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^.ecknical <v2©te 155

THE ENERGY PARAMETER B

FOR STRONG BLAST WAVES

D. L. JONES

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
National Bureau of Standar AU f 1965

130,746

HE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS

technical <Vi0te

155

JULY 1962

THE ENERGY PARAMETER B

FOR STRONG BLAST WAVES

D. L. Jones

NBS Boulder Laboratories

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, DC. - Price 25 cents

CONTENTS

Page

1. Introduction --------------------- 1

2. Procedures ---------------------- 3

3. Discussion ---------------------- 9

4. Application ---------------------12

5. References ----------------------17

Appendix A---------------------- 20

Appendix B----------------------21

Appendix C------------ ---------- 24

in
ABSTRACT

The energy parameter B used in the strong blast wave

equations is calculated for monatomic and diatomic gases.

Three geometries, spherical, cylindrical, and plane are

considered. Comparisons are made with previously published

values of B. Tables and curves of the distribution functions

are given for each case. The equations of the blast waves,

in the similarity solution, are compiled for the six cases.

An application of the analysis of a cylindrical blast wave

from an exploding wire is given.

IV
THE ENERGY PARAMETER B FOR STRONG BLAST WAVES

DONALD L. JONES

1. Introduction

The theoretical treatment of strong spherical blast waves,

assuming similarity, has been made by Taylor . This work was

extended by Sakuraiy to the case of cylindrical and plane blast

waves. Although Taylor developed some approximate solutions

for monatomic and other gases, the main emphasis in these analyses

has been on solutions for air, a diatomic gas.

In the present work, three ideal situations have been computed which differ in the geometry of their initial conditions.

Energy is instantaneously released: (1) at a point to produce a

spherical shock wave, (2) along a line to generate a cylindrical

shock, and (3) over a plane to yield a plane shock. The shock

disturbance is assumed to be similar at all times, changing only its

linear dimensions with increasing time. It is also assumed that

-1-
-2-

the gases are perfect, with constant specific heat ratios.

Energy losses from ionization and radiation are neglected.

Under these assumptions the distance R of the shock

front from its initial position is related to the time

by the expression

1 f E V2 c * -- ( — ) H° CD

c [ *0 J

where E is the energy released, p is the ambient density

ahead of the shock, c is a numerical constant equal to

5/2, h/2, or 3/2 for spherical, cylindrical, or plane shocks

respectively. B is a numerical constant depending upon the

geometry of the shock wave and the specific heat ratio y

In spite of the idealizations, equation (l) describes

2 3 real explosions of exploding wires '

, cylindrical charges

k 5 of high explosives , and even atomic bombs. In order to

compare experimental results with theory using equation (1),

the value of B is needed with a precision at least as great
-3-

as that of the experimental data. For the present work

three significant digits are adequate. The precision

claimed for the previously published values for B is

three digits also. However, it will he demonstrated that

few were correct to more than two digits and one was not

correct even in the first digit.

A table of values of B, calculated by the author,

with a precision of three digits, is given for each

geometry and for both diatomic and monatomic gases. The

blast wave expressions for each of the six cases are

compiled in Appendix A. This report constitutes the

more complete discussion referred to in an earlier publication *-

2 . Procedure

Taylor , in his original work, gives a thorough discussion of the similarity method. It is sufficient here to

* The author is indebted to N. Gerber, BRL, Aberdeen Proving

Ground, Maryland, for pointing out an error in the previous

publication.
-4-

indicate that the assumption of similarity is consistent

with the equations of motion and continuity and with the

equation of state of a perfect gas. The conditions at

the shock front are given by the familiar Rankine-Hugoniot

relations. Distribution functions are developed as a

convenient method for representing the pressure, density

and flow velocity at all points in the blast wave.

In the computation of the parameter B, it is first

necessary to integrate the differential equations of the

distribution functions given in Appendix C. Graphs and

tables of the solutions of these equations are shown in

Appendix B for each of the six cases. The abscissa 7] is

the ratio r/% where R is the distance from the origin to

the shock front and r is an intermediate point. The distribution functions f, to, and t are all dimensionless

functions of T. The function f is related to the pressure

ratio across the front; $ is the density ratio 5 and tp is

related to the radial velocity of the front. It should be
-5-

noted that the given boundary conditions in Appendix C

are correct only for strong shocks, i.e., for which the

pressure ratio across the front exceeds 10. As shown

in Appendix C, B is the integral of a geometry dependent

function of the distribution functions. Following Taylor,

B is found by evaluating first the differential equations step

by step and then numerically integrating from the table

obtained

7

A Runge-Kutta integration technique given by Gill

was used to evaluate the differential equations of the

distribution functions. Gill developed the method for

automatic computation, the main advantage being that only

one set of values at the boundary is required to initiate

the computation. The boundary conditions at the shock front

provide the necessary initial values. One change in the

method of Gill was required since the present computations

were performed with a floating point machine, whereas his
-6-

technique contained a means of reducing rounding errors

8 on a fixed point computer. Obitts has determined that

the use of floating point arithmetic reduces the accumulation of rounding error within a step, so that the

application of a bridging term as developed by Gill would

be unnecessary.

In computing the tables of the distribution, functions

the procedure followed was to select an arbitrary interval

for An and compute the table from *R = 1 to 11 = 0. Then

the interval was halved and a new table was computed. If

the new table agreed with the previous table to six significant digits the last table computed was accepted as

correct. If such agreement did not exist another table

was computed. This procedure was repeated until the desired

agreement was obtained.

The tables of the distribution functions in Appendix B

are not nearly as complete as the original tables calculated

for this work; many intermediate values are not listed. It
-7-

was necessary to use an interval of 0.0005 in T] to obtain

an adequate precision in the integration of the differential equations for the distribution function graphs.

The differential equations of the distribution

functions are quite well behaved, as evidenced by the

graphs in Appendix B. Also, the computer word length is

in excess of ten digits, while the distribution function

tables are only required to be accurate to six digits at

most. These two facts, when coupled with the relatively

* short length of the tables (~ 2000 entries), allow a

straight-forward evaluation of the distribution functions

without the serious loss of accuracy from truncation and

round-off that often plagues numerical evaluation of differential equations.

Preliminary computations of B were made on an IBM

650 computer, but the large number of calculations required in the step by step computations indicated the need

for a faster machine. Subsequently the program was placed
-8-

on a CDC 1906 computer and all computations were performed

with that equipment.

The computed values of B for the six cases considered

are shown in Table 1.

TABLE 1 - ENERGY PARAMETER B

Spherical Cylindrical Plane

v = 7/5 5-33 3.9^ 1.22

v = 5/3 3.08 2.26 O.678

Once B is known accurately it is a simple matter to

apply equation (l), its derivative, and the Rankine-Hugoniot

relations to compute the theoretical time, velocity, pressure,

and temperature for a shock front propagating in a known

gas. The theoretical values can then be used to compare with

experimental data or to predict experimental parameters.

Equations for these computations are given in Appendix A. These
-9-

equations are grouped according to the geometry of the shock

and further subdivided, as necessary, into monatomic and

diatomic gases.

If the distance of a shock front from the origin is

measured as a function of time, application of the equations

in Appendix A allow determination of the energy in the shock,

the front velocity, particle velocity immediately behind the

front, and the pressure and temperature in the shock front.

3« Discussion

Since the initial work of Taylor on spherical blasts,

several others have made further calculations on the spherical as well as cylindrical and plane blast waves. A list of

authors and the B values they have obtained is shown in

9 Table 2. The entries listed for Sakurai were calculated

from his published J values. Harris developed an approximate method for calculating B for any y, but even for

V = 5/3 his values are in error by more than 20 o/o. The

values listed for Sedov are obtained from graphs and the

* The author is indebted to Dr. H.T. Yang for calling his

attention to this work and also to reference 12.
-10-

spherical case appears closely related to the work of

Taylor

When the computed values of B in Table 1 are compared with the previously published values some discrepancies appear. For the plane shock wave with v = 7/5*

13

the value of 2.0U given by Lewis et al " disagrees with

our value by 67 0/0. This deviation probably resulted

from a mistake in their integrand of B. Their equation

y

(2k) contained a term ( ) which should have been

r ! 1

" 1^

. In the cylindrical case Lin's value of

LV(Y-1)J

3.85 for Y = 7/5 gives a disagreement of 2.3 0/0. Presumably this was caused by inadequate evaluation of the

distribution function differential equations. The agreement with the B values from Taylor for the spherical shocks

is remarkable in view of the fact that his were made without the aid of an electronic computer.
-lie/3 >

CM

pqEH

JT\ \ CVI tr\ ^ • • > OJ CM

JO\ «H CM -* OJ CM

fc- CM OJ o CM CM 1 • • • • • J- rH rH CM r-l t-i

0) JO oo ITN co

rH \ r— t— t^

PL, >

VO• O

VO• o

VO• o

JO\ O VO Mf

rH• rHr • >- CO CO

rH

05o JO IT\ •H \ -3- IfN rfN VO

O -3- -=J" 6 ii ON CO * • O• ON• O• ON• ON• c ^ WN N"\ -* K"N -=»- rfN K"\ •H

rH£ j°> \ CM o VO VO V

OJ• CM• OJ• CM• >- OJ OJ OJ OJ

j^ ON \ t- co ON

1? • o • o • o ? rH r-l rH

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CD

n • • Xi > fe- VO

ft CO JO\ vO

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fe- KN k>> H"N K> KN

ll • • • * • > ITN IT\ IA IfN ITN

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tfN o O rH

O ii • • • • ?- KN KN rr\ KN

ON CM VO

rH •H rH ro» ITN rH rH U ad CO rH rH rH VO

U o h -=*" U CO 0) > CO CO rH P H OJ •H CO O a> &

5? 5 c bfl

1-3

d TJ c U (0 Eh

to ih CO *1

O

O £ O 8
-12-

4. Application

We can now apply the equation for cylindrical shocks to the case

of an exploding wire in air. Radius- time observations of the shock wave were made simultaneously on three frequencies with the microwave

Doppler technique" as shown on the left in Fig. 1. The data for each

frequency, scaled from these traces, are given in Table 3.

TABLE 3 - Shock Wave Data

X = 3.0 cm X = 1.2 cm X = 0.81+ Cm

R cm t usee R cm t usee R cm t u sec

3.00 7-27

3.75 9.68

if. 50 12.82

5-25 21.14

• 329 1.38

.629 1.80

.929 2.24

1.23 2.78

1.53 3-21

1.83 3.43

2.13 4.49

2 A3 5.22

2.73 6.05

3.03 6.99

3-33 7.94

3.63 9-97

3.93 10.16

If. 23 11.27

4.53 12.42

4.83 13.96

5-13 15.60

5-^3 17.88

5-73 20.15

6.03 22.89

6.32 25.67

6.62 28.17

6.92 31.13

7.22 34.91

7.52 39.32

7.82 44.65

1.26 2.55

1.47 3.07

1.68 3.42

I.89 3.86

2.10 4.42

2.31 4.87

2.52 5.50

2.73 6.10

2.94 6.83

3.15 7.45

3.36 8.33

3.57 8.99

3.78 9.96

3.99 10.80

4.20 11.83

4.41 12.72

4.62 13.82

4.83 14.99

5.04 16.09

5-25 17-27

5.45 18.60

5.69 19.96

5.88 21.38

6.O9 22.90

6.30 24.56

6.51 25.86

6.72 27.51
•13-

The air density p , as determined from the ambient

k -3

pressure of 30 cm.Hg. is 4.63 x 10 gm cm . We now go

to Appendix A, to the column for cylindrical shocks in

a diatomic gas. The value of the parameter B is 3«94«

The time-radius equation is

1

1 . E x ~2 2

t = 2 fe~) R <A1 >

Since* the parameter B, the energy E, and the ambient

2 density p are all constants, a graph of R as a function

of time will yield a straight line with slope

1 » - 2(5-) • (2) ^0

The measured slope of the straight line portion of the

8 2-1 curve in Fig. 1 is 1.99 x 10 cm sec . Upon solving equation
14-

(2) for E, the energy in the shock is found to be

182.5 joules cm of wire length. Examination of Fig. 1

shows that the data follow the straight line for only part

of the shock trajectory. The positive curvature at the

beginning results from the finite time of the delivery of

energy to the shock during the explosion of the wire. The

negative curvature later represents departure of the shock

trajectory from the strong blast relation.

After finding the energy in the shock the velocity U

at any point can be calculated from relation (A3)

" = (—) R • CA3)

For instance, when the radius is 3-93 cm the velocity is

b -1

25.5 x 10 cm sec and the particle velocity immediately

behind the shock front is 5/6 U, or 21.2 x 10 cm sec
-15-

The expression for pressure at the shock front is

P = 7/6 R2 5- {Ah)

-1- vo

7 2 giving a value of 2.28 x 10 dynes cm" or 22.6 atmospheres

at the radius 3*93 cm. At the radius 5.5 cm where the shock

trajectory diverges from the strong blast relation the pressure

is 12.6 atmospheres.

Calculation of the temperature with expression (A5) gives

a temperature of 1130°K at the radius 3 -93 cm.

In assuming the specific heat ratio v to be constant, the

effects of excitation, dissociation, and ionization have been

neglected. These effects are such that they can decrease V

to below 4/3- If for a given geometry of shock the value of B

is available for several values of V, the correct value of B

could be estimated. Careful experiments could, at least in

principle, determine the value of B experimentally by making

use of equation (1) and the fact that quite precise values of

the energy E can be known.
-16-

ACKNOWLEDGEMENT

It is a pleasure to acknowledge the help of Mrs. J.

Herman, who wrote the computer program for the numerical

solution of the differential equations of the distribution

functions; M. Addison, who performed the numerical integrations from the distribution function tables; and particularly R. Gallet, whose suggestions and criticisms were invaluable

in the preparation of this report.

/
-17-

5. References

(1) Taylor, Sir Geoffrey, The formation of a blast wave

"by a very intense explosion, Proc. Roy. Soc . A 201 , 159 (March 1950).

(2) Bennett, F.B., Cylindrical shock waves from exploding

wires, Physics of Fluids 1, 3*+7 (1958).

(3) Jones, D.L., R.M. Gallet, Microwave Doppler measurements

of the ionization front in cylindrical shock waves from

exploding wires, Exploding Wires, Vol. II, Plenum Press,

New York (19^2).

(h) Birk, M. , Y. Manheimer, G. Nahmani, Note on the propagation- of explosion-produced air shocks, J. Appl. Phys.

25, No. 9 (September 195*0-

(5) Taylor, Sir Geoffrey, The formation of a blast wave by

a very intense explosion. II. The atomic explosion of

19^5. Proc. Roy. Soc. a 201 , 175, (March 1950).

(6) Jones, D.L., Strong blast waves in spherical, cylindrical

and plane shocks, Phys. Fluids h, 1183 (September 1961).

(Also see Erratum, Phys. Fluids 5, 637 (May 1962).

(7) Gill, S., A process for the step by step integration of

differential equations in an automatic digital computing

machine, Proc. Cambr. Phil. Soc. Vf_, 96 (1951).

(8) Obitts, D., National Bureau of Standards. Private Communication (i960).

(9) Sakurai, A., On the propagation and structure of the

blast wave I, J. Phys. Japan, 8, 662 (1953)-

(10) Harris, E.G., Exact and approximate treatments of the one -dimensional blast wave, Naval Research Laboratory

Report 4858, (1956).
18-

References ( continued

)

(11) Sedov, L. I., Similarity and dimensional methods in

mechanics, p. 210, Academic Press, New York (1959)*

(12) Rogers, M. H., Similarity flows behind strong shock

waves, Quart. J. Mech. Appl. Math. 11, l+ll (1958).

(13) Lewis, A.T., J.K. Oddson, H.H. Woodson, Unpublished,

MIT QPR %k, 36 (July 1959).

(Ik) Lin, Shao-Chi, Cylindrical shock waves produced by an instantaneous energy release, J. Appl. Phys. 2J3,

No. 1 (January 195*0-

(15) Rouse, C.A., Theoretical analysis of the hydrodynamic

flow in exploding wire phenomena, Exploding Wires,

p. 227, Plenum Press, New York (1959)-

(16) Gerber, N., J.M. Bartos, Tables of cylindrical blast

functions, BRL Memorandum Report 1376, Aberdeen Proving

Ground, Maryland (1961).
-19-

X= 0.84 cm

x=3.0cm

64

56

10 20 30 40 50

i i i

i i i /xsec

X= 1.2 cm

• x= 1.2 cm

- X = 0.84 cm

» X= 3.0 cm

I I L

16 24 32

t (/xsec.)

40 48

Figure 1. Oscilloscope traces of microwave Doppler measurements

of the expanding cylindrical shock front from a wire

explosion, are shown on the left. A plot of the square

of the radius of the ionization front with time is on

the right. The straight line portion of the plotted

points represents agreement with the theory. This

explosion was made in air at a pressure of 30 cm Hg.

The energyReleased into the k cm long, 18 mil copper

wire was 500 joules per centimeter of wire length.
-20-

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-21-

APPEND1X B

Distribution Function Curves and Tables

04

SPHERICAL SHOCK WAVE

7 ' 7/5 NBS

Ai) =00005 TAYLOR

SPHERICAL SHOCK WAVE y 7/5 SPHERICAL SHOCK WAVE

100 098

096

094

092 090

08B 086

84

82 080

78 076

74

7270 068

066

064

062

60

58 056 054

5250 046

042

38 034

30

26 022

18 0.14 010

006

002

f

I 6666667 03349096

08131 019 07 143236 06439796 03923293

03538214

03247000

03024335

04832673 04719533

04613936

04333121 04472068

04422924 04384709

04333093 04332247

043 147 16 04301 348 04291 229 04283631 04277976

042738 08 04270769 04268380

04263928 04264600 04264093

04263860

04263773

04263746

04263738 04263736 04263736

04263736 04263736 04263736

08333333

07983383 0766404 1 07368696

0709762 7 06848413

0661831 7 06404 381 620461 06016069

0583692 1 03665423

03300 I I 5 03339773

03183392

3030149 04879377

04730334

04383 190 04436998

0429 1686

041 47040 04002893

038391 18 0371561 3 3372309 03286081

3000 I 4 I 0271 4333

0242 6 58 7 02 142862 01837144 01371429

1 2857 1

01000 001 0071 4288 00428377

0014291 I

60000 000 = 100 40639 380 : 0.98

2 884 8390 : 096

2 1 197298 E 094

1 399 7640 E 032

1232 1 970 1 090

09637 827 : 088

07624734 86 608 1833 : 084

4878 446 : 082 03926863 E 080

03166384 : 078 02333826 E 076 02037690 E 074

1634446 E 72 1326 136 E 070

1058782 E 068 0084 131 3 E 066 00664839 064

00522 113 i 062 00407 186 060

0031 5129 0.36 02 4 1843 36 001 8 3904 034

001 38456 : 032

001031 1 1 050

003 5132 048

0027838 42

0001 3149 038

0000 3709 34 00002233 030 00000763 26 000002 16 E 022 0000048 : 018 00000007 : 014 00000000 10 oooooooo i 008

0000000 ; 02

f 12300000

1.07426 3 1 0.93828 44

0.63 16917 07478276

6806264 06270086

3834806

034 80582

051 90766 04952546

04755976

045932 7 1 04438281

4346 1 I I 04232826

041732 46 041 107 70 04057263

04OI2934

03978388

03946267 0392 1608 39013 09 03 8832 2

03672 I I I 03833329

384 I 74 I 03834892 03831034

03829042

03826072

03827833

382 749 6 03827432

03827442

382 7440 3827440

7500000 07205468

06922655

06652 500 06393580

6 I 5209 I 392186 3 03704420

0349903 4 33046 13 03120780 04943907

04779183 04819643

04466376

43 18349 04173408

4036 2 78 3900359 03767727

3637 320 03308939

03382241 03236931 03132736

03009304

02763085

02522570

022812 I 3 02040327

1800 206 1360070

O 132002 I 1060003 00840002

06O0O02 0038OO03

00120048

40000000

3171 1370 23506850

2 078 1430

1. 7124410

14231360

I 1962 170

101 14007 08603333

07335092

063 12624 03434 I 36 46878 20 040493 20 03499789 03024476

0261 1723 02232208

01938 380 1664063

0.1424218

12 14517 01031376

0087 1 703 00732833

00612432 0041902 1 002773 16 00176606 OO 1 06 966 00060876

0003 1963 OOOI5072

00006 109

00001 972 00004 34 00000044 OOOOOOOO
-22-

0.8

-eCYLINDRICAL SHOCK WAVE

r.7/5

AlJ- 0.0005

CYLINDRICAL SHOCK WAVE

1.00 0.98 036 034

032 030

088

036 034

082 030

0.78 076

0.74

0.72 0.70 038

066

0.64

0.62 0.60 038

0.56 054

052

030

046

0.42 038 034 030

026

022

18 0.14 0.10 0602

1.1666667 1.0087050 08899788

0.7990885

0.7284405 03728230

0.628 5670

0.5930338

0.5642897 0.5408940

0.5217575 05060450

0.4931084

0.4824382 0.4730630 04663 589 0.4603620

0.4554241

0.4 513680 04480468

0.4453378

0.443 I 384

0.441 3622

0.4399365

0.4387998 04379006 04366468

04359044

04354853

04352622

04351 517 04351019

04350820

43507 52 04350733

0.4350730 04350730 04350730

08333333

08034607

0.7749885

0.7479352 0.7222810

0.6979718

0.6749282 08530528 08322389

0.6123767 03933577

03750787 03574437

0.5403648 0.5237630

0.3075679

0.49 1 7 I 75 0.4761573

0.4608401 0.4457254 04307780

0.4159662

0.40 1 2707 03866642

03721309

0.3576561 03288348

0.300 1 272 02714857

0242 8806 02142943 01857170

0.1571436 01285716

1000000

0.071 4286 00428 572 00142858

60000000

43786812

3.5750860 23450400

23000562

1.8838684 13596000

I 3024355

1.0952829

0.926 1213

0.7 863344

0.6696371

0.57 I 369

488021 I 0.41691 13 0.3559606 03033349

02583322

0.2 1 9 30 1

1855819

0.1 56461 2

01 31 341

0.1097143

0.091 1445

0.0752541 00617126

0.0405442 00256782

0.01 55515

00891 26 00047653

00023297

0.00 10105 00003705

000 1 054 0000196 000000 1 5 OOOOOOOO

CYLINDRICAL SHOCK WAVE y 5/3

V

f + *

1.00 125000000 0.75000000 400000000

098 1.1 1889770 0.72406996 340401 587

0.96 1.00979580 0.6988 1 980 2 9 1 7 10821

0.94 091856441 067430297 23 1 606 1 38 092 084194494 0.65056232 2J 831 3064

0.90 0.777344 14 0.62762827 190463312

0.88 0.72268220 0.6055 1 77 1 186994251 086 067627944 0.58423 390 1 .47073,1 03

0.84 083677099 0.5637671 1 1.30044470 082 080304 183 0.54409599 1.15388876 030 0.574 1 7669 0.52518945 1.02692422

0.78 0.54942108 0.50700868 0.91623762

0.76 0.528 1 5056 048950933 081 91631 7 0.74 0309846 36 047264343 0.73354669

0.72 0.49407587 456361 27 0.65763965

0.70 048047696 0.4406 1 262 0.59001646

0.68 0.468 74527 0.4253461 0.32950987

0.66 0.46 1 1 58 1 041 41 6663 0.48821 598

0.64 044989525 0.39608280 042617275

0.62 0.44237382 038199378 0.38189071

0.60 0435901 08 0.36821 307 0.34176467

0.58 0.43034077 3 5470391 0.30533470

0.56 0.42557523 0.34143260 027221 346

0.54 0.42150242 0.32836847 0.24207297

0.52 0.418 03 344 0.31 548376 021 463385

0.50 041509033 01027 5355 18963663

0.46 041051802 027766983 14628772

0.42 0.40733001 0.25296749 1 1060634

0.38 040517079 022853072 008 1 54633

0.34 0.403761 30 20427231 0C582362I

0.30 0402883 48 1 801 285 1 003993098

026 040236896 1 5605447 0025965 13 022 040209061 13202000 0.01 372103

0.18 0401 95567 10800602 000860806

0.14 0401 89980 08400 136 000404967

10 040188285 0.06300027 001 70839 006 040187790 0360000 7 000031876

002 040187758 001200017 00001 18 1
-23-

PLANE SHOCK WAVE

12

1.0 PLANE SHOCK WAVE

r»5/3

Ai) 0005 f /

08

-6-

6.0 06 / V

-=>- - /

4.0 0.4

- V 2.0 0.2 yS /

/S -

PLANE SHOCK WAVE r • 7/3 PLANE SHOCK WAVE

1.00 98

0.96 094

9290 088

0.86

0.84

0.82 0.80 078

0.76

0.74

0.72 0.70

0.6866

0.64

0.62 0*0

038 036 034

032 030

0.46

0.4238 034

030

0.26 022

0.18 0.14

0.10 0.06 002

f

1.1666667

1.069 8629 09874815

03 I 70344

0.8563333 O.8043842 07592791 07201533

0.6861270

0.6364682 06303663 06079080

03880602

3 706360

0.3553829 05419740

03302000

0.3198636

0.5 I 07942 03028436

04958823

04897978

0.4844903

0.4798722 04736 659

0.4724023 04668633

0.4628337 04399789

045 80163

0.4567218

0.45391 19 01354394

455 1889 04550734

04550305 045S0I96

04550184

08333333

08083881

7843701

0.7607013

73 73980 071 50697

0.6 931 196 67 17444 06309348

0.6306762

0.6 109492 05917308

5729950 03347137

03368573

3 193937 03022985 04653358

046 90784 04528982

04369682

0.42 12632 04037591

03 904338

0.3732663 0360238 1 03 303302 030 I 1883

0.2 721 143 0*432292

02 I 44727

1837994

1371768 1283627

1000026

07 14290

0.042837 2 00 142837

6.0000000

3.1630443 45079704

3.94 32009

3472 1619

307 14360

2.72943 I 4 24335374

2 1812337

1.93 97920

1.7657772 13947 956 14432772 13082979

I 1874563 10787697

0.9803 945 089 13628

Oil 1093 I 3 07363423

0.6687 903 60639 30 034938 14 049666 14 04480193

0403 10 13 03232646

02331968

197431* 1488419 010833 10 00737301 0O498 33S 00301 3S9 00160837

0.0069333 000193 39 0O00I24I

V

100 12500000 098 1 1670709 096 1 0926 1 30 094 10257 123 092 09633646

090 0.91 14 602 088 8627706

0S6 0.8189384 084 07794676

082 07439163 080 07 1 188 98 078 06610333 076 06570371

0.74 063361 24

0.72 061 23077 070 03934939

068 03763732

066 03609369

64 03470833

062 03346033 060 03233909

038 031 3321 6 36 3042910 034 04962031

032 4869 71 7 030 048231 67 046 04716726

042 04*31 735 038 045662 09 034 04316668

030 04480219

026 044 34342

22 0443*839

018 0442977 1 014 04419430

10 04416323 006 04413172

002 04414946

075000000

072763399

0703631 39 068393360

066264734

064 I 72623 062 1 20926

0601 I 1062

38 I 4 4 1 2 7 036220877

03434 1 706 032306630

0307 13391 048967260

047261 360 043396403 043970947

042 383334 040831733 039314280

037828910

036373392 034946260

033344839 03216 7360 03061 1766 0281 36621 023371371 023037030 020343793

01606 1907 13643032

01 3220296 010806226

006402636 00*00038 5 003600060

001 200000

40000000

36393321 33348977

30817418 28362660

2* I 32428 24 137452 22352866

20716716

19229 733

I 7874989 16637377

c 3304372

I 4463804

I 3303667

I 2620938

I I 80 I 7 29

I 1040979

103 32 489 09670629

09031 I 72 08469269

7921377 074041 97 06914827

064307 16

3 589601 04806151 04089298

34 30*0* 02629166

022 6*669

17604 1

01300237

008*0836 00337490

0024973 1 00048062
-24-

APPEHDLX C

Blast Wave Distribution Functions and the B Integral

SPHERICAL

= 4n r r i . f + aj£ ^ ti2

dTi Jo V Y(v-l) 2 y

f 1-3T, + * (3 + i Y) - -^ 1 df

- f '

-

c fn - <p) - f}

[ 3

<p} dcp i

l-Y f

" 2

M 171 - cp }

dR " v " v. v

(l| - cp) J

with 71 = £

R

where R = distance from explosion to shock front

and r = distance from explosion to intermediate point

Boundary conditions at shock front

f = 2_v

= _2 ,

|f Y+l

1 Y+l * 1 Y+l ' 1 v-1
-25-

CYLINDRICAL CASE

B = 2n

o v y(y-i) 2 y

df » in

_ f

•g] (T1-cp) + V

- f - (11-cp) 2

*

Iff T

at

in

= cp = f ' -YJJP

Y 4 (l| - cp)

dt (n cp' + 9) an

=

('H - cp) i|

with \ R and r same as spherical case

Boundary conditions at shock front

1

" Y+l

^1 Y+l

V l

" Y-l
-26-

PIANE CASE

B

_ r ( i

Ju V v(v-l Y(v-l)

T 2 ) an

df _ fi

in - f - — v cp + 2 m - 9) ]

f-t 01 - q>)'

dcp _

i f ' - \ y jf dT] " * " 7 f tf-q>)

d* . » _ ^ cp

with \ R and r same as in spherical case

Boundary conditions at shock front

f

_ 2y

1 Y+l

<Pn =

1 Y+l

ilr = Hi

1 v-1

ft U.S. GOVERNMENT PRINTING OFFICE : 1962 O - 656929
U. S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

kA. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and

Boulder, Colorado, is suggested in the following listing of the divisions and sections engaged in technical work.

In general, each section carries out specialized research, development, and engineering in the field indicated by

its title. A brief description of the activities, and of the resultant publications, appears on the inside of the

front cover. WASHINGTON, D. C.

Electricity. Resistance and Reactance. Electrochemistry. Electrical Instruments. Magnetic Measurements

Dielectrics. High Voltage.

Metrology. Photometry and Colorimetry. Refractometry. Photographic Research. Length. Engineering Metrology.

Mass and Scale. Volumetry and Densimetry.

Heat. Temperature Physics. Heat Measurements. Cryogenic Physics. Equation of State. Statistical Physics,

Radiation Physics. X-ray. Radioactivity. Radiation Theory. High Energy Radiation. Radiological Equipment.

Nucleonic Instrumentation. Neutron Physics.

Analytical and Inorganic Chemistry. Pure Substances. Spectrochemistry. Solution Chemistry. Standard Refer- ence Materials. Applied Analytical Research. Crystal Chemistry.

Mechanics. Sound. Pressure and Vacuum. Fluid Mechanics. Engineering Mechanics. Rheology. Combustion

Controls. Polymers. Macromolecules: Synthesis and Structure. Polymer Chemistry. Polymer Physics. Polymer Characterization. Polymer Evaluation and Testing. Applied Polymer Standards and Research. Dental Research.

Metallurgy. Engineering Metallurgy. Microscopy and Diffraction. Metal Reactions. Metal Physics. Electrolysis and Metal Deposition.

Inorganic Solids. Engineering Ceramics. Glass. Solid State Chemistry. Crystal Growth. Physical Properties.

Crystallography.

Building Research. Structural Engineering. Fire Research. Mechanical Systems. Organic Building Materials. Codes and Safety Standards. Heat Transfer. Inorganic Building Materials. Metallic Building Materials. Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics. Operations Research. Data Processing Systems. Components and Techniques. Computer Technology. Measurements Automation.

Engineering Applications. Systems Analysis.

Atomic Physics. Spectroscopy. Infrared Spectroscopy. Solid State Physics. Electron Physics. Atomic Physics.

Instrumentation. Engineering Electronics. Electron Devices. Electronic Instrumentation. Mechanical Instru- ments. Basic Instrumentation. Physical Chemistry. Thermochemistry. Surface Chemistry. Organic Chemistry. Molecular Spectroscopy. Molecular Kinetics. Mass Spectrometry. Office of Weights and Measures. BOULDER, COLO.

Cryogenic Engineering Laboratory. Cryogenic Equipment. Cryogenic Processes. Properties of Materials. Cryogenic Technical Services. CENTRAL RADIO PROPAGATION LABORATORY

Ionosphere Research and Propagation. Low Frequency and Very Low Frequency Research. Ionosphere Research. Prediction Services. Sun-Earth Relationships. Field Engineering. Radio Warning Services. Vertical Soundings Research.

Radio Propagation Engineering. Data Reduction Instrumentation. Radio Noise. Tropospheric Measurements.

Tropospheric Analysis. Propagation-Terrain Effects. Radio-Meteorology. Lower Atmosphere Physics.

Radio Systems. Applied Electromagnetic Theory. High Frequency and Very High Frequency Research. Modulalation Research. Antenna Research. Navigation Systems.

Upper Atmosphere and Space Physics. Upper Atmosphere and Plasma Physics. Ionosphere and Exosphere Scatter. Airglow and Aurora. Ionospheric Radio Astronomy.

RADIO STANDARDS LABORATORY

Radio Physics. Radio Broadcast Service. Radio and Microwave Materials. Atomic Frequency and Time-Interval

Standards. Millimeter-Wave Research.

Circuit Standards. High Frequency Electrical Standards. Microwave Circuit Standards. Electronic Calibration

Center.
NBS
No. 15>6 cancelled