o UNDERWATER EXPLOSION

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1 -/ /,." 'NSW(/WOL MP SWAVE MAKING BY AN o UNDERWATER EXPLOSION By Gregory K. Hartmann DD- C Cý SEPTEMBER 1...

Description

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'NSW(/WOL

MP 76-15

MAKING BY AN o UNDERWATER EXPLOSION

SWAVE



By Gregory K. Hartmann



SEPTEMBER 1976

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FORM

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DI-TRISUTION

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bUPPLEMENTARY NOTES

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KEY WORDS (Cornirnue on reverse olde it necessary ald Identity by block number)

STATEMENT

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Underwater explosions Wave making Explosion bubble containment Explosion bubble blowout 20

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r-1ilrmm.di.ately after World War T1, from the smallest to the : .•[inclurlin-S the Atom Eaker Bikini. The various theori-as

c-f zpesivc wave making are discussed and comparisons are mq-e b-U.,--in thq ob.servations and the theoreticail expectations:. Scalinq lv -3-o,:rexarminei fuf the two distinct caF.s:s explosion bubble rontninment (deep cqse) and explosion bubble blowout (shallow ,as,). DD

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TABLE OF CONTENTS

Page

Foreword ......................................................

1

Chronology .....................................................

6

I

TESTS AT SOLOMONS,

Introduction .....................................

2.

Site ...................................................

3.

Experimental

8 a U

9

Arrangement ...............................

10

1 ................................................. and Details

.3

.........................

5.

Camera Arrangement

6.

Distances ..............................................

7.

Wavelength vs.

Velocity

(aerial data) ..................

21

8.

Wavelength vs.

Velocity

(surface data) .................

26

9.

Addendum

(1976) ..........

*

..

18

.

............

..

.......

Comparison of Surface and Bottom Measurements .........

10.

Summary of Data....

11.

I,

(1944) ..........................

1.

44. Shot

II

Maryland

.....................

29 37

......

DISCUSSION OF THEORY Historical Introo......

2.

General

Considerations ................................

3.

Cauchy,

Poisson,

4.

Penney's Crater Assumption ..............

A pH

48

.........................

1.

49

and the Explosion Problem ............. ........

51

56

o

-

5.

Kirkwood's

6.

Influence of Bubble Period .............................

7.

Arrival Times ...........................................

8.

Comparison of Theory with Experiment ...................

67

9.

Remarks on Scaling and the Influence of the Bottom .....

7

III

Basic Theory ................................

A2ALYSIS OF SOLOMONS'

DATA .f

1.

Bottom Pressure (nonblowout case) ......................

2.

Spitzer's Formula For Moderate Charges .................

3.

Bottom Pressure (blowout case); Other Estimates of Volume..............................

R7

........

4.

The Duration of the First Negative Phase ................

5.

Surface Amplitudes,

91

Trough and Succeeding Crest

(blowout cases).........................................

IV

THE BAKER SHOT AT BIKINI

1

Introduction ..

2.

Penney's Bore Theory ...................................

3.

An Energy Argument ....................

102

...................

103 ..

.......

Baker Data and High Explosive Scaling ..................

*4. 5.

A Speculative Adjustment to Make Scaling Applicable ....

6.

Use of Kirkwood and Seeger's theory To Make

106

108 .I.

Adjustments ............................................

116

7.

The Cavity at Baker ....................................

II

8.

Other Baker Predictions ................................

_220

ii ,

-

V

CONCLUSIONS

1.

Data Summary ...........................................

2.

Kirkwood

3.

Scaling ..............

121

and Seeger Summary

SImpulsive

..............

9

126

............................

...................

128

............................................

Boe.

129

.............

...

.............

130

Deep ..................................................

131

Blowout ................

0

............

132

4.

Energy Consideration for Deep or Shallow Scaling .......

5.

Prediction of Waves ..............................

6.

Correction Factor......

7.

Estimate for the Critical Depth Case for a

8.

...

*....

.

0

..............................

Ocean Inpact of an Asteroid ...........................

140 142

145

DISTRIBUTION LIST

160

...............................................

Cylindrical Coordinates ..........

147

APPENDIX B - Energy in Waves ..................................

151

APPENDIX C - Dispersive Medium,

156

Yes or No? ...................

APPENDIX D - Maximum and Minimum Values of G and Durations on

1

135

REFERENCES .....................................................

APPENDIX A - Gravity Waves in

Sthe

*

........................................... 138

.

Large Explosion ........

* ....

134

Bottom as a Function of T' .........

iii

ILLUSTRATIONS

Page

Title

Figure

1

Prints of Aerial Photos on Shot I ........................

14

2

Time-Distance Plot for Troughs A, B, C of Shot 2 .........

22

3

Print of Aerial Photo Shot 2,

34 Seconds after Explosion

with Reproduction of Hydrophone Record at Pole 12 ........ 4

Surface Amplitudes Compared with Bottom Pressures, Shot 2 .........................................

5

24

e.........

Depths ............

27

Period for Various

Wavelength vs.

Surface Waves:

...a...

31

.................

6

Prints of Aerial Photos On Shot 2 ........................

33

7

Hydrophone Records at Solomons ............................

40

8

Comparison of Surface and Bottom Waves,

9

Reduction of Pressure Fluctuation with Depth .............

44

10

Surface Wave G

63

11

Reduced Time of Arrival of Crests and Trough .............

66

12

Comparison of Theory with Charlesworth's Data ............

70

13

Bottom Pressure vs.

14

Extremes of G

S15

0

(10,l,t')

vs.

Distance

Shot 4 ...........

t'.........................

(Solomons) ..................

at the Bottom.............................

Maximum Bubble Radius Equal to Depth ...................

Distance ...................

16

Duration of First Suction vs.

17

Scaled Duration of First Negative Phase.................

S18

Surface Amplitude vs. 19

Extremes of G

Distance ...........................

at the Surface.............................

iv

42

82

83 92 94 95 98

99

20

Surface Wave Data Summary Scaled to Baker.................

21

Bubble Radius vs.

22

Correction Factor vs.- ..

1*

Depth ..................................

..

137 139

TABLES

Table

-

Title

Page

1

Charge

Schedule

2

Camera

Dctails...........................................

3

Distances................

4

Wavelengths

5

Surface and Bottom Amplitudes,

6

Data Summary:

7

Pressure on the Bottom,

Hydrophone Data ..................

88

8

Surface Amplitudes from Pole Photography .................

101

9

High Explosive Results Scaled to Baker...................

110

and

Details ..............................

Wave Height x Distance,

11

Data Summary:

I !V

.....

0..*.....

Shot 2

...

.......................

Shot 2

...................

Amplitudes and Periods ....................

10

4

19

0and Velocities,

11

Solomons ...........

Wave Height x Distance,

...

20

28

39

46

2............. 2 All Shots .........

123

WAVE MAKING BY AN UNDERWATER EXPLOSION

Foreword

Chapter I of this report was written in October 1944. had been analyzed and the results had joined the list

The data

of possible

countermeasures for use against the newly deployed German pressure mine.

However,

published,

the experiments and their results were not

partly because they were not very useful for sweeping

pressure mines,

and partly because of the constraints of security at

the time and the demands of other work.

During a trip to England in

late August 1944 to discuss the results of the experiments with Admiralty officials, J.

G. Kirkwood,

who was a member of the party,

put to paper his general theory of explosion wave making, was first title

published in the British Undex series as No.

This theory was immediately used for the

analysis of the experimental results obtained in

the Bureau of

Ordnance tests which had been conducted at Solomons, early August,

j

theory,

Maryland,

in

and in later tests conducted by the Underwater

Explosion Research Laboratory,

f

94 under the

"Memorandum on the Generation of Surface Waves by an

Underwater Explosion."

(

and this

at Woods Hole, Massachusetts.

The

containing numerical evaluations of the necessary integrals

made by the Mathematics Tables Project under the Applied Mathematics

Panel of the NDRC, Finkelstein,

J.

and certain suggestions made by R.

von Neumann,

and F.

J.

J.

Weyl of the Bureau of

Ordnance Research Group on the theory of explosions, was submitted to the Compendium of British and American Reports on Underwater Explosion Research in

1947.

The same article minus the tables was

published in the Journal of Applied Physics Vol. 1948 under the title J.

346-360,

April

"Surface Waves from an Underwater Explosion" by

G. Kirkwood and R. J.

Seeger.

The purpose of the present report is obtained in

19,

to describe the results

those early experiments which represented a considerable

effort and which would be difficult to repeat.

It

is

also of

interest that questions concerning the size of waves made by large explosions have arisen from time to tine, the Crossroads Baker shot at Bikini in herein reported were of use in

an early example being in

August 1946.

The results

the planning for Baker although the

data were not originally obtained for that purpose. of waves by explosions and their effects in

The production

harbours or ports will

doubtless continue to be a matter of tactical or strategic interest. The British researches dated early in 1945,

reported in

the LTER Compendium Volume II,

were designed in part to calculate or predict

the wave effects following the explosion of a ship-load of munitions.

In this report I have used the draft essentially as originally written for the description of the Solomons'

2

experiments and results

(i.e., Chapter I). data,

The discussions of other early experimental

and of the various early theories and scaling laws have been

added. Tech,

I have only recently seen some of the work done by Tetra Incorporated and by Scripps Institution of Oceanography

described in the "Handbook of Explosion-Generated Waves" TC-130, 1968.

Oct

I believe that the data given in the Present report may be

useful though belated addition to their worn

in

the shallow water

regime.

In those urgent days of World War II

it

goes without saying

that the Bureau of Ordnance had the advice and counsel of many distinguished men.

A meeting was called on 14 August 1944 to

discuss the results of the wave making work done up to that time for possible use in mine sweeping.

I have a draft memo of that date

entitled "Tentative Conclusions" which notes that the optimum charge weight would be such that the depth of water is times the radius of the charge, bubble radius;

approximately ten

this being roughly the equilibrium

that larger charges than that are wastefulr that

experiments on large charges indicate that distance has more effect on period than does weight; mask&

by errors; that the effect of depth is uncertain.

to the memo in longhand is (Commander S.), J.

that the effect of weight if

E.

the notation;

B. Wilson, J.

Keithly, J. Bardine, P.

Present:

von Neumann,

M. Fye,

J.

Appended

Brunauer

G. Kirkwood,

and G. K. Hartmann.

3

any is

There were many others who participated in NOL (hydrophones and photography); Applied Explosives Group,

from DTMB (photography);

Buord; from NfWTS Soloinons,

segments of the Navy; and from UERL Woods Hole. contributions,

this effort:

from from the

and other

Their

although perhaps forgotten by thaom,

may,

we hope,

be

recalled by this belated account and this belated expression of appreciation.

With regard to this current report-

I would like to express

appreciation and thanks to several at the Naval Surface Weapons Center for their help: publish the report,

to Dr.

to Drs.

W. C. Wineland for agreeing to

George Hudson,

Joel Rogers and

George Young for corrections and helpful suggestions, Couldren for administrative assistance, for placing its

splendid resources in

and to Grace

and to the Center generally

illustrations and typing at

the disposal of this work.

The urgency of these experiments made it them in

such a manner that subsequent tests could profit from

information learned in earlier tests. -Ilater

It

is

only in

the light of

insights (and in this case much later) that a reasonably unified view o

these complex phenomena has been achieved.

also remember that in

this practical world urgency is

spur to get something done. *

impossible to plan

If

a problem is

We must

frequently the

not born in a crisis it

frequently cannot command the priority to obtain the necessary resources.

But in a crisis there is

4

frequently no time to pursue

all .questions

:o a solution.

which remain open.

There are therefore some questions

The role of the bottom and its

characteristics

has not been theoretically dealt with when the charge is bottom or when the depth is

shallow.

on the

The phenomena shaping the

water cavity for shallow explosions in

either deep or shallow water

have been treated only in gross approximations.

The problem of

making reliable predicticns of wave phenomena caused by large explosions or of scaling from one experiment to another may still a subject of disagreement or at best of uncertainty.

The making of

unambiguous predictions should be a part of the repertoire of any explosion phenomena expert.

Perhaps in

these less urgent days it

will be possible to complete the missing information and put this subject to the continued rest that is

-

undoubtedly deserves.

p

5

J

be

..... , ,

,

ehzronorlhgy-

BuOrd Experiments at Solomons

22 Jul - 4 Aug 1944

Conference on Conclusions

14 Au% 1944

Trip to England

Aug - Sep 1944

Kirkwood's memo on "Generation of Surface Waves by an Underwater Explosion" written on this trip and published as Undex-94 by the British.

Shot #6 at Solomons in

100 ft

water

Writing on Experimental Results

(GKH)

6 Sep 1944

8 - 24 Oct 1944 (Interrupted)

Writing on Theory - Kirkwood & Seeger drafted between

Oct 1944 and Feb 1945

Production of Surface Waves by

UERL draft by

Underwater Explosion

R. W. Spitzer 29 Nov 1944

(Distributed and Lumped Charges)

"Gtavity Waves Produced by Surface

W. G. Penney

Underwater Explosions"

Imperial College of Science & Technology, London Mar 1945

6

"Waves in Baker" W. G. Penney

24 Jul 1946

(Joint Task Force)

Baker Event Bikini Atoll

25 Jul 1946

"Surface Waves from an Underwater

submitted to Underwater

Explosion" J.

Explosion Research

and R. J.

G. Kirkwood

Seeger.

British-American Compendium. 27 May 1947

Identical article minus tables appeared in Journal of Applied Physics Vol.

19,

pp 346-360 Apr 1948.

Writing on Experimental Results and Analysis (G. K. H.) completed Jan

i I7

-

Feb 1976.

resumed and

I.

TESTS AT SOLOMIONS,

1.

(1944)

Introduction

In the summer of 1944 tests were planned and conducted to produce gravity waves in water by explosions, suitability for sweeping pressure mines.

and to determine their

Although it

is

well known

from casual observations of underwater explosions that the detonation of convention.l charges,

say depth charges,

produces

practically no observable wave system, nevertheless it

was felt

desirable to try larger charges and to make specific preparations to observe whatever surface waves were formed.

Site

2.

A site for this series of experiments was chosen in

the

Patuxent River at the Naval Vine Warfare Test Station (NMWTS) Solomons,

Maryland.

The depth of water at this spot off Sotterley's

Point was about 40 feet over an area at least 2,000 feet by 1,000 feet. 4

The bottom was a soft mud into which for example a mine would

sink about three feet. of the waves,

This mud probably influenced the magnitude

but the first

requirement was to find a large uniform

area sufficiently remote to allow the experiments to be done.

'• '

,

,

,

,

,

I

i

i

] i...

"S

3.

Experimental Arrangement

Observations on the waves produced were made in

three ways:

aerial photography to determine wavelengths and velocities;

by

by

surface photography to measure surface wave amplitudes and periods; and by pressure

recording systems placed on the bottom.

photography was accomplished from a blimp.

In

The aerial

order to measure the

surface amplitudes a range of telephone poles was set up. was 30 feet long and was submerged in

Each pole

the water by a 300 pound

anchor so that about 7 feet of the pole extended into the air.

The

top portion of the pole was painted with alternate black and white strips 6 inches wide.

The range consisted of about a dozen poles in

a straight line about 140 feet apart.

The wave motion was found to have very little poles,

effect on the

except at distances less than about 300 feet from the

explosion where the outward rush of water caused the poles to sway, rotating more or less about their anchors and thereby submerging themselves.

The pressure recording

composed of a NOL Mk 1 hydrophone,

systems consisted of units each a bridge network and an Estraline

Ji

Angus recorder.

The hydrophone was protected from explosive shock

by a rigid brcnze cone which allowed slow seepage through a small hole but which screened out very sharp changes

in

pressure.

This

protective device was tried out in a preliminary series of shots made 22 July (reported by J. F. Moulton, BuOrd memorandum) in which it was found that the pressure sensitive diaphragm would operate

9

successfully

if

the shock impulse from the explosion,

making

was less than 0.25 lb sec/in

C-liowance for surface reflection,

The main shots were carried out under Explosive Investigation Memorandum No. July 1944.

62 under BuOrd forwarding letter

The schedule of shots fired is

S68 005316 of 22

shown in

Table 1.

Note to Table 1.

The bombs LC,

AN,

M56 were initiated

by filling

the nose fuze

seat liner with Comp C-2 and detonating this statically an Army Engineer Special detonator. were the 1U. booster,

13"

The Demolition charges used

14 Mod 1 approximately x 13"

x 6½" in

by means of

50 plus pounds Cast TNT no

cardboard box.

Mk 9 approximately 115

pounds cast TNT with 63 grams auxiliary booster Mk 4 (1.6" diameter, 3"

length granular TNT)

shot 4,

13"

x 13"

the charges were crated

feet on a side.

In

x 13"

in

in

steel container.

cubical boxes approximately 5

each crate a Mrk 9 charge was set in

and an electric detonator was used to initiate contained

in

each box.

4.

Shot 1

The size of charge for the first •that V

it.

the middle

Each crate

180 Mk 14 Mod 1 demolition charges and one Ilk 9,

about 9,200 pounds of charge

For

making

shot was chosen by considering

the bubble radius of the expanded gases should be at least

is

t ...

• I

••... i

••"• i

• w'-

-

Z7

4-3

' 4J '-4 444444

4--) 4.)

IV

z

0u

4J 0

4-

0

0 ()

W

U)

4

4-)

-)4

4-4

r0

o0

0 T$

>44

-)

41 Ln U) Lfl

4-)

0

0Q)

0

0(DC

>4

>4

4

CI

)U

U4E

E)

4J)

4j

0

1

--

4-

0

0

U

0

>4

>>

>

0 M

EI)

r4

0

0

(1)

Q)

,.C

to ý4

0

U

P-

U)

Q

0

0

-1

ul 0

ý4

"4.))

H

0 z

>4->i

4-4

-4

ko H -T

HN

Cl

C)C C

Hn v N

U)

0)

z (Nr

k.0(

equal to the depth of the water.

Por convenience the charges were

placed oh the bottom although it

was realized that this might not be

the most efficient use of the explosive.

Since the wave making

so inefficient from the standpoint of energy,

process is

the

question as to the best possible position for the charge does not seem to be of prime importance.

The maximum bubble radius for an

underwater explosior of TNT at depth D is

given in the absence of

free or rigid surfaces by

r

where W is

13.5

(33

W1/3

+ D)1/3

the weight of charge in pounds.

the total explosive energy is =

feet

This assumes that 45% of

retained in the bubble.

40 feet gives W = 1,900 pounds.

Putting rmax

The unit chosen for the first

shot was a 4,000-pound bomb containing 3,362 pounds of TNT.

This

choice yielded a charge which was presumably large enough and at the same time easy to handle.

For Shot No.

1 the range of poles was photographed by means of

especially mounted aircraft cameras having a field of view of 400, and capable of taking a picture every 2/5 second.

In order to save

film an estimate of the time of arrival of the waves at the various

12

IOR

poles was made using the velocity expected for waves of length great

"compared to the depth, i.e.,

V =

36 ft/sec.

For a pole at a

distance of say 1,000 feet from the explosion,

the earliest possible

time of arrival of waves would be 28 seconds.

At this time and

thereafter,

however,

on Shot 1 there were observed no waves at all

at these distances and consequently the cameras were turned off or in

some cases not started and consequently no records of any value

were obtained.

However,

a subsequent examination of the aerial

pictures taken from the blimp at altitude 1,500 feet showed unmistakably a system of ring waves extending at least 1,400 feet from the explosion and with wavelength increasing with increasing distances.

Consideration of these pictures shows that the long slow

swell of the outer rings would not be observable except under very calm surface conditions and only then by an observer with some experience 0,

27,

.

Figure 1 shows a sequence of photographs taken at t =

45 and 71 seconds after the explosion.

5.

Camera Arrangement and Details

Consideration of these aerial photographs made it

necessary to

investigate somewhat more in detail the wave system produced. Accordingly,

on Shot 2 the camera setup was changed so that long

focus narrow field lenses were used with one camera on each pole. The number of poles photographed was considerably reduced.

A

special 70 mm Mitchell camera was supplied and operated by the David Taylor Model Basin, which could photograph two poles simultaneously.

13

I0

NSWC/WOL/MP 76-15

MIN SEC

7 7f

-J

.

,....:TV,

.4.M,

-:.-

1

14

•Approximate

Ft.7.97

F .•-

3362.L.s.

o

Botm....

SPole

Number

Distance - 142 ..- +•.Ft.• .::-• 284

S3

3139

S4 S •5

546 714

10 11

.. +;--

L}

S•

~ ~.

i14 .. .. ...'• ..•• ...

Water.. : Det

0Ft

Pole Distances with Explosion at 0

Pole Number 7 8 9

S•- •1

.

14

.-

Figure.....l"Shot . Numbe.r 1

i

.. .. ..

. .+ .

-.

;,,

g'

I

WI

Figure Il(a)

....

, 2

...

Shot Number 1 -3362

°•-:•"•+,mmll•+,•.r•+•l•'+,•.=•~ ." • . .... ..- k

Distance 976 Ft. 1096 1264 1425 1575

Lbs. TNT on Bottom - Water Depth*,-40 Ft.

.

.- "<

:•I,.

'

NSWC/WOL/MP 76-15

3

919:917

-

to + 27 SEC

2

"Apoxmt

I4

Pol

Nubr Dsac

1

,

PoeDsacswthEpoina 51

ol

ubr

Dsac

14-t.797

t

I3

Figproiuae Iol(bDit

SPal

S1

Numer

ubr1

32 istace

1................

stace

wTNTBto Exon Ple

umbe

-Wate

Det04 Disanc

t

?NSWC/WOL/MP 76-15

"

10

"

t 0 + 45 SEC

-9 -7

-I" -4 -3

j' :

Approximate Pole Distances with Explosion at 0 Pole Number 1 2 3 4 5 6

I

Figure 1(c)

Distance 142 Ft. 284 389 546 714 849

Pole Number 7 8 9 10 11 12

Distance 976 Ft. 1096 1264 1425 1575 1669

Shot Number 1 - 3362 Lbs. TNT on Bottom - Water Depth'-40 Ft.

16

.

NSWC/WVOL/MP 76-15

20:01 to

i

0

R--•

. $U

12,..7

1

i

SPole S1

Number

Figure 1(d)

7



284 389 546 714

8 9 10 11 12

849

Shot Numbef 1

E'xplosion at

Pole Number

Distance

142 Ft.

2 3 4 5 6

A

76F.,-

.1,.

Pole Distances with

Approximate

l

6

Distance

976 Ft,

1096 1264 1425 1575 1669

3362 Lbs. TNT on Bottom - Water Depth -40 Ft.

17

71 SEC

The records obtained by this camera were used to determine wavelength by measuring the difference in phase between waves at the two poles.

In all the photograpiic work due attention was paid to

getting optimum resolution by reducing the circle of confusion and the optical diffraction to a value less than the resolving power of the film.

Details as to the various cameras used are given for the

sake of completeness in Table 2. to about

±

Wave amplitudes could be estimated

½ inch with the lenses of longest focal length.

6.

Distances

The range of poles was set out at the beginning of the series and the positions of the poles were determined before each shot by means of a range finder (1 meter base) and a crude azimuth (polaris) circle graduated in degrees.

Vie distances and angles were plotted

out for each shot and give rise to the following table of distances.

The various interpolar distances obtainable from this table permit an estimate to be made of the precision of measurement of distance.

It

turns out that if

the mean distance,

u2 is

m, between poles,

the variance associated with then a = ±0.085m.

This means

for example that the best distance between pole 11 and 12 on all shots is

109 feet ± 6 feet, using probable error equal to 2/3 a.

18

Table

Camera

in

F54 blimp

7"

film

K25

2

Details as to Cameras

Field of View

6" at this distance appears on film as

Focal lenath

Speed

Stop =n

p,

ltvery 10"

2 sec

..

..

2½ frames/sec

16

335'

400

3.5°

430'

.020"

19°

360'

.0069"

6

3/8"

flLstance

40

35 mm Mitchell

17"

24 frames/sec

16

2390'

70 itn Mitchell

6"

10 frames/sec

16

298'

2000'

.0021"

380'-680'

.0095".0049"

Where n = f/d Resolving power of the film Resolving power of lens =f lenses. C = Diameter of circle

of

Corresponding hyperfocal

50 lines/mm = .0008" = 1.22 Xn

confusion

distance

<

p P

4

1

19

=

.0004"

.00063" 2

/cn,.

if

n

16 for all

Table 3

Distances from Explosions to Poles in

Pole Number

Shot 1

0 1 2 3 4

Charge 142 284 389 546

5 6

7 8 9 10 S1575 12 15 16 17

(H) *•

Shot 3

Shot 4*

charge

-

-

-

Shot 5*

.... -... -... -...

-

714 849

charge 168

-

-

~

976 1096 1264

299 419 581

412 517 659

879 1028(11) 1170

636 771(H) 927

1425

753 895

801 928

-

-

1669

1007(1f)

1048(H)

1579

13

.14

Shot 2

feet

-

-

-

-

1330

...

. -

-

-

-

-

-

2000 214n(H) 2363

indicates hydrophone placed on bottom near pole. charge not at pole.

20

1760 1896(H) 2120

Wavelength vs. Velocity (aerial data)

7.

On Shot 2,

photographs from the air were also obtained.

average interval between pictures was 2.5 seconds.

The

From these

pictures in which a scale was provided by a ba -ge 110 feet long, distances between some of the outer poles was determined.

the

These

compare well with the average interpolar distances obtained by range finder and circle.

Thus:

Distance

Range Finder

Aerial

Between Poles

Method Ave.

Photograph

9 and 10

158 feet

165 feet

10 and 11

140 feet

138 feet

11 and 12

109 feet

112 feet

From these photographs a plot was made of the distance

Stravelled

i

troughs in

versus time since the explosion for the flist the wave pattern (Figure 2).

by the presence of shadow. whether the first

There is,

trough observable is

i 21

three

The troughs were identified however,

some question as to

really the first

trough in

NSWC/WOL/MP 76-15

DISTANCE FROM EXPLOSION SLOPE ASSOCIATED WITH CRITICAL VELOCITY v-gh

FT.

POLE #12

1000

/

Aoo /

POLE #11

/

Bo

POLE # 10

""X

POLE

#9

0

POLE

#8

40 30

40

50 TIME FROM EXPLOSION

FIG. 2 TIME - DISTANCE PLOT FOR TROUGHS A, B, AND C OF SHOT 2 (FROM AERIAL PHOTOGRAPHS)

22

60

SECONDS

the series since the glare from the sun makes that part of the water surface uniformly light in

the region into which any rapidly

travelling leading wave would advance. photograph,

Figure 3.

This is

illustrated in the

This is mentioned as a caution in the

application of the aerial technique for measuring wavelength. Indeed the hydrophone record appended to Figure 3 shows that the first section has already arrived at pole 12 before a wave disturbance shows itself from the air.

Various wavelengths in reveals that the first trough is

the pattern resulting from Shot 2

one has been missed.

The first

visible

called A, the second B and the third C.

In Figure 2 it

is

seen that the slopes of the ttree curves

increase with distance and that the velocity /7 approached.

It

is

is

also apparent that the separation between

successive troughs increases with distance, wavelength is

= 35 ft/sec,

increasing with distance.

which is

to say that the

Thus the separation between

A and B varies as follows with distance:

-.I 23

iX•I"U IIxMI•

I

I I

I •

NSVUC(WVOL /MP 76 15,

Fgr3

rcord at Pole. 1? showerd thE At th,, trýr the hvd raphrun-o Thp f,r,t withi. torqlr jir~tw.'rmi Kdels 10 and 11) het ivPe 5econrd -weCiwi (14beled Al to

Shot Nu mber 2 firsl

SICII ti-

ipp

IhCIi..,uI

42

MENO

Distance from

Velocity of

explosion of crest Pole

between A and B

9

581 ft

10

Distance from A to B =X

crest between V computed A and B

from X

95 ft

22 ft/sec

22 ft/sec

753 ft

112 ft

22 ft/sec

24 ft/sec

11

895 ft

119 ft

29 ft/sec

24 ft/sec

12

1007 ft

123 ft

31 ft/sec

25 ft/sec

The values in the last column are computed from:

V

I =

taking h = 38 feet.

(9X tanh

It

is

m),

(See Appendix A),

noted that at this depth and at these

wavelengths the value of V according to the ordinary monochromatic theory increases very slowly with X in

this range.

It

is

of course

not surprising that the simple theory does not agree exactly with *

the observed velocities.

25

8.

Wavelength vs. Velocity

(surface data)

On Shot 2 the 70 mm Mitchell camera was trained on poles 11 and 12.

The distance between these poles is

taken as 109 feet.

surface records obtained are reproduced in Figure 4. record obtained near pole 12 is

also shown.

The

The pressure

The correspondence

between the surface amplitude measurements and the bottom pressure measurements is

very good.

It

pressure peaks after the first the first

is

possible to number the positive

suction,

and the surface crests after

trough and to put these into one to one correspondence.

The camera was set to run at 8 frames/second,

but comparison of the

times of arrival of corresponding peaks at the bottom and at the surface,

assuming that the Esterline-Angus timescale was correct,

reveals that the camera was running a little

fast.

To correct

intervals the following factor must be used

Atrue= .87

(t

70

mm camera)

Even this does not provide a perfect correction because of local variations of speed in

the camera.

The wavelengths were measured from the film record as follows: Let At7 0

=

time of arrival at pole 12 --

The resulting velocities,

time of arrival at pole 11.

periods and wavelengths are listed in

Table 4.

26

UJ

oz

wI-

w.-o

0-0uj1

-u

0 CL

D

-

C

w CD-

~CA

0~0

to0 00 0to

Z Lo

w

c

wU

W CC

Uto

w

U 0 In

CY

N

I~wz

j

z

N

0

C-

4

t

N

(

*N

27

*1

Ln

CIA

m

H

N4 ('71 (N

C

0N

N*

0

c4

C4

04 N

co

C (N

co

00

4.)

0)

4

4-)

co

Lf

'.0

( (N-

LA

Nm0

C14

co

*NIDr

.4.J

C.)4

0.

0'0 0)C1

'U

0

CDL C11

4-)

(N

H

4

>i

0

¶n

a)

14

4-) IRV

I-

0.

(N

UlA

wOL u)

C')

0

41

(D

V

:d

to

mIC) 0.

-14 w

4 -

-

04

04J 4

go

>

coII

28

r-

0J

r-

1 0.

4-)

4 H) >

0)

0 H

44

NSWC/WOLIMP 76-15

These wavelengths are not comparable with those measured from since those were measured from trough to trough,

aerial photographs,

whereas these are measured from crest to crest.

"been mentioned, it

is

uncertain whether the first

Further,

as has

trough was visible

at all from the air.

thus:

Schematically the situation is

"4----214-o

1

A



S~This

might suggest that perhaps A is possibility is

•"

It

63

is certainly a4

S~measurements •

j1

.

16

really the third trough.

This

not ruled out by the comparison of the velocities. truc that difficulties of observation make the

from aerial photographs much less reliable than direct measurements on the surface.

In subsequent shots the aerial

photography was dispensed with.

1

9.

Addendum (1976)

The consistency of these measurements may be checked as follows:

If

At = time taken for a given crest to travel from pole

11 to pole 12,

i.e., 110 feet then,

29

-

velocity = 1--

T = period from one crest to the next at pole 12. velocity x T.

X

Having found the value for X we ask what velocity

does this require,

V2

Hence,

from

tanh --

(2)

--

(See Figure 5)

214 ft

162

116

108

97

92

94

88

87

88

95

26 26.5 26.5

26

25

26

26

22

21

21

21

21

average vel from (1) 36 ft/sec 31.5 29.5 27.5

velocity from (2)

30

27

24

23.4

21

21

The discrepancy in velocity can be largely eliminated by eliminating the correction made for the speed of the camera.

If

camera was accurate and the recorder was inaccurate,

in

fact the

(and there is

no way to be sure now) then the systematic bias can be relieved. This means that the values for the periods and durations as

30

0

o

b

0

CL

0

qr

3L

LLU

UL -m

ir

LU

L0

I0

LAa N.N -

-LII-31

'I

A~

determined by the hydrophone at least on this shot should be increased by about 15%.

Rather than indulge in such a correction

program I will leave the numbers as originally noted with a caution as to the general accuracy of all the measurements. the surviving aerial photographs from Shot 2, and 57 seconds after the explosion. waves had not appeared whereas in

however,

taken at 11, 27,

46

In the two earlier pictures the

the last picture the earliest

swells have gone beyond the range of poles. seconds,

Figure 6 shows

lets one with a little

The picture taken at 46

imagination list

the

distances from the outermost dark ring (beyond pole 12) to the next one inside and so on.

*

approximately 178, inside pole 9.

These distances are wavelengths and are

113,

This is

97,

86,

and 59 feet which brings us just

an instantaneous view of the wave pattern.

The longer waves travel faster than the shorter ones and consequently the pattern spreads out creating longer waves which then travel faster.

The whole pattern will spread out until all the

waves are long enough to travel at the same maximum speed. time however the waves will have vanished. at 46 seconds,

the "first"

It

is

this photograph

wave has a wavelength which is

shorter than the wavelength of the first ,photography.

Even in

By that

somewhat

wave obtained from pole

therefore concluded that the waves of very long

length (and hence very slight slopes) cannot be reliably detected by aerial photography.

I

I 32

•'

-

-

-

.c

-r

-

-'.-

.--

-

.

,-. --

NSWCtWOLiMP 76-15

c+ 11 SEC

=.A. a

V-

Approxim-ate Pole Distances with Explosion at 0 Pole Number 6168

•,.

i

8 9

Distance Ft. 299

Pole Number 10 11

419 581

Explosion Occurred at to

12

1007

16:37:09

Figure 6(a) Shot Number 2 -6724



Distance 753 Ft. 895

Lbs. TNT on Bottom

33

-

W4ater Depth -. 40 Ft.

j

NSWCIWOL/MP 76-15

•g27

6I 168I Ft 075 t t,

9 Fiur ShtNm e 581 .ib 62

:

b.T Ton_.on-W tmDph

w

4

t

-

SotNumber

Pole

2ist6724Lb.

to 1:70 Ocre ApoiaePlDitnewihExplosion 9

I34

54

NTmbe at

r

Diept..4

NMICAVOLIM11 7C IS~

~4

Appt oximate Pole D istan1ces with Expilosion at 0 Pole~ Ntirnh 6 7 8 9

Distance 168 Ft. 29i9 419 581

E xplosion Occurred mato t Fuqur, 6(c)

I3

'Ilat Numlher 2

Pole. Numheq 10 11 12

Distance 753 Ft. 89r, 1007

16:37:09 1524 lh.b. TNT Oi Uon B antm

'Lditi; Dcptt;

-

40 Ft.

NSWVCWQL/IMP 76. 1G

______ ___ ___ ___ ___ ___ ___

. ;7 Lx!.

•~~:

I,•."

': '

S•V':.

. ::

,,,

. •. .

. . . . . .. .. . . . ......

:"•"

r•'"2

-

..

• -•••,.

% , .•

' /

:

'"

'.

Approximatet Pole Distainces with Ex plosicm

8 9

Fx(plosulu

Distatictp 155 Ft. 299

Pote Numbur 10 11

419 58 1

12

dt

. - .

0 Distanc~e 753 Ft. 895 1007

Ocimiud at to16:37:09

F uit! 6(rl Shot Nutiihpr 2

-6724

Lbs. TNT on Bottoni 36

10

.

.•

-

Pole Numbe~r 6 7

t0 + 57 SEC

VV~7

___~

WAater Depth -40 Ft.

.

.

•Y ."

-.

-14

Comparison of Surface and Bottom Measurements

The pressure record obtained in the vicinity of pole 12 is shown in Figure 4.

also

A comparison between the surface and bottom

amplitudes can be made by use of the simple monochromatic theory. It

has already been seen that the surface and bottom amplitudes keep

in phase very well.

It in

can be shown

inches,

and Ap is

This is

to be expected from the simple theory.

(Appendix A) that if

n is

the excess pressure in

height z above the bottom,

the surface amplitude

inches of water at a

then

Scosh kh n =tsP cosh kz

where h = depth of the water and k = 2ir/X. either plane waves or cylindrical waves.

This relation holds for In the present case the

pressures were measured at a distance of 1.5 feet from the bottom. The depth of the water on Shot 2 of the hydrophone was 37½ feet. Hence z = 1.5 feet, h = 37½ feet.

SIn

order to apply this relationship it

is

necessary to know or

estimate X.

1 I

37

.''.....-].... ....l....-.....7...... ..r....

NEWTWe associate with each peak and crest a wavelength which is average distance to the two neighboring peaks on either side. Table 4).

In

the (See

Table 5 we compare the measured surface amplitudes

with those calculated

from the bottom amplitudes.

The agreement is

reasonable.

Figure 7 reproduces all the existing hydrophone records obtained in

the Solomons series.

Figure 8 displays the only other measurements of surface and bottom amplitude over a series of many waves.

(For Shot 4.)

Although there are no nearby measurements as in from which the wavelenqth may be inferred,

it

is

the case of Shot 2 possible here to

measure the periods between successive peaks and determine wavelength assuming that the wave train is monochromatic. is

at least locally

This assumption does not always apply.

The period P

given by

P

Furthermore, units is

[2A= tanh 27h] L27T

at the bottom the pressure change,

related to the surface amplitude

___________8

___ __ __

n by

Ap,

in

linear

4.4

0

LA. H .D.~

0

4 N N

*

U]

1.4

4J

0

0 4-J

mA

0o 1-i C

.

H -1

'4-4 0~ 44 a)

M

0)

.- 1

0

0

n

a)

4.i

()

M]

co-

4

H*

00

a)

u00

co)0

N

C14

0

4.) 04

H

'V

u

CH

in

0H

tyl

-ILn4.

*

0r 0

0

0

.- :

0

1-4

0) 41 0

4-

-

.H

1

14

C14

m

0) .4

4.) i

4-'Q

4-)

4

.Z

C)U)E)

C1

C)

a) '44

M 9

0

w.1-4

LA)

41 )-

0)~ >

M]

9

co

4.)

a)

0

0N1

4.)

~ o r-

LA

O

Ln

ý4

~0

N

*

U) .9: 41

L$

rd)

0)

0r .4.

C

0

(n)4)

0

41i Q)

4

44~

0

.-4

4.W

4.)

E,4 0~

(L)

4-4

r -~4

0

LAW C')~

H

**

P0

*

.)

4J)

0)

s t

0

0

04 4

-4 4J0L 0

4

.)

CM

40

r.

.

U]

0

4.)0 0

$

0 Q) 0

Ul

04

.11

0)

d

0d

4J

e

0s

Hd fo

a)

E-4

fell-

39

-40

to 0) 14

+H) 0

4

1.4

-)

4J

ý40 0L

NSWC/WOLYMP 76-15

g

--

z

Cz

a E.

C-

EE 00

N'.

(40

NSWC/WOL/MP 76-15

Shot No. 4

No. 8

I'hine 40'Pl

,

D a'c1008'

Distan -46,000

71,1.

,

*36 InchesOWIe

~p-,

W

MWi•

i_

SWater3

Phoner36'

Pole No. 16

.7'

0 Distance 2150' t52 Inch es Cfharge 46,000 Lbs. TNTZ .4= of Water

-77'

EUWater 42'SoN.5

Dlist ance 765' 93 Chups~ 2034 Lbs, TNT 3.6 Inci m

Shot No. 8

Pole No. 16

MO MIRM

j

5e 4

we~ e36 Wate38 U-MM

8

Charge 2034 Lbs TNT

Figure 1(b)

diSow Hydirophorle Recordsat

41

u

NSWC/WOLIMP 76-15

(0

0 L6 LL

0

-

a

(0

U

1%.

.

0

0) (00

wL 0 -

w

0j

(0 CL0 'La

44

U..

Iu

~~~Np cosh -it-

.

... .S~2h

using these relations it

is

possible to construct the curves of

Figure 9.

Referring to Figure 8, crests, P,

the times between successive wave

are:

Wave Number

1

2

3

4

5

6

7

Period (sec)

12

8

8

7

6.7

6.0

6.0

5.7

.83

.65

.65

.55

.50

.41

.41

.37

3.3"

2

5.5

5.5

5.5

5.5

5.5

3.5

Calculated Ap

2.7"

1.3

3.6

3.0

2.7

2.2

2.2

1.3

Measured Ap

2.8"

1.1

2.6

2.6

2.2

1.8

1.6

0.6

8

From Figure 9: Ap/n at 40 ft

n

½ (crest + trough)

The agreement is

reasonable.

The other cases where comparison is

possible between bottom and

surface measurements give similar results, namely for Shot 3 at 1048

43

NSWC/WOL/MP 76-15

0

I-

LU Lu 9-•

N

z'0 0

-

t

qj

C

44

c

C

(0•

0L

c

>

° L

feet and for Shot 4 at 1028 feet.

As already indicated,

slight elevation of the hydrophones above the bottom is then under the assumption that the wave motion is i.e.,

if

the

neglected,

monochromatic,

consists of a set of waves all having the same wavelength

(which is 40 feet,

not the case) then n = Ap cosh 2-th/X.

If

h is

taken to be

the factor to be applied to Ap in order to calculate n

depends on wavelength A in the following manner:

A

feet

n/Ap

If

A is

400

300

200

150

100

1.21

1.37

1.89

2.78

6.19

small,

the factor.

a small error in

80

11.7

X will produce a larger change in

For this reason and others, one would not expect very

close agreement between measured n and n estimated from bottom pressure measurements,

at short wavelengths.

11.

Summary of Data

The original data on the Solomons tests consisted of hydrophone - irecords

and films. disappeared,

but measurements were made from the films at the time.

These results,

Sall

The originals and the films have long since

wave heights, periods,

summarized in Table 6,

pressures and distances are

2 which pertains to the 40-foot sites.

45

Data Summary

Table 6

First Positive Fhasm

Suction Phase Die-

srm Shot

pol.

S

2

Records

from sxplosin

3SNO Fila

7'

20.2 sec b

2.8 sec

-3.6"

4.8"

21.3 sec b

581

10

3sm Fils

753 895

1007 FM Prasaure

3I5m FilA

10

Preesure

S9u le64th in ft.

Wave Velocity

80 C

-5.2'

6.0 sec

5.6* (Pressure Record) for 5.0 sac b

-2.5* to -3.81

4.8'

35.4 sec

4.0"

39.0 eec

22.1 d ft/sec

95 d 135 C

.1.9

inches 4.6" a

-2.2 inches

4.0 sec

112 d

22.0 d

119 d 185

29.0 d

177 172 d 195 c

30.6 30.7

2f.9 a

93.2 a

77.1 sec a

81 c

(70m)

Record S

Ties of Arrival

-12'

701: PFil

3

Sw face Asplitude

3.2 sec

35i ri

12

Duration

4W6wPee face Amplitude

419 ft.

9

11

sure on bottom

659

4.9 sec

-11'

15.5'

36.8 sec b

162 c

801

6.4 sec

-7'

12"

43.9 sec

160 205 c

921

4.0 eec

-7"

11.5-

49.4 sec

132 169 220 c

5.0 sec

-5.5'

70m 11 12

3Ism Film

b

1048 6.6 sac

MN Pres-

13' 4.5" a

sur* 4

879

7

210 c 55 aec

b 47.6 100.3 sec b 48.2 " 101.2 b

-5.064

4.5 sec 12.4 eec-

-9'

12"

48.8 see

6.8 sec 13.9 sec-

-6.5'

13.5"

51.0 see

3.2'

File 1028

-

5.0 eec 13.1 sec

PHt Pre*suZe Record 9

351: Film

1170

7.3 sec 15.8 seae

12

35Ia Film

1S79

6.3 sec 12.6 sac'

15

35am Film

2000

16

35n Film FM Free-

2140

sure

5

2363

9.1

240 C

30.4 0

67.8

536 b 240 c 3.0' +4.3'

3. 4

220 c

29.2 c

59.3 sec b

245

C

30.2 c

-4.5'

5.5'

53.9 sec

146

C

22

-5'

6.5' 4.0'

-2.8' -2.1'

5"

229.0 109.2 A

89.4 sec b 162.1 sec b 61.8 sec b 142.1 seac b

-4.5'

9.0 sec 18.2 see

160 c 55 aC

12"

-8'

9.1 sec 18.1 sec' 8.3 14.2 sec*

Record

48.5 sec b

-8.6' -4.310.0"*

25.4

+2.6" +3.6'

295 c 150 31.6 c

290 c

98.6 eeC b 173.9 sec b

24.1 10.8 a

17

35ma F11m

8

Pressure

771

7.8 sac

-. 94'

38 sec b

+.94

20 (dist/tie)

16

Pressure

1860

8.5 eec

-. 36'

66 sec b

+.38

28 (diet/time) b

SRecord

b

Record

Sthe

Surface "On Shot 4 the suction was divided into two shallow parts. Remark: Records x.-dicatc a brief positive phase between them, the pressure record does tarred times concern the duration of both parts, the unstarred that of not. firstThe -,art. Key:

(a) dats for some member of second wave group. 2h (b) unreliable dat 8 . teah (c) camputed from L Tz (d) bliap data.

A

46

a

I

I

I

I

I-

Before these data are subjected to analysis_(in Chapter III),, it

will be useful to review in

the next chapter some of the

theoretical concepts to be used.

Except for Figure 4 and Figure 8 there are no extant records from photography. photographs.

Shot 1 yielded no data except from aerial

Shot 6 done in 100 feet of water and only with

hydrophone data is

listed in Table 7 (Chapter III).

reproduces the only hydrophone data, Shot 3 at 1050 feet, 1860,

namely:

Shot 2 at 1007 feet,

Shot 4 at 1008 and 2150,

and Shot 6 at 1485 in

Figure 7

Shot 5 at 765 and

100 feet of water.

Figure 8 reproduces

film and hydrophone data from Shot 4 at 2140 feet.

44

47

S

.

. -.

.•

,. -

,...

.

.•.

.

.

•...... -

.

-• •

...-

--.. .

...

.

...

II

1.

DISCUSSION OF THEORY

Historical Introduction

The literature of gravity waves is

extensive starting in 1776

with Laplace who considered water motion in a rectangular canal. Results obtained by Lagrange a few years later for shallow water stated that the velocity of travel depended only on the water depth and not as Laplace found on the wavelength. "Problems in Water Waves " 1931(1) page 4,

"At the end of the

1 8 th

in the Historical Side Lights

Century there had been put forth two

different theories in regard to waves, which had never been explained,

K

As Thorade says in his

the mutual relation between

so in 1802 Gerstner put forth a new

theory which assumed that the water was infinitely deep, while the scientific study of waves was again promoted by Poisson and Cauchy (1815),

two savants of high rank.

Both blamed their predecessors

for having studied only fully developed waves,

and they dealt with

the creation of the waves by citing the following illustration: submerge a solid object, not too large,

in water of unlimited depth;

wait until the water has become calm and then suddenly withdraw the object.

*•

What kind of waves will be formed?"

and Lagrange were right. 1

If

the depth, Laplace was right.

the wavelength was small compared with If wavelength was long compared with

48

I

Of course both Laplace

I

.

depth, then Lagrange was right.

Poisson and Cauchy introduced

greater complexity as well as insight to the subject by initiating the wave motion with a mixture of wavelengths needed to describe Thorade's book contains much historical

their initial conditions.

The subject of waves is

information.

discussed in a few short

paragraphs by Landau and Lifshitz "Fluid Mechanics"'(2)

"The free surface of a liquid in

a deceptively simple introduction: equilibrium in

starting with

a gravitational field is

action of some external perturbation,

a plane.

under the

If,

the surface is

moved from its

equilibrium position at some point, motion will occur in

the liquid.

This motion will be propagated over the whole surface in

the form of

waves,

which are called gravity waves,

Gravity waves appear mainly on

action of the gravitational field. the surface of the liquid,

since they are due to the

they affect the interior also, but less

and less at greater and greater depths."

2. S(a)

The gravity waves considered by Cauchy,(

Penney, (5)

4

General Considerations

Kirkwood and Seeger(6)

irrotational,

nonviscous,

occur in

3

) Poisson,

a medium which is

incompressible and of uniform density.

very short and useful book by C. A. Coulson( 7 ) "Waves,

SMathematical i

Boyd,

Ltd.

two groups.

i

(4)

A

a

Account of the Common Types of Wave Motion," Oliver and 1941,

divides the types of wave motion in

liquids into

One group has been called tidal waves or better long

waves in shallow water and arises when the wavelength is much

49

greater than the-depth ofthe liquid.

With waves of this type the

vertical acceleration of the particles is with the horizontal acceleration.

neglected in comparison

Coulson refers to the second

group as surface waves in which the vertical acceleration is longer negligible and the wavelength is

no

much less than the depth of

the liquid.

The various treatments all use a linear equation of motion, neglecting the square of the particle velocity, amplitude is

and assume that the

small compared with the water depth.

treatment insists on the conservation of mass,

Of course each

and requires the

pressure to be constant at the free surface and the normal component of the velocity at a rigid boundary to be zero.

The differences

in

treatment then relate to the method of prescribing the initial conditions or of dealing with the explosion gas bubble. solution is

made up by a synthesis of individual solutions such that

at t = 0 the function is made to fit

I

the initial

surface contour

an initial set of velocities on a flat surface, Thereafter, fit

The

if

t is

allowed to vary,

initially continues to evolve its

happens which fits

the impulsive case).

the solution which was made to own description of what

all the conditions and is

produced depend on the volume of the cavity.

50

(or

also unique.

The waves

3.

Cauchy,

Note that in Poisson) it

Poisson, and the Explosion Problem

the first

memoires on the theory of waves (Cauchy,

was seen that a complete solution could be achieved from

one of two possible initial conditions. treated only for plane waves, example,

in

The problem was initially

i.e., waves that do not spread --

for

a canal.

The variables are distance,

height, and time.

For these first

papers the medium was infinitely deep and infinitely extended in directions + x.

Case 1.

See Lamb,8)

sections 238 and 239.

Initial elevation of the free surface around the

origin.

_

_"

SURFACE

_FREE

}0

1of

The initial elevation is the origin.

confined to the immediate neighborhood

The initial elevation is

but infinitesimal values of x, but

51

given by f(x)

0 for all

J

f(x)

a so called 6 function.

dx

=

1,

The subsidence of this initial elevation

produces a train of waves at a distance, is a positive wave,

a crest.

the first

arrival of which

The assumption of a delta function

here is

mathematically the simplest but physically quite unreal in

that it

calls for an infinitely tall infinites simally thin column

of water at the origin which descends under gravity with constant acceleration to feed the wave system.

Poisson preferred to start

with an initial depression in the water formed by a paraboloid which at t = 0 was suddenly removed.

He solved this problem for the case

of propagation in two dimensions.

If

-

one were to start from rest with a crater in the surface,

which is

I

otherwise at the undisturbed level,

arrive would be a trough.

However,

the first thing to

an explosion near the surface,

blowing out, cannot produce a pure cavity.

There has to be an edge

of water piled up above the undisturbed level at the same time the cavity reaches its maximum.

Further, at this instant the maximum

radius of the cavity may be at rest, but the lower parts of it

5

already filling in and the outer parts of the annular edge are moving outward.

It

might be possible to obtain a solution using the

Cauchy-Poisson method if i

are

4

icontour

Icavity

f1

one could assume the proper "stationary"

for the water surface in the blowout case.

This would be a

surrounded by an annulus all taken to be at rest at a time

52

t

zero.

It

would be necessary to obtain an analytic expression

for this contour, and z,

assuming cylindrical symmetry,

as a function of r

and depending on the parameters charge weight,

and water depth. its validity is

Case 2.

Penney,

in

fact, achieved this approximately,

but

limited to depths just short of blowout.

The other solvable situation is

flat surface with a limited part of it distribution of vertical velocities,

that of an initially

endowed at t = 0 with a i.e., initial impulses are

applied to the surface supposed undisturbed. explosion,

charge depth

the underwater shock wave is

In the case of a deep

reflected almost immediately

from the free surface imparting upward velocity to successively deeper layers.

The resulting spray dome is

descends much later, left the area.

in some cases,

flung into the air and

after the waves have already

Consequently the velocity imparted upward has

negligible effect on wave formation.

The removal of water in

the

form of spray by the shock wave reflection leaves a slight depression in the remaining surface which could contribute to wave formation but will be neglected. formation is

then the expanding gas globe which increases to a

maximum size and then decreases in period. and is

The only remaining cause for wave

This situation is

a time equal to the bubble

treated in Kirkwood and Seeger's paper

not applicable to the blowout situation.

an explosion in

On the other hand,

air over water at rest does reproduce the condition

pertaining to the second Cauchy calculation.

The initial impulse is

downward into the water as in Cauchy's case.

The resulting wave

53

train again begins with a positive pulse.

The water surface

initially having to move downward requires the adjacent surface to move upward,

the water being incompressible.

It

is

this elevated

annulus which again causes the initial wave train to proceed. Because of the poor impedance match between air and water, even for air compressed in

shock,

the fraction of the air blast energy

impinging on the water surface which could be taken up by the water in

kinetic energy is

small, probably less than 4% or perhaps 1% of

the total explosion energy. Waves.)

On the other hand,

(See Appendix B for Energy in Surface the energy in the nonventing underwater

explosion retained in the gas globe is total explosion energy, moving the water.

approximately 45% of the

and all of this energy is

available for

One therefore expects that an underwater

explosion would be more efficient at making waves than an air burst. However,

if

a charge is

will have very little

exploded deep enough,

effect on the surface height.

produced only by a local variation in

*

it

As the deep gas globe

emits pulses at each minimum,

turbulence and otherwise dissipates its

causes

energy so that no surface

Swaves

are made.

As we shall see l•er, process is

Waves are

surface height, not by a

gradual or general slight increase in height. oscillates and rises,

the bubble expansion

very low even in

the efficiency of the wave making the underwater case where the actual

t *

wave energy is

only a fraction of a percent of the total

54

A&i

energy.

Clearly a key question is

at what position above or below the

surface are the greatest waves made. is

It

seems reasonable that this

at some point below the surface rather than above.

It

is

important to see how the cavity or crater formation varies with depth near the surface.

This question will be considered in a later

section.

It

is

apparent from Lamb's discussion of wave propagation in

two dimensions

(reference

(8),

Section 255)

that Cauchy and Poisson

Sworked this problem and also that the latter considered the formation of waves from "an initial

paraboloidal depression."

start with a limited initial displacement,

we

then the description of

this contour will be a superposition of all wavelengths. waves travel outward,

If

As these

the longer ones will travel faster than the

shorter ones so that after a while the original harmonic content of the disturbance is This is

spread out and displayed on the water surface.

true as long as the medium is

dispersive,

waves which are short compared with the depth.

i.e.,

However,

for those the

asymptotic solution for diverging (cylindrical symmetry) waves in unlimited sheet of water of uniform depth (reference (8), 194,

195)

Section

shows that the amplitude of these waves ultimately varies

inversely as the square root of the distance from the origin. is

an

This

readily seen from the fact that at a large distance the

wavelengths are large compared with the depth and consequently all travel at the same speed.

Therefore,

the total energy of a wave is

proportional to the amplitude squared and to the circumference of

55

th-e' circle which the wave has reached, which now is there is

constant as distance is

no energy dissipation,

mentioned because close in

but noc to' the wavelh.n.th

further increased.

the result follows.

Assuming

This is

to explosions the wave amplitude

decreases inversely with distance, not with the square root of the distance.

This is

consistant with the dispersive mode of

propagation in which the wavelength is with distance. gradual.

not constant but increases

The transition from one mode to the other is

Also, see brief discussion of dispersion in Appendix C.

4.

Penney's Crater Assumption

Penney in his paper on Gravity Waves description of the surface crater.

has tried an ingenious

The wave system is

released from

rest at time zero from a configuration given by

=•~~

~-

2 1-i 2Df { D 2(r 2 37"+2• / ' 2_ 3D

5%4T/'2'

(This configuration applies to only one position of the explosive charge,

namely that depth, D, at which the ensuing maximum bubble

just reaches the plane of the free surface above it.)

56

The first

term in 4(r) describes the maximum contour of the d

-

expanding bubble. surface is

.me formeff by- the .

The volume of this dome above the former free

equal to the volume of the spherical cavity beneath it, 3

namely 4/3nD3.

The second term replaces the spherical cavity with

another one of the same volume and of the same class as the surface dome.

If

r = D/7,

r being horizontal distance from a point in

undisturbed plane directly over the charge, greater values of r the value of 4 is

then 4(r) = 0.

small but positive,

the expression for 4 describes an open crater if second term from the first.

In practice it

to fall back into the bubble, filling in

from beneath.

For so that

we subtract the

takes time for the dome

and during that time the bubble is

However,

closed cavity or an open one; its same if

the

we can look on the crater as a mathematical description is

we neglect the time of collapse.

the

Using this and other

considerations Penney calculated that the explosion of 2,000 tons at optimum depth would create a wave system, the leading part of which was a trough that would be roughly 30 feet deep at a distance of 1,000 feet.

The optimum depth was described as the depth at which

the maximum bubble became tangent to the plane of the original undisturbed surface.

I

The optimum depth for 2,000 (long)

tons is

approximately 300 feet depending on the fraction of the total energy which isassumed to be retained in

the bubble.

We shall assess in a

later section (Conclusion) how good an estimate this was.

This paper also contains the suggestion that the explosion of a charge at a depth D less than its

optimum depth will produce a wave

57

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

,"-.

-..

system -whichis exactly the same as a charge of less weigh-, which the optimum depth is optimum depth or less,

D.

This implies that if

for

a charge is

at

the wave system cannot be enlarged by

increasing the charge weight at the same depth.

The bigger the

charge the more blows out, and the wave system is

the same.

This

statement neglects the effect of increasing air blast on the wave formation.

5.

Kirkwood's Basic Theory

The Kirkwood and Seeger theory(6)

is

also plagued by the bubble

behavior near either rigid or free surfaces. maximum radius is

invalid in

estimating bubble volume.

The expression for

these cases but is

However,

in

used as a means of

treating the case of a charge

on the bottom, the calculated bubble volume is

arbitrarily divided

by two to compensate for energy loss into the bottom. volume of gases is bottom),

the same in these two cases

Although the

(free water and

one must remember that the volume of the bubble is

thousands of times greater than the original charge volume and is more dependent on the distribution of energy than on the original gas volume.

In the case of free water,

the theory proceeds quite

elegantly from a simple spherical source and its bottom,

image in the rigid

to a solution for a complete potential function P which The strength of the

satisfies the free surface boundary condition. source is

dV/dt where V is

function of time.

the volume of the spherical bubble as a

The initial

configuration of the sea is

flat and

58

-•

i

,

I

I

I

I=--

l---i..

.

. . ... i"

..

...-- i-"

a rest. ....

otin

from KirkWdod-and Seeger

integrals involved in but lengthy.

It

is

"The evaluation of the

f for an actual gas globe is

convenient to introduce,

simplifying assumption,

straightforward,

therefore,

a

the value of which must be tested by

analysis of the experimental data.

If

the period,

T,

of the first

pulsation of the gas globe is much less than the time interval after the explosion,

it

is

reasonable to suppose that V(t) = V for O

0OX-

00

gI o -

o

01

!

' .x .to ,

ZO

I0

>0

-

co

C-4

ox

I•

,

13

124





....

...

.

'

-

ree water -a

give a value of between 80,000 and 90,000 feet

when

scaled up to Baker.

An estimate is

made using the Kirkwood and Seeger theory with

cylindrical volume calculation for the three Solomons' were blowouts,

i.e.,

for which L>D.

shots that

We have:

Theory*

Theory

HR(spherical

HR

Measured

volume)

cylindrical

HR

S~4L Shot No.

L

2

61.6

2.0

1820 ft

3

96.5

3.2

4

115.0

4.0

4 L

2

Ratio

910

630

1.44

6500

2040.

1470.

1.38

11600.

2900.

1740.

1.67

*From Table 8 calculated values.

It

can be seen that the cylindrical volume overestimates the results

by only 38 to 67%.

Perhaps these remaining differences can be

attributed to the uncertainties of the mud bottom. assumed,

for example,

should be 71,000,

that were

then one could claim that the blow out results

should be given by the cylindrical theory. HR values for Shots 2,

If

3,

and 4 if

Then for Baker scale the

they had been done at middepth

84,000 and 82,000.

give a mean value of about 30,000.

The nonblowout, The conclusion is

bottom, shots indicated that

charges on the bottom even at optimum depth produce waves which are

125

1/2 to 1/3 the height of waves produced by the same charges off the bottom in

....

the same water depth.

2.

.

Kirkwood and Seeger Summary

According to Kirkwood and Seeger, 1equal to =1hV x related G value.

the amplitude of a wave is

V is the time average of the bubble volume.

For deep explosions not on the bottom,

v

4i

(.4819)

where

13

a

M(t

.4819 L

3

L3 .'.Amp=

.321

G ft.

For nonblowout explosions on the bottom, that the value of V to use in

calculating amplitude is

126

M

Kirkwood and Seeger say half as

23 lt.rge,-i.e.- ¶W.----.4#19} Si3.

Henee for eharqe -on the bottom the

amplitude is

amp = .1606

G ft.

-

h

If

charge is

at a blowout depth but not on the bottom,

approximation is

the volume of a cylinder if

"a better

a height equal to the

depth of the water and a radius slightly less than the maximum spherical globe in an unbounded liquid,"

(6).

Actually one should

life. consider the time average of the volume averaged over its This is hard to do in the blowout case because we don't know the volume mode of expansion or collapse.

At any rate take the volume

expression as = h L . If L I sin ot for example, then V = .5 2 7rhL One could take this time average factor to be the same, i.e., .4819. If the charge is on the bottom one could, to be consistent with the previous calculation, it

i

would be if

assert that the volume was half what

the charge were at the same depth in deep water.

Then ;hL2

i -

V = .4819

and

•:

1 12 7

MWW 21ch Half cylindrical on bottom.

also

amp

=

1

2 3 -h , .4819 L G;

Half spherical on bottom.

tio spherical

2 L

cylindrical

3=

4L

( 7L

The volume for Baker and hence the predicted wave height is overestimated by a factor of 7 (h = 180,

L = 930) by using the

spherical rather than the cylindrical expression.

3.

Scaling (Impulsive, Bore,

Deep, Blowout)

Surface waves in water are a gravity controlled phenomenon. When we change from one scale to another, different charge weight,

for example by chosing a

the value of gravity is

unchanged.

linear dimensions change in the ratio of n = (t_)/113 1

128



, '

a

a

go

oI

This

The

includes the charge radius.

charge depthý,_waterdextbhanrlsdifatar,,a---

Time however goes as AT in order for g to remain constant from cne scale to another.

This requires wave velocity to scale as AT.

Wave displacement and the particle velocity associated with it, however,

may have various scaling laws depending on the method of

wave formation,

To see how this comes about let us start with the

case of waves produced by a downward impulsive loading of the water surface

(such as would be caused by an airburst over water).

been shown by the Applied Mathematical Group N.Y.U. Penney (Gravity Waves,

etc.)

(%1946),

that the wave amplitude in

velocities are equal. as n initial

n = Aw cos wt.

This is

by

This can

Let the

vertical wave displacement be represented by n = A sin wt. particle velocity then is

The

On the two scales these

because the tmpulse to an area varies

and the mass of ixater affected also varies as n3. velocities are the same.

Hence all

Note that for the initial loading

phase gravity is not a part of the process and time scales in same manner as distance.

it

is

then,

the same on all scales.

to the ensuing wave notion we have,

t

Awn

the

for this reason that the initial

particle velocity acquired by the water is Returning,

has

this case is

proportional to the sixth root of the charge weight iatio. be illuminated by the following simplified argument.

It

= Aw

129

-

"The wave velocity is proportional to V-ep, uncnlanging,

and hence scales as

/F. Therefore,

sciac-,

n.

Hence it

w scales as 1

relatio'm A scales as yE.

gravity being follows that time must

and from the above

Note that wave velocity being

proportional to wavelength times frequency requires that X scale as n.

Therefore,

goes as

.n

all lengths scale as n except the wave height which

or w

It

is

this very fact which allows the particle

velocities on different scales to be equal, as /n.

For the case of impulsive loading,

since time also scales if

the charge weight is

increased the wave amplitude increases as the square root of the linear dimension or the sixth root of the charge weight.

If we now put the charge into the water in a blowout position, we see that the bore forming mechanism is

somewhat analogous to the

previous instance except that the loading is vertical.

Water is

pushed outward by the explosion gases,

outward velocity impulsively acquired is scales.

horizontal instead of and the

the same on the different

This outward movement causes the bore front to form in

same fashion as a shock front forms in

a shock tube.

the

The

conservation laws (mass and momentum only) applied to this formation require the bore height and hence ensuing wave heights to vary as rT, when we go from one scale to another. formed, is

After the wave system is

time scales as the square root of the linear scale factor as

generally required for all gravity waves.

130

Let us now consider the nonblowout case and ask how wave .. heights should change when charge weight is blowout case to another.

changed from one non-

We take the Kirkwood and Seeger theory to

be a valid expression for wave height for this situation and can write

Amplitude % V

h

If

3

WG

x related G function % LG h 11,

h

we change charge weight,

G is

(r1z1i D ,zl,t) h (D+33) T

unchanged except for the effect of

W on r. Consider the critical case in which L = D and, they both are large compared with 33.

W

or n 3/4

Then L3 0 , W or D4

Let m be proportional to WI/

according to m, time as /rm,

in

4

.

If

addition, %

depth is

W and D

scaled

wave velocity as /r, distance as m,

then,

Amplitude

We have seen that if of L

is

r1.

I!

the charges are not too large,

limited (less than about 2),

distance is

mn m

proportional to T1:.

G is

i.e.,

the value

then the G function at a given also inversely proportional to

This leads to an expression for wave height such that

4

; ~.

[2

-.--- ... .. ... ... .

-

~

............. ... ... ..... h

WI/

Tih•

h

Since

Hr lo m

r

3

W

51/3

(D+33)

we find that

and wave heights vary as m = W

for deep explosions

at corresponding distances

(neglecting 33 compared with D) where W is

the order of one ton or less. scales

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

---

All distances

of

scale as W1/4 and time

as W

Let us now look at the blowout case or the relatively shallow

*

explosion in a cylindrical

which the spherical bubble volume is crater of radius L and depth h.

In

to be replaced by this case for

large charges we have

3

2

amplitude

G

2•

h (D+3 3)

If

G is

unchanged,

according

i.e.,

if

to this,

T is

very large and if

the amplitude

"

n n .*

132

D>>33,

/

then

nl/3

If

D

C14

tnto%

0 H-

a (A

0

in

4

r

N

C4

n-

in

4

o

en ,-

1,

0

C

0

o

WD

Lo

.4-)

a

0

4

CU0 •-4

0;

N

')In

0;0

C1 ON-

0

(A

M

40

ýj

c

0-1

6

1'

-m4

%0

! 44

4-4

o

4o

0M

N

4

(a-1

N

m

O

C')*

1.4

C14

U)~(

4-)UN 4

01% 0

-I

0

_•

4.x

0)

15 2

41

x

'4

-----

~. .

. . ....... ..

..

~-tm-

Tckta1 w.av-e-ýery--in -1~2 waveas-

6 Charge wt of 46,000 lbs has total energy of 46,000 x 1.48 x 10

ft/lbs. or 68,000 x 10 6 .

.

Total wave energy is

approx 143.8

61006

.21% of total

energy.

A similar calculation for Shot 2 at pole 12:1007 ft. from 6700 lbs.

I•

1

153

CID

LIA 0ý

IA

IA in I- *

N

It,

0

N4

r:

.

D

L*

en

(4

inO

in *

4~0

%.0

Ito

0l

40

v,0

C-4

H0

OL'

0

4

IA

0)

IA

'.0'

*

vr-4

4-

in0

(4

co

44-)

*

n Iw

U) OD

(~

OD

IA

0

-r-I

*

C" LA

4-)

0

tp

r4

C <

41 4)

> H

nv

.-4

Q r

N
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