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=1-I2944 NUCLEAR WERPON EFFECT RESEARCH AT PSR (PACIFIC-SIERRA vRESEARCH CORPORRT 10.. (U) PACIFIC-SIERRA RESEARCH CORPLOS ANGELES CA L SCHLESSINGER 29 SEP 94 PSR-1422-YOL-9
UNCLSSIFIED DN-TR-84-369-V8 DNASS±-93-C-SS15 F/O 19/3 N
Lm hEhh
1-0-
125 111111-4 1
NATIOMA 9LUEAI Of STANDARSWKSOCOPY ESWLUTMC TEST CMAT
9AA NA-TR-84-308-V8
NUCLEAR WEAPON EFFECT RESEARCH AT PSR-1983Volume Vill-Effect of Shock-induced Ground Conductivity on KratzParticle Velocity Gauge
* Leonard Schiessinger* Pacific-Sierra Research Corporation
12340 Santa Monica BoulevardLos Angeles, CA 90025-2587
* 28 September 1984
Technical Report
CONTRACT No. DNA 001-83-C-0015
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11. TITLE (Include Security Classification)NUCLEAR WEAPON EFFECT RESEARCH AT PSR-1983. .
Volume VIII-Effect of Shock-Induced Ground Conductivity on Kratz Particle Velocity Gauge -
12. PERSONAL AUTHOR(S)Leonard Schl essi nger
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18Underground Test Nuclear Test Instrumentation 419.TCTCtnuo rees if1 Craterino MINI JADE
19.ABSRAC (Cryinu onrovrs, i neessryand identify by blocknubr
This report details the effect of shock-induced conductivity on the operation of the .Kratz velocity gauge, which measures the ground particle velocity induced by the shockwave due to a large-yield high-explosive or underground nuclear detonation. Such shockwaves are typically strong enough to induce significant ground conductivity, which canaffect the operation of the gauge. Three models of increasing fidelity to the physicsof the advancing conductivity front are presented and analyzed. Numerical resultsindicate that, for typical gauge parameter values, shock-induced ground conductivitiesof 100 mhos/m or less do not affect gauge response.
20 DISTRIBUTION/I AVAILABILITY OF ABSTRACT I21 ABSTRACT SECURITY CLASSIFICATIONUNCLASSIFIEOUNLIMITED fl SAME AS RPT 03 OTIC USERS IUNCLASSIFIED
22a. NAME OF RESPONSIBLE INDIVIDUAL 22b. TELEPHONE (include Area Cod*) 22c OFFICE SYMBOL . -.
Betty L. Fox (202) 325-7042 DNA/STTID0 FORM 1473.684 MAR 83 APR edition may be used until exhausted SECURITY CLASSIFICATION OF THIS PAGE
All other editions are obsolete.UNLSIIE UNLSSFE
SUMMARY ,..
This report details the effect of shock-induced conductivity on the
operation of the Kratz velocity gauge, which measures the ground particle
velocity induced by the shock wave due to a large-yield high-explosive or
underground nuclear detonation. The Kratz gauge operates by sensing the
time rate of change of the magnetic field produced by a field coil embedded
in and ideally moving with the medium. The shock wave from a large-yield
detonation can induce significant conductivity in the ground, which can in
turn affect gauge response both before and after the shock wave passes the
field coil.
Three models of increasing fidelity to the physics of the advancing
conductivity front are presented and analyzed, the third being the most
complex and most accurate. Numerical results presented for model 3 indi-
cate that, for typical gauge parameter values, shock-induced ground con-
ductivities of 100 mhos/m or less do not affect gauge response.
'"' ."-",-4
7--11
. . . . .
.- .-'..'-. ,.j,'-. -...-. .- °- .- °-,.,, .. =¢ .,..'°- .'.' .. " -.-.... . '. '.. -'...-'.' .'.. -, .'.- . -. -. -.- , -. . -. -. ,-,-.. -
WR A - - 51 - - C -1. W 1707 -3. 7..-kIL- RM . .-* , .
%. "o '. ,eop
PREFACE '"
V
. q lThis report is one of a multivolume set comprising the Pacific-Sierra
Research Corporation (PSR) final report on Defense Nuclear Agency (DNA)
contract DNA001-83-C-0015. The work done under this contract spans a
wide range of nuclear weapon effect research covering airblast, cratering
and ground motion, intermediate-dose radiation, underground test design .- '.
* and development, fire research, and electromagnetic pulse research.
This volume details the effect of shock-induced conductivity on the
operation of the Kratz particle velocity gauge. The project officer was -
Michael J. Frankel, and the technical monitor was David L. Auton.
2
P.. ..-. ... '--"-".
a.. - -. o .- . .
*.,.~p j.,.O'o-.* -.. -.... °<°
... .. ........... .. ............... °,'.°°
TABLE OF CONTENTS
Section Page
- SUMMARY ......................................................
PREFACE.....................................................2
1 INTRODUCTION .............................................. 5
2 KRATZ GAUGE OPERATION AND THREE MODELS ... o..................6 - :
3 ANALYSIS OF MODELS 1 AND 2.....................................Derivation of basic equations................. o.............9Green's function ....o........................0..............10Evaluation of voltage in pickup coil ..................... 13
4 ANALYSIS OF MODEL 3 ...................o.........o.............15
5 NUMERICAL RESULTS.....................................o......23
REFERENCES ...........................o.........o..............27
APPENDIX
LIST OF SYMBOLS.............................................29 '
3
w J.....................- 2
SECTION 1 .. -
INTRODUCTION
The Kratz particle velocity gauge [Kratz, 19761 is designed to mea-
sure the particle velocity in the ground induced by a shock wave due to
chemical or nuclear explosives. It has been successfully used in several .
high-explosive tests as well as the nuclear underground tests (UGTs)
HURON LANDING and MINI JADE. It is to be deployed on several other UGTs,
including MISTY JADE. The Kratz gauge is typically used to measure ve-3 2
locities ranging from 3 x 10 m/sec (peak) to 8 x 102 m/sec (peak), decay-
ing with a time constant of 10 to 100 psec. The period over which mea-
surements are made is a few time constants.
Both Kratz [1976] and Schlessinger [1981] presented and evaluated
the basic theory for calibrating the gauge and quantifying its operation,
but neither completely modeled the electrical conductivity induced in the
medium by the advancing shock wave. Schlessinger demonstrated, though
only approximately, that the shock-induced conductivity would not signifi-
cantly influence the gauge response for a wide range of relevant gauge
parameter values. However, the high quality of data recently obtained
from the gauge (Coleman, 1982, 1983] may permit us to identify and quantify
even the small changes in gauge response caused by the shock-induced con-
ductivity. A better theory to describe those small signals could explain
the data more thoroughly.
This report details the effect of shock-induced conductivity on
Kratz gauge operation. Because the problem is so complex, we present and
analyze three models of increasing fidelity to the physics of the moving
conductivity front, the third being the most complex and most accurate.
For model 3, we give numerical results that can be used to quantify the
effect of the shock-induced ground conductivity. .- "
5
. . . . . . . . . . . . . . ..°. " .i
SECTION 2
KRATZ GAUGE OPERATION AND THREE MODELS
,~ .' %'
Detailed descriptions of the gauge and its operation are given by ->.•-.
Kratz [1976] and Coleman [1982]. Basically, the gauge consists of a
field coil and several pickup coils. The field coil comprises several
turns of wire embedded in an aluminum disk. Current is run through the .
coil. At the time of the explosion, the external current is turned off
but, because the coil is in aluminum, current persists for a time long
compared with the measurement times of interest. Thus, the current in
the field coil can be considered constant in time. The shock wave ap-
proaches and overtakes the field coil as shown in Fig. 1. After the
shock wave passes, the field coil ideally should move with the particle
velocity of the medium. A voltage proportional to the rate of change of
the magnetic field and related to the field coil velocity is induced in
the pickup coils, located some distance from the field coil.
Ground.. . . . '-: . .. S** ~ *~ ...... ..
z= flt) Field coil/current" "~~ -- loop embedded,n aluminum
ZOsheI Shock front z = g(t)
Pickup coils
Figure 1. Gauge geometry.
6
/ -~~~ . -.. ........... ...,.-. ,.........- .,. ,.....,. . .. . , ,. .:::;, ~~~~~~~~.*. . .. . .. .. ,-. . -. ...-. . . . . . --. - -. o.°.'-- . ,.o '% - -,-. ."° .. ,. ...
• " ---"'-;'" -"'".' ".":'- "''.-': -'>"- -"-"- ., "'',("-",""'' ,,'",,""," - "'""-",'-v-..,."".- ."..r.-.. ." . , '., - ..... ,.".-" -Y
The relationship between the field coil velocity and the voltage in
the pickup coil is modified by the ambient and shock-induced conductivities
of the .intervening medium. The ambient conductivity allows eddy currents a.
induced by the changing magnetic field of the field coil to flow in the
medium. The currents produce magnetic field changes that tend to cancel
" those produced by the moving field coil. If the cancellation is signifi-
cant, the signal recorded by the pickup coil will be badly distorted.
The shock-induced conductivity can have two effects. First, the
pickup coil responds to the changing magnetic field produced by the ad- " ""
vancing conductivity front even before the shock wave arrives at the field
coil. Second, the shock-induced conductivity can modify the gauge re-
sponse after the shock wave passes the field coil. The shock wave produces
conductivity between the field coil and the pickup coil, thus modifying
the response as described above by allowing even greater eddy rurrents in
the medium.
We have developed three models to describe those effects. The models
differ only in their treatment of the moving conductivity front; they
share the same description of the field and pickup coils. The field coil
is taken to be a current loop of radius a centered at z 0 0, r = 0 of a
cylindrical coordinate system. The field coil carries a constant current
I and moves in the z direction with velocity v(t) at time t. The position
of the field coil is given by
f(t) = v(t') dt'
The pickup coil is a single loop of radius R centered at z = zO, r - 0
in the z, r plane
Model 1 assumes that the coils are embedded in a medium of constant
conductivity a and that the medium is moving with a constant velocity v0
in the coordinate system stationary with respect to the pickup coil. The
results of model 1 were presented previously [Schlessinger, 1981] and are
only summarized here. The model demonstrated that the effects of theambient conductivity were negligible for all parameter values of interest
7
. . . . . . . . ... . . . . . .. .. 2.. . . ....-.
.. :.:.:.:..:.:
and that the shock-induced conductivity was not crucial (for typical
values) until it reached about 100 mhos/m.
Model 2 resembles model 1 but more completely describes the moving
conductivity front. It assumes that the medium velocity is a function
of time--vot). Both models assume that the medium conductivity extends
throughout all space and is constant in space, and that the entire medium
is moving with respect to the pickup coil. The results of model 2 are
the same as those of model 1 when v0 is held constant. Model 2 shows that
when v (t) is equal to v(t)--that is, when the field coil velocity and0
the medium velocity are the same--the medium conductivity has no effect. -
Further, when there are small differences in vo(t) and v(t), the medium0
conductivity begins to have a noticeable effect, for typical gauge param-
eter values, when it reaches 100 mhos/m."
Model 3 was devised to overcome the deficiencies in models 1 a-id 2
and to be used for quantifying the small but perhaps noticeable effects
of shock conductivity. In model 3, the medium conductivity and velocity
are allowed to change as functions of space and time. The medium has
conductivity a for positions z < g(t) and conductivity 0 for positions
z > g(t), where the function g(t) represents the position of the .hock
wave. The medium is moving with velocity v0 (t) for z < g(t) and is at -.-
rest for z > g(t). This model describes the physics of the moving con- ."
ductivity front much more accurately, but is considerably more difficult
to describe and analyze mathematically.
8
S i!!iii
SECTION 3r.. . -. .
ANALYSIS OF MODELS 1 AND 2
Models 1 and 2 are so similar that they can be analyzed together. .. .
The only difference between them is that the medium velocity v0 (t) is a
constant for model 1 but can vary over time for model 2. The major steps
in the analysis are as follows. First, we derive the equations of the
vector potential A generated by the current density J in the field coil.
Second, the equations for A are solved by finding their Green's function.
Third, we evaluate the voltage in the pickup coil generated by the vector
potential A. We use mks units throughout this report.
DERIVATION OF BASIC EQUATIONS
The Maxwell equations for the electric field E and the magnetic
field B in a medium of conductivity a moving with velocity v0 (t) are
and
X B 1i:.+v::B](2)
where p is the medium permeability. In our case, the current density J
of the field coil has a component only in the 1 direction of a cylindri- - -
cal coordinate system and is given by
J = I6(r - a)6[z - f(t)] , (3)
where I is the constant current, 6(x) is the Dirac delta function, r and
z are the radial and axial coordinates, respectively, of the cylindrical
coordinate system, a is the radius of the field coil, and f(t) gives the
position of the field coil at time t.
"9
- L
I--
To solve Eqs. (1) and (2), we introduce the vector potential A,
satisfying
B= ×I. "-'v'B X.A
It follows from Eq. (1) that
E- =- ."-
where 'D is the scalar potential. We choose the gauge such that -
A + ji¢ =0
Because the current flows only in the * direction, A has only a com-
ponent that satisfies
L2 A AV - p - -A1jv 0 (t) L = - .IJ (4)
r
Here we have assumed that the velocity vn (t) has a component only in the
z direction. Equation (4) must be solved for A, which can then be used
to find the voltage in the field coil.
GREEN'S FUNCTION
A standard method of solving an equation like Eq. (4) is to find
its Green's function G. For Eq. (4), that is the solution of
v -- l- ao T- G(r, r', z, z', t, t')I"~( r "[ '
= (r - r')6(z - z')6(t - t') (5)
Given the Green's function, the solution to Eq. (4) is
10
. ._ .
46 *5 . 4-
A(r, z, 0)= lf dr' dz' dt' G(r, r', z, z, t, t')J(r', z', t') . (6)
To find G, we express the delta function in a Fourier-Bessel
representation:
Sr- r')6(z -z') = A-ffdk d eikZ' J( r)j (&r) *(7)
where J I(x) is a first-order Bessel function. We also express G in the
same form as
G(r, r, z, z',t t') = ffd k d~ ei(Z)i. iThe equation for G follows from Eqs. (5), (7), and (8):
(k2 2A )a , t0 + ipakv G + pia 2G 6 (t -t,) .(9)
Equation (9) can be solved to obtain
G(k, ~,t (-t)exp f { + ikv 0 (tu)] dt" } (10)
where
e(x) =0 x < 0
Thus, the Green's function can be written in integral form using Eqs. -
(8) and (10), which can be further reduced by performing some of the
integrations. The integration can be performed using the result -e %i -e.
[Gradshteyn and Ryzhik, 1980) ]__
(2 Fj ~ (r 2 + v2lcEj (&r)J (Er') exp (t e)exp~2( t .0 - = -* x 4(t -t') .
1 1 L1C
Il4(t t')
where I (x) is the Bessel function of the second kind and first order.
Also, the k integral can be done using the result
2 dk exp [+. ik(z -z' -D)i
- 1/L4rt t,) ex 4(t t ') - - ~
where
t
D - v(t"l) dt"
t
Thus, the Green's function is
G(r, r', z, z', t, t') et- 1/
4(t - )
exp- t--9 -- )~ [(z- z' D) + (r -r)1
exp~ ~~~ Irla I~ r1c7 .(1x ex r IL2(t t') 2(~t t'
12
EVALUATION OF VOLTAGE IN PICKUP COIL
The voltage V in the pickup coil is determined by the rate of change
of the magnetic flux linking the coil. In terms of the vector potential
A, the voltage is given by ..
at
VVt)) 27rR A p (R o, rf, ) ., (12)
V x) 2nr , z' tffj iiGr, z0d ,rz' , t')t
It follows from Eqs. (3) and (11) that V(t) is A
- f ~ (~1/2V(t) -27rRaIJI a (d- 32\i/-Bt f 4(t - t')3- 7
exp L 4(t - ' [1z -f(t') D] 2 -R a)j
x exp a~l~a ] 1 aRIJ ] 13x exp 2(t t'- ( -t)(3
Taking the indicated derivatives and changing variables results in the
final expression for the voltage:
V(t) = RaviI f dx [z 0 f(*Y) -D][v(y) + v 0 (t) v v0(y))
0
exp [-~{zO f~y) -D1' ( a)J exp 1- x) x~~~
(14)
13
where
x
D J v - dyxL '
and.*.*,
df~tv (t) =
14
SECTION 4
ANALYSIS OF MODEL 3
• ' . "
Model 3 encompasses models 1 and 2 but also includes the effects of
the moving shock-induced conductivity front. In model 3, the field coil
is located at position z - f(t), as before, and the shock front at
z - g(t). The medium has conductivity a and velocity v0(t) for positions
behind the shock wave [z < g(t)]; the conductivity and velocity are zero
for positions in front of the shock wave [z > g(t)]. We label as region 1
the shocked region, where a # 0; and as region 2 the unshocked region, .
where a = 0. The fields that exist before the interface (shock wave) has
passed the field coil are labeled with a superscript b. V.
Before the interface reaches the field coil, the field in region 1 r
is a solution to
V2 1 0bi-[- "":
2 - v0 1 .r
This solution can be represented as "..-..
Ab i4I0 JdkI dk2 J11(kr)
x exp ik2D - ik2 z _k ^b " ") ] a (kip k o
In region 2, before the interface reaches the field coil, the field
is
b b bA2 = A0 + a2 , (15)
bwhere A is the field produced by the field coil,
15
•. -• . , .. °
Ab lbA 1= j ir dr' dz' ,(16) A
and a~ is a solution of
(2 b )a 0This expression can be represented as
a PI f dk e J,(kr)a (k)
bThe Green's function G is given by
be ::r,~::i r'-,z)d x z'j) Jl(Cr)Jl( r') . (17)
Tefntosaand a are determined by requiring that A and DA/3z --
be cntiuou atz gt)-tha is acossthe shock front. Imposing
those conditions yields the following equations for al. a:
2dk 2)k +k ik\
f kexp [--ik D -ikg - 1 2 1 2ab k) k)
-a exp k (~f -g)] J (k a) (18)
and -
k7 2~ - .
Jdk2 exp [+ ik D -ikg 1 +-,
=2 exp k- k9) a2(k 1 ) .(19)
1.6
Equation (18) can be solved for a and then used in Eq. (19) to obtainsba.
After the interface has passed the field coil, a slightly different
set of equations applies. Now the field in the conducting region includesthat of the field coil so, in region 1, we have ~
A-A + A, A.~ -A
where
AO p -r dr' dz' dt'
with G given by Eq. (11), and
A1 P- 10fdkl1 dk 2 1 (k 1 r)
K: ~x exp ik D ikz- (kl* jc k)
The field in region 2, after the interface has passed the field coil,
is given by
a 2 = - dk ek 1 (kr)32 (k) .(20)
The quantities a, and a are determined, as before, by requiring A and3A/az to be continuous at z -g(t). imposing those conditions yieldsthe following equations fr aa 2
17
TV V - Z . -. ;.N.. .V.- M 7 =
* j,- .° ° • o
2 1 1-dk 2 i -1 exp ikD - k - (ki k2)
f 2( 2 2 )ICY J1 2
1/2
-fdt' k Ii(kI a ) 1 -2(t - t')k 1 (g - - D (-t') la
k~x - (t - t') - f D) = 0 (21)
and
A k 2 + k 2)t+ exp ik 2 D -ik - a ( kfdk 2 ( ik2) ekp (2 - 2) - al , k2I) ....1/2
dt klJl(kla) 1 + 2(t - t')k f D)]
2(t~~. - .r~ 4T ( -t',c
2. -k
X exp - 4(t) t ) (g-
f- D) J 2(k t) e
(22)
Equation (21) can be solved for al(kl, k2) and used in Eq. (22) to ob- S
tain a2 (kI , t).
The voltage in the pickup coil is then obtained using Eqs. (12) and
(15), (16), (17), or (20).
Numerical solution of Eqs. (18), (19), (21), and (22) is difficult
be a 1 i sharply peaked function having a maximum a k i
Thus, an appropriate method of solution is used here. Equation (18) can
be satisfied only if
18
..........................-.-..-. "
--'._' '- '.~-- -.~~- -%- "
or :
p ik2 P.a (g D) +$~ U~-)2 aD2tk1 1 + 2ti D)~ 1
For small values of a, this expression is
k2 - iaD
k ~2k t
For large values of a, it is
ik2
ki g-D
Assuming a (k., k) -a 6(k2 k1) then Eqs. (18) and (19) can be solved
to yield
2a aR11I paD1
a f-g) (23a)
r for a small and
2aa2 42g-D) 1 (23b)a2 4(g - (z + f -2g)
3
for ay large. For our problem, a is almost always small (<10 3mhoslm).
Thus, using Eq. (23a), we evaluate the voltage in the pickup coil before
the field coil starts to move (i.e., g < f) as
19 '
V 7R2 a2 r t Dt 4v s(t)D(t)](4a16t7ra 2 0( Dt) (z +$ f-(2a
Here v (t) is the shock velocity, v (t) 3 g/3t. For very large conduc-S s
tivities, the voltage is
2 2 F(gv +Dv - Dy 0 3v D7R a UI 0 l s ]V .2g - D . 3I 2g -D ~f + z-2b
(f + z-2g)L0 g
For the case when the shock wave has passed the field coil, the con-
tribution of a to Eq. (22) is negligible. Thus,1
2 a 2~7Q dt'Jdk
x exp [ 2 t- ')+ -~---(g -f -D)
2 + k(z -g)l-L a 4(t -t')
UJ (ka)J1 (kr)r
1+ t - t - k
After the small-argument approximation for the Bessel functions has been
used, one integration can be performed and the result put into a more
convenient form:
2 1/a2 f(g3 Jdx e-' 2 1/ [2 (2, 1/2, x)
2/ 3[ 7rk
1(25
202
4 2E X......3/2..).+.... 2x (X 1 + x. ..3
7~~~~ V.. t 111 - .
where q)a, b, x) is the confluent hypergeometric function of the first
kind (regular at x =0) and X is given by
g(t) f ~ t - -d(t, x)
z -g(t)
with 72 Via
Y [z- g(t)] -4-
and
Y/
d(t, x) v (t -y) dy
C 0
Equation (25) can also be written as
aRI dx e-X(~/ - U(2, 1/2, x) + AxU(2, 3/2, x)] (26)2 8(z - g) 7r 2T
where U(a, b, x) is the confluent hypergeometric function of the second
kind (singular at x -0). The voltage is obtained from the vector pa-
tential using Eq. (12):
V 3 dy y2 exp ~ )-2X Xy2 Ar (z -g)
0
x U(2, 1/2, y)+AU(2, 3/2, y2) +XU(2, 3/2, y')~ (27)
where '.-
21
1I3.17. .1.7----7-67
wi th A'. N
-a ( g)
2
In the limit as a -~0, we have d -~0, y -~0 and Eq. (27) reduces to
2 2v 37pIa R 4 V(t)
2(z 0 f)C0
22
%
SECTION 5
NUMERICAL RESULTS
To obtain numerical results, we evaluate the expressions in Eqs. (14),
(24), and (26). The velocities were parameterized as zero for t < 0 and as
V(t) { exP [( t) exp (t -to)]
3 2-
v (t) xep
for t > 0 and all velocities are zero for their arguments less than zero.
The distances are taken as zero for t < 0 and as
D(t) vnmI expY~)~ exp(k)2 '
f) f0 + vr 2 x t~&} x t ]~
(t t
. ... 0
.°-L
We evaluated Eqs. (14), (24), and (26) using the following values
for foo z0 9 v, Vo, Vs, T1 through T4, to, and a:
fo = g(t0) = 0.25 m,
z0 = 0.75 m,
3V = 103 m/sec, "-.3,.nsc
v0 = v exp (t Pr) 1.63 x m/sec,
3v = 3 x 103 m/sec,
-4T1 = T = 2 x 10 sec,3T= 10 sec,2
T4 =10- sec,
-5to 9.8 x 10 sec,
0 0, 10, 100 mhos/m.
These parameter values are similar to those appropriate for MINI JADE
and MISTY JADE. The results fcr these values are plotted for models 2
and 3 in Figs. 2 and 3, respectively.
Knowing the value of the shock-induced conductivity and using the
theory developed above, it is possible to determine the particle velocity
from the gauge response. However, the value of the shock-induced conduc-
tivity is rarely well known. Thus, the gauge should be used to measure
particle velocity only in situations where the effect of the conductivity
does not depend strongly on the value of the conductivity.
Figures 2 and 3 show that, for typical gauge parameter values, shock-
induced conductivity just begins to affect the gauge response at 100 mhos/m.
We thus conclude that, for measurements of MINI JADE and MISTY JADE phe-
nomena, shock-induced conductivity has little effect on gauge response as
long as it is less than 100 mhos/m.
Figure 3 shows the magnitude and time dependence of the precursor
signal caused by the advancing conductivity front before the field coil
begins to move. It has been suggested that the precursor signal could be
24
. . .... '...- -
12- %
10-
8I
6-
2-
0 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72
Time (mme)
Figure 2. Gauge output versus time for model 2. :
8-I
4-I0
cc2-
0a 1 o100 mhos/m
0 0.9 0.16 0.24 0.32 0.40 0.48 0.56 0.6 0.72
Time (msec)
Figure 3. Gauge output versus time for model 3.
25
used to deduce the medium velocity before the field coil begins to move.
However, Fig. 3 indicates that, when the gauge is designed to measure the
signal from the moving field coil (and consequently a < 100 mhos/m), the
precursor signal is probably too small to be reliably used to measure the
medium velocity.
If one is willing to design the gauge to measure only the precursor *signal, then, according to Eqs. (24a) and (24b) for the gauge response,
in certain ranges of conductivity, the gauge could measure either the
shock-induced conductivity or the medium velocity. That is, Eq. (24a)
shows that, for a < 1000 mhos/m, the gauge response is linear in the con-
ductivity, so precursor signal measurements along with independent mea-
surements of the medium velocity and shock velocity could be used to
determine the medium conductivity. Further, Eq. (24b) shows that, for
a > 1000 mhos/m, the precursor signal is once again independent of the
shock-induced conductivity. Thus, for very large conductivities (>1000
mhos/m), the precursor signal could be used to measure the medium velocity.
We conclude that the Kratz particle velocity gauge is not sensitive
to shock-induced conductivity for conductivities less than about 100 mhos/m. "
For gauges designed to measure the velocity of the field coil, the pre-
cursor signal is probably too small to be useful. However, gauges can be
designed to take advantage of the precursor signal and measure either
the shock-induced conductivity (for a < 1000 mhos/m) or the medium ve-
locity (for a > 1000 mhos/m).
- -
.'.,. -~ • -
26.
............................
.........................................
.....................................................
REFERENCES
Coleman, P., briefing presented at the HURON LANDING review conducted at
the Defense Nuclear Agency Field Command, Albuquerque, New Mexico, -
1982.
-..... briefing presented at the MINI JADE review conducted at the DefenseNuclear Agency Field Command, Albuquerque, New Mexico, 1983.
Gradshteyn, I. S.. and I. M. Ryzhik, Table of Integrals, Series, andProducts, A. Jeffrey (ed.), Academic Press, New York, 1980.
Kratz, H. R., HUSKY PUP Debris Impact Experiment--Development and Fieldingof an Electromagnetic Particle Velocity Gauge, Systems, Science and
Software, La Jolla, California, SSS-R-76-2950, 1976.
Schlessinger, L., Analysis of Particle Velocity Gauges for MISTY JADE,Pacific-Sierra Research Corporation, Note 357, February 1981.
27
............................................................. ......- ,
. .. . . .
......
', .
47.%
28
APPENDIX ." ....
LIST OF SYMBOLS 6."6,
a = radius of field coil .. --
a2 = vector potential in region 2 (which has no shock-induced." -
conductivity) after shock wave has passed field coil .
b~a2 = contribution to vector potential in region 2 (which has
no shock-induced conductivity) caused by presence ofconductivity front before shock wave has passed fieldcoil, i.e., Ab = Ab + ab
2 0 2,,b ,b b b
al. a 2, al, a2 Bessel transforms of a1 , a2, a1 , a 2
A = vector potential
A0 = vector potential of field coil in homogeneous movingmedium with shock-induced conductivity
A = contribution to vector potential in region I (which has
1 shock-induced conductivity) caused by presence of con-
ductivity front after shock wave has passed field coil
bA 0 vector potential of field coil in homogeneous medium of
zero conductivity
bA - vector potential in region 1 (which has shock-induced
conductivity) before shock wave has passed field coil F -.\.
bA= vector potential in region 2 (which has no shock-induced
conductivity) before shock wave has passed field coil
b superscript indicating fields that exist before shockwave has passed field coil
B magnetic field
D -time integral of medium velocity
E = electric field
f(t) - axial position of field coil
g(t) - axial position of shock wave
G -Green's function
bG - Green's function for field coil in medium of zero
conductivity
29
. *.~**.C.................
~~~. . .....,. . . ......
w ,-.. . -%
_VV71•- r'.V
I-current in field coil (assumed constant)J - current density
r radial coordinate of cylindrical coordinate system
R = radius of pickup coil
U(a, b, x) = confluent hypergeometric function of second kind(singular at x - 0)
v(t) - field coil velocity
v W - medium velocity
v- axial velocity of shock wave
V - voltage in pickup coil
z =axial coordinate of cylindrical coordinate system..---'.--. .I..T . -- '. :
z axial position of pickup coil
6(x)- Dirac delta function
azimuthal angle of cylindrical coordinate system
scalar potential -
x)- confluent hypergeometric function of first kind (regu-lar at x - 0)
a - medium conductivity '
U - medium permeability (taken to be that of free space)
30
...........
. . .. ..~~ ~. . . . . . . . ° . .. . . ° .. . . . . . o °
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