THE RESPONSE OF A POINT SOURCE IN A LIQUID
LAYER OVERLYING A LIQUID HALF SPACE
by
SINSR TECA
A
"'%~~LRn
ROY J. GREENFIELD
S. B. Massachusetts Institute of Technology'
(1958)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF
SCIENCE
at the
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
June, 196Z
Signature of Author..
4.
--- -------. ..
. . . . . .
iences, May 19, i96Z
Department of Earth
-
Certified by ..
-
%
--
I
Ther~s ~Sprvisor
Accepted by .....
Chairman, Departmental Committee
on Graduate Students
,
C~,
The Response of a Point Source in a Liquid Layer
Overlying a LAquid Half Space
Roy J. Greenfield
Submitted to the Department of Earth Sciences on June 196Z
in partial fulfillment of the requirements for the degree of
Master of Science
ABSTRACT
The response to a harmonic point source in a liquid layer
overlying a liquid half space is computed as a function of frequency.
Inclided are the contributions frorm all normal modes that occur,
and the branch-line integral representing the refraction arrival.
The value of the refraction arrival is given in terms of the complex
error function.
The effect of different velocity and density contrasts are considered, and the effect of source depth and the distance to the receiver
are investigated.
The results giving the behavior of the magnitude of the branchline show that it is much larger at the mode cutoffs than at other
values of frequency.
The total amplitude of the response shows a
regular oscillation in the frequency range in which two modes are
present, and somewhat irregular high and iow values over the
range in which three modes are present. This behavior reflects
the difference in amplitude at frequencies for which unodes reinforce
each other.
or interfere with each
Thesis Advisor:
Professor S. M. Simpson, Jr.
Assistant Professor of Geophysics
ACKNOWLEDGEMENT
The author wishes to thank Professor S. M. Simpson Jr.
for discussion towards formulating the thesis topic, and for his
criticism during its writing.
The IBM Corporation supported the research on this thesis
for which the author is indebted.
In particular, he wishes to
thank Dr. J. P. Rossoni for his encouragement.
The author also wishes to thank Mrs. S. F. Martin for the
care she exercised during the typing of this work.
TABLE OF CONTENTS
Introduction
4
Derivation of Formulas
6
Computations
Zz
Results and Discussion
Table I
Appendix A
U
31
4i
Flow Chart of Program
isting of Program
42
43
Appendix B
Figures i-27
Bibliography
47
73
4
INTRODUCTION
The problem of a point source in a liquid layer over a liquid
half space was solved in terms of normal modes by Pekeris (1948).
The solution, good for horizontal distances large with comparison
to the depth of the layer, was expressed in terms of normal mode
contributions.
phase velocity.
Each of the modes travels with frequency dispersed
The arrivals of each mode at a given point were
overlapping in time, but the frequencies for low frequency cutoff
of the modes increases with mode number.
By using recording
systems that accentuated with low frequency part of the spectrum
the arrival of the first mode could be seen on the time recording.
In this paper the point of view will be to add the contributions
from the several modes in the frequency domain.
The results of the
computation will give some idea how propagation through a layer
affects the shape of the power spectrum of the recorded signal.
The power spectrum of the motion is given by
I g(w)f
Id(t)(t, r, Z)[
Z
where g(w) is the Fourier transform of the source and
(w, r, Z)
is the response at the observation point to a harmonic point source.
5
In his original paper, Pekeris (1948) derived an expression for
the branch-line integral, representing the refraction arrival, which was
valid away from the mode cutoff frequency.
Officer (1953) showed that
a different expression that give large contributions must be used at
the cutoff frequencies.
In this study an expression for the branch-
line that is valid for all frequencies is derived.
6
DERIVATION OF FORMULAS
In this section will be presented a brief outline of the
approach used by Pekeris (1948) to solve the problem of a point
source in a liquid layer over a liquid half space.
The model is
shown in Figure 1.
b
p
d=
depth of source
Za
depth of observation point
C
velocity in layer
C =
velocity in half space
H=
layer thickness
r=
horizontal distance from source to observation point
1
/ p
The density constrast
(2. i)
The problem was solved in terms of the velocity potential
S(r,Z, W,t) a
it
(r,Z, )
(2.2)
The pressure is given by
Pa p
at
(2.3)
The particle velocity is given by
VI n
V B
(.
The velocity potential obeys the wave equation.
4)
The source
is taken to be a simple harmonic source of the form
(2z. 5)
The solution must obey the boundary conditions
i)
at Z=O
The pressure is
ii)
0
at Z=H
The pressure and the vertical particle velocity are
continuous
iii)
Energy does not come in from -
)Onor does
the velocity become infinite as Z goes to - C
.
In the layer above the source the solution to this problem written
in its integral form is
o
f
with the following definitions
b= p1
P2
J (Kr) is the zeroth order Bessel function
F
The period equation
The
K
V
XA-
x4
V KL
P cos
H + ib
K<
j
1H
rea.
2=)
Lt"II"
's introduce branch points at KK
The integral for
sin
iw/C
l
and KiK a
will be evaluated by contour integration,
Without proof (see Ewing et. al.,
1957, p. 135) we state that all
real poles are between KI and K..
just below the real axis.
They may be considered to be
Using the identity expressing the Bessel
function in terms of the Hankel functions
ZJ (Kr) a H' (Kr) + H
o
o
(Kr)
The integral for T is broken into two ingals
the one containing
HI is deformed upward in the K plane; the other is deformed downward as shown in Figure 2.
None of the arcs contribute because of
the behavior of the Hankel functions.
/C 2
10
The integral along the real K
dxis has been replaced by
the contours shown, plus the residuais that arose by deforming
the contour containing E and D.
using
-H
C)
o0
(iq) = H
(2)
the integral is even in
branch cut.
(-ig).
Lines A and B cancel each other
Lines D and C also cancel because
and so equal on ooth si4es of the
The contributions from lines E and F will be con-
sidered following the discussion of the poie contributions.
The residues of the poles contribute what is
the normal modes contributions.
of the period equation.
For K
known as
The poles arise at the zero's
< K< K.
The period function
may be written as
tan
A(,
b
01.1- C2
z
tan H
f
-9(Nj
-(K)
0
1
(2.8)
The contribution from each pole is given by the Cauchy
residue theorem as
-
I
VKL~
p- el R~s
HH-b.9)
(2. 9)
4N)is the Nth root of the period equation and
is evaluated
for K=eN).
The Hankel function has been expanded in an asymptotic
form good for large r
H(2
(K
(N)
r)
.
r
r.4 - K(N)r
(2. 10)
The total from all the poles is obtained by sumrning the contributions from each pole.
T{JPOLES
7
JiVw)
Ni=
The number of poles that occur increases with w. The number
of poles N(w) for a given w is given by the smallest integer
greater than
/"c L) - (/iC
)
H ix + 1/2.
(2. 11)
There is no pole contribution if the fre uency is below the cutoff
frequency of the first mode.
12
The branch line contribution is an integration path that
comes in from (K , -iLN
) to KL on the left side of the branch
line then returns on the right of the branch cut as shown in Figure
2.
The value of
is the same on both sides of the cut, but
changes sign acr- ss the cut.
The branch line contribution
is then
where
L
Officer (1956, p. 19o) shows
4 b inay be combined into a
single integral.
1
3
IM
IA/
SVH
6/rd
eL
kc/H
13
A new variable is introduced by the relation
The asymptotic expansion of (
10)
0. is again used for the Hankel
Function.
H(Kr) = H(K r
)
_.,
4
e
-i(K r -f4)
e
-K_ rX
(.
15)
Because the Hankel function vanishes exponentially for
negative inmaginary parts of K, for large r almost all the contribution to the branchline integral occurs in the vicinity of the
point K=K
.
At this point X=0.
In the change of variables
only terms of the first order in X have been retained.
this the denominator of F is written
Cu
Lc,
bH
--
K 1 4 ccs4l-f-
A+15<
13X
Doing
14
where
A
Bt
cos (K MH)
K
M
Lb
K L(
ij
C
) sin K MH
C
The branch line integral may now be written as an integral over X.
b
TY
ITK,
._,
IC-X Cl.
k
I/TI2. s(KI A
Kl
.I(O<A
4C()
4
+-2 8 x.
/4-~(X
* -- <K K
dX
4'?. IB)
4
b K/ - S/M K )A
S In K4$
OT
Pekeris (1948) evaluated the integral (2. 18) under an assumption
equivalent to setting B=0. For much of the frequency range this is
a reasonable procedure, because most of the contribution occurs
for X small so
A>> BX
15
Following from this the denominator may be approximated by A.
Under this condition
j=.
312
A 2 (KI
Ir)
(2. 19)
AI-- oA(4)-
U
r-e
(2z.
0)
Officer (1953) pointed out that a different evaluation was necessary
at frequencies for which A= 0.
The frejuencies for which this
occurs are given by
These are the cutoff frejuencies for each mode.
With A=0O
o-Ulv
o~I
(~
[)j'~~-,
(2. 2l)
and
/-;v ,)
2. cii(Kid)
(e. 2)
16
At these fre uencies the magnitude of the branch line contribution
decreases as I/r rather than as ir Z in the case of frequencies
away from the mode cutoffs.
Officer (1953) explains the significance of this as constructive
addition to the refractive wave by the wave reflected in the layer.
Consider a plane wave in the layer, travelling at the critical
angle as in Figure 3.
At D energy is put into the refraction wave
The wave is reflected then travels
travelling in the half space.
to C where it is then reflected downward to D'.
adds to the refracted wave.
At D' it again
The phase of the refracted wave that
reaches D' will be the phase that was at D at a time L/C, earlier.
For constructive interference, the wave in the layer and the refracted wave must differ in phase by ZrN at D'.
The phase change
over the path DCD' is given by
CI
CS &-
+
r(2.23)
The change for the refracted path is
2_
TA A/-
C .
(2. 24)
17
Their difference is
w 2-HH
C, Cos -0
C1
T4+V
(2.25)
For the critical angle
sin 0 = C / C
So the condition that the phases differs by 2 w r
(2z. 26)
(
ct
c
-I
-
gives
I
C
H-Q
_N
-
-
)-_
7
"rr(N- "x-
(2.27)
(2. 2)
Comparing with (2. 17) this is seen to be the necessary condition
for A=O.
The contribution from the branch line integral is large
at the cutoff frequencies but falls off away from them.
An ex-
pression is desired that gives the value of the integral good for
all values of frequency.
18
The following presents an evaluation of the branch line good
for all values of A and B.
The integral in question is from (2. 1c).
J
=
(2. 29)
4- -Xjg
~k27 Q
=
0
1-L
(2
(2. 30)
£S
'"
(Z. 31)
Y
OL
-
d-
;1
28
,A
---
A
-+ 8
jB"~Y'Sd~
**5
(?. 32)
;8 v y
~~04
Be
(L. 33)
The emIuation of I follows.
Consider I as a function of S.
sP
'
0o -/
cLZ
C
+~y
S2
Tf 1- I
+
0
d
e)y
5
(2. 34)
2_._.
19
assume a solution to (2. 34) of the form
1= e
DS
U(S)
oij
(*'
d6
E-p-)t~
+
0~-Pd
A(o)
+M/1
77 t
2.
.Q4W+ L((O)
S
WI
Sdw+ (A(o)
2.
rp
U
T(' ) +( (0)
(2. 35)
to get
U(O)
to get U(O) we have, using erf (0) = 0,
t(O )
=
(b)
(Z. 36)
Substituting (2. 36) in (2. 35)
"
C
02.
(2. 37)
1.
Equation (2. 37) is substituted into (2. 32) giving
r='.
-f
I
~w(V$b)I~]
(2.3b)
20
Using this evaluation of J, the general expression for the branch line
integral is
Ci Kt_)Y
-Q
9jb K. stw(4Md) )W (k1AI)
i;
lSIrvyi(
II
(2. 39)
43J
This reduces to the previous formulas 2. 19 or 2. ZI for J if B or A
is 0.
For A=O
D=O
and
s= re VV S
which is the result in (.
19)
To compare the result for B=0O with (Z. 2I), we nust find the
linit as B 4- 0.
Js4-
I11-
-,7rTF
(
PS I.-M
ALVR
POS-) 3
The error function is expanded in its asymptotic form good for
large agreement.
~
gLV~
.. - rreL-P1.
+S
+
+ pn.
'OT
I
-
iOs
t'
1.1
24S
Or .OMMW.W
vFIFS "gob,
k( Y~
(. 4o0)
'
21
This is the vaiue of the integrai found for B=0 and given in (2.
i).
The expression for the branch line given in (L. 39) is a
good approxinmation at all fre juencies, and could be used in a
nucerical integration to obtain the time response from the refracted
wave.
COMPUTATIONS
COMP UTA TIONS
Computations were made to obtain the vertical velocity of the
liquid layer at the surface for a simple harmonic source, the response
at the surface is itself a sinusoidal motion.
The veritcal velocity
is obtained by differentiating the veocity potential with respect to
Z and letting Z=0.
Previously the velocity potential was derived as
the sum of the normal mode contributions (2. 9) plus the branch-line
contribution. (2. 36) Although it is the magnitude of the response
that is to be considered, the phase of each contribution must be
used in combining the terms to find the total response )
(0).
The contribution of the poles to the vertical particle velocity is given by (1. 9) as
?- :H
6--
r
G
-j)
SN
H
g-s (sl
(Pd)
(3-1)
- b(4
,
~
k,:~~c?~7_
The contribution from the branch-line is given by (2. 36) as
,-T{d)hVU
~Lc
A
[bf~
r
;:r
A
#--
,
-
(VP"
24
A and B are both real and positive so that D is negative imaginary.
The complex quantity
K rD
has a phase of - r/4. In evaluating
the branch line integral use was made of the Share subroutine BS WOFZ.
'The subroutine calling sequence has been changed to allow its use by
FORTRAN.
This subroutine computes
From this the complex error function is obtained from
For each frequency the contribution from the poles and from the
branch-line are computed and added together to give
result was printed as output.
)VI
. The
In addition, the amplitude and phase
of each pole, the total arnptade of all the poles, and the amplitude
of the branch-line I
bJ were printed.
The computations were coded in FORTRAN and run on an
IBM 7090 computer.
of the program.
Appendix A gives a listing and a flow chart
25
Solution of the Period Eluation
Before the value of the pole contribution can be calculated,
the values of K ( n ) must be obtained by solving numerically the period
equation
TBVAN(0 1) t-
7
> K>
This form of the period equation is valid for K
The graph iL given in Figure 4.
is solved in terms of
plotted in the graph.
e I"
A graph
139) suggests a method of solving
given by Ewing et. al. (1957, p.
this equation.
K .
The period equation
Both sides of the period equation are
Where they are equal give values of
that satisfy the period equation.
in terms of
Smay be epressed
(l,
rather than in
terms of K.
On eliminating K
.,,o
I,
o
.
r
do
The period equation is then expressed in terms of
nN(PI):
-I~iI
as
26
The number of roots for a given frequency is given by
() the smallest integer, greater than
(N
l)es
( ftK
From Figure 4 it is seen that the Nth root
in the interval between (n -
1
) r and nr.
(K(n)) lies
With this knowledge, the
equation is solved by starting with the end points of the interval
enclosing the root, then dividing it in half.
Evaluating the period
equation at the midpoint deterrmines the half of the interval that
contains the root.
This opertion i is repeated 15 times so that
the error in H (3 is less than
The method of false position is then applied once to further decrease
the error. 'The value of K
is then obtained from the value of
K(n)
The computations have all been performed with C = 1 and
Hl1.
H
This allows the results b be used for other values of CI and
if the distance unit is taken as H and the time unit is taken as
H/C .
The model is then defined by giving r, d, and C2 in terms of
these units.
In the computations the model is defined by giving r, d,
27
C and b.
As an example of how the program could be used to calculate
the response in a model in which C1 or H are not one, consider the
model,
C1
Z
km/ sec
CZ
5
km/ sec
d =
km
H = 3
km
r a 100 km
Use a distance unit of 3.
The time unit is
H
C
1
2
3 sec.
In these
units
HO I
CI= 1
C
d
-
5/2
11/3
r = 100/3
If the response were desired at w = 10 radiani sec, the computation
would be made at the value of
-
3
RESULTS AND DISCUSSION
29
Branch- Lane
As was expected, the magnitude of the branch-line integrated
was much higher at the mode cutoff frequencies than at other points of
the frequency curve.
This is shown very strikingly in Figure 5.
In
this figure as in all the amplitude curves, the cutoff frequencies
for four modes appear as vertical arrows.
In Figure 6 a plot is
presented for the same model, showing the behavior of the branchline contribution, near the cutoff of the third mode, using a smaller
difference in frequency between points.
In the derivation of the branch-line conitribution it was shown
that at the mode cutoff frequencies the branch- line falls off as 1/r
while away from the branch-line the frequency falls off as 1 /r
Table I has the value of the branch-line contribution
distances and also the ratios of the contributions.
the general formulas does indeed give 1/r and 1/r
I
b
for three
This table shows
2
depe
ence at the
limits of frequencies close to and far away from cutoff frequencies.
As a check on the computations, the magnitude of
L
has
been computed using the formulas (2. 20) and (2. 22).
This was done for the case C= 1. 5, b= .5,
da .5,
and Rz 50,
which was plotted in Figure 6.
At
= 10. 53, the formulas (2. 39) used in the computed cal-
culations gave /
/
= . 36Z7.
The cutoff frequency is w = 10. 535.
At
30
this frequency formulas (Z.22) gave j
the rapidity at which
1IV
seems satisfactory.
At w = i
=I.44381.
Considering
drops away from cutoff, this agreement
. 31,
a frequency far from cutoff
equation (2. 20) gave . 0010596, while (2. 39) gave .001060.
In formula (2. 40) the general formula was shown to go to
Pekeris' formula (2. 20) away from cutoff by using a limit process.
In the program this limiting process was not carefully programmed
so that far from cutoff frequencies the branch-line contfribution is in
error due to loss of significance in the machine computation.
This
is seenin Figures 5 and 6, by the uneveness of some of the values
having small magnitude.
is
However, the magnitude of the contribution
so small in these ranges that it was not -elt worthwhile to improve
the program for the case of small values of the parameter B.
31
TABLE I
Model
C = 1.5
b= .5,
(20)
d= .5,
(50)
Rm 50
(100)
(zW
(100)
(5o/
(100)
10.51
.6836
.Z09z
.07812
8.
68
10.53
.9756
"3627
.1670
5.
17
10.55
.8936
.3141
.1360
6.
31
10.57
.6384
.1856
.06588
9,
82
10.59
.4609
.1138
.03499
13.
25
o10.61
.337b
.0733
.02060
16.
56
10.71
.09628
.01612
. 004062
23.
97
10.81
.04207
.006791
.001700
24.
99
5.
0
25.
0
__________
__
I________________
Expected Ratios for I/r
Expected Ratios for
0
-
Cutoff frequency for the third mode is 10. 535.
-
.
1
32
Looking first at the modes individually we will consider
The term
several parts of the expression for particle velocity.
G(K( n ) in equation (Z. 9) has been discussed by Pekeris (1948).
He noted that for each mode this factor is 0 at its low frequency
cutoff. This occurs because at the low frequency cutoff of the
nth mode
(n- 1l/)
(n=
H
makes the term tan
r
iH in the denominator infinitely large.
This
behavior is seen in Figures 7, 8, and 9 where the amnpiare of each
mode is 0 at cutoff; then rises rapidly as H 81 goes from (n-i )
toward nr with increasing frequency.
(K(n)) was made for the first 5
In Figure 10 a plot of
modes.
9
Notice in each case
its cutoff value of (n--)
.
i( K( n ) ) initially rises rapidly from
The second derivative of
respect to w is negative; the slope of the
increasing w.
nr
As H
1
curve decreases for
the value of G(K(n)
/
The factor 1/ K( n ) causes the mode amplitude to fall off slowly as
the frequency increases.
V
(3.1)
1Cj,
(3.2)
33
This is expanded by the binomial theorem as
2
Kn)=
C
_+
1to
2
W
higher orde r in
Therefore
(3.3)
\fW
when w >>
.
The frequency where this approximation is valid
modes is plotted in Figure I1.
K(n)= t.
(n)
K(n ) is
seen to go towards the line
The fact that phase velocity curves go to C1 for large w
reflects this.
A more important factor in determining the amplitude of the
mode is the factor sin
d.
This gives the amount that the mode
is excited by a source at depth d. In Figure 9 a plot has been mnade
of the amplitude versus frequency for the first six modes.
depth in the case is .5.
The source
The amplitudes of the second and fourth
modes start to fall off faster than that of the first and third modes.
occurs because
P 1d goes towards
2Tr
This
(. 5) = nr for the even modes
while going toward ( 2 n+l) r (. 5) = (n + l/2) r
for the odd modes,
Several other plots were made to illustrate the behavior of this factor
in controlling the mode amplitude.
In Figure 7 the amplitude of the
first mode was plotted against frequency for the three source depths
d= . 25, . 5, . 75.
At the low frequency part of the mode the deepest
source gives the largest amplitude.
As the frequency increases the
source.near the bottom become less important.
The curve for the
source at d = . 5 gives the largest amplitude at high frequencies.
The
variance of the response with depth for the first two modes is examined more closely in F;igure 12.
In this plot the amplitude
of the mode is plotted against depth of source.
This has been done
for four frequencies in the range between the cutoff frequencies
of the second and third mode.
For both modes the behavior of the
amplitude is similar at each of the four frequencies.
The amplitude
for the first mode varies fairly smoothly with depth and has its
maximum value near d = . 65.
shallow source.
d.
The amplitude is small for a
The second mode has a small amplitude for small
This amplitude reaches a maximum near .33 then falls to a
value of 0 between . 6 and .7.
occurs is that where
FPd is equal to r.
As an example, at
(see Figure 10).
be 0 at
The depth at which the minimum
= 7. 9,
for the second mode is S.
Then the amplitude of the second mode should
d'=
5
/
.63
This apparently occurs as seen in Figure 12.
At depths greater than this minimum, the amplitude rises as
the source is moved toward the bottom of the layer.
To see if important changes occur when the velocity C2 in the
lower layer is varied, the value of the amplitudes of the first two
modes versus depth of source were plotted in Figure 13.
By com-
paring this curve for which C z 2 with Figure 12 in which C t
1.5,
it is seen that the behavior of the mode amplitudes are almost identical.
In this figure the mode amplitudes for three depths are plotted
for a density contrast of b = .8 with C a 2.
These points seem to
indicate that this model has the same behavior as the models with
density contrast of .5.
The phase of each mode contribution is determined by
i
r
4 i If the first M modes are present with amplitudes
Ai (w) and phase of e
the square of the amplitude of their
sum is given by
A 1
4.II
A/Ycs(-&)
1)
K -VL
.)
(3.5)
For Mr2, this formula gives for the square of the amplitude
A
+ A2 + AIA 2 cos (
-
(3.6)
2
The maximum value then taken is
(A
+ A2 )
(3.7)
occurring when the modes are in phase.
The minimum is
(A i - A2 )
(3. 8)
which occurs when the modes are r radians out of phase.
If the
phase difference 0 - 92 is changing fairly uniformly with
a
then
the difference between frequencies at which successive maximuma
of the amplitude of the sum occurs is
AW=
2V / d(9 - 02
dw
(3.9)
This alternating constructive and destructive interaction between
the modes is the most striking feature of the curves computed.
In
the range above the low frequency cutoff of the second mode but below
the cutoff of the third mode this behavior is seen to match the curves
almost exactly.
(see figures 14 through 24).
The phase of the modes
are given by
e i
LK )r + i 4J
(3. 10)
The rate of change is given by
d(K( 1
r
which is proportional to r.
KZ))
(3. 11)
dw
It would then be expected that the
frequency difference wald change much faster with increasing frequency.
This is born out by comparing Figures 15, 17, and 18.
The same model and source depth are used in all three curves.
The horizontal distances from the source are 20, 50, and 100 respectively.
The frequency difference between success maxima are
approximately 2. 20 for distance 20, 90 for distance 50, and 45 for
distance 100.
These differences are roughly inversely proportional
to the horizontal distance.
It is noticed that the frequency difference
between successive minima or maxima increases with frequency.
dK(n)
This occurs because as the frequency increases the value of (see Figure 11).
frequency.
Therefore
d(K )
dK
)
decreased with
= I
Plotted in Figure 19 is a curve for a small density contrast
of d= . 8.
This curve is quite similar to the curve in Figure 15.
However, the maxima are slightly further apart.
In Figure 20 the velocity contrast in the model is large;
CZ 2. 5.
Here the distance between maxima is very small when
compared to Figure 17.
Both of these curves are for the amplitude
response at a distance 20.
By a knowledge of the size of the mode amplitudes it is
possible to make some surmise about the depth of source. However,
the modes overlap in the frequency domain, so the magnitude of the
contribution to Jw)) from the different modes cannot be measured
separately.
Still the low excitation of a mode does show up in a
curve of amplitude of I ql(wl versus frequency.
As was discussed,
the amplitude of the second mode was very dependent on depth
(see Figure 12).
If the source depth is near the depth at which
the second mode is not excited then the magnitude of
t (W)does
not start to show fluctuating variation in amplitude as the frequency
rises above the cutoff of the second ciode.
In Figure 22 the oscilla-
tions of the amplitude are small at frequencies above the cutoff for
the second mode.
The source in this case is at depth d = . 65. At
this depth the second mode is not greatly excited until well above
the cutoff of the second mode, which is w = 6. 32, for the velocity
of C 2 = 1. 5,
As the frequency gets higher the second mode does
start to contribute and the amplitude starts to oscillate.
If the amplitude of the first two modes are about equal, as would
occur if the source were at a depth near . 54 or . 70, then the modes
would cancel each other ~am out of phase.
These would then be
values of w for which the amplitude of their sum would be very
small.
In Figure 22 a plot made for depth . 55 shows this to
occur.
For very shallow source depth the amplitude of the first mode
is small compared to that of the second.
Figure 12 shows that for a
depth of . I the ratio of the second mode's amplitude to that of the first
is approximately 4.
For such a shallow source
greatly above the curoff of the second mode.
) /I
would increase
The difference between
successive maximum and minimums would be approximately twice
the magnitude of the first mode.
In Figure 24 the magnitude of
has been plotted.
From
the computer output the first mode is seen to have amplitude of . 12.
Unfortunately, a run was not made that covered the region below the
second mode cutoff.
However, considering other results, the curve
was drawn in for the section between w = 5. 5 and w = 6. 1. The first
40
mode amplitude of. 12 is seen to be very close to half the difference
between successive maxima and minima.
The amplitude of the first
mode changes very slowly at frequeny well above its cutoff.
The first
mode amplitude, then, would have about the same value below the
cutoff frequency of the second mode as it has in the frequency range
where the second mode is also present.
Considering this a shallow
source could be detected by a sudden jump from low amplitude to a
much higher amplitude with fluctuations between maximum and minimurm of about twice the amplitude of the first mode.
Three plots have been made for the frequency %rangeabove the
cutoff frequency for the third mode.
25, 26, and 27.
These are presented in Figures
The three magnitude curves show an irregular pat-
tern of highs and lows.
There is a certain regularity between the
local maxima of the curves, but the size of the maxima varies.
It
is probable that reflects the behavior that formula (3. 5) predicts.
The maximum that could be obtained would be the sum of the amplitudes of the three modes.
were in phase.
This would occur if all three modes
By comparing Figures 26 and 27 it is observed that
the frequency difference between the maximat is smaller for the
larger value of r, as was true in the frequency range where only
two modes were present.
41
APPENDIX
A
FLOW CHART AND
PROGRAM 1USTING
'
42
s TA RT
AD
,CON*
I
*
)DEL
iLST
Coq.puTe
phnch
o SN
CNIIiTrib.i
l
S
Coer e
!te'rch
SCor4ri 'uT'er
S0
o1- oiA
Co
co
vuTo
o rT os
f
cF pT prRTs
pNdga Ou
CompTe T-t-J
Pol
CJTvl bT
for
Co'
t/on"'
-
C!7-Te ToTl
bcVa NC-h
Pro
computations
p,.;T
To t 1
ro
doo
MAiN PROGRAMV
0110) ,W(juO) 9Ai4I .(ju) ,v4(lU020)9,PiP(10O) ,
DIMEN61ON
IRPOLE(100,)20),Fr-PUE(lU0,2.U), PQLEMv(2U),PLC.G(1L0U,2O),Rb(1.OU),
2FB(10O) ,XK1(lUO),BivV(lU'2Q),RP(2U),FP(2O)PULi~i(2O)
I
DiMEN,;,ION 6RNCri( iv
I
1001
ONL=I.*Uo)
N1=4
N2=2
PIE =3. 1415926
TPIE=2**PIE
R EAD I NP UT FAP E NlLIOU tC I C,0 EN H
100
FORMAT(7F10.7)
lol
RE\D INPUT TAPt. i~lvlOlqNwqNR'NU)D
FORMAT(515)
XMU=.-QRTF( 1.-(C1/L2)**2)
i
2
3
4
5
k( 10)
vkRi<( I),.~wti<( 1
9Y( L)qEFCN( I)
9vLERO DEL.W
READ INPUT TAPE. N.LIUi00(RJ)~jj=iNRud)
NR=NRL)
IF(NDD) 3vk,3
vii v100, (UJ) ,.J,i NUL.)
READ INPUT TAP.
NU=NDD
DU 5 JW=1,NW
Lw
W(JW)W.t.RO+FLUA[F(JW-I)*U.JW=IPNW
DO 9
WW=W (JW)
CALL ROOT$(WWgrUENc1,C2,JWNP XNPIL)
XK1 (JW)=1AW/C1
NPO=NP (JW)
DO 8 JP1,tNPD
CS=COSF ( bl*H)
POLEG(JWJP)=B1**2* H/(DE*.'Qi-TF(XK(JWJP)))
POLEG(JWJP)=PULL(JW9JP)l 5e001326
BIW (JWgJp)=B1
6
9
C04T INUE
CONTINUE
DO 50 JR=19NR
SQR=,'QR-TF(R(JR))
DO 23 JW=19NW
XK2=W (JW) /C2
NPL)NP (JW)
DO 22 JP=19NPD
PHASE=X(JWJP)*k(JR)-PIL/4o
PStEJWJP)=MODF( PHA$E,-TPI'r')_____
RPOLE (JWtJP) =PUL&3( JW JP ) C~j. F (PHASL) /$Qwk
22
L.
6EC
CONTINUE
TO EVAL bRANCm
A=IXKI(JW)*XMU* CUQDF(XKI(Jw)*AMU*Hi) )-k*
rTF (Xi,
14+
GFAC=4**DEN*XK*AviU*,-)
I.
CG=GFAC*XI*EXP'4HAI*XK2*<(Jk))
S=XK1 (JWfl*R(JR)
I
Y=-XI*A/b
(Jw(HPI E*k (VJR)-kX&I.(JW)
k*
SQRY=SQRTF(Y*S)*XI
XX=SQRY ( 1)
YY=SQRY (2)
CALL CERR(XXYYo.RR1,E'RR2,,NO)
ERR (1)=ERR1
ERR (2)=ERR2
E FCN=ONE-EXPF(-S*Y)*ERR
FI=EFCZN-QNr'_
I
QY=6QRTF (Y)
I
F2=QY*P E*F 1*XPF (Y*-z)
IF3=(Q'QRTF(PIE/6)iF2)/ (XI*b)
I BRNCH =F3*CG
RB (JW) =5 RNCH ( 1.)
FB (JW) =BRNCH (2)
GO TO 23
CFOR 6=0
ia REL=2 *.*DEfi*XK2
)*R('JR[**Z*XMU
DEiNNXK1IJW
L)LNN
RL3(JW)=k-EAL*C.F(Pt/2+r 2*k(JR)/
FB(JW)=-REAL*SINF(PI/2+K.*rd JR))/ DLNN
23
-'-ON TINU L
D0 40 JD=J.PND
NZ29112 'C.1 ,C2ZjDEN 9H t R(JR)
WRITE OUTPUT TAPL
110
32
U(JU))
I 0*4,4X6HDEPTH=Fl10.4//I)
WRITE OUTPUT TAPE N2 f131
FORMAT(IH F8*597X6F15*6)
DO 39 JW1'sNW
TRP=0.
TFP=O*
NPU=NP ( JW)
DO032 JPitNPD
DFP=,.'INF( 1W (JvwjP )*D( J0)
RP (JP)I=RPOLE(JUw jP )*Dt-P
FP (JP)I=FPULE (J0# JH )*UFP
(JP) ; *2+FP (~)*~
POLM ( JP) =.$w RTF (RH"%
TRP=TRPi-RP (JP)
TFP=TFP+FP(JP)
TOTP=SQRTF (TRP**, rTFP*,*2)
DFt3=bINF!XK(JW)*XMU*oiJD))
TR3=RB (JW)I*DFB
TFB=Fb (JW) *DFB
BRMAG=SQRTF (TRb**?-+TF6**2)
XMAG=6QNTF((TRP+fti~B)* ',2+(T'FP+iFb)**e-)
WRITE OUTPUT TAPE. N2,119vj4,)(PLv(JP),P6E(JWJP)-JP=iNPD)
TOT (K) =TOTP
BR (K)= BRMAG
XMG (K)=XMAG
:3) CONTINUEN42'112'Ci<C2'DENqH iR(JR)qD(JD)
WRITE OUTPUT TAPE
N29135
OUTPUT
TAPE
WRITE
DO 42 K1,PNW
42
WRITE OUTPUT TAPE N2,i1Uv4(s),TOT(NW3,R(KXG(K)
21,2.X6-iNU0E 3i2XbHMODE
FO' MAT(9HOANG9 Fk.11X6HrAQUE ii2X6riiMQDL0
.L31
1HMODE 512X6HMOu)L 6)
X5HTUTAL/)
135
FORMAT(T)H0ANGo Ft-.jiXiIHWkQTAL PU)L,-%~HBkACH
412A6
45
-40
50
CONTINUE
CONTINUE
GO TO 1001
FNU
TOTAL
4 ft
SUbROUTINERO
D IIAEN6 ION XK( lUO,2O),NP(I UO) YR(I0OU 20)
PE-DF(6ET)=8T
/zR(iK-tL*2)+TAg4F(6ET*H-)*WEN
SK1XKi**2
NP (JW) m*RTF
NPO=NP (JW)
DO 9 JP=IsNPD
3
i-
/iL499~99
.NiT=0
RH=FL0ATF(JP)*PIL
kL=RH-*49999*PI
RHMAX=SQRT (SKi-z)k2)
10
RH=RHMAX-OOOO.L
il
CQNTINUE.
DO 36 IIER=1i1it)
RT=*5*(RH+RL)
IF(PIERDF(RT))
31
36
8
3O93Ov3i
GO TO 36
RH=RT
CONTINUt.
PERL=PERD (RL)
PERi-1PERDF (RH)
SLJAP=(PERH-PERL) / (H-RL)
TEM~4PR =Ru-PE~R L/ - LU AP
YR(JWPJP)=PERDtr([ fvPR)
XK ( J W PJP = OR T (~- r MPkR
Rt TURN
LIN L
T0T AL
L 2 - T SXff MIJ
9
XICINICICTV
MODE L
VACUu M
d
14
,
gIOUYCE
-
H
C2
C2 )C,
FI G URE
--
-
I ~
--1-
K PL KN E
FIGURE 2
- ----
'
49
DIAGRAM TO EXPLAIN
REFRACTION PHENOMENA
C-
K
D'
L
3
FIruRF.
__ __ ____.
~_-___
_,___
GRAPH ILLUSTRATING SOLUTION
OF THE PERIOD
EQUATION
i
I
FIGURE
4
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IIU-~
-- UI
5.0
;
1-=
-,,....,,_..,----r~
+
5-
-
T I-T-;.-T
4"__- "! ! -4-1
--i-::.
_
FIGURE -5
-Magmtude of theb-fanch-
T
.'
_
__
T
__
line versus frequ-nc
50-,
..
I
T
il
- -:
ii-----I-
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FORM
1
H
40 MASS. AVE, CAMBRIDGE, MASS.
TECHNOLOGY STORE, H. C. S.
'ij
Magitude
of
vs.
.3
,
S
frequency.
1 5, b= 5,
2
4'
s 5 ,
25
I,
I
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1
1
7 77. . ..
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TECHNOLOGY STORE, H. C. S.
FIGURE
m
agnitude of
C2
CA__E
*1OMASS. AVE.,I___
STRE H._C. S.
___ TEHOLG ___
I_ H~
FOR
FORM 1 H
MAS
40 MASS. AVE., CAMBRIDGE, MASS
15
Yvs,
frequency
1. 5, b=. 5, r= 50, d= .5'
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40 YASS. AVE., CAMI~RI03E, MASS.
STORE, H. C. S.
TECHNOLOGY
TECHNOLOGY STORE, H. C. S.
FORM I1 H
40 VASS. AVE., CAMBRIDGE, MASS.
:
--
,.
. ',,I
RJ
'
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FIGURE 16
Magnitude of
Wvs.
frequency
= 1.5, b=.5, r= 50, d= .75
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FOR
FORM 2 H
.40MSAECMRDE
.C
EHOOYSOE
2
TECHNOLOGY STORE, H. C. S.
40 MASS. AVE., CAMBRIDGE, MASS.
-mcr
t
vs~egency,
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FORM I H
40 MASS
TECHNOLOGY STORE, H. C. S
AVE, CAMBRIDGE, MASS.
A3
l~~t:I.:
1'."' :; '
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17
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4
FIGURE
1
t
18
t
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Magnitude o
vs. frequency
I -::-i7i-C = 1.5, b= 5, r= 100, d= .5
2
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FORM 2 H
TECHNOLOGY STORE, H. C. S.
40 MASS. AVE., CAMBRIDGE, MASS
,,
7.4
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40 MASS. AVE., CAMBRIDGE, MASS.
TECHNCLOGY STORE, H. C. S.
FORM 2 H
1
T-1RE 7-11
-
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1
Magnitude of
vs. freuenc
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40 MASS. AVE., CAMBRIDGE, MASS.
TECHNOLOGY STORE, H. C. S.
!: :: F IG U R E .
V.
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't~it
K
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II
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TECHNOCLOGY STORE, H. C. S,
I'
40 MASS. AVE., CAMBRIDGE, MASS.
T
12,~
Magnitude,6
off 1 s
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t
' . . .: :' ...+
I+
I
71
''1
' +
tt
""
...
. ... .; + . .. ''4
. . .i. ...
',
j
it Ii
tI
t i''f
t
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+ . '.,
I :
.
...
.
| <,,
+ i1 ' t4 ; :+ : '!,4.
': + ++:," t
"
. .. ..
_
_
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_
_
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'++
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.
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.
.
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.
+
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,. . . ... +. + . . [.
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4
.
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+
+. ' .
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.
+
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i t'
4
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73
Ii
FIGURE
Magnitude of
C
I k .1
'f
vs. frequency
1.5, b= .5,
'
I
25 ,
"
1
r= 100, d= .75
'"
,
.
•
.
.
I
2
,A\ .
..
WI
,-*.
+
:/
/
rV
,-
.
..
..
358-14
10 X 10 TO THE CM.
K'UFLL
•
LSS
CO.
V:
---__-_--------^----r
-----
-7---1--7- -- -~--------~*-I-----
4'
lip'
II~
S
:
-
ga
itw' .....e&
. IiiJ.
uec
.
-"
r 100 4= .. 5
Verical arrqw-indicat
b
1.~
1
I
4
{
I..
rnde c toff
/9'
.-.---1.
I
t
I,
I
I
I
~
..
I
I'
i
.
I
...
..
.
. . ......
I.
~.
'-~1
~1
.
t.
-
I
i
I,
lo
,
/
.i/
,7
,4/
F1
-
'/
//.P/
I//
121
--
./
___ill_
___
, /
.3-f
~--~----~
-~-I- -
,e7/
,?/
, i/
12(/
#I
,.1
.. i
11_
0 OWN__FIFI
Kiv.:17
:1 IL
i7
K1FIGi
-7-
Magnitudeo
iLd'±calar4,
:4:;.
T
7-t
I
Z7
LJRE
1*
T
*1
71
*I..
I,
-~
.1
Ll
:1
:1
-77i
It'.31
1015'1
10,7 1
73
BIBIOGRAPHY
Jardetzky, W. S., and Press, F, Elastic
Waves in Layered Media, 1957, McGraw-Hill.
1. Ewing, W. M.,
2. Officer, C. B., 'The Refraction Arrival in Water Covered
Areas", Geophysics, Vol. 18, pp. 805-819, 1953.
3. Officer, C. B., Introduction to the Theory of Sound Transmission,
McGraw-Hill, 1958.
4. Pekeris, C. L. "Theory of Propagation of Explosive Sound in
Shallow Water found in Propagation of Sound in the Ocean,
Geol. Soc. Amer. Mom. 27, 1948.