FREQUEt
)NALYSIS OF
T
D
AN UNDERWATER EXPLOSION
BY
HARRY B. MCCURDY
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
BACHELOR OF SCIENCE
From the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
1944
Signature
of Author.--.
Signature of Professor
in Charge of Thesis
.
.-
,
-*.*.
a..
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38
TABLE OF CONTENTS
Pg.
I.
II.
INTRODUCTION...........................,. 1
PRELIMINARY STATEMENT OF RESULTS......o,
3
DISCUSSION OF DATA...................... 4
III.
IV.
V.
VI.
DETERVINATION OF ENERGY DISTRIBUT ION.....5
APPLICATION OF METHOD TO DATA..........0.9
CONCLUSIONS............................. 11
TABLES AND GRAPHS.......................13
DATA AND ANALYTIC APPROXIMATTONS........
APPENDIX A.........
BIBLIOGRAPHY
.....................
............................
280944
0
18
26
27
ACKNOWLEDGEMENTS
The author wishes to express his sincere
appreciation to Professor R.D.Fay of the
Massachusetts Institute of Technology and of
the M.I.T. Underwater Sound Laboratories, to
Dr. Maurice Ewing of the Woods Role Oceanographic
Institution, and to Mr. isdward Colson for their
many valuable suggestions and criticisms.
Acknowledgements are also accorded to the
U.E.R.'Laboratorfy at Woods Hole for the use of
their data and to the M.I.T. Underwater Sound
Laboratories for sponsoring this thesis.
1
I. INTRODUCTION
The object of this thesis is to determine the
distribution of energy in the frequency spectrum of an
underwater explosion from a pressure vs time curve, and
to present a preliminary study of this distribution as a
function of the size of the charge.
In order to completely cover the question of energy
distribution in the frequency spectrum it is necessary
to consider the energy expressed as a percentage of the
total, and also the distribution of absolute energy
expressed in milliwatt-seconds. It is the absolute energy
which is of greatest interest but the percentage
distribution is useful in correlating the results.
The need for this frequency analysis arose as a
result of recent investigations in the transmission of
sound in sea water conducted by the Woods Hole Oceanographic
Institution.
A knowledge of the energy distribution will
fascillitate the study of long range signalling and will
also aid in. the study of absorption and reflection phenomena.
Former work on this problem is non-existing since
heretofore underwater explosions were not used as a method
of signalling and thus the frequency spectrum was of no
significance. However, as soon as the tifansmission of sound
2
becomes the objective, a knowledge of the frequency spectrum
of the source is of importance.
The ultimate objective of this work is to know, the
energy-frequency charactaristics of an explosion, given
the size, type and depth of the charge. Future work to
obtain this objective may be divided into two parts:
1.
The taking of a sufficient amount of data to
make a systematic study of frequency distribution
as a function of type, depth and size of charge.
2.
Apply the method presented in this paper to
determine the energy distributions.
Sufficient data should be taken to obtain reliable
results.
3
II.
1.
PRELIMINARY STATEMENT OF RESULTS
The conditions giving the maximum absolute energy of
12.9 milliwatt-sepnds
occured for the largest charge,
460 lbs. of TNT, in the 400-800 cps. octave.
2.
Increasing the size of the charge tends to increase
the percentage energy in the low frequency end of
the spectrum and diminish the percentage' energy in
the higher frequency end (above 500 cpa.).
3.
The mximum percentage energy for the three charges,
(60 lbs,
225 lbs and 296 lbs
of TNT),are very
nearly equal and occur in the vicinity of 400 cps.
4.
40 % of the total energy occurs between 100 and 800 cps
for the three charges mentioned above.
4
III.
DISCUSSION OF DATA
The data for this Thesis was supplied by the U.E.R.L.
Laboratories at the Woods Hole Oceanographic Institution,
Woods Hole, Massachusetts.
This data consisted of pressure
vs time photographs of explosions of 2 lbs, 60 lbs,
225 lbs,
296 lbs,
except
and 460 lbs of TNT.
All of these records,
the 2 lb charge were taken at a distance of 20 feet from
the explosion and at a depth of 41 feet below the surface.
The 2 lb charge record was taken 4 ft from the explosion
and also at a depth of 41 ft.
As is usually the case.in underwater measurements,
a large amount of data should be observed and averaged to
eliminate the possibility of using erroneous data if only
a few records are taken for a given set of conditions.
Unfortunately sufficient data to obtain averages was not
available so that it is felt that the actual numerical
results obtained in this thesis may be considerably in error.
Quantitative results will be stated but the emphasis will
be placed on general trends.
In one case where tvo pressure-time curves were obtained
for identical conditions, the results calculated from the
two curves differed by 100%.
This emphasises the need
for taking sufficient data to obtain a reliable average.
5
IV.
DETERMINATION OF ENERGY DISTRIBUTION
From a pressure-time curve of an underwater explosion
it is desired to obtain the distribution of energy in the
frequency spectrum.
The method presented in this paper
consists of obtaining the magnitude of the Fourier Integraly,
F,, as a function of frequency, using this function to
obtain the percentages of the total energy in octave
bandwidths and then returning to the pressure-time curve
to obtain the total
energy.
Undoubtedly the most desirable method would be one
which uses the pressure-time curve directly.
the Cinema Integraph,
A machine,
is capable of performing the
desired Fourier Integral, but unfortunately this machine
is not in operation at the present time.
A graphical
method employing the original curve is quite unsatisfactory
since it would eonsist of graphically integrating under
curves of the type,
ce4 wft
would be a cumbersome task.
a-wA
(*V*MLe
which
Thus it is highly desirable
to obtain an analytic expression which closely approximates
the actual curve.
From a glance at a pressure-time curve it seems
logical to center the attack on the exponential curve.
See Reference 1.
6
A single exponential can be found which closely approxinates
the initial slope of the curve and another exponential can
be found which approximates, the final
slope.
Thus it
seems logical that the sum of two exponentials might
closely match the curve at all points.
the case.
This is actually
The method used to match the sum of two exponentials
to the actual curve is illustrated in Appendix A.
F(t= Cae
Thus, choosing
+a,e-
it is possible
to obtain the Fourier Integral of this function by
standard methods.
The majority of the straight-forward
mathematics will be omitted, but it seems advisable to
include some of the intervening equations which might
be helpful in interpreting the energy equations.
where
Fey>
=
a, e-b+t
a,e-' *
=5
ft
Ff
F
=
I.p/
0 Fur+
A +- 49 771,
2+r 17' 2.
{ bf
a,
b,
a.,
C
cA
b- .
are determined by the
method presented in Appendix A.
a t
here
A
+ 4.7;Lf'
A =(a~s
t
a
+ aL)
=
acLb,
1 Only the initial pulse of the explosion is considered.
7
The pereentage of the .total energy which occurs in
an octave bandwidth is given by the expression:
5'"|
f
/*a
Expanding the expression for /Ff/,obtained on the
preceeding page, as a sum of partial fractions and
then performing the integration indicated- above, the
expression for percentage energy becomes:
b,
[36obi 180(bibx)
oh, 1&(biMb,)
b,
a,
+
4FA
aL,
b.
+ Ja
b.*.
//b
where the inverse tangents are expressed in degrees.
It now remains to obtain the expressions for total
energy. The analytic approximation will also be used in
this derivation which consists of evaluating the expression:
Total Energy
4
8
10
C x 10
411R
C X10,
8nR'
Ft is in cgs units.
7
t
a^'
+
aae~
LCJ FLI0
a
&oCXto4b74x
a
.,+
.
.b
Q~
bi.
8
The expression in parenthesis in the last equation is
recognized as the same expression which appears in the
denoml.nator of the percentage energy equation. Thus,
the absolute energy in the octave fl to 2f, can be
expressed as:
~"'~'
X1b7
/*caxo
L
;"%
t___
+ 180(b,+b
2)/)
+o.a
I o(b&+
bC
where R is the distance of the pressure element from the
explosion in dentimeters.
* .o x
cgs units.
It is convenient to evaluate the energies for
defined octaves.
For this purpose the octaves 25 cps
to 50 cps, 50 to 100, 100 to 200, 200 to 400, 400 to
800, 800 to 1600, 1600 to 3200 and 3200 to 6400
chosen.
were
9
V.
APPLICATION OF METHOD TO DATA
Having developed a method of attack, the remainder
concerned with applying it
of this paper is
to the data
and then correlating the results as far as is consistent,
as previously discussed.
Photographs of the actual data and the comparisons
of the actual and the approximate curves are found on
Pages 18 to 25
inclusive. 1
every case the analytic curve is observed to compare
favorab y with)the actual curve except in the vicinity of
t= 0.
To attempt to match in this region would be unwarrented
since it is very likely that the initial peak teading is in
error due to the poor response of the hydrophone at very
high frequencies.
The resulting percentage energy distributions and
absolute energy distributions are tabulated in. Tables I
and II respectively, on Page 13
Once again it should be mentioned that the results
hardly warrent plotting and the connecting lines on Graphs
C and D merely serve to indicate the general trend.
The plot of percentage energy distribution for the
2 lb oharge is observed to possess
maxima.
Physically there is no explanation for this,but mathematically,
assumming the actual curve can be represented by the sum of
two exponentials, the reason is obvious.
As the difference
Photographs of the 2 lb and the 225 lb charge. were not available.
between the values of the slopes of the two component
exponentials becomes greater, the effect is to increase
the spacing between the maxima of the two inverse tangent
curves in the resulting expression for percentage energy
distribution. In the 2 lb case this difference was
sufficient to cause the sum of the two inverse tangents
to have two maxima. Thus it appears as though this method
fails under such conditions.
11
VI.
CONCLUSIONS
The percentage energy distributions for the 60 lb,
225 lb, and the 296 lb charges can be correlated very
effectively, but the 2 lb and the 460 lb charges do not
seem to follow the general trend. Since this lack of
consistency also appears in the data it is probably a
combination of two teasons. First, the possibility of
erroneous data and secondly, the effects of the nonlinear properties of seawater.
For the three percentage energy distributions
plotted on Graph A, the following observations should
be noted:
-In each case the maximum percentage energies
are very nearly the same and occur in the vicinity of
400 cpsi .At higher frequencies the percentage energy for
the largest charge decreases most rapidly, while thothe
low frequency range the peraentage~ energy for the Amallest
charge decreases most rapidly with decreasing frequency.
It is also interesting to note that approximately
40 % of the total energy is in the range between 100 and
800 cps.
Another point of interest is the rapidity with
which the percentage energy decreases after the maximum
has been reached.
The percentage energy in the last
three Octaves during which the energy is increasing is
approximately 1.5 times the percentage energy in the first
three octaves during which the energy is decreasing. This
indicates that the energy fa more concentrated in the
12
lower portion of the spectrum.
Alth,,ough the percentage energy correlation is interesting
it is really the absolute energy which is of interest.
Graphs C and D show that the maximum energy is obtained in
the 400 to 800 bps octave for the largest charge.
The
real value of this plot would only be reilized if the
calculated points were sufficiently reliable to produce a
smooth curve.
Thus it would be possible to give the
effectiveness of any size charge between 2 lbs and 460 lbs
at a depth of 41 ft below the surface, in any frequency
range.
However, as the graphs stand now, they ari notni
sufficiently accurate to indicate more than the order of
magnitude and the general trend.
13
Size
of
Charge
225 lbs
296 lbs
460 lbs
8.4 %
10.5 %
9.2 %
10.3
10.6
13.5
11.5
10.8
12.6
12.6
15.5
12.1
200 - 400
11.0
13.6
15.8
15.7
13.8
800
9.7
15.1
16.2
13.6
14.9
800 - 1600
10.7
14.1
11.7
8.6
11.9
3200
12.5
9.6
6.6
4.6
7.2
3200 - 6400
11.9
5.3
3.5
2.3
3.8
Octaves (cps)4.
2 lbs
60 lbs
25 -50
4.5 %
7.0 %
50
100
7.7
100-- 200
-
400--
1600--
PERCENTAGE ENERGY DISTRIBUTION
TABLE I
Octaves(cps)+
2 lbs
60 lbs
225 lbs
296 lbs
460 lbs
25 - 50
2.46
104
395
765
1542
50-- 100
4.22
153
495
986
1925
100 - 200
7.45
187
593
1128
2030
200 - 400
6.06
198
742
1143
2320
400-.- 800
5.37
224
764
994
2500
800 - 1600
5.87
209
564
625
1990
1600 - 3200
6.92
142.
312
337
1198
3200
6.56
163
165
632
-
6400
78.6
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26
APPENDIX A
METHOD FOR MATCHING THE SUM OF TWO
EXPONENTIAL CURVES TO TEE ACTUAL CURVE
The method of matchting used in
this thesis consisted
of the following procedure:
1.
Two poirits were plotted on semi-log paper
which define the tail of the actual curve.
i.e. "a" and "b" in Fig. 1
Two other points were chosen at positions
2.
approximately those of "c'' and "d" Fig. 1.
From these values the ordinates of Fig. 2,
at the corresponding times, were subtracted.
3.
Plot these differences on the &emi-log paper
thus defining the second exponential.
J/09
s I.,.
b.
ActuntI~
A chl-ve;
Q
F1 ,
b
i
b
jp p3
Thus the actual curve is matched at the points a,b,
c, and d.
The slope of the second exponential is usually
so steep that it does not effect points "a" or "b".
A
pvoper selection of points is essential to give a good
approximation, but after a few attempts the proper choice is
usually obvious.
27
BIBLIOGRAPHY
1.
Hazen, Barold Locke and Brown,
Cinema Integraph,"
G.3.,
"The
Journal of the Franklin
Institute, July (1940), v. 230 no. 1.