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L I B R‘A R Y
Michigan State
~ University

THE?“

This is to certify that the

thesis entitled

Coupling Between Pressure and Temperature

 

Waves in Liquid Helium
presented by
Garold F. Fritz

has been accepted towards fulfillment
of the requirements for

& degree in

Physics

 
  

. QQQQJQ

Major professor

0469

  
     
 

  

V manna-v ' 4
Ill“ & SII' ‘ -
‘um ‘.

 

ABSTRACT
COUPLING BETWEEN PRESSURE AND TEMPERATURE
WAVES IN LIQUID HELIUM
By
Garold F. Fritz

When the thermal eXpansion coefficient (1 is retained
in the linearized hydrodynamic equations of the two-fluid
model of superfluid liquid helium, pressure and temperature
waves are not independent. Thus a periodically varying
temperature source produces not only temperature waves in
' and p ',

1 2
which prOpagate at the velocity of first and second sound,

liquid helium, but also two pressure waves, p

respectively. Similarly a vibrating diaphragm produces
not only pressure waves but also two temperature waves,
T1' and T2', which also prepagate at the velocity of first
and second sound, respectively. Lifshitz has shown that
the amplitudes of these cross-modes should be prOportional

to CL . This thesis is an investigation of this coupling

by observing and studying the two pressure waves p1' and

p2 produced by a heater and the temperature wave T2
produced by a capacitor micrOphone. The temperature
dependence of the amplitude of these waves has been studied
using both pulse and standing wave techniques in the tempera-
ture range from 1.2 K to the ).-point. Good agreement

with theory has resulted only for the p1 mode.

COUPLING BETWEEN PRESSURE AND TEMPERATURE
WAVES IN LIQUID HELIUM
By
Garold anrritz

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Physics

1970

I I
I .
£17 ‘4‘
-'\
I 1,.
(.

To Carol

ii

ACKNOWLEDGEMENTS

This experiment was suggested by Professor G. L.
Pollack. I would like to thank him for his guidance and
encouragement during the course of my work and for his
advice in the preparation of this thesis. I would also
like to express my gratitude to Dr. T. H. Edwards for
the use of his vacuum coating equipment and to Mr. Carl
James Duthler and Mr. Charles W. Leming for their assist-
ance in the laboratory. Also, I wish to thank my wife,
Carol, for typing the manuscript. I am also indebted
to the U. S. National Aeronautics and Space Administra-
tion for support in the form of a three year fellowship.
Finally I would like to acknowledge the financial support

of the U. S. Atomic Energy Commission.

111

TABLE OF CONTENTS

Page

I. INTRODUCTION . . . . . . . . . . . . . . . . 1
II. THEORY . . . . . . . . . . . . . . . . . . . 5
A. First and Second Sound . . . . . . . . 7

B. Coupling Coefficients . . . . . . . . 10

C. Vibrating Solid Surface . . . . . . . 13

D. Fluctuating Temperature Source . . . . 15

E. Vibrating Porous Surface . . . . . . . 17

F. Energy Considerations . . . . . . . . 18

III. EXPERIMENTAL . . . . . . . . . . . . . . . . 21
A. Pressure Waves p1' and p ' Produced by
a Heater . C O O O C O .2. O O O O O . 21

1. Pulse Technique . . . . . . . 29
2. Standing Wave Technique . . . . 35

B. Temperature Wave T2' Produced by a
MicrOphone . . . . . . . . . . . . . . 41

l. Porous Diaphragm Second Sound
Transducer . . . . . . . . 41
2. Apparatus and Procedure for
Measuring T2' . . . . . . . . . 44

IV 0 RESULTS 0 O O O O Q 0 O C O O O O Q 0 O C O 49
A. Pressure Waves p1' and p ' Produced by
a Heater . . . . . . . .2. . . . . . . 49

1. Output Amplitude Data From

MicrOphone . . . 49
2. Temperature Amplitude To Produced

by the Carbon Disk Transmitter . 57
3. Output Amplitude Data Corrected

for Variation of To . . . . . . 64

B. Temperature Wave T ' Produced by a

Microphone . . . .2. . . . . . . . . . 7O

1. Sensitivity of the Porous Dia-
phragm Transducer . . . . . . . 7O
2. Measurement of T2' . . . . . . . 74

iv

V. DISCUSSION AND CONCLUSIONS . . . . . . . . . . 78

A. Pressure Waves p '

and p ' Produced by
3 Heater . . . .1. . .2. . . . . . 78
B. Temperature Wave T ' Produced by a

Microphone . . . . . . . . . . . . . 81

LIST or REFERENCES . . . . . . . . . . . . . . 88

Figure

10

ll

12

LIST OF FIGURES

EXperimental chamber.
Carbon disk transmitter.

Biasing circuit for the capacitor micro-
phone used as a receiver.

Schematic diagram of the experimental
chamber and electronics used in the pulse
measurements of p1' and pe'.

Relative voltage gain of the frequency
selective amplifier versus normalized fre-
quency for various Q values.

Some typical oscillosc0pe traces obtained
in the pulse measurements of p1' and p2'.

A typical frequency response curve of a
standing wave resonance of p2' at a

temperature of 2.06 K.

Some typical OSCillOBCOpe traces illus-
trating the response of the cavity to a
long tone burst driving current at the
resonant p2' frequency.

Biasing circuit for the capacitor micro-
phone used as a transmitter.

The pressure wave amplitudes p ' and p '
produced by the carbon disk trdnsmitte ,
plotted versus temperature.

The p ' standing wave resonance amplitude
plottgd versus temperature.

The damping constant of p ' standing waves
plotted versus temperatur .

vi

Page

23
24

27

30

32

34

38

4O

46

51

53

56

Figure
13

14

15

l6

17

18

A schematic diagram of the apparatus used
to measure the second sound amplitude T2'
produced by the carbon disk transmitter.

The second sound amplitude T ' produced
by the carbon disk transmitter, plottes
versus temperature.

A plot of the p ' data on Figure 10
divided by the 2' data on Figure 14.

A plot of the p ' data on Figure 10
divided by the 32' data on Figure 14.

The output voltage from the porous dia-
phragm transducer used as a receiver of the
second sound waves produced by the carbon
disk transmitter, plotted versus tempera~
ture.

The second sound amplitude T ' produced

by the capacitor microphone, plotted versus
temperature.

vii

Page

60

63

66

69

73

76

I. INTRODUCTION

Liquid 4He cooled below 2.17 K, which is called the
lambda temperature, exhibits a number of peculiar properties
which have been successfully explained by the introduction
of a two-fluid model.1’2’3 In this model it is assumed that
the ordered motion of the elementary excitations in the
liquid is associated with only part of the liquid. This
part is called the normal component and is characterized
by the normal density F%. The normal component has viscosity
and its flow is described by a velocity field vn. The
remaining part of the liquid, which is called the superfluid
component, is characterized by the density F: = F5- F£ ,
where F>is the total density of the liquid. The superfluid
component has no viscosity and performs an independent motion

which is described by the velocity field v No explicit

S.
assumptions are made about the temperature dependence of

5% and F% except the following:

P. 0 Ion

 

 

 

P .___. p :1 atT:T>\:2.17K
ps :1 .8120 atT:O.
P p

2

With these assumptions a complete set of hydrodynamic
equations can be derived for superfluid liquid helium.4’5
One consequence of these hydrodynamic equations is the exist-
ence of two wave modes which prOpagate with different veloci-
ties. One of these wave modes is ordinary sound or pressure
variations which propagate with the velocity u1= Bbp/bp)s]%,
where p is the pressure and S the specific entrOpy. In the
other wave mode the temperature is the fluctuating thermo-
dynamic quantity. This mode is called "second sound". The
velocity of second sound is u2=[3T82/0)(F%/F%i]%, where C
is the Specific heat. The velocity u2 is an order of mag-
nitude lower than u1 and vanishes at T = T)p

When terms containing the thermal expansion coefficient
((13 ~(1/pXPP/5T') are neglected, first and second sound
are independent. In this approximation a vibrating dia-
phragm would produce only pressure waves which prOpagate
at u,; that is, pure first sound. Similarly a fluctuating
temperature source would produce only temperature waves
which prOpagate at u2; that is, pure second sound.

When thermal expansion is taken into account pressure
and temperature waves in liquid helium are coupled. In
this case there exists both pressure and temperature fluc-
tuations which prOpagate with each velocity u1 and u2.

The original theoretical treatment of this coupling
is due to LifshitzfS He has shown that the relative ampli-
tudes of the pressure and temperature waves produced in
liquid helium depend upon the boundary conditions at the

transmitter.

 

3
The object of this eXperiment was to investigate this

coupling by observing and studying the pressure waves pro-
duced in liquid helium by a fluctuating temperature source
and the temperature waves produced by a vibrating diaphragm.
The amplitudes of the two pressure waves produced by a fluc-

tuating temperature source are denoted by p1 and p2'. The

subscripts 1 or 2 indicate that the wave prOpagates at u1

or u2, respectively. Similarly the two temperature waves

produced by a vibrating diaphragm are denoted by T1"and
I

T2 .

The pressure waves produced in liquid helium by a
periodically heated resistor have been studied by other

investigators. Jacucci and Signorelli7 have Optically ob-

E
served p2' by the Debye—Sears effect. Eynatten, et 3;.

have also observed this wave mode with a magnetic micrOphone.

Hofmann, g; 3;.9 have observed and studied both p1' and

p2 . All of the above results agree neither with each other
nor with the theoretical results calculated by Lifshitz.6
The results of the independent measurements of this investi-
gation are in substantial but not total agreement with those
of Hofmann, gt 3;.9

Historically the first attempt to observe second sound
was made by Shalnikov and Sokolov who attempted to generate
this wave with a piezoelectric crystal.1O They were unsuc-
cessful. This negative result was, in fact, the motivation
for the above mentioned investigations by Lifshitz.6 He
showed that a much more effective method of producing second

sound would be by periodic heating. When this was pointed

4
10
out second sound was soon observed and has been compre-

hensively studied since that time."2
No successful attempts have been reported, since the
work of Shalnikov and Sokolov, in which second sound pro-
duced by a vibrating mechanical transducer has been observed.
In the present investigation second sound, produced by a
capacitor micrOphone, has been observed. A new type of

second sound receiver whose active element is a metalized

porous diaphragm has been used to observe this wave mode.

I I . THEORY

When dissipative terms are neglected, the complete
set of hydrodynamic equations based on the two-fluid model
of liquid helium can be determined from conditions imposed
by the Galilean relativity principle and the necessary
conservation laws.5 The derivation is similar to that
for an ordinary fluid except that an additional relation,
which takes into account the superfluid prOperties, must
be included. Most often this additional relation is an
equation of motion for the superfluid component. This
choice, however, is not unique.11

When dissipative processes are considered, additional
terms must be included in all the hydrodynamic equations
except the continuity equation. There is considerable
disagreement among the various authors on the prOper form
of these terms.2 For the purpose of studying first and
second sound the hydrodynamic equations constructed by
Landau and Khalatnikov5 are the most useful.

In our investigation the amplitude of the sound waves
is small enough that only terms linear in the normal

fluid and superfluid velocities, v and vs, need be retained.

n
In this case the hydrodynamic equations due to Landau and

Khalatnikov take the following forms:

5

1.8.. 4—6.? :: O (1)

 

 

 

‘o
‘03: ,b2 = ), ‘ovni *.Wk 38 2311)
bt bri brk brk bri 11‘ br

owl-5 VG Pvn) +C2_V>~?n] (2)

4.

 

 

B? A A .5 A a. AA
bts 4V“: V[§3V-(J -pvn) + §4Vovn] (3)
2353?. +PS$R§I :lg—VgT (4)

In Eq. (2) summation over the repeated indices is implied.

The various symbols in these equations are as follows:

A A
j : mass flux = ngs 4 f%;; (5)
{3: total liquid 4He density : (g +f% (6)

ll

superfluid density

505D
u

normal fluid density

<:
I

_ superfluid velocity

4
(D
II

normal fluid velocity

p : pressure

r1 : i-th component of the position vector
S : specific entrOpy

fiL: free entrepy

T

temperature
i<= thermal conductivity coefficient
77: ordinary viscosity of the normal fluid

€1,€2,é3,£4 : coefficients of second viscosity

7

Equation 1 is, of course, the continuity equation
representing conservation of mass. Equation 2 is the
equation of motion of the fluid as a whole. Equation 3
is the equation of motion of the superfluid component.
Finally Eq. 4 represents the balance of entropy. The
coefficients of second viscosity represent energy losses
associated with changes in density which occur rapidly
in comparison with the relaxation time required for restora-
tion of thermodynamic equilibrium.

If all dissipative terms are neglected and use is

made of the thermodynamic relation‘e:

1
d = - Sd’l‘ _.._...dp (7)
H + P

then Eqs. (1) - (4) take the following form:

 

1341+ I75?” (8)
g+$p=o (9)
his - SGT 4 _.F‘.J..$p = o (10)
\O‘otS) +ps$mén=o (11)

A. First and Second Sound
Plane wave solutions in which all quantities are
prOportional to exp [10(t—x/ui], will now be considered.
If the symbols F“, p', T', and 8' represent the variation
off), p, T, and S from their equilibrium values, Eqs. 8 - 11

become:

up' — (vaE) + ,Onvn) = 0 (8a)
u( Jag“) - p' .-= 0 (9a)
upv8 4 SpT‘ - p' = 0 (10a)
u(Sp' +ps') -5pvn=o (11a)

It has been assumed that the fluid is stationary at
equilibrium so that vn and v8 were not denoted with primes.
In obtaining Eqs. 8a - 11a the expression for j in Eq. 5
has also been used. Eliminating J = Svs + F$vn from
Eqs. 8a and 9a yields

u2p' — p' = o (12)
Solving Eq. 11a for S' and then substituting expressions
for F)and Ff obtained from Eqs. 6 and Ea, respectively,

gives the following result:

5 5 SP
3 __ _ 8
Similarly, solving Eq. 10a for T' and then substituting
expressions for F>and p' obtained from Eqs. 6 and 9a,

respectively, gives the following result:

I _ u u __ uPn
T " (Pave * ann) " “vs "

sp 5 sp

Eliminating (vn-vs) from Eqs. 13 and 14 then gives

2 210

uS'-S——8—T':O (15)

81

(14)

 

 

In Eqs. 12 and 15 pressure and temperature will be used

 

as the independent variables. The density and entrOpy
will then be eliminated by writing:

9' = F—fjp' + Pi) 1" (16)
br>q, ‘bT p

S' = (b8 p' + (ES T' (17)
2>p T Y>T

The thermal expansion coefficient<1.is defined as follows:

as 1 (\OP\ (18)

1° \M/p

Also the following thermodynamic relations can be shown

to hold12:
be _ (‘Ov __lz(\o1°) __ _g_.__ (19)
bPT bT P ET P ’
p p
and
‘05 C
(w) 2—2— . (90)
‘oT T
In Eq. 19, v : —l— is the specific volume and in Eq. 20,

{3

CF) is the specific heat at constant pressure.
Substituting Eqs. 16 - 20 into Eqs. 12 and 15 then

gives:

[112(BP ~1]p'- quQT' : O (21)
2>p T

2
C
" Ltd. 13' '0' [u2—p— - 52£]T' :: O (22)

and

 

 

10

The coupling, due to the thermal expansion coefficient,
between pressure and temperature waves in liquid helium

is apparent in Eqs. 20 and 21. If'a.is neglected in these
two equations, pressure and temperature waves will be

independent and will prOpagate with the respective velo-

 

cities:
a
U1 :: (2P (2})
t
2
1-2-8— P81

C R.
In Eqs. 23 and 24 no distinction has been made between

(hp/hp)T and (hp/bps or between GP and CV because of

the following thermodynamic relations12:

a - “1“"
pr Cv hp;

C 2
p : + C1. T (‘08.) (26)
cv op bp 8

 

 

A

 

Thus in the approximation that (1-.- o, (hp/bp)T a: (hp/25mg

p v

B. Coupling Coefficients
The coefficients needed to determine the relative
amplitudes of the pressure and temperature waves produced
by a given transmitter can now be calculated as follows:
In a plane wave the velocities v8 and v and the variable

n

parts p' and T' of the pressure and temperature are

11

prOportional to each other. Upon introducing the prOpor-

tionality factors a, b, and c according to6

vn : av8 (27)

p' : bVq (28)

T' : cv (29)
s

Equations 10a, 14, 21, and 22 can be written as follows:

 

 

b - 8pc = up (30)
a - 1 : SF) c (31)
1181
2 2
u 33-2— -1b-aupc:0 (32)
3P
T
2 2
(lTu b - (u - T82 PS )0 = C (33)
PCp CD pm
There will be two sets of the coefficients a, b, and
0; one for the case when u : 111 and one for u : u2. When

u : u1, Eqs. 30, 31, and 33 can be solved for a b and

 

 

 

 

c‘. The results, to first order intl, are
2 2
u u
a :: 1 4- up 1 2 (34)
1 Sf) u 2 - u 2
s 1 2
u 2
(ITS 1
b1 :pu1 1 4 C ‘9 2 2—) (35)
u1 -u2
u 3
CLT 1
c1- 2 T . (36)

 

 

 

12

In Eqs. 35 and 36 the subscript has again been left off

the specific heat C because no distinction is being made
between the specific heat at constant volume and at constant
pressure. Doing so would merely introduce terms of higher
order in a.which have been neglected anyway.

When u : u Eqs. 30, 31, and 32 can be solved for

 

 

 

 

 

29
a2, b2, and cg. These results, to first order in a,, are
“ 2 2
Pg CLP L11 L12
a? = -— + 2 2 (37)
Pn Sel u1 -u2
2 3
CLP u1 112
b = -—- ~ (38)
2 S u 2--u 3
1 2
u u 2n 2
C2 = ' 2 1 - SE é 2? (39)
S S u1 --u2

In the approximation,<:.: 0, Eq. 34 gives the result,
vngévs. Thus, as expected, in a first sound wave in this
approximation the liquid vibrates as a whole in each volume
element. For a second sound wave with. 0,: 0, Eq. 37 shows
that vn’}: - (Fa/9+8. From this it follows that
3 2 Eve 4 ann: 0. Thus in a second sound wave in this
approximation the superfluid and normal fluid components
vibrate out of phase in such a way that the center of
mass of each volume element remains stationary.

The coefficients given by Eqs. 3h - 39 can now be

used to analyze radiation of pressure and temperature

waves from various types of transmitters.

 

13
C. Vibrating Solid Surface

The first type of transmitter which will be considered
is a plane solid surface lying in the y — 2 plane and
vibrating in the x-direction with velocity given by
u : uoexp(iUIQ. The superfluid velocity will consist of
two components; one associated with waves which prOpagate
at u1 and the other with waves which prOpagate at u2.
Solutions in the form of plane waves will be sought.

 

 

Thus
v : v (1) + v (2) (40)
s s s

vs(1) : A1exp iUJt - —§_W (41)

u i

1

2 - -
V<>=AexP1wt-.—2<_ . (42)

S 2 L ug-

In general, the boundary conditions at a solid sur-
face at rest in liquid helium require that the tangential
component of the normal fluid velocity-$5 and the normal
component of the mass flux-F : fgv; + F£3n vanish. Since
only the normal fluid component is associated with the
excitations in the liquid the heat flux will befDSTvn.

The normal component of this heat flux at the surface
must equal the heat flow into the surface due to thermal
conduction. However, due to the high rate of heat transfer

in liquid helium compared with that in a solid, the latter

can be neglected. The boundary conditions then reduce

 

14

to the vanishing of the normal components of each velocity

A

A
vS and vn. For the vibrating surface the normal components

of R78 and RTn must then equal the velocity of the surface.

From Eqs. 27, and 40-42 these boundary conditions give
A + A = u (43)

: u (44)

a1A1 + a_A2 o

9

From Eqs. 43 and 44 it then follows that

 

1 - a2
A : _______ u (45)
1 a - a O
1 2
1 - a1
A2 : - no (46)
a1 - 32

The amplitudes of the pressure and temperature waves pro-
duced in the liquid helium by this vibrating surface are

then given, from Eqs. 28, 29, 45 and 46, by

 

 

 

 

b1(1-a2)
p ' : b A : u (47)
1 1 1 a —a o
1 2
c (1-a )
T ' = C A : 1 2 u (48)
1 1 1 a -a o
1 2
' b?(1-a1)
p? : b?A? : - A -8 uo (49)
1 2
c2(1-a1)
T?' I 02A? 2: - uo (50)

 

 

15

Substituting the expressions for the coefficients from

Eqs. 34-39 into Eqs. 47-50 gives

2
p1':(pu1_ &)uo (51)

 

S
CLTu1
T1' = T 1.10 (52)
2 3
CL Tu
p2' = (—30—2—4110 (53)
CLTu2
T2' : C 110 (54)

The velocity u2 is an order of magnitude smaller than
u1 for the temperatures between 1.2 K and T)\which are
of interest here. Thus in obtaining Eqs. 51-54 the

2
approximation (u12 - u22)§éu1 has been used.
D. Fluctuating Temperature Source

The second type of transmitter which will be consid—
ered is a surface in the y-z plane whose temperature varies
as T : To'expfliut). The boundary conditions in this case
require that the normal component of the mass flux—F
vanish at the surface and that the temperature of the
liquid adjacent to the surface equal that of the surface.

From Eqs. 27, 29 and 40—42 these boundary conditions give
ps(A1+A2) + 90(81A1-a2A2) : O (55)

_ 1 ,
01A1 + 02A2 _ TO . (5C)

16

Solving Eqs. 55 and 56 for A1 and A2 yields

a
A : ps+2eq T'

 

 

1 o (57)
°1(Ps+a281)'°2(%*a1pn)
A = _ P848181 T I
2 c1(ps+a2%)-c2(g+a1pn) o (58)

The amplitudes of the pressure and temperature waves
produced in the liquid helium by this fluctuating tempera-

ture plane are then given, from Eqs. 28, 29, 57, and 58

 

 

by
b
p1':b1A1= 1(gm2a) T ' (59)
C1‘B*3291)-°2(%*a1%) O
T '.-:c A -..-.- c‘(ps+a28‘) T ' (60)

1 1 1 c1(pg+a2RI)-c2(ps+a1%)0

b2<Fé+a1F%) T ' (61)
(. 2 01(p84a2Pn)—c2(%+81el) O

 

Tq'=c A = - 62(gm1ex) T ' (62)
c1(%+a2%)-02(pe+a1el) 0

Substituting the expressions for the coefficients from

Eqs. 34-39 into Eqs. 59-62 gives

p" =(apu1u2)TO' (63)

2
(1 Tu u
T1':( 12)TO' (64)

 

C

 

17

 

2
v _ _ t
92 -( (1qu )TO (65)
agTu u
T ' = 1 - 1 9 T ' (66)
2 C O

E. Vibrating Porous Surface

A third, rather interesting method of generating
pressure and temperature waves in liquid helium is that
of a vibrating porous surface. The pores are assumed
small enough so that, because of its viscosity, the nor-
mal fluid cannot flow through them. The condition which
must be satisfied in order that this be true is that the
normal fluid shear wave penetration depth, SAM/81w )%-’ 13
be much larger than the pore diameter. The superfluid,
having zero viscosity, can flow through the pores unimpeded.

The plane will again be assumed to lie in the y - 2
plane and to be vibrating in the x-direction with velocity
given by u : uoexp(iUJt). The boundary conditions at this
surface require that the normal fluid velocity vn equal
the velocity of the surface and the superfluid velocity v8
be zero. From Eqs. 27 and 40-42 these boundary conditions

then give

A1+A =0 (67)

a A1 + a A : u (65)

 

18

Solving Eqs. 67 and 68 for A and A2 gives

1

Az-Az—i— (69)

To lowest order in Q,the amplitudes of the pressure and
temperature waves produced in the liquid helium by this
vibrating porous surface are then given, from Eqs. 28, 29,

69, and 34-39, by

 

 

 

b
u _ 1 _
p1 — —;—:;—— uo — (F%u1)uo (7O)
1 2
c (1Tu F)
T ' :: -—""-'L"“' U : (fl—L n 1.10 (71)
1 a1-a O C P
3
b u
p9: = __ 2 u0:_(.g'_P.E__2__-)uo (72)
a1-a2 S
02 u f)
T?' = - u = (—3- --9- uO (73)
- are2 O S P

F. Energy Considerations

From Eqs. 51-54 it can be seen that the predominant
wave mode produced by the vibrating surface is p1'. To
lowest order the amplitudes T1' and T2' are prOportional
to Cl while p2' is proportional to 0.2. The amplitude
p1', to lowest order, is not proportional to<l . Thus
most of the energy radiated by the vibrating surface goes
into the p1' wave. This can also be seen by calculating

the intensity of each type of wave. For small oscillations

the average Kinetic and potential energy densities are

19
1
equal. Thus the total energy density in a wave is

E : psvs2 4 filing = ’5 A2<Ps 4' {13132)0 (74)

The intensity is given as usual by the product of energy

density and velocity:

I -.= Eu. (75)

From Eqs. 34—39, 45, and #6 the ratio of the intensities

I2 and I1 of the waves which prOpagate with the velocities

uQ and u1, respectively, is

-

I 2 2) 2 3

2 _ u2A2 (339182 :O'Tus (76)
’ 2 2 ‘—

11 1119.1 (g+ga, ) Cu

 

1

Since Q,is small, I2 is much less than I1. For example,
at T = 9.0 K, 12/1121. 2x10“6.

One tends to regard this result as obvious since a
vibrating surface is the basis of any kind of transmitter
of ordinary sound. However, as was already mentioned,
these results were not known to the early investigators
who first attempted to generate second sound with a
piezoelectric transducer.1

Equations 63-66 indicate that the fluctuating tempera-
ture source is a much better transmitter of second sound.
Here the amplitude Te', to lowest order, does not depend
on 0., while the amplitudes p1', p ', and T1' are preportional

2
to either the first or second power of'a,,. Moreover if

20

one calculates the ratio of the intensities of the second

and first sound waves as before, the result is:

I C

_’“_ 2 (77)

I1 (I Tu1u2

 

At T : 2.0 K this ratio is 5X103 and increases with
decreasing temperature. The first successful observation
of second sound resulted when the transmitter was a

periodically heated resistor.1O

 

 

III. EXPERIMENTAL

A. Pressure Waves p1' and p2' Produced by a Heater

Figure 1 shows the eXperimental chamber used to inves-
tigate the two pressure waves produced in liquid helium
by a fluctuating temperature source. It consisted of a
cylinder which was closed at one end by a carbon disk
resistor and at the other end by a capacitor microphone.
Most of the experiments were carried out in a chamber
which was 5.7 cm long with a 2.0 cm id. The cylindrical
chamber was made by drilling a 2.0 cm hole through a 1.5
inch diameter nylon rod.

The carbon disk resistor, a detailed drawing of which
is shown in Figure 2, served as the transmitter. It was
made by cutting a circular section out of a commercially

15

available carbon resistance strip. This resistance
strip consisted of a paper base phenolic laminate with a
thin carbon resistance layer permanently bonded to it.
The resistance of the carbon layer at room temperature
was 1000 ohms per square. A low resistance electrode
strip was painted around the periphery of the circular
section with silver paint. This electrode was grounded

by means of a small screw to the outer conductor of a

miniature coaxial cable.16 The inner conductor of the

21

$13?

If"???

22

Outer bross housing

Brass ring which holds the Mylar diaphragm
down

Nylon insulator
Bross back plate
Aluminized Mylar diaphragm

Set screws to adjust the tension in the
diaphragm

Screws holding the whole chamber
to the support rod

. Stainless steel support rod
. Copper electrical lead which is connected

to the back via a metal spring

. Carbon disk
. Nylon cylinder
. Electrical leads to carbon disk

(miniature coaxial cable)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7/2 Nylon

 

 

 

 

 

 

 

H.
3 _____., I /
m3 mag/MW

 

 

 

 

 

 

Figurel Experimental chamber

 

 

 

 

 

 

 

 

 

 

 

tter

nsmi

Figure 2 Carbon disk tra

 

25

coaxial cable was connected to a similar electrode at the
center of the disk. The resistance of this carbon disk
transmitter at liquid helium temperatures was approximately
600 ohms.

The details of the construction of the capacitor
micrOphone can be seen in Figure 1. A sheet of 0.00025
inch thick Mylar,17 E, aluminum coated on one side, was
stretched over a brass back plate, D. It was held in
place with a brass ring, B, which was screwed securely
to the brass outer body, A. This also served to ground
the aluminum film to the outer body. The back plate was
separated from the outer body by a nylon insulator, C.
The back plate and aluminum film formed two plates of a
capacitor with the Mylar dielectric in between. Small,
1/32 inch wide, concentric grooves were turned in the
surface of the back plate. It was found that doing so
would increase the sensitivity of the microphone. The
capacitance C0 of the micrOphone was measured with a
capacitance bridge and was found to be 275 picofarads.

The chamber was immersed in a bath of liquid helium
and supported by a 0.25 inch od, thin walled, stainless
steel tube, H. This stainless steel tube also served as
the outer conductor of a coaxial cable. The inner conduc-
tor was a no. 32 capper wire, I, which was connected to
the back plate of the micrOphone via a metal spring. The
space between the capper wire and stainless steel tube

was completely filled with styrofoam spacers. This was

26

done to reduce noise due to vibrations of the wire. If
the wire vibrated, the capacitance of the coaxial cable
varied and this would introduce noise in the received
signal. The capacitance of this cable was 60 pf.

Small radial grooves were made in the flat surfaces
of the brass ring which held the Mylar diaphragm down.
This provided passages for the liquid helium to fill the
chamber.

The temperature of the liquid helium bath was measured
by measuring the vapor pressure and using the 1958 Helium
Temperature Scale.18 The vapor pressure was measured
with two Wallace and Tiernan model FA160 absolute pressure
gauges. Pressures above 20.0 mm Hg (T = 1.942 K) were
measured with a gauge which had a 0 - 50 mm Hg scale.
Lower pressures were measured with a gauge which had a
0 - 20 mm Hg scale. Both of these gauges were calibrated
against a Hg manometer.

The temperature of the liquid helium was reduced
below its boiling point (T = 4.2 K) by pumping with a
Heraeus-Engelhard, Model E-225, 147 cfm vacuum pump.

The lowest temperature which could be reached in this
manner was approximately 1.2 K.

Figure 3 is a schematic diagram of the circuit used
with the capacitor microphone. A dc bias voltage of 270
volts was applied, through a series resistor R, between
the back plate and aluminum film. When pressure waves

impinge upon the diaphragm, the resulting displacement

27

 

 

R= 2.0 M
1:;- Co= 275 pf
:— V= 270 volts -—‘—
__ fl..— 9o .
MICROPHONE TO
AMPLIFIER

 

 

 

Figure 3 Biasing circuit for the capacitor
microphone used as a receiver

28

of the latter alters the capacitance Co of the micrOphone
and causes a signal voltage e0. Resistance R is made

large enough so that the capacitor cannot charge and dis-
charge rapidly enough to follow the alternations of capaci-
tance caused by the pressure waves. The condition which
must hold for this to be true is that the time constant,

R0 of the circuit be long compared to the period of the

0’
pressure wave.

Assuming the diaphragm to be replaced by an equiva-
lent piston of area A at a distance x0 + xsincot from
the back plate, where x0 is the equilibrium distance, the

capacitance will be

 

 

:- A 22’ A 1-3—sinwt : C + C'sinwt , (78)
477(x +xsinwt) 41Tx x O
o o o
where 00 : A/A7Txo and C' : -Cox/xo. It has been assumed

that x/xdGKi.

In the circuit of Figure 3

t

v - iR =—1— idt (79)
c

Substitution of Eq. 78 into Eq. 79 and differentiation

with respect to time gives:
I d1 ' |
(CO+C sin(ut)R——+(1+RC a)cosc0t)i-VC u)cos(ut = 0 (80)
dt
Assuming a solution to Eq. 80 of the form

i : Acoswt 4- Bsinwt (81)

 

29

gives

-1
A = case B = wvnc' R2 + (-1—? (82)
0 (000

By using Eq. 82, Eq. 81 can be rewritten as

= VC' sin(LUt+d» ‘ (83)
Co [32 e (1/wao)2]§

tang} : (84)

 

Equation 83 implies that the capacitor micrOphone may
be considered as equivalent to a generator having an Open-
circuit voltage amplitudejVC'/Co and an internal capacitive
impedance 1/1L000. The amplitude of the signal voltage
eo will then be

-19: a .. .YE.’ 1
° 00 [R2 + (Max:321; - co E + (1/wRCo)2:I%

Both pulse and standing wave techniques were used to

e

 

 

. (85)

study the pressure waves produced by the heater.
1. Pulse Technique
Figure 4 shows a schematic diagram of the experimental
chamber and electronics used in the pulse technique. The
input to the carbon disk resistor was a pulse of current
consisting of two cycles of a 5.0 kHz signal. This pulse
was produced by gating the signal from a General Radio

1310-A oscillator with a General Radio 1396-8 tone burst

generator.

30

 

 

 

‘b

OSCILLOSCOPE

n

 

 

 

 

 

 

 

 

 

 

FREQUENCY
SELECTIVE
E: AMPLIFIER
a: CAPACITOR
g MICROPHONE ‘WW‘WIH
E \ 1::1|

 

   

 

R\\\\\\\\\\V
k\\\\\\\\\\‘

   

 

 

 

TONE BURST __¢\N\/\/\.___,

OSCILLATOR GENERATOR f

 

 

 

 

 

 

 

 

 

a: CARBON DISK _:
— TRANSMITTER —

Figure 4 Schematic diagram of the experimental

chamber and electronics used in the
pulse measurements of pl’ and p3.

 

31

Since heating occurs on both the positive and negative
half cycles of the pulse, the carbon disk transmitter is
a frequency doubler. Thus a heat pulse consisting of four
cycles of a 10.0 kHz wave was transmitted into the liquid
helium. The input power to the heater during a pulse was
25 mW/cmg. This was the maximum power input which could
be used in order to remain in the linear region of received
signal versus input power.

The output signal from the micrOphone was amplified
with a Princeton Applied Research, Model 110, variable Q,
frequency selective amplifier. Figure 5 shows a plot of
relative voltage gain versus normalized frequency for
various Q—values of the amplifier. The amplifier was
tuned to 10 kHz and the Q-value set at 1. The signal was
then fed into a Tektronix Type 531A oscilloscope with a
Type 1A1 plug-in amplifier. The oscilloscope trace was
triggered simultaneously with the application of the input
signal. Figure 6 shows some typical oscilloscope traces
obtained in this manner. The times of flight of the

pulses labeled p1 and p2' verify that they prOpagated at
the velocities of first and second sound, respectively.

The photOgraphs of Figure 6 were taken with a Tek-
tronix 0-12 trace recording camera. The tone burst genera-
tor was Operated in the single burst mode. A fraction of
a second before the application of a single driving pulse

to the carbon disk transmitter the camera shutter was

Opened. A single oscillosc0pe trace was recorded and the

 

32

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LC

05 I
z 0"
‘ 0.2
4
° 0.. / \
u: / \
‘9 I
g .05
a); Q=I0 i

.02
“‘ /
E .Ol / \
3’ .005 / \
a: Q=|OO

.002

001

 

 

 

 

' .Ol .02 .05 DJ 0.2 0.5 LO 2 5 IO 20 50 IOO

NORMALlZ ED FREQUENCY if.

0

Figure 5 Relative voltage gain of the frequency
selective amplifier versus normalized
frequency for various Q values.

33

Figure 6 Some typical oscilloscope traces
obtained inthe pulse measure-
ments of pi and pg.

34

 

 

 

 

 

 

11 1111 1111
VVTY YYYY VTYY

 

 

“a

 

 

 

 

 

 

L111 1111 11
IVY YrT' 7

V

 

 

 

 

 

 

 

x n..N dth

.2336 0.. 32. 06:.

 

 

_.

TUV' 'TTV

:

 

 

 

 

11 1111 1111

 

 

 

 

 

 

'MP/A'r! as

 

m 952m.

'AlP/A‘rl 92

v. 0b.. . Eu...

«a

 

.333... so :2. 25»

'MP/A

  

 

:.

 

 

 

 

1 1111 1111

1' 'V"

 

 

 

 

 

 

 

1111 LL

“Wm" 09

 

1" 'YTY

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IIYT YVTV '1'

 

 

1
l
1
1
L
1
1
1
1

 

 

 

i. ii .

 

 

 

x 2... asfi guess no 32. as: x 8.. .33 22082 no as. as:

35

camera shutter closed. All pulse data were taken in this
manner.

With the Q of the tuned amplifier set at 1, the
received pulses were distorted only slightly and enough
filtering was obtained to eliminate most low frequency
noise due to vibrations. Subsequent reflections of the
p1' pulse were reduced considerably by drilling many small
holes through the carbon disk transmitter and backing.

’ would

This was necessary because the reflections of p1
interfere with the initial p2' pulse. The smaller pulse
appearing after the p2' pulse in some of the photographs
of Figure 6 was due to first sound or p1' waves being

produced at the receiver diaphragm when it was caused to

' pulse.

vibrate by the p2
2. Standing Wave Technique

In the standing wave technique the carbon disk trans-
mitter was driven continuously with ac current. In this
technique a solid transmitter with no holes drilled in it
was used. The frequency was varied until a standing wave
resonance of either p1' or p2' was observed. All measure-
ments were made with the fundamental resonance, which
corresponds to a wavelength equal to twice the length of

the cavity. Since u1 and u are temperature dependent,

2
the fundamental resonance frequency was not constant in
these measurements. For waves prOpagating at u2 the fre-
quency ranged from approximately"75 Hz to 190 Hz while

the range was approximately 1950 Hz to 2090 Hz for waves

 

36

propagating at u

1
It turned out that this method was satisfactory for

making quantitative measurements of p2' but not for p1'.
This is because other resonances, which are believed to
be higher harmonics of p2', were observed at frequencies
very close to the fundamental frequency of p1'. At some
temperatures it was difficult to distinguish these reso-
nances from one another. This also made measurement of
the Q of the cavity for the p1' resonances impossible.
As will be discussed below the measurement of the Q of
the cavity at each resonance is necessary in this tech-
nique.

When the frequency of a sound source in a small en-
closure equals one of the normal mode frequencies of the
enclosure, the amplitude of the pressure variations will
be directly prOportional to the output of the source and
inversely prOportional to a damping constant k which is

19 When the

due to losses from absorption and radiation.
driving frequency does not coincide with, but is close to,
the normal frequency the pressure variations build up
according to a standard resonance curve. The width of
this curve is prOportional to k.

In order that the temperature dependence of the out-
put of the source could be studied, the amplitude of the
fundamental resonance and the width of the frequency

response curve around each resonance were measured as a

function of temperature. The width of the resonance

 

37

curve was obtained by measuring the amplitude at points
off resonance and simultaneously measuring the frequency
of the driving oscillator with a Monsanto Model 100A
Counter/Timer. A typical resonance curve obtained in this
manner is shown in Figure 7. The resonance amplitude in
this case was 0.575 mV at a frequency fo : 127.0 Hz.
The ordinate of this graph is relative amplitude; that is,
the ratio of the amplitude at frequency f to the amplitude
at the resonant frequency f0. This method of determining
the damping constant k was rather tedious. A less time
consuming method, which will be described below, was there-
fore devised.

In this method the driving oscillator was tuned to
the resonant frequency but then gated on and off as in
the pulse technique. Now, however, many more cycles were
included so that the resonance could build up to its maxi-
mum amplitude before the driving signal was turned off.
Figure 8 shows some typical oscillosCOpe traces illus-
trating the response of the cavity to this type of driving
current. When the driving current is turned off the ampli-
tude decays with a characteristic exponential envelope
e'kt. From this type of photograph the damping constant
k and also the amplitude can be measured.

The signals shown in Figure 8 were the result of
amplification of the micrOphone output by the frequency

selective amplifier tuned to the resonant frequency and

 

38

 

 

 

|.0 —- o _
0.9 - .—
O.8 —- ' —
O

3

3 0.7 *- .1

t

_.l

0.

E 0.6 — —+

“J

?_

l- _ _

4 0.5 . o

_l

LIJ

O:
0.4 - —
0.3 —- _

O O

0.2 J l l l J

 

125.5 126.0 |26.5 122a 127.5 123.0 l28.5
FREQUENCY (Hz)

Figure 7 A typical frequency response curve of a
standing wave resonance of p’z at a
temperature of 2.06K.

39

Figure 8 Some typical oscilloscope traces
illustrating the response of the
cavity to a long tone burst

driving current at the resonant
p’z frequency

m 332“.

._. u: 0.09 now

 

QZ'O

'MP/Aw 93'0

  
 
   
 

A: as 2. 1..., so

  

._. u: 0.03» 0.0

 

41

with the Q set at 1. Higher amplifier Q-values could

not be used because of the longer time constants associated
with the filtering circuitry. However, in the first method
in which the width of the resonance curve was measured,

the Q of the amplifier could be set at 100.

B. Temperature Wave T2‘ Produced by a Microphone

Initially an attempt was made to observe the tempera-
ture waves produced by a micrOphone by merely interchanging
the transmitter and receiver in the apparatus already
described (Figure 1). The capacitor micrOphone was driven
with ac voltage and the carbon disk resistor was used as
a second sound receiver. When used for this purpose a
constant current I z 1.0 ma was maintained in the carbon
disk. This was accomplished by connecting it in series
with another resistor, R, much greater than the disk
resistance, and a dry cell. The temperature variations
in the second sound wave cause the resistance of the disk
to vary and this results in a signal voltage.

Extremely weak signals, barely visible above the
noise, were observed with this arrangement by using standing
waves. The signals were too weak for quantitative measure-
ments to be made. However, these waves were able to be
studied quantitatively by using a new type of second sound
transducer as receiver.

1. Porous Diaphragm Second Sound Transducer
This transducer was identical to the capacitor micro-

phone already described except that the aluminized Mylar

 

42

diaphragm was replaced by a diaphragm made from a porous
filter material.20 A conducting layer of aluminum was
vacuum deposited on one side of the filter.

This aluminum film was deposited on the filter with
a standard vacuum deposition apparatus consisting of a
high vacuum bell jar chamber at the base of which was
mounted a tungsten filament. The filaments were Edwards
High Vacuum Inc., Type A10 tungsten filaments. The filter
was mounted by means of a cardboard ring and masking tape
on a glass microsc0pe slide. The slide was then mounted
approximately four inches above the tungsten filament
with the filter facing the filament. Another microsCOpe
slide was mounted next to the one which was holding the
filter. The purpose of this second slide was to provide
a means of observing how much aluminum was being deposited.
A few strips of high purity aluminum wire were hung from
the tungsten filament.

The procedure used to deposit the aluminum film on
the filter was as follows: 1). The bell Jar was pumped
down to a vacuum of 10"4 micron. 2). The current to the
filament was switched on and increased to approximately
30 amps. At this current the aluminum strips would melt
and wet the whole filament in about 15 - 50 seconds.

3). The current was then increased to approximately 40
amps, while observing the plain micrOSCOpe slide through
the bell Jar. At a current of 40 amps this plain slide

would be coated with a totally reflecting layer of

43

aluminum in 5 - 10 seconds. 4). As soon as it was observed

that this layer was totally reflecting the current was
switched off. This provided a satisfactory conducting
layer of aluminum on the filter.

When the leads of an ohmmeter were applied at opposite
points of the periphery of a 47 mm diameter filter coated
in this way the measured resistance was 10 - 20 ohms.
Sherlock and Edwards21 have conducted tests of the flow
rate and bubble point pressure of the same type of filters
on which a conducting layer of gold had been vacuum
deposited. They found no significant reduction in either
the number of Open pores or the pore diameter. It is
assumed that this is also the case with the filters used
in this investigation.

The pore size of the filter material was Specified
by the manufacturer to be 0.45 1 0.02 micron. The capi-
llary pores occupy approximately 80% of the total filter
volume. If the normal fluid shear wave penetration depth,
8: (2'77/e1m))%,13 is much larger than the pore size, then
flow of the normal fluid through the pores is impeded.
However, the superfluid component, having zero viscosity,
can flow through the pores unimpeded.

As was shown previously, in the first approximation,
second sound consists of the normal fluid and superfluid
components oscillating out of phase in such a way that
the net density or pressure fluctuation is zero. Thus a

second sound wave impinging on the porous diaphragm will

 

44

exert a force on it.

It should be pointed out that this transducer will
also be sensitive to pressure waves. This must be taken
into account when interpreting signals received with it.

This type of transducer has been used to study the

velocity of second sound close to the >\,-point22

3

and also
to study second sound in He — 4He mixtures below 0.3 K.23
In these experiments the principle advantage of this type
of transducer was that it was an efficient transmitter and
receiver of second sound without introducing heating.
A detailed study of these transducers has been made by
Sherlock and Edwards.21
An experiment was devised to test the sensitivity of
this transducer. The details of this experiment will be
described in detail in Section IV-B below. The results
were that, in addition to the above mentioned desirable
prOperties, this transducer was also much more sensitive

to second sound temperature fluctuations than the carbon

resistance receiver described above.
|

T
2. Apparatus and Procedure for Neasurins 2

The investigation of the temperature waves produced
by a vibrating diaphragm was carried out with the experi-
mental chamber previously described (Figure 1) except that
the regular capacitor micrOphone replaced the carbon disk
as transmitter and the porous diaphragm transducer was
the receiver. The biasing circuit shown in Figure 3 was
also used with the porous diaphragm receiver in this experi-

ment.

45

Even with the increased sensitivity of the porous
diaphragm receiver the second sound waves produced by
the micrOphone could only be observed with the standing
wave technique. Again the amplitude of the fundamental
resonance and the damping constant were measured as a
function of temperature. The damping constant was deter-
mined by measuring the width of the frequency response
curve around each resonance. The alternate method of
observing the decay of a long tone burst could not be
used because the Q of the tuned amplifier had to be set
at 50 or higher in order to make accurate measurements
of the signal amplitude.

The regular capacitor microphone transmitter was
driven with 30 volt peak ac voltage. In addition a
1f0 volt dc bias voltage was applied in order to increase
the output. The fact that this dc bias will increase the
output can be seen as follows. Under the action of an
applied force F : Fosin(00t) the diaphragm of the micro—
phone will execute harmonic vibrations of frequencycu with

24
its velocity amplitude uO given by :

u : O (86)

where Zm is the mechanical impedance of the diaphragm and
Z is the acoustic radiation impedance of the liquid

r
helium. The dc bias voltage was again applied through

a large series resistor, as indicated in Figure 9, so that

46

 

 

 

R=2.0 M 0.22 ,
——"WW\v "
If
C: 7 f
-:_-.—- _i_ 2 5" OSCILLATOR
__ ’-
PVo= |80 volts MICROPHONE V ' 30 volts
TRANSMITTER

 

 

 

Figure 9 Biasing circuit for the capacitor
microphone used as a transmitter.

47

charge could be assumed to be constant. The purpose of
the 0.22}Lf capacitor in the circuit in Figure 9 is to
block the dc voltage from the oscillator. The impedance
of this capacitor is negligibly small compared to that of
the micrOphone. The force between the aluminized side of

the diaphragm and the back plate will be

2 2
F=__217_9—= ”TOO v2 (87)

A A

 

where Q is the charge, C the capacitance, A the area of

o
the diaphragm, and V the potential difference between the
aluminized side of the diaphragm and the back plate. In

this case V : VO+ V'sincdt so that

 

ancoe 2 2 2
F : ——-— [VD-t 2VOV'sin(wt)+(V') sin (wt)]
A
2 2
2W0 3 2 t
F : ____9_[V?+_(_\_,_)_ + 2V V'sin(wt)- (V ) cos(2wt):l,(88)
A O 2 O 2

2
o

The term 1TC§(V')2/A is just a constant and therefore

If v0: 0 then F :‘WCS(V')2/A - (’1TC (V')2/A)cos(2wt).
the micrOphone will produce pressure and temperature waves
whose frequency is double that of the driving voltage.

This is physically clear because both the positive and
negative half cycles of the driving voltage produce attrac-
tion between the diaphragm and back plate. However, if

V0 is several times larger than V' the coefficient of
sin(UJt) in Eq. 88 will be much larger than that of

cos(2cut). The microphone will then produce waves whose

 

48

frequency is equal to that of the driving voltage and whose

amplitude is much larger than those produced when vozo.

IV. RESULTS

A. Pressure Waves p1'

and p2' Produced by a Heater
1. Output Amplitude Data From MicrOphone
In the pulse technique the amplitudes of the pressure

pulses p1'

and p2' (Figure 6) were measured as a function
of temperature. The corrections to theaadata due to the
attenuation of first and second sound were negligibly
small. A plot of the data is shown in Figure 10. On

Figure 10 the squares represent p1 pulse amplitude measure-

ments and the circles p '

pulse amplitude measurements.
Also included on this plot are data from the standing
wave measurements of p2'. These are represented by the
triangles. The ordinate of this graph is the amplitude
of the ac output voltage of the micrOphone from the pulse
experiments. As will be discussed below, the standing
wave data have been reduced by a constant factor in order
that they could be displayed on the same graph as the
pulse data on Figure 10.

Figure 11 shows a separate plot of the actual pg'
standing wave resonance amplitude as a function of tempera-
ture. These amplitudes are, of course, larger than the

amplitudes of the p2' pulses. The damping constant of

each resonance was measured both by measuring the width

49

 

50

Figure IO The pressure wave amplitudes p; and p'2
produced by the carbon disk transmit-
ter, plotted versus temperature.

51

o. 8:2...
3: 823358.

N._

 

 

 

 

 

 

10¢
sow p...
a
10m. m
M.
10m_ ‘1
d
A
100N(
JO¢N
0mm

 

52

Figure II The p'zstanding wave resonance amp-
litude plotted versus temperature.

53

__ 2.6a

 

 

 

c: 3:25.“.sz
Nu o.~ B B E N.
a _ _ d _ _ _ a _ o
as
100
10..
d
a
3
12 w
n
a
3
1 ca
m
A
L3 (
1 o.»
1m.»
_ _ _ _ _ _ _ _ 0+

 

54

of the frequency response curve around each resonance
and by the decay of a long tone burst driving signal at
the resonant frequency. Figure 12 shows a plot of the
damping constant k measured by both of these methods.

The circles represent the experimental values of k
obtained by measuring the width of the frequency response
curve, (see Figure 7). In this case k =7T(f" - f') where
f' and f" are the frequencies at which the amplitude is
1K/2'times its value at resonance. The squares represent
the values of k obtained from the decay of a long tone
burst driving signal as illustrated in Figure 8. The
time it takes for the amplitude to decay to 1/e of its
maximum is equal to 1/k.

The resonance amplitude is directly preportional to
the source output and inversely prOportional to the damping
constant.24 Thus the amplitude data of Figure 11 must
be corrected with the data of Figure 12 in order to obtain
the temperature dependence of the source output from the
standing wave measurements. The data represented by tri-
angles in Figure 10 are the result of correcting the ampli-
tude data of Figure 11 for the variation of the damping
constant data in Figure 12 and then dividing by a constant
factor.

The values for the attenuation of second sound derived
from the damping constant data of the standing wave
measurements, (see Figure 12), are larger than the published

values.25-27 This is not surprising, however, because

 

55

Figure 12 The damping constant of p'a standing
waves plotted versus temperature.

56

 

 

l

 

l I

 

 

o
(I_oos)

«3 ¢ l0
.LNVLSNOO SNIdWVCI

LG LG 2.0 2.2
TEMPERATURE (K)

L4

|.2

Figure 12

 

57

in the standing wave experiments there are losses due

to reflection and radiation in addition to absorption

in the liquid. Experiments whose purpose it is to measure
the attenuation of second sound are specifically designed
to minimize the first two types of losses.

In Figures 10 - 12 the solid lines are merely smooth
curves drawn through the data points. In Figures 10 and 11
the units on the ordinate are microvolts and millivolts,
respectively, because the capacitor microphone was not
calibrated. The output voltage will be prOportional to
the pressure wave amplitude.

2. Temperature Amplitude To Produced by the
Carbon Disk Transmitter

Before the experimental p ' and p '

data of Figure
10 can be compared with the theoretical expressions of
Equations 63 and 64 the following must be considered:

As previously indicated in Section II, the heat transport

1 2
is associated with the normal component and is given by ’ :

i = pervn . (89)

The units of Q are energy per square centimeter per second.
An approximate relation between d and the amplitude of the
temperature variation in a second sound wave will now be
derived.
Solving Eq. 14 for (vn-vs) and letting u = u2 gives
SP

”231

-V :
Vn S

 

"32' . (90)

 

58

In the approximation that CL: 0 it was shown, in Section II,

that for a second sound wave

2 T82 Pg

, 1
2 C 1% <9)

 

 

and

j: PSVS + nVn = O o (92)

Solving Eq. 92 for v8 and substituting this into Eq. 90

gives
Ft ) Sf)
v 1 + = T ' . (93)
r1( F¥ nefg 2

Solving Eq. 93 for vn and using p 2 pa + 81 and Eq. 91

 

 

 

gives
S Cu
Vn : i T. : 2 T2' . (94)
'uZfia TS
Substituting Eq. 94 into Eq. 89 then gives
° __ I
Q, _pCu2T2 , (95)
or
T ' : Q . (96)

2
1° cue

In the measurements of p" and p2' the input power

to the carbon disk transmitter was constant. Therefore

Q was constant. In the earlier analysis of the pressure

waves produced by a fluctuating temperature source the

 

 

59

transmitter was considered to be a plane surface whose
temperature varied as T = Toexp(icut). However, because
the power input to the carbon disk transmitter was constant,
and because of Eq. 96, To is itself a function of tempera—
ture in this eXperiment. It is given, from Eqs. 66 and
96, by To = 6.430112.

The theoretical expression of Eq. 96 was verified
by doing the following experiment. In the experimental
chamber of Figure 1 the capacitor micrOphone was replaced
by a second carbon disk resistor identical to the trans-
mitter. This resistor served as a receiver of the second
sound wave T2' produced by the carbon disk transmitter.
A schematic diagram of the experimental apparatus in this
case is shown in Figure 13. A constant current I : 1.0 ma
was maintained through the receiver R by means of a dry
cell in series with a resistor R1 which was much larger

than R. The temperature fluctuation T2' causes the resis-

tance of the receiver to vary and this produces the output

voltage
dB
8 t: I“ T ' (97)
0 dT 2

The resistance of the receiver as a function of tempera-
ture, dR/dT, was obtained by measuring the dc voltage
across R due to the 1.0 ma dc current. This voltage was
measured with a Leeds and Northrup type K—3 potentiometer.

The result was that the average value of dR/dT from 1.2 K

6O

eTo AMPLIFIER
A

CARBON DISK
RECEIVER

TONE BURST
OSCILLATOR GENERATOR

'—__—:' CARBON fDISK “'2‘:
TRANSMITTER

 

 

 

 

       

A\\\\\\\\\\‘X
L\\\\\\\\\\\

 

 

 

 

 

Figure 13 A schematic diagram ot the apparatus
used to measure the second sound

amplitude Tg' produced by the carbon
disk transmitter.

61

to TX was approximately 70 Q/K.

Both the previously described pulse and standing wave
methods were used to investigate the temperature dependence
of T2'. The input power to the carbon disk transmitter
was again 25 mW/cm2 which is the same as was used in the
measurement of the pressure amplitudes p1' and p2' produced
by this transmitter.

The results of these measurements are shown in Figure 14.
The solid line is the theoretical curve of Q436u2 with
Q = 25 mW/cm2. The published measured values of density,
specific heat,29'30 and velocity of second sound31”33 were
used. The data represented by circles in Figure 14 are
measurements of T2’ obtained by the pulse method and the
squares represent standing wave measurements. Again, as
in the standing wave measurements of p2', the amplitude
of the fundamental T2' standing wave resonance, and the
damping constant k were measured as a function of tempera-
ture. The standing wave data on Figure 14 have been
corrected for variation of the damping constant k and have
been reduced by a constant factor so that they could be
plotted on the same scale with the pulse data.

' from the pulse measure-

The experimental values of T2
ments on Figure 14 were obtained from Eq. 97 and the measured
values of eo and I(dR/dT). The scale on the right hand
side of Figure 14 is labeled with the corresponding values
of eo in microvolts. Good agreement is indicated between
both the pulse and standing wave T2' data and the theoretical

curve.

 

62

Figure l4 The second sound amplitude Tg'
produced by the carbon disk .

transmitter, plotted versus
temperature.

S

e. 23:.

 

 

 

 

 

Ax. mmah<mwm2wh
%d _N “3“ a. O; .5 Q. n; *3 Q. N; ;v
. _ _ A _ a _ _ _
2H ml-dro 0..

leod
m... n
O 1onW
n d
I. 3
m . w
as... a
D 126.:
V .l
m m
n. IONOM
m w.
M icwom
N
ONT imde

INQO

m?" p _ _ l_ _ _ _, _ _ _ mno

 

64

3. Output Amplitude Data Corrected For

 

Variation of To.

Figure 15 shows the results of dividing the p1' data
in Figure 10 by the T2' data in Figure 14. Since the
temperatures of the data points in the two experiments
did not coincide, the points used for Figure 15 were taken
from smooth curves drawn through the data. In Figure 15
the solid line is the theoretical curve of p1'/TO-.:(1,pu1u:2
from Eq. 63. Since the capacitor microphone was not cali-
brated, the experimental data have been normalized to agree
with the theoretical curve at T : 1.4 K. In this way the
temperature dependence of the data could be compared with
the theory. Reasonably good agreement between the eXperi~
mental data and the theoretical curve is indicated.

Figure 10 shows a sudden sharp increase in p1' just
below the eroint. The temperature measurements in these
experiments were not sophisticated enough to determine
with great accuracy at what temperature this sharp increase

' occurred. However it was definitely below the

in p1
X-point. This phenomenon is not predicted by the theory
based on the linearized two-fluid hydrodynamic equations.
No satisfactory explanation for this phenomenon has been
found. It is believed to be associated with nonlinearities
near the )\-point.

The p1' amplitude continues to increase through the

‘proint and then levels off at a value which is much

higher than that below T): Thus a fluctuating heat source

65

Figure IS A plot of the p'I data on Figure IO
divided by the Tg' data on Figure l4.

66

NN

QN

C: 0.20.368.

m. use:

 

 

a _

4

1

Q.
_

Q.
_

 

 

 

 

67

is useful as a transmitter of ordinary sound in liquid
helium both above and below the :K-point. The generation
of sound waves by a fluctuating heat source has long been

38

known in gasesja’B? and liquids and is known as the
thermOphone effect.
Figure 16 shows the results of dividing the p2' data

in Figure 10 by the T ' data in Figure 14. Again the data

2
points in Figure 16 were obtained from smooth curves drawn
through the data in Figures 10 and 14 because the tempera-
tures of the data points from these two experiments did
not coincide. In Figure 16 the solid line is the theoretical
curve of p2'/To :: -apu22 from Eq. 65. The dashed line
represents a correction to Eq. 65 due to the viscosity
of the normal fluid and the coefficient of second viscosity
£2, This correction will be discussed in greater detail
in Section V-A below. Here the data have been normalized
to agree with the theoretical curve at the maximum in
order that the temperature dependence of the two could
be compared. From Figure 16 disagreement between the
experimental data and the theory is apparent.

This discrepancy between the temperature dependence
of the experimental p2' data and the theoretical curve
was not unexpected for the following reason. An inspection
of Eqs. 63 and 65 shows that the ratio of p2' to p ' should

1
be

 

68

Figure 16 A plot ofthe p'2 data on Figure IO
divided by the Tg' data on Figure l4.

69

Nu

m. 23E

C: 232359?

 

 

 

d

O.N m.—
_ a a

Q.
_

 

 

 

70

p u
1,23. . (98)
p u

l 1

In the temperature range of these investigations u2<:O.1 u
so that p2' should be less than 0.1 p1'. However, as can
be seen in Figure 10, the results of this investigation show
that p2' is the same order of magnitude as p1'.

In addition, the thermal eXpansion coefficient of

liquid helium changes sign at T = 1.17 K.28 Therefore,

according to Eqs. 63 and 65, the amplitudes p1 and p2'
should be zero at this temperature. Although this tempera-
ture could not quite be reached in the cryostat used in
these experiments, the results on Figures 10, 15 and 16
indicate that p2' is not zero at this temperature but
that p1' is.

Thus it appears that the experimental p1' data follow
the theoretical predictions but that there is a contribution

' which has not been accounted for.

to p
2 i
B. Temperature Wave T2 Produced by a MicrOphone

1. Sensitivity of the Porous Diaphragm Transducer

 

As previously indicated, the temperature wave TP'
produced by the vibrating diaphragm of a regular capacitor
micrOphone was observed by using, as a receiver, a trans-
ducer similar to a capacitor micrOphone. The active ele-
ment of this transducer was a porous diaphragm made from
metallized filter material. The sensitivity of this trans-

ducer to temperature waves was investigated by using it as

 

71

a receiver of the second sound waves produced by a carbon
disk transmitter.

The experimental chamber used in this experiment was
identical to the one used in the investigations of the

pressure waves p1 and p ' and described in Figure 1,

2
except that the diaphragm of the receiver was made from
the porous filter material. The same carbon disk transmitter
and electronics as described in Figure 3 and 4, respectively,
were used. The power input to the transmitter was again
25 mW/cmg. The amplitude of the second sound waves pro-
duced by the carbon disk heater was again measured in this
experiment by both the pulse and standing wave methods.

The results of these measurements are shown in Figure 17.
The solid line is again the theoretical curve of Q4bCu2
with i = 25 mW/cme. The data represented by circles in
Figure 17 are measurements obtained by the pulse method
and the squares represent standing wave measurements.
As in Figure 14, the ordinate on the left hand side of
Figure 17 is labeled with temperature fluctuations in
millikelvins. The ordinate on the right is labeled with
peak output voltage in millivolts from the porous diaphragm
transducer. The standing wave data in Figure 17 have
again been corrected for variation of the damping constant
and reduced by a constant factor so they could be plotted
on the same graph with the pulse data.

Comparison of the ordinates on the right hand sides

of Figures 14 and 17 shows that the porous diaphragm

 

72

Figure I? The output voltage from the porous
diaphragm transducer used as a
receiver of the second sound waves
produced by the carbon disk trans-
mitter, plotted versus temperature.

73

(AW) gOVl'IOA 1.11:!an

 

 

 

 

 

m
'0 (I: 3 g; N 2 E. a) e ON
I T l l l 1 I IF<_¢\i
*' oi
— 0.
N
- cg
- 0.
3'2
._ Ii :-
”we:
ms.
3:
cot—0
— -.<o-
cc“-
3.’
Lu
.—
t— ‘2
-— "2
p. 9.!
J l L 1 1 1 i 1 ..,
cc N a) o (D N 0 ¢ -
n. '0. «a 3’4 N. 3 3 0. 0 °
0 o o o o O o

()l‘“) NOILVOLOO‘IJ 3801V83dW31.

 

74

transducer was approximately 1000 times more sensitive

to second sound temperature fluctuations than the carbon

disk resistor. The data from these two experiments do, in
fact, constitute a calibration of the porous diaphragm
transducer. Thus the sensitivity in Volts/Kelvin of this
transducer can be obtained from the data in Figures 14 and 17.
Only a slight variation with temperature of this sensitivity
is indicated from these data.

' ,

2. Measurement of T2

 

Figure 18 shows the results of the standing wave
measurements of the second sound amplitude T2' produced
by the regular capacitor micrOphone and received by the
porous diaphragm transducer. The solid line is the theo-
retical curve of T2'/uo :ClTue/C from Eq. 54. The magni-
tude of u0 was not known so the experimental data have
been normalized to agree with the theoretical curve at
T : 1.4 K. In this way the temperature dependence of the
experimental data could be compared with that of the theo-
retical curve.

The dashed line in Figure 18 is the theoretical curve
of T2'/uO which results when the contribution from the
viscous forces at the chamber walls is taken into account.
These forces are due to the viscosity 7? of the normal
fluid component. This correction will be discussed in
more detail in Section V-B below. There is fair agreement
between the experimental data and the theoretical curve

at lower temperatures but a large discrepancy at higher

 

 

75

Figure I8 The second sound amplitude Té
produced by the capacitor micro-
phone , plotted versus temperature.

76

 

m. mesa
C: msaaosmafio...

m.
__ my.

 

 

 

 

 

 

 

 

77

temperatures.

In this experiment plane wave resonances corresponding
to waves which prOpagate at the velocity of first sound u1
were also observed. The amplitudes of these resonances
were an order of magnitude larger than the T2' resonances.
Since the porous diaphragm receiver will also respond to
pressure waves, it is not clear whether these are resonances
of p1' of Eq. 51, or of T1' of Eq. 52, or both. To answer
this question the relative sensitivity of the porous dia-
phragm transducer to pressure and temperature waves would
have to be known. This same question must, in fact, be
considered in connection with the T2' resonances. There
may be a contribution to these resonances from p2' of
Eq. 53.

This question of the relative sensitivity of the
porous diaphragm transducer to pressure and temperature
waves, and the contribution of the various wave modes to

the resonances observed in this experiment will be dis-

cussed in detail in Section V-B below.

 

V. DISCUSSION AND CONCLUSIONS

A. Pressure Waves p" and p2' Produced by a Heater

In the measurements of the pressure wave amplitudes

p ' and p

*1 ‘2
dence of p1' was in good agreement with the theoretical

’ produced by a heater the temperature depen-

expression of Eq. 63. However, these eXperimental results

 

indicate that there are contributions to p2' in addition

to that obtained by merely retaining the thermal expansion
coefficient in the two-fluid hydrodynamic equations.

Recall that the theoretical expressions, Eqs. 63 and 65,
were obtained from the set of hydrodynamic equations, 8—11,
in which no dissipative terms were included. The effect
of retaining some of these dissipative terms will now be
examined.

6
39 using the method of Lifshitz, has calcu-

Dingle,
lated the coupling coefficients when terms due to the
viscosity‘n of the normal fluid are included in the hydro-
dynamic equations. He took into account both the viscous
forces in the bulk liquid and also the forces at the walls
when the liquid is confined in a cylindrical tube. The
resulting contribution to the coupling coefficients due
to the viscous forces in the bulk liquid is negligibly

small compared to that due to 0.. This is also true for

78

 

79

the contribution due to the viscous forces at the tube

' waves at low frequencies and tempera-

walls, except for p2
tures below about 1.4 K.

We have extended the calculations by also including
in the hydrodynamic equations terms due to the coefficient
of second viscosity Q2 and the thermal conductivity
coefficientl<. The coefficient of second viscosity is an
order of magnitude larger than the normal fluid viscosity‘n

and contributes the largest term in the eXpression for

the attenuation of first sound.5 The attenuation of second

 

sound is due mainly to the thermal conductivity coefficient.5

The results of these calculations are that the con-
tribution to the coupling due to the thermal conductivity
coefficient is negligibly small compared to that of 0;.
This is true for both p1' and p2' waves at all temperatures
in the range of our investigations. The combined contri-
bution of 77 and £2 is also negligibly small compared to
that of 0., except for p2' waves at temperatures below
about 1.4 K.

When 7] and g2 are retained in the hydrodynamic equa-
tions l—A and the viscous forces at the walls of the tube
are taken into account, Eqs. 63 and 64 for the pressure

wave amplitudes p1 and p2' produced in liquid helium by

a fluctuating temperature source take the following form:

80
. c “2 1w 2(1+1)77P

10‘ :53“ [2172+ MW)" pr (eavfl pu‘u T° (99)

1 2(141) 7781 2

p2 GAVE“? the) Pr (2w)]pu2T° ' “00)

In Eqs. 99 and 100, r is the radius of the cylindrical

 

 

chamber. These results were obtained from the hydrodynamic
equations by the same procedure as was used in Section II
to obtain Eqs. 63 and 64.

Both terms which are added to a.in Eq. 99 are neg—
ligibly small compared to<1.. However in Eq. 100 the
first term, (Ciao/Bysug)(4nyj +Q2), is the same order of
magnitude as<:.for temperatures below about 1.4 K and at
the higher frequency of 10 kHz used in the pulse measure~
ments. Similarly the second term, (20(1+i)/qur)(th/2L0)%,
is the same order of magnitude as<1 below about 1.4 K and
at the low frequencies used in the standing wave measure—
ments.

The correction due to the first term at a frequency
of 10 kHz is represented on Figure 16 by the dashed line.
The correction due to the second term is about the same
order of magnitude at frequencies in the range from 75 Hz
to 190 Hz and has thus not been shown on this figure.

The combined effect of T) and Q2 does therefore pro-
vide a mechanism which results in the production, by a
fluctuating temperature source, of larger amplitude p2'

waves while not affecting the p1' amplitude. This

 

 

81

contribution increases with decreasing temperature and

is not zero at the temperature whereia,vanishes. Thus

the corrections due to this mechanism do shift the theo—
retical curves in the right direction; that is, toward
better agreement with the eXperimental data. However,

as can be seen in Figure 16, the magnitude of these cor-
rections is still much too small to bring about good agree-
ment between the experimental p2' data and the theory.

The values of'7)used in the above analysis were those
measured by Tough g; 9;.40 and the values of' €2,were those
calculated by Khalatnikov and Chernikova.41

Hofmann gt g;.9 have also observed a larger amplitude
p2' wave produced by a heater than predicted by the theory
based on the two-fluid hydrodynamic equations with CL
retained. They suggest that this might be due to the

' pressure

micrOphone receiver being more sensitive to p2
waves than to p1' pressure waves. This appears unlikely
since both waves were prepagated at the same frequency.
The micrOphone sensitivity would therefore depend upon
wavelength. We have not been able to devise a technique
whereby this speculation might be checked.

T

B. Temperature Wave 2 Produced by a MicrOphone

There are a number of factors which might explain

the discrepancy between the experimental T2' data and the

theoretical curve on Figure 18. We have found that the

contributions to T2' due to the normal fluid viscosity

and the coefficient of second viscosity in the bulk liquid

 

82

are negligible. However, the correction, calculated by
Dingle,39 due to the viscous forces at the walls of the
chamber should be considered. When this is done, the
eXpression for the temperature wave amplitude T2' pro-
duced by a vibrating diaphragm (see Eq. 54) takes the
following form:

QC 1 l M Tu
T2' = [on (+ )(7791 fl—g—u . (101)

s'rpr 2w 0 °

 

In Figure 18 the solid line is the theoretical curve
of T2'/uo =(1Tu2/C from Eq. 54 while the dashed line is
that of Eq. 101. As can be seen, the correction, although
not negligible, is much too small to explain the discrep-
ancy.

Another possible source of the discrepancy is the
fact that it has been assumed that the velocity amplitude
uO of the transmitter diaphragm is constant. This may
not be the case. As previously indicated, for a driving
force F : Foexp(icdt), the velocity amplitude of the dia-

24
phragm will be :
u =.______— (102)
In Eq. 102, Zm is the mechanical impedance of the

diaphragm and Zr is the acoustic radiation impedance of

the liquid helium.

 

83

The mechanical impedance of the diaphragm is given

2
21m : R+i ], (103)

where M is the mass of the diaphragm, S is the stiffness,

 

S (00
) R 4 10JM[1 (

 

R is the mechanical resistance which results in dissipation,
and uJ02 : S/M. The radiation impedance is prOportional
to pul.

The quantity/Du1 is not strongly temperature depen-
dent. It decreases monotonically from T = 1.2 K to TX.
with the total variation being about 8%. Thus if Z§$>Zm,
the velocity amplitude of Eq. 102 would vary little with
temperature.

We do not have quantitative knowledge of the parameters
of the microphone. Thus the relative amplitudes of Zm and

2 cannot be calculated. However some qualitative argu-

r
ments can be made which would indicate that the variation
of 2m in this experiment is also small. Investigations by
Kuhl, 23 31. 2 of the prOperties of capacitor microohones
similar to the ones used in these investigations indicate
that the resonant frequency, ujo, was 10 - 15 kHz or
higher. The frequencies used in the standing wave measure-
ments of T2' were in the range from 75 Hz to 190 Hz. It

is therefore probably safe to assume that¢u<31¢ooin these
measurements. At frequencies far from resonance it is

also usually possible to neglect the mechanical resis-

24
tance, R.

 

 

84

In this case Eq. 103 gives 2mg iMwog/w. If it is

also assumed that Zfii>zr then, from Eq. 102, u would be

0
prOportional to a). However, the resulting correction to
the T2' data due to the variation of frequency in the
standing wave measurements is still much too small to
substantially reduce the discrepancy between the data and
theory.

A final consideration which might be the cause of
the discrepancy between the T2' data and the theoretical
expression of Eq. 54 is the fact that the porous diaphragm

transducer will also be sensitive to pressure waves. This

 

was already mentioned in Section IV-B-2 where it was
pointed out that resonances of waves propagating at u1
were also observed in this experiment. The question is:
Are these resonances of p1' of Eq. 51 or of T1' of Eq. 52
or both? Actually a better way to put the question is:
What is the relative sensitivity of the porous diaphragm
transducer to pressure and temperature waves?

The same question is involved in the observation of
resonances corresponding to waves prOpagating at u .

2
Could there be a significant contribution to the received

signal due to the p2 waves of Eq. 53? Since Q.is very
small, Eqs. 53 and 54 would indicate that pg', which is
prOportional to C12, could be neglected in comparison to
TQ', which is prOportional to 0.. However this may not

be true if the porous diaphragm transducer is more sensi-

tive to pressure waves than to temperature waves.

 

85

We have not been able to arrive at a definite quan-
titative answer to these questions. However, certain
results obtained in our investigations and also the results
of the investigation by Sherlock and Edwards21 would seem
to indicate that the porous diaphragm transducer is a
more efficient receiver of temperature waves than of
pressure waves.

Recall the following two experiments which were per-
formed in this investigation. First a regular capacitor
microphone was used as the receiver in the experiment in
which the transmitter was a periodically heated carbon
disk resistor. In this case two pressure waves, one
prOpagating at u1 and the other at u2, were observed. In
a succeeding eXperiment, described in Section IV-B-l, the
same carbon disk transmitter was used but the receiver
was now the porous diaphragm transducer. In this case
only waves prOpagating at u2 were observed. Moreover the
output signal from this wave was two orders of magnitude
larger than that from the u2 wave in the first eXperiment.

In the first experiment the two waves were p1' and
p2' of Eqs. 63 and 65 because the regular capacitor micro-
phone is sensitive only to pressure waves. In the second
experiment the received signal could be due to either

' or T ' of Eqs. 65 and 66 respectively. However, if

2 2
the contribution of p2

P

to this signal were significant,

then we should also have been able to observe p1 waves

in this experiment.

 

 

86

The fact that we did not observe any waves prOpagating
at u1 in the second eXperiment, but did observe very strong
signals corresponding to waves prOpagating at u2, would
indicate that the porous diaphragm transducer is a more
efficient receiver of temperature waves than of pressure
waves. This conclusion is also indicated from the work
of Sherlock and Edwards.21 They used a configuration in
which both transmitter and receiver were porous diaphragm
transducers. They also observed only waves prOpagating
at u2.

These results indicate that the discrepancy between
the T2' data and the theoretical expression of Eq. 54 is
not due to a contribution from p2' of Eq. 53. However
it is still not clear what the relative contributions of
p1' and T1' of Eqs. 51 and 52 are to the u1 resonances
which were observed. In this case T1' is prOportional
to a.while p1' is not. Thus these resonances could still
be due mainly to p1'. The fact that the amplitude of these
resonances varied little with temperature would seem to
indicate that this is the case sincepu1 in Eq. 51 is
relatively independent of temperature compared to the
corresponding coefficient in Eq. 52.

A qualitative study of these u1 resonances was hin-
dered by the same problems (see Section III-A-2) which

prevented the use of the standing wave technique to study

the p1' waves produced by a heater. The exact nature of

 

87

the u1 waves in this experiment might be better understood

if they could be observed and studied with pulses.

 

 

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11.

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13.

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