o UNDERWATER EXPLOSION
November 8, 2018  Author: Lauren Whitehead  Category: N/A
Short Description
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 7615
MAKING BY AN o UNDERWATER EXPLOSION
SWAVE
•
By Gregory K. Hartmann
Cý
SEPTEMBER 1976
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FORM
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INTRUC.TiNSg
RECIPIENT'S 1CATAi.0 .
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17.
DITRISUTION
IS.
bUPPLEMENTARY NOTES
19
KEY WORDS (Cornirnue on reverse olde it necessary ald Identity by block number)
STATEMENT
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sheabe•
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II
Underwater explosions Wave making Explosion bubble containment Explosion bubble blowout 20
A'kSTMA CT fContlnue on reve.prsi olde
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If1reCsaar, y end Idonwlby by blockm nrtritt)
narper is a. historical account of wave makIng ••perimeit,
made
r1ilrmm.di.ately after World War T1, from the smallest to the : .•[inclurlinS the Atom Eaker Bikini. The various theorias
cf zpesivc wave making are discussed and comparisons are mqe bU.,in thq ob.servations and the theoreticail expectations:. Scalinq lv 3o,: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
TimeDistance 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.
346360,
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 shipload 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 ExplosionGenerated Waves" TC130, 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 Undex94 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.
BritishAmerican Compendium. 27 May 1947
Identical article minus tables appeared in Journal of Applied Physics Vol.
19,
pp 346360 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
Cliowance 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 C2 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
43
' 4J '4 444444
4) 4.)
IV
z
0u
4J 0
4
0
0 ()
W
U)
4
4)
)4
44
r0
o0
0 T$
>44
)
41 Ln U) Lfl
4)
0
0Q)
0
0(DC
>4
>4
4
CI
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U4E
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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
44
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 4,000pound 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 7615
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 7615
3
919:917

to + 27 SEC
2
"Apoxmt
I4
Pol
Nubr Dsac
1
,
PoeDsacswthEpoina 51
ol
ubr
Dsac
14t.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 7615
"
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 7615
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 7615
DISTANCE FROM EXPLOSION SLOPE ASSOCIATED WITH CRITICAL VELOCITY vgh
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 raphruno 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 EsterlineAngus 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
00uj1
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 7615
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
"4214o
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 
LII31
'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 7615
c+ 11 SEC
=.A. a
V
Approximate 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 7615
•g27
6I 168I Ft 075 t t,
9 Fiur ShtNm e 581 .ib 62
:
b.T Ton_.onW 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 4J
mA
0o 1i C
.
H 1
'44 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
14
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.14
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)
44
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)
E4
fell
39
40
to 0) 14
+H) 0
4
1.4
)
4J
ý40 0L
NSWC/WOLYMP 7615
g

z
Cz
a E.
C
EE 00
N'.
(40
NSWC/WOL/MP 7615
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 UMM
8
Charge 2034 Lbs TNT
Figure 1(b)
diSow Hydirophorle Recordsat
41
u
NSWC/WOLIMP 7615
(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 7615
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 2th/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 40foot 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 thedepth 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
CauchyPoisson 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
the' 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 1i 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=
li..
.
. . ... i"
..
... i"
a rest. ....
otin
from KirkWdodand 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
C4
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 Vep, 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
01
6
1'
m4
%0
! 44
44
o
4o
0M
N
4
(a1
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.aveýeryin 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
C4
H0
OL'
0
4
IA
0)
IA
'.0'
*
vr4
4
in0
(4
co
44)
*
n Iw
U) OD
(~
OD
IA
0
rI
*
C" LA
4)
0
tp
r4
C <
41 4)
> H
nv
.4
Q r
N
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