Explosions due to pressurized spheres at the ocean surface

Explosions

due

to

Pressurized

Spheres

at

the

OceañSurf

_ace

B. C. Ca&N, M. HOLT, _AND R L. WELSU

Aeronautical Sciences Division, Univérsiiy of California, Berkeley, California

(Received 7 September 1967)

The effect of _an explosion due to the rupture of _a pressurized sphere, centered at theocean surface,

s determined. The field of disturbance near the vertical axis through thecenter of the explosion is

determined by Friedthan's analytic approach, based _{on a} perturbation of the linear-shock-tube sol-

ution. The interction of the main blast wave with the ocean surface is determined by fitting a lOcally

plane shock with _a PrandtlMeyer expansion, which reduces the shock _pressure to atmospheric value..

The main object of the calcu1ation is to determine the shape of the distorted ocean surface (or crater).

Calculations _are made for initial sphere _pressures of 619 and 7000 atm, respectively. The shape of the

disturbed ocean Burface flattens at the base _as the strength of the explosion increases, in agreement

with the behavior found for intense point _source explosions.

L TRODUCTION

The effect of the _ocean surface on the behavior of

a plane or spherical blast wave can be determined

by existing shock-interaction theories' when the

center of the explosion is below the surface. In this

case the explosion is symmetrical for initial times,

and unsymmetrical effects oniy develop after the

main blast wave first strikes the ocean surface to be

reflected and refracted there. The effect of placing

the center of this explosion _on the _ocean surface is

more difficult to determine, since the flow field is at

no time spherically sym±netric, not even at the

instant of initiation. This problem has recently

been investigated2 in the _case of' _a point _source

explosion, when similarity properties can be used

and the flow field depends _on only two independent

variables, namely the ratio of the local radius (meas-

ured from the center of the explosion) to the shock

radius, and the angle between the radius and _a

vertical axis through the center.

In the present _paper the explosion centered _on

the _ocean surface is investigated for _{a source} of

finite dimensions when similarity properties _no

longer apply. The explosion results from the release

of gas in a pressurized sphere, corresponding to a

spherical shock tube, _so

that

_no detonation phase

need be considered. This problem has been investi-

gated extensively for the symmetrical _case of release

in _a single uniform medium by Brode,3 who used _a

Lagrangian numerical method, and by Friedman,4

who used _an approximate analytic approach. The

* Present address: University of Strathclyde, United

Kingdom.

1 R. H. Jahn, J. Fluid Mech. 1, 547 (1956).

2 R. Collins and M. Holt, Phys. Fluids 11, 701 (1968).

'H. L. Brode, Phys. Fluids .2, 219 (1959).

M. P. Friedman, J Fh.iid Mech. 11, 1 (1961).

714

pressurized sphere. model _can be used to simulate

different types of explosions ranging from those _on

the acoustic scale, when the _pressure differences

across the sphere are sniall, to point, nuclear ex-

plosions, when the confined _gas is kept at extremely

high _pressure and temperature.

The _ocean surface explosion is treated by _an

extension of Friedman's solution. Two simplifying

properties _are used. Firstly, the effect of the _ocean

surface is most marked _near the intersection of the

main blast _wave with the surface. Near the vertical

axis through the center of the explosion the surface

effect is negligible, and the flow field _can be calculated

as though the shock were propagating in an infinite.

medium. Secondly, although the blast _wave resulting

from the explosion propagates into both air and

water, the water shock is much stronger and _moves

much faster than the air shock, _so

that

the influence

of the air _on the propagation of the water shock _can

be neglected. This simplifies the _- treatment of the

shock-ocean surface interaction.

Friedman's4 solution for _a spherical explosion in

air is extended to apply to propagation in water.

This solution applies in the neighborhood of the

vertical axis through the center.

It

is also used to

determine the speed and strength of the shock

moving along the ocean surface. At the ocean surface

the shock is reflected _{as a}

PrandtlMeyer

expansion

which reduces the _pressure behind the water shock

to atmospheric conditions. The shock-ocean surface

interaction is determined using _a steady-flow model

and the analysis is based _on the work of Grib et al.6

The _ocean surface interaction analysis determines

the slopes of the shock and _ocean surface (behind

-

'A. A. Grib, A. G. Riabinin, and S. A. Khristianovich,

Prik. Mat. i' Mekh. 20, 532 (1956).

the shock). This is combined with the solution _near

the vertiéal axis to fit curves for the water shock and

ocean surface.

Results _are worked out for two pressurized sphere

ratios representing a medium and strong explosion,

respectively.

II. EQUATIONS OF MOTION

The equations governing unsteady,

- spherically

symmetric, adiabatic flow _are

äp au .3p_

+p+u

2pu

--i-,

t3u &u

+u+

at ox pOX

as

as

where _p denotes density, _u fluid velocity (in the

radial direction), _{p pressure,} S specific entropy,

x radial coordinate, and t time. The third equation

is the condition

that

the entropy is constant on a

particle path _so

that

the effects of viscosity and

heat conduction _are neglected. This equation is in

characteristic form since it involves the derivative

of S in _one direction, dx/dt _{= u,} i.e., the particle

path. By taking an appropriate combination of the

first two equations these _may also be expressed

in

characteristic form, the resulting derivatives being

taken in the directions dx/dt _{= u}

_±

_a, where _a

is the sound speed defined by a2 ₌ (Op/Op)

. These

characteristic equations can be written

i3u lap

+

2pua2 ₌ 0, (4)'

_pa+(ua)(Pa).

9nna2

+

=0.

(5) X

We shall _assume the _gas to be polytropic _so

that

a2

₌

-yp/p, where y is the ratio of the specific heats.

If, in addition, we assume the flow to be homen-

tropic, i.e., 5 is uniformly constant, which will be

true for _a region which has not been traversed by _a

shock _wave of varying strength, then the motion

is governed by the following two characteristic

equations for _{u, a:}

FIG. 1. Initial paths of boundaries between sectors radi-

ating from the origin of blast, for explosion in _a single medium

(M.S., main shock; C.F., contact front; S.S., secondary shock;

E.W., expansion wave).

01

1

±(u±a)--(!±

1

a')±=O.

Ox2

-yi

/

X

The flow resulting from the release of _{a pres-}

suri.zed sphere in _a uniform medium has been cal-

culated by Friedmn,4 and is sketched in Fig. 1.

There are five distinct regions, 0 and 4 being _Un-

disturbed gas in the initial states outside and inside

the sphere, respectively. The main shock propagates

into region 0 with decreasing strength due to the

increase of its surface _area, setting the _gas behind it,

region 1, iii _an Outward expanding motion. The _gas

initially inside the sphere is _set in motion (region 3)

by _a simple _wave propagating inwards. The contact

front separates the _gases which _were initially inside

and outside the sphere. A fluid element is compressed

on traveling through the main shock and expands

behind it. However, the increase in _pressure due to

the contraction of the stream tubes towards the

center causes a secondary, inward-facing, shock

to be formed behind the contact front. This shock

is _very weak initially and _moves outwards, _{but later}

strengthens and _converges towards the center of

the sphere with increasing strength The region 3 is

evidently homentropic whereas regions 1 and 2 _are

not.

The two main difficulties in finding _a complete

solution tO this problem are the complexity of the

equations and the fact that the positions of the

boundaries between adjacent regions (on which

boundary values would have to be taken) _are un-.

known functions. The governing equations _{are non-}

linear since the disturbances _are not small.

In Order to obtain a solution it is essential to make

(6)

EXPLOSIONS

AT THE OCEAN SURFACE

116

CHAN,HOL.T,

various assumptions and simpliflcations. This has

been done 'for _a pressurized sphere in' _a single uni-

form nedium by Friedman,4 who finds his reu1ts

to be in surprisingly good agreement with the _ex-

perimental data' by Boyer,6 _{and a} complete numeri-

cal solution-by Brode,3 in view of the simplifications,

employed. His results are used where applicable and

the solution extended. to the _case of _a pressurized

sphere, the center of which is _on the surface of _a

water-air interface. Our main interest is in the

effect _on the main shock due to the interaction be-

tween the air and the water,. and the position of the

disturbed water surface. Firstly, the solutions _cor-

responding to the rupture of _a pressurized sphere in

a single uniform medium _are calculated in the cases

of air and water, respectively. These _are then modi-

fied to obtain the solution when the sphere is

centered _{on an} interface between air and water. In

solving the spherically symmetric problems it will

be necessary to develop the solutions _on either side of

the contact front and match the fluid velocity and

pressure there.

A. The Expansion Region 3

The fluid in region 3 has not been traversed by _a

shock _{wave so} that the flow there is homentropic,

governed by Eqs. (6). The simplification employed

here is to _assume, following Ftiedman,4

that

the

flow is _.a linear perturbation of the one-dimensional

point-centered rarefaction _wave, the solution for

which is known analytically. We write the fluid

velocity and sound speed in the form

U3 = 1 4- 'us',

a3=a+a',

where the first term is the one-dimensional solution

governed by Eqs. (2) (6) with the inhomogeneous

term omitted, and the second is the spherical term,

which _we take to be _a perturbation. At the initial

instant t ₌ 0 the second term is in fact zero and is

small provided is sufficiently smalL We, therefore,

assume

that

Iu'/uI

<< 1,

Ja'/a

_<< 1 and neglect

terms involving the squares of these quantities.

Let _us introduce the Riemann invariants in the

form r4 1

71

a'

±

y

1 6 D. W. Boyer, J. Fluid Méch. 9, 401 (1960). AND WELSH

It

follows

that

r4 _are constant along the character-

istics dx/dt _{= u}

_±

_a,

respectively, and are given by

u

=2r+±(i_)X,l

,

a3=p2r

(+

x-1

+

r = (1

2)r

(1 ,.)

where _{i =} (y,

_-,

1)/(73

_±

1), and _{z =} 1 is the

initial radius.

The linearized equation for R is

(where _a stiffix denotes _a partial derivative) w.hich

has the characteristics

-

dt dx

xdr

=

_4irt

₊

(1 2z)(x 1)

tua

(7)

The first of Eqs. (7) gives

at2 ₌

Kj,

K const

which _we simplify by assuming

that

't(2r'

Kt2)

₌

0

or

x

-

1 ₌ 0.

This implies that the characteristic remains close

to _x

₌

1. Under this assumption we obtain

1 _a2 2r '

a]

+

f(at2),

(9)

IL(1-41h)

,

1-2k

where

/

is _a function which is constant on a positive

characteristic, determined -by the condition in

region 4, which is uniform, and _we find R to be

i ,\1/2u IL 1 j 1G3 ₂ t (1

4)

[a3 \a4) 2r

r

,

.[a__)

_a4 a

An analogous solution _may be found for R 'but

will not be needed, here, _as will be. _seen later. A

knowledge of R would be _necessary for predicting

the iiotion of the secondary shock.

(8) - (10) so that dR _24[2r (1 dt Li

+

t(2r

-

Kt_26)

-. B. The Main Shock and Region 1

The flow variables immediately behind the main

shock _may be readily evaluated, from the _conserva-

tion equations, in terms of the shock speed, which

is unknown and could only be determined by inte-

grating the equations of motion _as far _as the contact

front and matching with the solution behind

The

pathof

the main shock _wave will be evaluated

by application of the rule due to Whitham,7 which,,

although based _{on an} approximate method, is

kno*n to give excellent _accuracy in _many problems.

In this _case the main _cause of the change in shock

speed is the increase in surface _{area as}

it

_progresses,

so

that

the approximate result will give good ac-

curacy. The method is simply to apply the char-

acteristic condition to be satisfied _on the incoming

characteristic (in this _case the positive one) to the

boundary values

at

the shock, and _so obtain _a dif-

fereritial equation for the shock speed in terms of the

shock radius.

-

The flow variables behind the shock _are

2a0 (M

M),

Po _(2't0M° 'to

±

1), 'to

±

1 2 Po(7o

+

l)M ('ta 1)M°

_±

2 -2 a1 Pi

where the constant C

is

determined from the initial

Mach number.

The initial value of _any quantity _may be deter-

mined from one-dimensional shock-tube theory,

since the initial radius _can have no effect in the

limit _as t tends to zero. From this we obtain the

initial Mach number from

G. B. Whitham, J. Fluid Mech. 4, 337 (1958).

EXPLOSIONS

AT

TUE

OCEAN SURFACE 717

where M is the shock Mach number equal. to the

shock speed/a0 and here _we will _use the nondinien-

sional variables UI IL1 a0

_i!.

Pi Po

The motion behind the shock is not homentropic

and the appropriate characteristic equation is

p,+pau,

or

dp+padu-

dx

_=u+a.

Application of Whitham's rule gives the following

equation for the shock Mach üumber M in terms

of the shock radius:

2dZm 4M Xm dM 2't0M2

-.

't

±

1. 2(111°-I-

fl

+

_{M{[2 M°}

-

't0±1][M°('t0 2M i('to 1)M°

_±

2i

+

_{(M2 L270M°}

_1]

which integrates to give

x(2't0M°

-

_'to

₊

1)"°(M2 1)

M[2('t0

+

1){[M2('t0

1)

+

2][2't0M2 _To

+

1])*

+

2('t

+

1)

+

(M2 1)[2'yo('to

+

1)

+

_'t

+

1]]

[2[2't('t 1)]{(2yoM° _'to

₊

l)[('yo 1)M2

+

2]}'

+

4'yo('to 1)IW°

+

[4't ('t° 1)2J]7/[21

'''exp

[2('t 1)] 1 12(2'to1112

-

'to

+

1)

-

.('t l)[('tO - 1)M2

+

2]

sm

_M2('t0+1)°

r

i

2Ii-(-

i\

I

\

_{J.)(ao)(MM_i)}

Po L

\'t

+

li\a4i

+

2't (MO _1). (14) -

70±1

The point centered expansion _{wave causes} the

pressure initially driving the shock to be con-

siderably less than

_p.

Since the initial radius _{Z) was} chosen to be unity,

(13)

p

AND

WELSH

and the contact front motion _can be shown to satisfy.

the differential equation

ai1{1

_a22

1

±

1 (2Q2

=

{x,.

+

w(l.

t,,,)

-

(15),

where

_w=u2a2,

_w1=u,a,,

Q2=a2/(y2

-

1)+u2.

When the secondary. shock is neglected,

a,

-

1

a4

±R.

-

1

In Eq. (15) the primes denote differentiation with

respect to M.

D. Gas and Water

Since the Tait equation,8 rather than the perfect

gas equation of state, applies in. water, the above

analysis has to be modified for _a contact front

separating _gas and water. The Tait equation _may be

written

or

p+B

('

₁₆

B

.\poI'

( )

where B is _a slowly, varying function of- entropy

having the dimensions of _pressure. When _p has the

finite value

_p,

the pressure is zero. The constant n

evidently plays _a role equivalent to

that

of. for

a polytropic gas. The values of n 'and .B (assumed

constant) will be taken to be 7 and 3260 atm, re-

spectively.

The above equation of state differs from

that

for

a polytropic gas in term B in 'the numerator. The

relation along _a particle path in _water is

.p

+

Bap.

Denote p

+

B by and n by ,. The speed of sound

in water is. given by

2

=

_\äpis

'fiP

p

I R. 11. Cole,- Unde'rwater Explosions (Dover Publications

Inc., New York, l965, p. 38.

718

CHAN,

HOLT,

x,,, is the dimensionless radial coordinate of the shook

position. Let

L

be the time

at

which the shock is in

position _x,,,, and introduce the dimensionless variable

L defined by La0

=

xo then

MdL=dx,,,,

where dx,,. is' given by (12).

In performing the matching at the contact front

we require information concerning only the position

of the negative characteristics there. Since the

contact front follows closely behind the math shock,

we will approximate the negative characteristics

in the region I by straight lines. Thus the equation

of the characteristics is

x = x,,,

+

w,(M)(t

-

Im),

where _{w, =} u1

-

a1, evaluated

at

(x,,,, L).

C. The Contact Front between Gas and Air

The contact front travels with the local fluid

velocity _u and is met by the positive characteristics,

of slope u

+

a, from region 2 behind

it

arid the nega-

tive characteristics, of slope _u

-

_a, traveling from

the main shock through region 1. The positive

characteristics in region 2 originate in region 3 and

their slope is altered _on crossing the secondary shock.

However, in the early stages of the motion this

shock is _very weak, and _we shall _assume

that

the

slope of the characteristic is unaltered _on crossing

this shock, i.e., dx/dt

₌

_u2

+

a2 u3

+

a3.

The conditions to be satisfied

_at

the contact front

are continuity of pressure and fluid velocity. The

latter condition is

U2.

On a particle path we have (in the case of a poly-

tropic gas)

p.

\a,/

where i denotes the initial state

(a')27_

a,!,

If (x,, L,) is the point _on the contact front met by

the negative characteristic coming from the main

shock when its Mach number is M, then

- x, = x,,,

+

w,(M)(t,

-

or

(a'\2 (7.-i)

/

_1a

\2,/(-1)

/

_1a8

\2/(-y,-i)

-

-

where _aB is the speed of sound in water

at

zero

pressure

a,

is the speed of sound in water at pressure

p., and

a,

is the speed of sound in air at pressure _,.

In this

cue

the differential equation for the motion

of the contact front is

1 ('12

-

1

a1i

-'

\'12

+

1 27/(7,-1) . (2Q2

-

c112)] -

+

(:)}

dM , ,

=

-j-

{x,,

+

wi(t.

-

-

E. The Interaction between Air and Water

OCEAN SURFACE

PRANOTL- MEYER

EXPANSION WAVE

N.

T..

SHOCK WAVE

Fic. 2. Water shock _on the oceansurface.

métry. Since the water shock _wave is much stronger

than the air shock, _we shall consider only the effect

of the interface _on the water shock and neglect

that

of the air shock.

The effect of _an air-water interface _{on a} plane

shock traveling in. water has been studied theo-

retically by Grib, Riabinin, and Khristianovich.K

Since _{we are} considering the flow 'only in the neigh-

borhood of the surface- _we can take the water shock

to be plane_in _a vertical plane. We shall _assume

steady-state conditions, i.e., neglect the fact

that

the shock _wave is decelerating and neglect the _curva-

ture of the shock in the horizontal plane. Under these

assumptions the problem is _very similar to

that

mentioned above. - -

In

Fig. 2, A is the point of intersection of the

dlined at angle

_a

_to the vertical: If the normal shock

interface and the water shock, the, _latter being in-

speed is N, then the point A _moves along the inter-

face with velocity N/cos _a. Immediately behind the

shock the flow is uniform but there is _a (one-dimen-

The aialysis given _so far relates only to spher-

sional) Prandtl-Meyer expansion _wave centered

at

ically symmetric flows. We _now consider the release

A reducing the _pressure in the fluid. The angle /3

of _a pressurized sphere the center of which lies _on

measures the inclination of the water surface to the

the plane of the horizontal surface between air and

vertical.

water. In this _case the floW is not spherically _sym-

If the angle

_a

is less than the critical value

_a,

metric

but

we shall assume

that

the flow on the _{the expansion}

wave catches up with the shock wave

vertical, line through the. center of the sphere is

which is then distorted: We shall _assume

that

the

unaffected by the interface, _so

that

_on this axis the

equilibrinm state has been reached in which.a

₌

previous spherically symmetric solutions hold in the

where _a* is given by.

air and in the water.

The flow is expressible in terms of two coor-

_.f'7+1

dinatesthe

radial distance _r and the angle. t' made

-

_\

2B'y

'

(19)

with the vertical axis, the solution being known _on

0

and'

_=-w. We shall not attempt to find the _{which is}

a smi'll quantity since

p'

<<B [See Collins2

positions of the main shocks and contact fronts for _for

a fuller 'discussion of this point.]

general values of &. Instead _we shall evaluate them _{The linearized' solution (neglecting higher-order}

at

1'

=

_r, i.e., on the interface only. _{terms in}

a

and /3, which are both small) gives the

At t

₌

0, the instant Of release, the air and water _{relation between-fl and}

a

as

shocks will be vertical. As they propagate with dif-

ferent velocities _any interaction between the two

/32 _a

+'1

-(

+

media will _cause departures from spherical _sym-

EXPLOSIONS

AT TH O:çEA, :.SURFACE 719

Thus,

p. 'ca,

(a)27R7_

where

, _Continuity

=

p

±

B and p. is reference pressure.

of _{pressure across} the contact front _may

10

720 ' CHAN,

HOLT,

AND. WELSH

Fio. 3. Paths of main shock and contact front in water with

the _ocean surf ace

where _p is the pressure

at

the cOntact front. between

the _gas and water. Thus

=

a{2[1

_.-

(p/p1)J}. (20)

Taking the _pressure to. be uniform along the

contact front, _so

that

the value of _p in Eq. (20) _can

be found from the solution calculated _on the vertical

axis, _we then evaluate fi

F. The Shock and Contact Front Profiles

To determine the

.

profiles of .the sliock _wave and

contact front we have theiI positions on the vertical

axis and their inclinations _to the free surface at A.

The position of the point' A is unknown but, since cv.'

is small, _we shall _assume

that

the distance traveled

'by the water shock horizontally is the _{same as} that

vertically. Thus, in Fig. 3, OA

₌

OC

₌

_-a, which is

known, and the shock profile is hemispherical _apart

from the slight modification

_at

the surface.

Let us assume a profile for the contact front äf the

form-.

r'=D+Ecos±Ccos2#.

(21) CONTACT FRONT 30 _{CONTACT FRONT} MAIN SNO MAIN SH0

Fio. 4. Paths of the main shock and contact front for

single medium explosion in air and water, initial _pressure

P4 619 atm.

20

30

50

Fia. 5. Paths of the main shock and contact front for

single medium explosion in air and water, initial _pressure

p4 = 7000 atm.

Then the conditions to be satisfied by this _{curve are}

r=a,

Oir,

rrsb,

_O=,

tan(r

_- 0 ₌

the condition

that

the slope is _zero for U _'r being

satisfied automatially. Thus,' _we obtain

E=$a,

F=

(b

-

_a

±

pa),

this result being linearized in terms 0(8.

50 CONTACT FRONT d(r _cos d(rsin MAIN SHOCK P4 7000 ATM AIR 60 80 . 00 .120 140 160 CONTACT FRONT, MAIN SHOCK

FIG. 6. Nondimensional, variation of main shck pressure

P/Po and contact front pressure P,/Po for single' medium

explosion in water, initial pressure p4 = 619 atm. _{p., pressure}

behind the main shock; Pa, pressure ahead of the main shock:

p,

pressure on the contact front.

AiR 20 . 40 60 80 IO0 WATER 120 40 160

-

WATER x

Ftc. 7. Nondimensional vañation of main shock _pressure

p./po and contact frofit _pressure pa/p,, for single medium

explosion in water, initial _{pressure 4 =} 7000 atm.

p,

pressure

behind the main shock; _{Po, pressure} ahead of the main shock;

p,

pressure on the contact front.

III. RESULTS AND DISCUSSION

The jnethod described is applied for two dif-

ferent values of the initial gas-phere pressure (1)

= 619 atm, (2) 4 = 7000 atm. The values of

the specific heat ratio in air and within the _gas

sphere _are both taken _{as y =} 1.4. In water the Tait

equation of state is used. Figures 4 and 5 show the

time histories of the principal boundaries along the

vertical axis above and below the center of the

explosion. These emphasize the much higher speeds

of shock and contact surface in water compared with

corresponding speed in air. Figures 6 and 7 show

the variation of _pressure behind the main blast

wave and the contact front in water (along the verti-

cal axis). For Case (1) the contact front _pressure Pc

is given by PC 411 12 Po

-

U.88 (2Q2

-

u1)} where a4

_+R

-

1 I 20 40 60

t

80 00 120 U1 =

-

0 20 30

t

40 50 60

Fn. 8. Incidence angle of the main shock _a, and the angle

between the contact front and vertical direction

. Initial

pressure 4 = 619 atm.

10 0 30 40 50 60 70 80 90 00 10 120

Fia; 9. Incidence angle of the main shock _a, and the anle

between the contact front and vertical direction

, initial pressure _p4 7000 atm.

=

1.635

+

t(15.7a2 4a

-

177a),

=

0.091(3.27

:

1)

It

is found

tMt

the pressure on the contact front

decreases _very rapidly initially, becomes zero, and

then increases, which is presumably due to the for-

mation of the secondary shock. Since the analysis

here is for small times, when this shock is weak, the

results are presented only for the initial, decreasing

phase.

The pressur&bhind the shock _p., is given by

p

7W

-

3

Po.

4

and decreases with time.

The angles of incidence of the main shock and the

contact front at the _ocean surface _are given by

Eqs. (19) and (20) and _are plotted in Figs. 8 and 9.

The angle of incidence of the main shock _a decreases

with time and approaches _zero for large times _as the

shock becomes progressively weaker. The angle of

- the contact front increases initially and then de-

creases. For large times $ tends to

Va.

The shapes of shock _wave 'and disturbed _ocean

stirface in the lower (water) half-plane _are given in

Figs. 10 and 11. In both _cases the shock boundary

Fin. 10. Paths of main shock and contact, front in water

in ocean sOrface explosion for 4 = 619 atm, I = 10, 1 =

0.0895

eec, ii _{= 320} ft, _{a =} 390 ft.

722

20 40 601

-CONTACT -

FRONT

MAIN SHOCK

CRAN,

HOLT,

AND -WELSH

.35.

ocean surface. The present approach could not be

extended to higher-order approximations (taking

account of departures

.

from spherical symmetry

away from the- _ocean surface), since

it

would not be

feasible to carry out nonspherical perturbations of

Friedman's sblution. A good method available for

higher-order calculations is the method of _near

characteristics.9 This is entirely numerical and

would, therefore, require considerably _{more com-}

puting time. than

that

used in the present aØproach.

ACKNOWLEDGMENTS The authors

. are

indebted to. Mr. Edward A.

Schuert of the United. States Naval Radiological

Defense Laboratory for suggesting the investigation

and for arranging support for the work. They also

wish to thank Professor Samuel.S. C. Lee, _now at the

University of Miami, for his assistance in the early

stages of the investigation.

This work was supported by the United States

Naval Radiological Defense Laboratory.

°.M.

bit,

Aeronautical Sciences Division, University of

California Report AS-63-2 (1963).

FxG. 11. Paths of main shock and contact front in water

in _ocean surface explosion for 4 7000 atm, I = 10,

1 _{= 0.0895 sec,} b 350 ft, _{a =} 508 ft.

and disturbed _ocean surface _are always close to the

vertical

at

the- surface intersection point.

It

should

be noted

that

the disturbed _ocean surface is much

flatter for the strong explosion than for the moderate

one. This is in agreement with the behavior found

b-y Collins2 for the intense explosion calculation.

The combination of Friedman's analytic solution

with the shock-ocean surface interaction analysis-

provides _a good first approximation to the descrip-