/ AW'DNA453

QUICK ESTIMATES OF PEAKOVERPRESSURE FROM TWO

<t SIMULTANEOUS BLAST WAVES< [EVEr '-R &'D Associates

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4' .r : December 1977

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QICKESTIMATES OF PEAK OVERPRESSURE FRO 0O Topical Xepget,

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B310078464 P99QAXDBOO136 H2590D.19. KEY WORDS (Coagrlnut, an revere# side if neceay and identify by block numb er)

Multibursta Simultaneous Blast WavesPeak Overpressure Target CoverageReflected Pressure Blast WavesShock Interactions-Shock-on-Shock20 ABSTRACT (Contimn aon reverse mide It necesaryt and Identify by block tnumber)

Oeveral simple approximations to the peak overpressure between two interacting4 (simultaneous) blast waves are explored and compared to the LAMB approximation

method for multibursts. The best estimates suggest about a 30Z (10%-50%)increase in line-target coverage over that covered by two nonsimultaneousbursts, Coverage is restricted to strong shocks (AP > 100 psi).

DO " 47 EDITION OF I NOV 46 It OBSOLETE 5UTY NCSLID4',SEUIYCLASSIFICATION OF THIS PACE (*W

- - 78 09 "08 Y~

Table 1. Conversion Factors for U.S. Customaryto Metric (SI) Units of Measurement

TO CONVERT FROM TO MULTIPLY BY

FOOT METER 0.3048

MEGATON TERAJOULE 4183

MILLISECOND SECOND 0.001

PSI (POUNDSOFORCE/INCH2) KILO PASCAL (kPa) 6.894 757

fort

S White soctwasWC BUuff sedlo 0UOWffUED C3I

JWTWICATIOI' .'..-...-...----..-.-.

c.

I ~LL

TABLE OF CONTENTS

Page

1. Introduction . . . . . . . . . . . . . . . . . . . . 5

2. Several Estimates . . . . . . . . . . . . . . . .. 6

3. Interaction of Unequal Shocks . . . . . . . . . . . 18

4. Comparison with the LAMB Procedure . . . . . . . . . 27

5. Summary and Conclusions .............. 34

References . . . . . . . . . . . . . . . . . . . . . . . 36

'S T,

LIST OF FIGURES

Figure Page

1 Two 1-MT Surface Bursts (Simultaneous)reflecting to 600 psi .......... 8

2 Geometry for Two Simultaneous Blast WavesInteracting ... .... ........ 9

, 3 Peak Overpressure vs Range for Two 1-MTSimultaneous Surface Bursts Separatedby 6860 ft . . . . . . . . . . . . . . . 11

4 Pressure Profiles at Early Times for 1-MTSurface Bursts ............. 12

5 Approximations for the Peak OverpressureBetween Two Simultaneous 1-MT SurfaceBursts 4500 ft Apart .......... 14

6 Shock Interaction Notation for UnequalShocks . . . . . . . . . . . . . . . 18

7 Resultant Peak Overpressure from Two Shocks(One at 100 psi) . . . . . . . . . . . . 22

8 Shock Interaction Between Unequal Shocks . 23

9 Further Approximations for the Peak Over-pressure Between Two Simultaneous 1-MTSurface Bursts 4500 ft Apart . . . . . . 26

10 Comparison of Reflection Factor (APR/APS)for Normal Shocks and by the LAMB Method;Comparison of Reflected Density and Densityby LAMB Method ............. 28

11 Approximations for the Peak OverpressureBetween Two Simultaneous 1-MT SurfaceBursts 4500 ft Apart Compared with theLAMB Model ............... 32

3

LIST OF TABLES

Table Page

1 Conversion Factors for U.S. Customary to Metric(SI) Units of Measurement...... . . . . . 1

2 1-MT Simultaneous Burst Interaction (Separatedby 4500 ft) Approximations to Peak Over-pressure . . . . . . . . . . . . . . . . . . . 16

3 Comparison of Range and Area Coverage Increasesfor Five Different Approximations to thtInteraction of Two Simultaneous 1-MT SurfaceBursts Separated by 4500 ft ........ 17

4 1-MT Simultaneous Burst Interaction Peak Over-pressure App:oximations Using Unequal ShockResults . . . . . . . . . . . .. . . .. . . . 25

5 Range Coverage Comparisons (1-MT Bursts 4500 ftApart) . . . . . . . . . . . . . . . . . . . . . 33

bF 4.=77

'4'.-'-'I

SECTION 1. INTRODUCTION

Strong shocks in air reflect off rigid surfaces at many

times (5-13 times) their original pressure. Opposing shocks

of equal strength behave just as a reflected shock, leading

to very high pressures at the initial point of contact. If

two shocks of 100 psi collide, the resulting peak pressure is

around 500. For two 1000-psi shocks, the peak pressure jumps

to 8500 psi. This suggests that the area covered by a given

overpressure from two simultaneous blast waves may be consid-

erably larqG:: than twice the area of a single burst.

Yet, while strong shock reflection factors are impressive,

there is reason to believe that such high values do not extend

far beyond the initial point of contact, and, further, two

unequal shocks may interact in a much less impressive way.

Strong blast waves are extremely transitory, and pressures,

densities, and flow rates behind each blast front drop off

exceedingly rapidly.

Careful considerations of such blast wave nteractionslead directly to three-dimensional geometries which very muchinhibit the accuracy and practical resolution achievable with

canonical numerical methods. The following series of estimates,

without benefit of rigorous modeling and detailed nuak-1ical

calculations, are intended to bound the expectations for enhancedcoverage by means of simultaneous blasts. Conparison is also

made with the LAMB procedure, as applied to two simultaneousbursts and the overpressures along the line joining their centers.

A

Low-Altitude Multiple Burst.

5 ,r

2l

SECTION 2. SEVERAL ESTIMATES

The peak overpressure from a single burst on the surface

is well approximated with the formula '1]

3300 W 192Np (1)R R

with W the yield in m egatons and R the distance in kilofeet.

A normally reflected shock reaches a reflected pressure

enhanced by the stagnation of the flow, and results in pres-

sures for a strong shock much more than double the incident

shock pressure. An approximate shock reflection factor for

an ideal gas of specific heat ratio y is given by

"Pr 4yP0 + (3y- 1) AP

AP 2yP + (y-1)AP (2)

where AP is the incident overpressure, APr is the reflected

overpressure, and Po is the ambient pressure (14.7 psi). Forsea level air, 1.1 < y < 1.7 (y = 1.3 for AP = 1000 psi)[1].

A more exact fit (within 3 percent) to this reflection

factor is provided by tile formula below, which accounts for

the nonideal gas properties of sea level air [2].

This formula agrees to within 10 percent with the acceptedaverage of the nuclear test data which in turn are 90 percentcontained by a spread of ±50 percent at pressures above 40 psi.

6

RF= 0.002655AP + 2

1 + 0.0001728AP + 1.921 x 0-AP

0.004218 + 0.04834AP + 6.856 x 10 -6AP2

+ (3)1 + 0.007997AP + 3.844 x 10-6AP2

At the point where two simultaneous spherical blast waves

meet, the peak overpressure can be fairly precisely described

with tkese two approximations (Equations 1 and 3). For

example (as in Figure 1), for two 1-MT simultaneous surface

explosions separated by 6860 ft, the blast waves meet when

each has a peak overpressure of 116 psi (Equation 1), and

these shocks reflect at the point of contact to a pressure of

600 psi (a reflection factor of 5.18)(Equation 3). The value

of 600 psi from a single 1-MT surface burst occurs at 1840 ft.

For nonsimultaneous bursts, a separaticn of 3680 ft would

cover a line target with more than 600 psi everywhere. The

separation distance at which the interacting shocks reflect

to 600 psi (6G6C ft) is nearly twice as long.

The area associated with the region of enhanced blastpressures is a thin lens about the point of contact between

spherical (or hemispherical) shocks, and is only a small

fraction of the area covered by the individual blast wr--es.

However, the use of the larger distance (6860 ft), where

the shocks reflect to just 600 psi as an effective kill distance

for a line target, is quite incorrect. The geometry of the

situation (Figure 2) suggests that as the two shocks pass

It is very wrong to ansume that if the peparation distancewere increased by a factor of two, the area coverage would be Icorrespondingly increased by a factor of four.

7

06

LOA 04J

4J

4J,

4J

.9-go

aI

A'U10, 7-

I.

I

p Ptncl / 3 P s to 1/lk P

i s

* I S

0 RREINOF REFLECTION

Figure 2. Geometry for Two Simultaneous Blast Waves Interrcting

I.-

9~ KY-. *;I v ~ ---.-

- - - - - -- - - - - - - - - - - - - - - - - - -

througn Pch other, they continue to expand spherically, andso dr.,p rapid, ".n pressure from that peak value of first con-

tact (1:. . 1). If their peak inturaction gives only 600 psi,

then all L.iv region between burst points that lies beyond a

radiAo.s from each burst point of 1840 ft would see less than

600 " (Figure 3). That is, as in Figure 3, a total of3200 ft ,ould experience two shocks--both less than 600 psi.

rit creite,': interest is the sepe ration distance that exposes

the nt' :ire target to more than 600 psi (or some other peak.vei'prC3s.re of interest). For this we need to know how rapidly

these diveraing shocks decrease in overpressure as they expand

into each other. Since, on first contact with the opposingblast w,,,ves the transmitted shock starts at the peak reflection

v,:lue, the initial value "; well defined (Equation 3), but theree-s not exist an equally simple or direct formula for predicting

the subseqaent decay rate as the two blasts penetrate each other.

Numerical blast calculations provide detailed descriptions

of the pressure field into which each transmitted shock is

expanding and how it behaves in both space and time [3,4,5].For most applications to hard targets (e.g., for 600-psi

trenches), a simple strong shock model would suffice. In any

case, while peak pressure decays with distance (as in Equation 1)

approximately in proportion to the inverse cube of the distance,

the interior of the fireball/blast-wave follows with a pressure

about one-third to one-half of the shock value (Figure 4). Infact, both peak pressure and interior pressure decrease in pro-portion to the inverse of the time measured from the instant

of explosion. An approximate relation is given by 12]

Two simpler but less exact forms for the pressure-time rela-tion in a nuclear blast wave may be found in Reference 1,pp. 180-181.

WIAw" 10

PEAK REFLECTED PRESSURE EQUALS

4'icc

j / ~%(APPROXIIWATE)A

- SINGLE001 (UNREFLECTED)

SHOCKS

___________________REGION______WHERE______AP<___600_________

Figure 3. Peak Overpressure vs Range for Two 1 MT SimultaneousSurface Bursts Separated by 6860 ft

-1777-

'I-U

co-

Uj U

CD

4-

I--121

_ _ _ _ 4.- F7

AP(t,T) 8.00 0 t) 0 4 1 7 + 0.583 ]1 t-T f(t) (4)

in which t is the time in msec (after burst), T is the shock

arrival time (msec) for 1 MT, (t > T), D is the positive phaseduration (msec) (the time during which the blast pressure is

greater than ambient), and f(t) is an empirical adjustment fit

of secondary importance.k2t 00 + 6.72t + 0.00581t2flt M (5)100 + 18.8t + 0.0216t

2

The essential time behavior of the pressure is illustratedby this dpproximation: The dominant behavior is a decay almost

linear in time (more precisely as A- t - ' 15) with a very sharp

drop just behind the shock front to a value of about 40 percent

of that at the front ( t - 6 ).

One possible approximation is to assume that the transmitted

shock continues to generate the same peak reflection factor as

it continues to expand and decrease inside the other blast wave.

This is likely a gross overestimate of the off-peak pressures,

since one might better use reflection factors appropriate to

the transmitted shock as it continues to expand and decrease.

The original blast that it is running into also continues to

expand and decrease in pressure. A more correct approximationwill account for this double decay, but, for the moment, consider

several simple approximations for the transmitted peak over-

pressure.

Figure 5 shows several choice* for these transmittedpressures in the particular case of two simultaneous surface

. 13

2400

2200

'CO'

1800

CI'

~1600I

1400

00

8 100L- S

1000 3

200

40-

0 1200 1400 1600 1M8 900 2200200 60 2800DISTANCE FROM BURSTS (ft)

3100 290 2700 2500 2300 -2100 1900 1700

Figure 5. Approximations for the Peak Overpressure betweeinTwo Simultaneous 1 MT Surface Bursts 4500 ft Apart

14

bursts (1-MT) separated by 4500 ft. Table 2 illustrates the

development of numerical values for each case.

(1) The peak interaction is defined as the localpressure in front of the transmitted blast wave

multiplied by the initial reflection factor for

the colliding shock fronts (reflection factorRF = 7.0).

(2) The peak interaction pressure is defined as the

local pressure in front of the transmitted blast

wave multiplied by the reflection factor appropriate

for a shock of strength equal to that of the second

shock if it had expanded (unreflected) to that dis-

tance beyond the initial contact point.

(3) The peak interaction pressure is defined as the

local pressure in front of the transmitted blastwave multiplied by the reflection factor for a

shock of peak pressure equal to that pressure.

(4) The peak interaction pressure is defined as the

unreflected (incident) shock pressure multiplied

by the reflection factor appropriate to the timeof first contact (RF = 7.0).

(5) The peak interaction pressure is defined as thieunreflected (incident) shock pressure multipliedby the reflection factor appropriate for thatreduced shock strength as it expands in an undis-turbed sea level atmosphere.

-Z

15

'77777

II

( x LO II r. ls GO f r.

-C * V-4 - "4 v-4 V-4 "

0N0 0 4)a

.0c 040 a "4a a

4 J -4-

Z; w w. 00 0 O 0Y C~4w r-4 1-0. 40 Ln0 r M0 M

ww

4"4

2

(A o " < 4'0 " - PI n v v aIt 5 -t " 9-4

Of-~

., "4 @ O 5. 0 U44 N:1,"4 " 4 4 "

4 ~ 4J

2. ~ c r- 4

t ,ctr I, r. U;.f LA~ rr'A~f 4 -. 4 4.. ,

None of these approximations account for the expansion

of both shocks as they pass through each other. The most

reasonable approximations are (3) and (2), but the others

are shown for comparison (Figure 5).

The extra line target coverage for this example (1-MT

bursts separated by 4500 ft) is listed in Table 3.

Table 3. Comparison of Range and Area Coverage Increasesfor Five Different Approximations to the Interaction

of Two Simultaneous 1-MT Surface Bursts Separated by 4500 ft

PEAK SINGLE DISTANCEOVERPRESSURE BURST ADDED BY RANGE

APPROXIMTION COVERED RANGE INTERACTION INCREASENUMBER (psi) (ft) (ft) (%)

3a 540 1915 670 17

2 610 1830 840 23

1 640 1800 910 25

5 990 1540 1410 46

4 1120 1470 1550 53

_ __ _

aMOST PLAUSIBLE APPROXIMATIONS.

The last two are clearly overestimates of the effect,

but the first three may also predict more enhancement than is

real.

Further estimates are given in the folloving sections.

17 *

SECTION 3. INTERACTION OF UNEQUAL SHOCKS

The preceding discussion has dealt with the case of two

simultaneous bursts (equal strength shocks) interacting.

Interactions between shocks of unequal strength are also of

interest, since timing of multiple bursts may not be exact

and the first blast may be considerably weaker by the time

the second blast meets it.

Suppose two shocks meet when their respective peak over-

pressures are AP1 and AP2 , and suppose the first is stronger

than the second (AP1 > AP2). These shocks, on interacting,

lead to an overpressure APR, a density PR and a resultant

particle velocity UR. Figure 6 identifies the nomenclaturein which two initial shocks have met and have resulted in two

transmitted shocks which have velocities URl and UR2 directed

away from each other, and a particle velocity UR which ispositive to the right if AP1 AP2.

-I I I2

A1 uR 2U 1 2

U p2

URl PRI PR2 UR2

Figure 6. Shock Interaction Notation for Unequal Shocks

Actually, two states of density and temperature separated bya contact discontinuity exist in the shock interaction region(shaded area, Figure 6), but pressure and velocity are thesame in both states, viz., APR, .

-'- A

From standard shock relations, Ulf U2, Pl. and P2 are

expressible in terms of AP1 and AP2 and are considered as

known here.

U =API((XYA.)AP 1 + YPo)

U Al 2 1AP + YP 0) vPO

AP1 = p [(4)AP1 + YP]((~) A715- +

1I\ + 1[P2~ AP' + 1PPi -Po 2, Yo] [(Yo)

= o [ AP2 AP - 2 + P (6)

Expressing the conservation of mass flow rate through each

shock, one may write

PRl(URl + UR) = P1 (Ul + URi)

PR2 (UR2 - UR) p 2 (U2 + U R2) (7)

The momentum change related to the pressure jumps at the

shocks can be expressed as

APR - A 1 = PlU (U1 + URI) - PR1UR(UR + URi)

AP R - AP- P2 U2 (U2 + UR2) + PR2UR(UR2 - UR) (8)

19

Eliminating URI and UR2 from Equations 6 and 7, one

can write

I2APR = Ap1 + - (U1 - UR) 2

PRI"Pl

and

APR = AP2 + PP 2 (U2 + UR) 2 (9)

The init 4al energy density (E) jump conditions at each

transmitted shock require

ER - E 1 2 . P-

and

E - E 2 2 ( 2 -R2

If one assumes an ideal gas with ratio of specific heats (y),

then E -P/[(y-l)p]. Solving for pR leads to

PR (Y+l) APR + (Y-1)API + 2yP0

71 (Y-11)APR + (y+l)AP 1 + 2yP0and

PR2 (y+l)APR + (y-l)AP 2 + 2yP O-R2 yI)AP R + (Y+1)AP 2 + 2P 0 (11)

Using Equations 10 and the Hugoniot relations to express

Pit P,,, U,, and U2 in tezms of AP1 and AP2 (Equation 6), one

can derive

20

TMA,

APR A _A 2 + C (12)R|

and

APR = D ±D + F (13)

in which

A AP + Pl(Ul - UR) 2

C-E-API12 + (Yi API + yPo) P1(UI - US)2

_AP 2 2+(X AP2 +Ypo) P2 (U2 +UR)

These equations can be solved for APR by iterative

selection of the velocity UR (all other values being pre-

determined by the values of yPo' AP1 ' and AP2).

A set of example values of the resultant peak overpressure

APR for two shocks meeting when one of them is 100 psi is

illustrated in Figure 7 for two values of the specific heat

ratio. For example, a 100-psi and a 500-psi shock meet togive more than 1500 psi for y - 1.4 and about 2000 psi for

y- 1.2. 1

Figure 8 illustrates the peak pressure amplification jfrom the meeting of two shocks with a plot of the ratio ofthe resulting peak overpressure to the sum of the two incident

peak pressures. The amplification approaches unity if one of

.if

3000 ]

N

2500 -

2000-

w.J

1500

1000 AP-100 psi

1!

500 2I I ,, iI I I I , I300 600 900

INITIAL OVERPRESSURE (psi)

Figure 7. Resultant Peak Overprnsire from Two Shocks(One at 100 psi)

22

____ _T__777_

- 2C- (D 9n

00 1 *

'4 0

41

p..

idl

0

4n

m cmU) 41t 1

23

i 1Ji0

14 ,I Iv '.401

i (Zd' LdUi d0

i the shocks is weak, and seems to reach a peak when the ratio

of the two incident overpressures lies between 3 and 5.

This solution for the resultant peak pressure from the

collision of two unequal shocks suggests yet another approxi-

mation to the peak overpressure distribution for our example

of two simultaneous 1-MT bursts separated by 4500 ft (Figure 5).

The interior overpressure (behind the shock front) that a

transmitted shock encounters can be assumed to act similarly

to a shock of that overpressure, and the resultant overpressure

computed (by means of Equations 12 and 13). This approximation

is tabulated in Table 4 and illustrated in Figure 9 (labeled

Case 6).

This approximation, using unequal shock properties, is

not very different from the simple average of the peak over-

pressures from the earlier bounding cases--Cases 3 and 5--

which use the interior blast wave pressure multiplied by the

normal shock reflection factor for that pressure (Case 3) andthe single shock peak overpressure multiplied by its normal

reflection factor (Case 5). This average is also shown in

Figure 9 as Case 7.

24 . .1. ,-

-4 U)I 0 U) Le) LO 0 In)w *r ix L 10 9-4 r.. r-. im co mc C.V + -j 411- U) "4 Om e% to %0a I

0 am. C N .e"

C0C r%. V4 "-4 M0 Ina. A 0o W- CO1 n W* m~ r%

C. r.. 4Y4 C% jN N

4A

Q>4w 1L w

%I n In P%.C

Eu m m m m

.4 .4 4" " 4"

4J:3

cm 5 0 0 0D (C 0 in f%.0.. A U P. N 4 % 0 In

IA

40

14

NY w 04 04 04 (m "4

44'

25

Nix~-'I

"I00

i /

2200

/ \I

1800I

CL 16 00 0 0 0

100010 20 70250 20 10~

,. ue .FuterAprx /in fo h I Ovrrsuebte \ I

Boo

1010

T00-e s

206 .010I

200: I!'' "

I . .I . .i _ 1 _ . _ .i , i , ,

TDISTANCE PAN7 MR$TS (ft)

Figure 9. Further Approximations for the Peak Overpressure Ibttwwnt'g. 7JTwo Simultaneous I14lT Surface Bursts 4500 ft Apart -.

l2

- ---- --' -, " " ..

' " " ,<. ' : ... '< "**'" " ; ''k' h/,"., .r +< , -.,">. ".< l " , ., / .'7 ,, -"V. T.7'7--=77--f, "''

SECTION 4. COMPARISON WITH THE LAMB PROCEDURE

A more careful accounting for the usual conservation of

mass, momentum, and energy during these shock interactions

requires some further assumptions and more geometry and arith-

metic. Such an attempt is the basis of extensive calculations

and predictions at the Air Force Weapons Laboratory with aprogram referred to as LAMB [6].

The essential features of this procedure are as follows:

Mass conservation; assume:

npc + i Api, (p 0.5 p0 ) (14)

in which p is the air density, p the ambient (pre-shock) air

density, and Api is the over-density (pi - pc) in each blast; wave,

wv.Recognize that densities in strong shock fronts may rise

to more than ten times the ambient density, but may also fall

to less than one-tenth the ambient value inside the fireball.

This prescription predicts the peak density of two equal

crlliding strong shocks to be only about half of the correct

Hugoniot value. (See Figure 10.)

The program, originally designed to approximate over-

lapping bursts at altitudes between 10 and 30 kft (in a

missile defense role),, quite arbitrarfly restricts the under-

denpities (negative overdensities) to half (or more) of the

ambiant density. Such a restriction provides a reaxonable,

.... if unsatisfactorily empirical accounting of other disnersive

27

VWF

S,.

o PR/pO

co Q-j

a- S.

0 wR

..

° Cc

ILA .0

i!in

~V.

028

i:+,++'",+.0. W,4 M 11 1.I

4, Q4J "2

effects likely to occur in the high temperature (low density)

interior of a nuclear fireball, but it cannot be justified

rigorously (from first principles). Actually, some such limit

is needed to prevent an undesired increase in net kinetic

energy for a shock in the very low density interior of a strong

blast wave (fireball) that would result otherwise from this

prescription. Recognize also that this expression is not one

of mass conservation, but prescribes density, or mass-per-unit

volume. It has long been a favorite piece of magic for simpli-

fied blast solutions (many published as serious contributions)

to make seemingly innocuous assumptions about density distribu-

tions and then proceed to unfold a marvelously consistent

picture of some blast wave. The rabbit is always already in

the hat here, however, since density distributions are, infact, integral representations of the entire movement history

of the blast, and the movements so described are a direct

consequence of the acceleration or force (pressure gradient)history. So any density profile contains the blast wavehistory to that instant, and is not a trivially adjustable

parameter of small consequence.

Conservation of momentum; assume:

n

i-I1

where pi is the density in the ith blast wave, vi is the

particle velocity of the ith blast wave# and p and v are theresultant density and particle velocity in the blast interac-tions. This represents a local momentum density vector sum

without consideration of pressure impulse contributions, which

is not quite consistent with the usual assumptions of inviscid

gas dynamics for blast wave characterization.

294°

Conservation of eaergy; assume:

in which APi = Pi - Po is the overpressure in the ith blast,and AP is the resulting overpressure in the interacting blasts.

This prescription purports to convert excess kinetic

energy from the opposing flows into pressure Energy, a proce-

dure consistent with normal hydrodynamic flow characterizations.

However, it is inexact as an energy equivalent to add 1/2 pv2

terms to overpressure. The dimensions are correct, but the

compressibility factor 1/(y-1) is missing. As a consequence,

the reflection factor for two equal shocks resulting from

this prescription is slightly in error. For an ideal gas:

(r =2yAP + 4yP0\ P/. (-1) AP + 2yPo

AP LAMB (yy (7

More rigorously (as in Equation 3),

LP (3y-l)AP + 4yP(

AP R = (y-1)AP + 2yP0 (18)

These two formulae are compared in Figure 10 for 1 < AP < 104,

showing the LAMB .procedure to be low by a factor of 7/8 at

high overpressures (y - 1.4).

1LThe LAMB procedure, when applied to the previous example

(two simultaneous 1-MT surface bursts 4500 ft apart), gives

even less coverage with high pressure than the previous

30"

. 30

estimates using peak reflection factors. As mentioned, the

LAMB method misses the peak pressure by a factor approaching7/8, and drops away from that peak value very rapidly (Figure 11).

Table 5 compares the range enhancements from the approximations

previously discussed with that for the LAMB model. The curves

in Figure 11 and the basic data i.n Tables 2 and 4 are derivedfrom Reference 1, but any of the descriptions in References 3,

4 or 5 would serve as well.

The added line coverage for this example with the LAMB

procedure is about 11 percent, a smaller effect than any of

the approximations given in the previous section. Good agree-

ment with experiments is claimed for the LAMB model when applied

to HE tests or to the shock reflections on a nuclear test such

as PLUMBBOB-PRISCILLA. However, the arbitrary assumptions in

the model are neither intuitive nor physically correct, and

their effect on results far from the point of initial shock

contact remains unclear.

~* 1 31

-JJ

2400

2200

2000

1800

~14005

1200 -

100 / I IIII I

1000 I LMSt

I APPROXIMATION600

400 %

2W0

1 i 14 m W U I= = 0Z 2400 260 aSISTAKNI FIN MSTS (Ift)X, ZW U N 250 230 0 2100 1900 1700

Figure 11. Approxlmttons for the Peek Overpressure between"Two Slultanemus 1 MT Stface lursts 4500 ft ApartC ared with the LM Obdl

32

Table 5. Range Coverage Comparisons(1-MT Bursts 4500 ft Apart)

PEAK SINGLE DISTANCE PERCENTAPPROXIMATION OVERPRESSURE BURST ADDED BY RANGE

CASE COVERED RANGE INTERACTION INCREASENUMBER (psi) (ft) (ft) (1)

LAMB 450 2030 430 11

3 540 1915 670 17

6 710 1 1730 1040 30

5 990 1540 1410 35

33

SECTION 5. SUMMARY AND CONCLUSIONS

A reasonable lower limit approximation to the combined

overpressure would appear to be to multiply the pressure ata point inside a single blast and in front of the transmitted

shock from a second burst by the reflection factor for a shock

of that interior pressure (Case 3). A similarly simple upper

bound should be provided by multiplying the single blast peakoverpressure by the normal reflection factor for that pressure

at distances beyond the point of first contact for the two

shocks (Case 5). A further procedure, almost as simple, pro-

viding an intermediate value, uses the pressure predicted for

the interaction of unequal shocks, based on the pressure aheadof and behind the transmitted shock (Case 6).

The LAMB procedure is not rigorously correct, but itprovides a more general and a more detailed treatment of

interacting shocks. Unfortunately, it is not clear whether

it overestimates or underestimates the resulting pressures,

although all the predictions for this example lie above theLAM-derived values. Accepting all the apparent uncertaintyin these approximations to multiple shock interactions, one

can still conclude that:

* The region of enhanced overpressures from twosimultaneous separate blast waves is a small

fraction of the area ccvered by each individual

blast.

9 The extra coverage of a line target with high

overpressures (AP5 > 300 psi) is of the order of10-50 percent, and, ,more probably, less than30 percent.

34

The LAMB procedure may, in some cases, underpredict

the peak pressures for interacting blast waves.

F

35

" :.';,'35

REFERENCES

1. H. L. Brode, "Review of Nucle-r Weapons Effects," AnnualReview of Nuclear Science, Vol. 18, 1968, p. 180.

2. H. L. Brode, Height of Burst Effects at High Overpressures,DASA 2506, The Rand Corporation, RM-6301-DASA, 1970.

3. H. L. Brode, Point Source Exlosion in Air, The RandCorporation, RM-1824-AEC, 1956.

4. H. L. Brode, The Blast Wave in Air Resulting from a Hi hTemperature, Hih Pressure Shere in Air, The RanCorporation, RM-1825-AEC, 1956.

5. C. E. Needham, M. L. Havens and C. S. Knauth, Nuclear BlastStandard (1 kT), Air Force Weapons Laboratory, KrtlandA-r Force Base, New Mexico, AFWL-TR-73-55, (Revised) 1975.

6 C. E. Needham, Private Communication, Air Force WeaponsLaboratory, Kirtland Air Force Base, New Mexico, November1977.

36i

If1I

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