and above the λ-transition. Izzy’s first ex-
Pg 13, column 1: 2/3
The maximum second sound speed is an order of magnitude
periments with superfluid helium began with quantitative
tation (i.e., a heater is not cooled when the electrical cur-
Pg 14, column 1, equation 2:
Pg 14, column1⁄2 1, equation 1:
rent through it is reversed), it would be difficult to study the
measurements of the speed and attenuation of first sound
)* # ∂ ρ / ∂ ( v! − v! ) 2 & ,-
speed of second soun%d close tno thse λ(-transition if a heater is
near the superfluid transition temperature, with microkelvin
These first sound measurements, made with Marty Barmatz,
$'
+ T ,P .
22
[ρs; c =(ρ /ρ )s (∂s/∂T) ].
c =(∂p/∂ρ) ≅ 220 m/s
used to exc1ite second sound. Izzy realized that second sound
2sn
temperature resolution at frequencies in the gigahertz range.
Pg 14, column 2, equation 1:
Pg 14, column 1, equation 2:
Pg 14, column 2, equation 3:
P
Jim Imai, Richard Williams, and Dan Commins, pushed for-
rr2
[ρs; c =(ρ /ρ )s (∂s/∂T) ].
ward our understanding of the application of equilibrium ther-
2/3
the Buckingham-Fairbank-Pippard relations at the λ-transition.
PSgec1o4n, cdolSuomunn2d, equation 2:
Because there are two independent flow fields, there can be
ing “piston” can be used to excite first sound. It is possible
to have antisymmetric mode where the two fluid compo-
while the superfluid would be sucked in and vice versa. Us-
[ρ ∝(T −T) ]
mos dynaλmics to quantum fluids and demonstrated the utility of
ing that mecha2nical excitat2ion technique and the high sig-
)2,22
# !!&
two modes of oscillation. First sound is the symmetric mode
*%$∂ρ/∂(vn−vs)(' -
+where the flow fiT,ePl.ds move in phase so an ordinary vibrat-
4s1
If nonlinear interactions are included, first and second
nents move out of phase, as shown in Figure 2b; this mode Pg 14, column 2, equation 3:
beams interact at an angle such that their intersection travels
rr2
an antisymmetric mode might be excited. (v −v )
),
at the speed# of firs!t sou!nd2 &. This “mode conversion” is similar
is called second sound. It is interesting to examine how such
*%∂ρ / ∂(vn − vs ) ( - Pg 13, column 1:
to three-ph$onon interacti'on, which is the nonlinear process
ns
vn and vs ,
When a boundary that is in contact with superfluid helium
is heated, the superfluid component is attracted to the heat
rr2 vn −vs
Pg 15, column 1:
Izzy recogPngiz1e4d, ctohlautmtnhe2,weaqyuatotioanch3:ieve precise control over
source. When the superfluid component reaches the heater, th2e absorption2 of thermal energy converts the superfluid
Pg 14, column 1, equation 1:
[c4 = ρs /ρ c1] ()
second so(und wa)ves were planar was to use the low1⁄2 est fre-
c =(∂p/∂ρ) ≅ 220 m/s
component to the normal component that leaves the vicin- ity of the heater and carries away the heat. Because there are no mechanical forces communicated by this conversion, there can be no acceleration of the fluid’s center of mass. As a result, the second sound wave’s motion corresponds to a counterflow of the normal and superfluid components.
In regions of the wave where the superfluid becomes con- centrated due to the counterflow of the two components, the average entropy of the fluid is diluted, corresponding to a local decrease in the fluid’s temperature. In regions of the wave where the normal fluid becomes concentrated, the av- erage entropy of the fluid is concentrated, corresponding to a local increase in the fluid’s temperature. Second sound is a propagating wave of temperature and entropy; it is impor- tant to note that a detailed theory (Rudnick, 1976) would
1
16 | Acoustics Today | Winter 2017
[ρs ∝(Tλ −T) ] slower tharn the firrst sound speed.
vn and vs ,
Because electrical heating provides a positive-definite exci-
Pg 14, column 2, equation 2:
could be excited mechanically by using a porous “piston” so that the normal-fluid component would be pushed out
2sn
(v −v )
P
nal-to-noise ratio it produced in a resonator, it was possible
ns
for Mary Beaver, Richard Williams, Jim Fraser, and Reynold
KagiwadaPtgo1d4e, tceorlmumine2t,heqsucaatiloing1:of the superfluid density Pg 15, column 1:
as a function of temperature very close to the transition tem-
perature [ρ ∝(T −T)2/3].
s λ [c =(ρ /ρ)c ]
sound can be coupled in a process where two second sound
Pg 14, column 2, equation 2:
+ T ,P .
of two slower shear waves in a solid that generate the faster
longitudinal mode.
the second sound interaction angle and to ensure that the
quency higher order mode of a waveguide. Because the larg- est helium Dewar in his lab had only a six-inch inner diam-
Pg 15, column 1:
eter, Izzy suggested the use of a spiral waveguide to provide
a meter-long “end-fire array” interaction length, based on an
[c2 =(ρ /ρ)c2]
4 s 1 [ρ;c2=ρ/ρ s2 ∂s/∂T ] anechoic termination that B ob Leonards develo(ped to) sup( - )
2sn press reflections in a probe-tube microphone.
P
Measurements of the absolute amplitude of the mode con-
version by Steve Garrett was enabled by a reciprocity cal- ibration that was made in situ at temperatures below 2 K
[ρs ∝(Tλ −T)2/3]
and demonstrated Seth Putterman’s calculation predicting
the largest component of the nonlinear coupling coefficient
depended on the thermodynamic derivative of the fluids’
density with respect to the square of the difference between
superfluid and normal-fluid velocities )*#∂ρ / ∂(v! − v! )2 & ,- . +%$ n s ('T,P.
rr
Pg 14, column 1, equation 2:
Pg 14, column 2, equation 1:
Pg 14, column 2, equation 2:
Pg 14, column 2, equation 3:
rr
.