Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses
Yuk Tung Liu and Kip S. Thorne
Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125
共Received 16 February 2000; published 20 November 2000兲
An analysis is given of thermoelastic noise 共thermal noise due to thermoelastic dissipation兲 in finite sized
test masses of laser interferometer gravitational-wave detectors. Finite-size effects increase the thermoelastic
noise by a modest amount; for example, for the sapphire test masses tentatively planned for LIGO-II and
plausible beam-spot radii, the increase is ⱗ 10 percent. As a side issue, errors are pointed out in the currently
used formulas for conventional, homogeneous thermal noise 共noise associated with dissipation which is ho-
mogeneous and described by an imaginary part of the Young’s modulus兲 in finite sized test masses. Correction
of these errors increases the homogeneous thermal noise by ⱗ 5 percent for LIGO-II-type configurations.
PACS number共s兲: 04.80.Nn, 05.40.⫺a
I. INTRODUCTION AND SUMMARY
Internal thermal noise is one of the most dangerous noise
sources in a laser interferometer gravitational wave detector
in the frequency range ⬃10 Hz to ⬃200 Hz. It is caused by
a fluctuational redistribution of thermal energy inside each of
the detector’s mirror-endowed test masses. This energy re-
distribution produces a fluctuational change of the test
mass’s shape and thence a change of the position of its mir-
rored face, which in turn mimics a gravity-wave-induced
motion of the test mass’s center of mass 关1兴.
The fluctuation-dissipation theorem 关2兴 describes a rela-
tionship between thermal noise and the energy dissipation
共entropy increase兲 that occurs inside the test mass, when the
front of the test mass is subjected to an oscillatory driving
force 关Eq. 共3兲 below兴. There are various types of internal
thermal noise, each one associated with a specific dissipation
mechanism. Until recently, gravitational-wave experimenters
have focused almost exclusively on homogeneous thermal
noise 关1兴—i.e., noise associated with all forms of dissipation
that are describable by an imaginary part of the Young’s
modulus which is homogeneous inside the test mass 共e.g.,
dissipation due to homogeneously distributed impurities and
dislocations兲. Thermoelastic dissipation 共dissipation due to
heat flow down temperature gradients, which are produced
by inhomogeneous compression and expansion of the test-
mass material兲 is not homogeneous; but until recently it was
thought that thermoelastic noise 共thermal noise associated
with thermoelastic dissipation兲 would be negligible in Laser-
Interferometer Gravitational Wave Observatory 共LIGO兲 test
masses, compared to homogeneous thermal noise.
Indeed, this is so in the fused silica test masses of LIGO-I
detectors—i.e. of the first detectors that will operate in LIGO
关3兴. However, a careful analysis late last year by Braginsky,
Gorodetsky and Vyatchanin 关4兴共BGV兲 showed rather con-
vincingly that for the sapphire test masses currently planned
for LIGO-II 共the second generation detectors in LIGO兲, ther-
moelastic noise will be significantly larger than homoge-
neous thermal noise, and in fact will be so large as to sig-
nificantly constrain the performance of LIGO-II detectors in
the frequency band between ⬃10 Hz and ⬃200 Hz.
The BGV computation of thermoelastic noise was based
on an idealization in which each test mass has an arbitrarily
large radius and length compared to the size of the light’s
beam spot on the mirrored test-mass face. In this limiting
case, BGV showed that the spectral density S
h
(f) of the
thermoelastic gravitational-wave noise scales as the inverse
cube of the beam-spot radius r
o
, S
h
⬀1/r
o
3
, so it is desirable
to make r
o
large. However, when r
o
is no longer small com-
pared to the test-mass size, the BGV analysis breaks down.
The principal purpose of this paper is to explore, quanti-
tatively, the sign and magnitude of that breakdown. As we
shall see, that breakdown 共i.e., finite size of the test masses兲
increases the thermoelastic noise; but for expected beam-
spot radii (r
0
ⱗ 3/10 the test-mass radius a), the increase is
modest (ⱗ 10 percent兲.
A second purpose of this paper is to show how the BGV
analysis of thermoelastic noise can be simplified consider-
ably; and 关adapting techniques due to Bondu, Hello and Vi-
net 关5兴共BHV兲兴, to show how to generalize the BGV analysis
to finite sized test masses.
A third purpose is to point out and correct errors in the
BHV formulas for homogeneous thermal noise in finite sized
test masses 共formulas that are currently used in designing test
masses and predicting the performance of gravitational wave
detectors兲. The corrections of the BHV formulas increase
homogeneous thermal noise by ⱗ5 percent for beam-spot
radii ⱗ 3/10 the test-mass radius a, and thus are primarily of
conceptual importance, not practical importance.
In Sec. II, we outline our method of computing ther-
moelastic noise, in Sec. III we use our method to verify the
BGV result for thermoelastic noise in the limit of arbitrarily
large test masses, in Sec. IV we compute the thermoelastic
noise in finite sized test masses and estimate the accuracy of
our analysis, in Sec. V we correct the errors in the BHV
computation of conventional, homogeneous thermal noise,
and in Sec. VI we make some concluding remarks.
II. METHOD OF CALCULATION
Our analysis of thermoelastic noise is a simplification of
one of the procedures developed by BGV: Appendix C of
Ref. 关4兴. The foundation of the analysis is Levin’s 关6兴 ‘‘di-
rect’’ method of computing thermal noise 共of which ther-
PHYSICAL REVIEW D, VOLUME 62, 122002
0556-2821/2000/62共12兲/122002共10兲/$15.00 ©2000 The American Physical Society62 122002-1
moelastic noise is a special case兲:
Levin begins by noting that the gravitational-wave detec-
tor’s laser beam reads out a difference of generalized posi-
tions q(t) of the detector’s four test masses, with each q
given by an average, over the beam spot’s Gaussian power
profile, of the normal displacement
␦
z⬅u
z
of the test-mass
face:
q⫽
冕
0
a
冕
0
2
e
⫺ r
2
/r
o
2
r
o
2
共
1⫺ e
⫺ a
2
/r
o
2
兲
␦
z
共
r,
兲
rd
dr
⯝
冕
0
a
冕
0
2
e
⫺ r
2
/r
o
2
r
o
2
␦
z
共
r,
兲
rd
dr. 共1兲
Here (r,
) are circular polar coordinates centered on the
beam-spot center 共which we presume to be at the center of
the circular test-mass face兲, r
o
is the radius at which the
spot’s power flux has dropped to 1/e of its central value, and
a is the test-mass radius. 共The factor e
⫺ a
2
/r
o
2
must be Ⰶ 1in
order to keep diffraction losses small, so we shall approxi-
mate 1⫺ e
⫺ a
2
/r
o
2
by unity throughout this paper.兲 Levin then
appeals to a very general formulation of the fluctuation-
dissipation theorem, due to Callan and Welton 关2兴, to show
that the test-mass thermal noise can be computed by the fol-
lowing thought experiment:
We imagine applying a sinusoidally oscillating pressure
P⫽ F
o
e
⫺ r
2
/r
o
2
r
o
2
cos
共
t
兲
共2兲
to one face of the test mass. Here F
o
is a constant force
amplitude,
⫽ 2
f is the angular frequency at which one
wants to know the spectral density of thermal noise, and the
pressure distribution 共2兲 has precisely the same spatial profile
as that of the generalized coordinate q, whose thermal noise
S
q
(f) one wishes to compute.
The oscillating pressure P feeds energy into the test mass,
where it gets dissipated by thermoelastic heat flow. We com-
pute the rate of this energy dissipation, W
diss
, averaged over
the period 2
/
of the pressure oscillations.
1
Then the
fluctuation-dissipation theorem states that the spectral den-
sity of the noise S
q
(f) is given by
S
q
共
f
兲
⫽
8k
B
TW
diss
F
o
2
2
共3兲
†Eq. 共2兲 of Ref. 关6兴‡; here k
B
is Boltzmann’s constant. The
interferometer’s gravitational-wave signal h(t) is the differ-
ence of the generalized positions q of the four test masses,
divided by the interferometer arm length L. Correspondingly
the contribution of the test-mass thermoelastic noise to the
gravitational-wave noise is 1/L
2
times the sum of S
q
(f) over
the four test masses 共which might have different beam-spot
sizes and thus different noises兲:
S
h
共
f
兲
⫽
兺
A⫽1
4
S
q
A
共
f
兲
L
2
. 共4兲
The rate W
diss
of thermoelastic dissipation is given by the
following standard expression †first term of Eq. 共35.1兲 of
Landau and Lifshitz 关7兴, cited henceforth as LL‡:
W
diss
⫽
冓
TdS
dt
冔
⫽
冓
冕
T
共
ⵜ
ជ
␦
T
兲
2
rd
drdz
冔
. 共5兲
Here the integral is over the entire test-mass interior using
cylindrical coordinates; T is the unperturbed temperature of
the test-mass material and
␦
T is the temperature perturbation
produced by the oscillating pressure; dS/dt is the rate of
increase of the test mass’s entropy due to the flux of heat
⫺
ⵜ
ជ
␦
T flowing down the temperature gradient ⵜ
ជ
␦
T,
is
the material’s coefficient of thermal conductivity, and
具
•••
典
denotes an average over the pressure’s oscillation pe-
riod 1/f ⫽ 2
/
. 共For conceptual clarity we explicitly write
the average
具
•••
典
throughout this paper, even though in
practice it gives just a simple factor
具
cos
2
t
典
⫽1/2.兲
To compute the thermal noise, then, we must calculate the
oscillating temperature perturbation
␦
T(r,
,z,t) inside the
test mass, perform the integral 共5兲 over the test-mass interior
and the time average to obtain the dissipation rate, then plug
that rate into Eq. 共3兲 and then Eq. 共4兲.
The computation of the oscillating temperature perturba-
tion is made fairly simple by two well-justified approxima-
tions 关4兴:
First: The radius and length of the test mass are a⬃H
⬃14 cm and the speeds of sound in the test-mass material
are c
s
⬃5 km/s, so the time required for sound to travel
across the test mass is
sound
⬃30
s, which is ⬃300 times
shorter than the gravitational-wave 共and pressure-oscillation兲
period
gw
⫽ 1/f⬃0.01 s. This
sound
Ⰶ
gw
means that we
can approximate the oscillations of stress and strain in the
test mass, induced by the oscillating pressure P,asquasi-
static. It seems reasonable to expect this approximation to
produce a fractional error
quasistatic
ⱗ
sound
gw
⫽
f
f
sound
⬃
1
300
共6兲
in our final answer for the thermoelastic noise S
q
(f). Here
f
sound
⫽
1
sound
⬃
c
s
min
共
a,H
兲
⯝30 000 Hz 共7兲
for the currently contemplated LIGO-II test masses: sapphire
with a⬃H⬃14 cm.
Second: The time scale for diffusive heat flow to alter the
temperature distribution,
T
⬃C
V
r
o
2
/
⬃100 s, is ⬃10
4
times longer than the pressure-oscillation period
gw
共here
C
V
⯝7.9⫻ 10
6
erg g
⫺ 1
K
⫺ 1
is the specific heat per unit
1
It is here that our analysis is simpler than that of BGV. Instead of
computing W
diss
and using Eq. 共3兲 for the thermal noise, BGV com-
pute the imaginary part I(
) of the test-mass susceptibility
共which is much harder to compute than W
diss
) and then evaluate S
q
in terms of I(
) 关their Eq. 共14兲兴.
YUK TUNG LIU AND KIP S. THORNE PHYSICAL REVIEW D 62 122002
122002-2
mass at constant volume,
⯝4.0 g/cm
3
is the density, r
o
⬃4 cm is the spot size and
⯝4.0⫻ 10
6
erg cm
⫺ 1
s
⫺ 1
K
⫺ 1
is the thermal conductiv-
ity, and our values are for a sapphire test mass兲. This
T
Ⰷ
gw
means that, when computing the oscillating tempera-
ture distribution, we can approximate the oscillations of
stress, strain and temperature as adiabatic 共negligible diffu-
sive heat flow兲. The only place that heat flow must be con-
sidered is in the volume integral 共5兲 for the dissipation. The
dominant contribution to that volume integral will come
from a region with radius ⬃r
o
and thickness ⬃r
o
near the
beam spot. The region of the integral in which the adiabatic
approximation breaks down is predominantly a thin ‘‘bound-
ary layer’’ near the beam spot with radius r
o
and thickness of
order the distance that substantial heat can flow in a time
⬃
gw
⫽ 1/f, i.e., thickness of order
r
heat
⫽
冑
C
V
f
⯝0.4 mm
冑
100 Hz
f
for sapphire.
共8兲
This region of adiabatic breakdown is a fraction ⬃r
heat
/r
o
of
the region that contributes substantially to the integral, so we
expect a fractional error
adiabatic
⬃
r
heat
r
o
⬃0.01 共9兲
in S
q
(f) due to breakdown of the adiabatic approximation.
The quasistatic approximation permits us, at any moment
of time t, to compute the test mass’s internal displacement
field u
ជ
, and most importantly its expansion
⌰⫽ ⵜ
ជ
•u
ជ
, 共10兲
from the equations of static stress balance †Eq. 共7.4兲 of LL
关7兴‡
共
1⫺ 2
兲
ⵜ
2
u
ជ
⫹ ⵜ
ជ
共
ⵜ
ជ
•u
ជ
兲
⫽ 0 共11兲
共where
is the Poisson ratio兲, with the boundary condition
that the normal pressure on the test-mass face be P(r,t) 关Eq.
共2兲兴 and that all other non-tangential stresses vanish at the
test-mass surface. Once ⌰ has been computed, we can evalu-
ate the temperature perturbation
␦
T from the law of adia-
batic temperature change †Eq. 共6.5兲 of LL 关7兴‡
␦
T⫽
⫺
␣
l
ET
C
V
共
1⫺ 2
兲
⌰; 共12兲
here
␣
l
is the linear thermal expansion coefficient, E is
Young’s modulus and C
V
is the specific heat per unit mass at
constant volume.
2
This temperature perturbation can then be
plugged into Eq. 共5兲 to obtain the dissipation W
diss
as an
integral over the gradient of the expansion
W
diss
⫽
T
冉
E
␣
l
共
1⫺ 2
兲
C
V
冊
2
冓
冕
共
ⵜ
ជ
⌰
兲
2
rd
drdz
冔
.
共13兲
This W
diss
can be inserted into Eq. 共3兲 to obtain the ther-
moelastic noise.
III. INFINITE TEST MASSES
A. Dissipation and noise computed via BGV techniques
We illustrate the above computational procedure by using
it to verify the BGV 关4兴 result for thermoelastic noise in the
case where each test mass is arbitrarily large compared to the
spot size.
Following BGV, we approximate the test mass as an in-
finite half space. Then the solution to the quasistatic stress-
balance equation 共11兲 is given by a Green’s-function expres-
sion 关LL Eq. 共8.18兲 with F
x
⫽ F
y
⫽ 0, F
z
⫽ P(r,
)兴,
integrated over the surface of the test mass. Taking the di-
vergence of that expression 关or, equivalently, taking the di-
vergence of Eq. 共39兲 of BGV兴, we obtain the following equa-
tion for the pressure-induced expansion:
⌰⫽⫺
共
1⫹
兲
共
1⫺ 2
兲
F
o
2
r
o
2
E
cos
共
t
兲
⫻ z
冕冕
⫺ ⬁
⫹ ⬁
dx
⬘
dy
⬘
e
⫺ (x
⬘
2
⫹ y
⬘
2
)/r
o
2
关共
x⫺ x
⬘
兲
2
⫹
共
y⫺ y
⬘
兲
2
⫹ z
2
兴
3/2
,
共14兲
where we have converted from polar coordinates to Cartesian
coordinates. Following a clever procedure implicit in the
BGV analysis 关in going from their Eq. 共39兲 to 共40兲兴,we
insert into the integral 共14兲 an integral of the Dirac delta
function written as
冕
⫺ ⬁
⫹ ⬁
␦
共
x⫺ x
⬘
⫺ x
⬙
兲
dx
⬙
⫽
1
2
冕冕
⫺ ⬁
⫹ ⬁
e
ik
x
(x⫺ x
⬘
⫺ x
⬙
)
dk
x
dx
⬙
共15兲
and a similar expression for
兰
␦
(y⫺ y
⬘
⫺ y
⬙
)dy
⬙
, and we re-
write x⫺ x
⬘
and y⫺ y
⬘
in the denominator as x
⬙
and y
⬙
,
thereby obtaining a new version of Eq. 共14兲 with integrals
over k
x
,k
y
,x
⬘
,y
⬘
,x
⬙
,y
⬙
. The integrals over x
⬘
,y
⬘
,x
⬙
,y
⬙
are
then readily carried out analytically 共they are well-known
Fourier transforms兲, to yield Eq. 共40兲 of BGV:
3
⌰⫽⫺
共
1⫹
兲
共
1⫺ 2
兲
F
o
2
2
E
cos
t
⫻
冕冕
⫺ ⬁
⫹ ⬁
e
⫺ k
⬜
2
r
o
2
/4
e
⫺ k
⬜
z
e
i(k
x
x⫹ k
y
y)
dk
x
dk
y
, 共16兲
2
LL use the volumetric thermal expansion coefficient
␣
⫽ 3
␣
l
and
the specific heat per unit volume C
v
⫽
C
V
.
3
Note that our notation differs slightly from that of BGV: Our x is
their z, our z is their x, and they have factored out the cos
t, which
they write as e
i
t
.
THERMOELASTIC NOISE AND HOMOGENEOUS THERMAL . . . PHYSICAL REVIEW D 62 122002
122002-3
where k
⬜
⬅
冑
k
x
2
⫹ k
y
2
.
It is straightforward to take the gradient of this expres-
sion, square it 共with one term an integral over k
x
,k
y
and the
other over k
x
⬘
,k
y
⬘
), and integrate over x and y 共from ⫺ ⬁ to
⫹ ⬁) and over z 共from 0 to ⬁); the result is
兰
(ⵜ
ជ
⌰)
2
dxdydz
expressed as an integral over x,y,z,k
x
,k
y
,k
x
⬘
,k
y
⬘
. The inte-
grals can be done easily, first over z to get 1/(k
⬜
⫹ k
⬜
⬘
), then
over x and y to get Dirac delta functions, then over the k’s.
The result, when inserted into Eq. 共13兲,is
W
diss
⫽
共
1⫹
兲
2
␣
l
2
T
冑
2
C
V
2
2
r
o
3
F
o
2
. 共17兲
By then inserting this into Eq. 共3兲, we obtain the BGV result
for the thermoelastic noise 关their Eq. 共12兲兴
S
q
ITM
共
f
兲
⫽
8
共
1⫹
兲
2
␣
l
2
k
B
T
2
冑
2
C
V
2
2
r
o
3
2
. 共18兲
Here the superscript ITM means for an ‘‘infinite test mass.’’
B. Derivation via BHV techniques
Equation 共17兲 for W
diss
can also be derived in cylindrical
coordinates (r,z,
) using the techniques of BHV 关5兴: The
displacement u
ជ
has components 关BHV Eqs. 共5兲 and 共6兲 with
the denominator in Eq. 共5兲 corrected from
to
⫹ and
with

⫽
␣
; see passage following BHV Eq. 共8兲兴
u
r
⫽
冕
0
⬁
␣
共
k
兲
冉
1⫺
⫹ 2
⫹
⫹ kz
冊
e
⫺ kz
J
1
共
kr
兲
kdk,
u
z
⫽
冕
0
⬁
␣
共
k
兲
冉
1⫹
⫹
⫹ kz
冊
e
⫺ kz
J
0
共
kr
兲
kdk,
共19兲
u
⫽ 0,
where
␣
共
k
兲
⫽
e
⫺ k
2
r
o
2
/4
4
k
F
o
cos
t 共20兲
关BHV Eq. 共11兲, with the overall sign corrected from ⫺ to ⫹ ,
with w
o
⫽
冑
2r
o
cf. BHV Eq. 共2兲, and with F
o
cos
t inserted
because our method of applying the fluctuation-dissipation
theorem is dynamical while BHV’s method is static and
BHV set F
o
⫽ 1]. In Eqs. 共19兲 the J
n
are Bessel functions
and and
are the Lame
´
coefficients 共and
is also the
shear modulus兲, which are related to the Young’s modulus E
and the Poisson ratio
by
⫽
E
共
1⫺ 2
兲
共
1⫹
兲
,
⫽
E
2
共
1⫹
兲
. 共21兲
The divergence of the displacement 共19兲 is
⌰⫽⫺
2
⫹
冕
0
⬁
␣
共
k
兲
e
⫺ kz
J
0
共
kr
兲
k
2
dk. 共22兲
The nonzero components of the gradient of this expansion
are
⌰
r
⫽
2
⫹
冕
0
⬁
␣
共
k
兲
e
⫺ kz
J
1
共
kr
兲
k
3
dk, 共23a兲
⌰
z
⫽
2
⫹
冕
0
⬁
␣
共
k
兲
e
⫺ kz
J
0
共
kr
兲
k
3
dk.
共23b兲
By squaring the gradient, integrating over the interior of the
test mass, and using the relations
冕
0
⬁
J
n
共
kr
兲
J
n
共
k
⬘
r
兲
rdr⫽
␦
共
k⫺ k
⬘
兲
k
共24兲
共which follow from the Fourier-Bessel integral兲, and by re-
placing the Lame
´
coefficients by the Poisson ratio and
Young’s modulus 关Eqs. 共21兲兴, and inserting the resulting
兰
(ⵜ
ជ
⌰)
2
rd
drdz into expression 共13兲, we obtain the same
result 共17兲 as we got using BGV techniques. By inserting this
into Eq. 共13兲, we obtain the thermoelastic noise 共18兲.
IV. FINITE SIZED TEST MASSES
A. BHV solution for displacement
Consider a finite sized, cylindrical test mass with radius a
and thickness H, and with the Gaussian shaped light spot
centered on the cylinder’s circular face. For this case, Bondu,
Hello and Vinet 共BHV兲关5兴 have constructed a rather accu-
rate but approximate solution of the static elasticity equa-
tions. Unfortunately, their solution satisfies the wrong
boundary conditions and thus must be corrected:
The error arises when BHV expand the Gaussian-shaped
pressure 共2兲 as a sum over Bessel functions. BHV incorrectly
omit a uniform-pressure term from the sum. As a result, the
pressure that they imagine applying to the test-mass face
关their Eq. 共18兲兴,
P
BHV
共
r
兲
⫽ F
o
cos
t
兺
m⫽ 1
⬁
p
m
J
0
共
k
m
r
兲
共25兲
关where J
0
is the Bessel function of order zero, k
m
is related
to the m’th zero
m
of the order-one Bessel function J
1
(x)
by k
m
⫽
m
/a, and p
m
are constant coefficients given below兴,
has a vanishing surface integral
冕
0
a
P
BHV
2
rdr⫽ 0. 共26兲
In other words, their applied pressure 共25兲 is equal to the
desired pressure P(r) 关Eq. 共2兲兴 minus an equal and opposite
net force F
o
cos(
t) applied uniformly over the test-mass
face:
P
BHV
共
r
兲
⫽ P
共
r
兲
⫺ p
0
F
o
cos
t; 共27兲
p
0
⬅
1
a
2
. 共28兲
YUK TUNG LIU AND KIP S. THORNE PHYSICAL REVIEW D 62 122002
122002-4
关Recall that we are approximating 1⫺ e
⫺ a
2
/r
0
2
by unity; see
discussion following Eq. 共1兲.兴
It is evident, then, that to get the correct distribution of
elastic displacement u
ជ
inside the test mass, we must add to
the BHV displacement a correction. This correction is the
displacement caused by the spatially uniform pressure
p
0
F
o
cos
t on the test-mass face. That uniform pressure
causes the test mass to accelerate with acceleration a
ជ
⫽
关
(F
o
cos
t)/M
兴
e
ជ
z
, where M⫽
a
2
H
is the mass of the
test mass and
is its density. In the reference frame of the
accelerating test mass, all parts of the test mass feel a ‘‘gravi-
tational’’ acceleration ge
ជ
z
equal and opposite to a
ជ
, i.e. g⫽
⫺ (F
o
cos
t)/M 共which can be treated as quasistatic, though
it oscillates at frequency
). Thus, the displacement is the
same as would occur if the test mass were to reside in the
gravitational field ge
ជ
z
with a uniform pressure on its face
counteracting the force of gravity. The solution for this dis-
placement is given by LL 关7兴共problem 1, p. 18兲.
4
Translating
into our notation and converting from the Young’s modulus
and Poisson ratio to the Lame
´
coefficients via Eq. 共21兲,we
obtain
␦
u
r
F
o
cos
t
⫽
p
0
r
2
共
3⫹ 2
兲
冉
1⫺
z
H
冊
, 共29a兲
␦
u
z
F
o
cos
t
⫽
p
0
r
2
4
H
共
3⫹ 2
兲
⫺
共
⫹
兲
p
0
共
3⫹ 2
兲
冉
z⫺
z
2
2H
冊
.
共29b兲
The total corrected displacement, in cylindrical coordi-
nates, is
u
r
⫽ u
r
BHV
⫹
␦
u
r
, u
z
⫽ u
z
BHV
⫹
␦
u
z
, u
⫽ 0, 共30兲
where u
j
BHV
is the BHV displacement 关their Eqs. 共15兲 plus
共25兲 and 共17兲 plus 共26兲兴:
u
r
BHV
共
r,z
兲
F
o
cos
t
⫽
⫹ 2
2
共
3⫹ 2
兲
共
c
0
r⫹ c
1
rz
兲
⫹
兺
m⫽ 1
⬁
A
m
共
z
兲
J
1
共
k
m
r
兲
, 共31a兲
u
z
BHV
共
r,z
兲
F
o
cos
t
⫽⫺
共
3⫹ 2
兲
冉
c
0
z⫹
c
1
z
2
2
冊
⫺
⫹ 2
4
共
3⫹ 2
兲
c
1
r
2
⫹
兺
m⫽ 1
⬁
B
m
共
z
兲
J
0
共
k
m
r
兲
, 共31b兲
u
BHV
共
r,z
兲
⫽ 0. 共31c兲
Here the coefficients c
0
and c
1
are 关equations following Eqs.
共24兲 and 共26兲 of BHV兴
c
0
⫽ 6
a
2
H
2
兺
m⫽ 1
⬁
J
0
共
m
兲
p
m
m
2
, c
1
⫽
⫺ 2c
0
H
, 共32兲
and A
m
and B
m
are the following functions of z 关Eqs. 共19兲
and 共20兲 of BHV兴:
A
m
共
z
兲
⫽
␥
m
e
⫺ k
m
z
⫹
␦
m
e
k
m
z
⫹
k
m
z
2
⫹
⫹ 2
共
␣
m
e
⫺ k
m
z
⫹

m
e
k
m
z
兲
共33兲
B
m
共
z
兲
⫽
冋
⫹ 3
2
共
⫹ 2
兲

m
⫺
␦
m
册
e
k
m
z
⫹
冋
⫹ 3
2
共
⫹ 2
兲
␣
m
⫹
␥
m
册
e
⫺ k
m
z
⫹
k
m
z
2
⫹
⫹ 2
共
␣
m
e
⫺ k
m
z
⫺

m
e
k
m
z
兲
, 共34兲
where
␣
m
,

m
,
␥
m
and
␦
m
are constants given by 关Eqs.
共21兲–共24兲 of BHV兴:
Q
m
⫽ exp
共
⫺ 2k
m
H
兲
共35a兲
␣
m
⫽
p
m
共
⫹ 2
兲
k
m
共
⫹
兲
1⫺ Q
m
⫹ 2k
m
HQ
m
共
1⫺ Q
m
兲
2
⫺ 4k
m
2
H
2
Q
m
共35b兲

m
⫽
p
m
共
⫹ 2
兲
Q
m
k
m
共
⫹
兲
1⫺ Q
m
⫹ 2k
m
H
共
1⫺ Q
m
兲
2
⫺ 4k
m
2
H
2
Q
m
共35c兲
␥
m
⫽⫺
p
m
2k
m
共
⫹
兲
⫻
关
2k
m
2
H
2
共
⫹
兲
⫹ 2
k
m
H
兴
Q
m
⫹
共
1⫺ Q
m
兲
共
1⫺ Q
m
兲
2
⫺ 4k
m
2
H
2
Q
m
共35d兲
4
LL seek to solve a problem in which 共in the presence of the
uniform gravitational acceleration兲, instead of having a uniform
pressure applied to the face of the cylindrical test mass, the face has
vanishing displacement. Their solution actually satisfies our desired
boundary conditions but not theirs; therefore, they comment on it
being inaccurate near the test-mass face. For our problem it is ac-
curate.
THERMOELASTIC NOISE AND HOMOGENEOUS THERMAL . . . PHYSICAL REVIEW D 62 122002
122002-5
