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Calhoun: The NPS Institutional Archive
Theses and Dissertations Thesis Collection
1968-06
Spectral analyses of waves and associated bottom
pressures off Point Sur, California
Steele, Boyden Tracy
Monterey, California. Naval Postgraduate School
http://hdl.handle.net/10945/13082
SPECTRAL ANALYSES OF WAVES AND ASSOCIATED BOTTOM PRESSURES
OFF POINT SUR 9CALIFORNIA
by
Boyden Tracy SteeleLieutenant, United States Navy
B. S., Naval Academy, 1961
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN OCEANOGRAPHY
from the
NAVAL POSTGRADUATE SCHOOLJune 1968
ABSTRACT
A time series of pressure variations was obtained by digitizing
analogue records of bottom pressure, recorded at Pt. Sur, California,
during the periods November, December, and January, 1947-1948, and
June, July, and August, 1949. The power spectrum of the pressure vari-
ations was found via the amplitude spectra obtained by means of the
fast Fourier transform. This spectrum was then smoothed by averaging
over several frequency bands and confidence limits were assigned.
Surface wave power spectra were inferred from the pressure power
spectra and significant wave heights were estimated from these, after
applying a procedure designed to remove noise.
Values of significant height on the bottom and at the surface
are displayed in tabular form and plotted versus time for the selected
periods January, 1948, and August, 1949. The spectral patterns of wave
activity show the transient nature of the energy due to storm swell
and local winds.
TABLE OF CONTENTS
PAGE
11
13
14
15
19
23
31
32
1 Components 33for Selected Periods
II. Graphic Display of the Spectrum for 58Selected Periods
III. Summaries of H3B and H3S 79
SECTION
I. Objectives
II. Related Work
III. Data Used in This Study
IV. Spectral Analysis
V. Performing the Analysis
VI. Discussion of Results
VII. Conclusions
Bibliography
Appendices
I. Tabular Display of Speci
LIST OF ILLUSTRATIONS
FIGURE PAGE
1. B. , Representing Ae in Band Width m Af 17
2. Esterline Angus Co., Inc. Chart 4313 20
3. Steplike Approximations to the Recorded Trace 19
4. Geometric Relations for Conversion to t,R9 21
Co-ordinates
5. Comparison of Height - Period and Spectral 22Values of H3S for January, 1948
6. H3S and H3B vs. Time for January, 1948 26
7. H3S and H3B vs. Time for August, 1949 27
8. Frequency (CPS) vs. Time (Days) with Contours 28of Equal Power Density
9. Frequency (CPS) vs. Time (Days) with Contours 29of Equal Power Density
Symbol
TABLE OF SYMBOLS AND ABBREVIATIONS
Definition Unit
max
t
A t
A f
E
\M
HARM
A(f.) = A.J J
f .= j Af
j
t, = k A tk
B.J
m
sCfj)
A x
A y
the maximum frequency about which the
analysis can gain information
time
sampling interval
1/ At, elementary spectral band width
energy
pressure amplitude
number of observations
IBM computer subprogram, a fast Fouriertransform
complex spectral amplitude
frequency of Fourier amplitude
time of the pressure data point
Energy in band width A f
arbitrary number of band widths
average spectral amplitude in bandwidth m A f
approximate class frequency
/o 3 m 5m m- m/2, —£— , -y, .«<>,M- —
average energy density in band widthm Af
incremental horizontal value of the
digitizer
incremental vertical value of thedigitizer
sec
.
sec.
sec.
sec.
2ft.
ft.
-1
-1
ft.
sec
sec.
ft.2
ft
-1
-1sec
.
ft. -sec
summation of the A x for the i
position in time
th
ft
ft.
ft.
Symbol Definition Unit
iy. summation of the ^ y for the i ft
position in time
R radius of the ordinate on the wave ft.
record
RO measure of arc length ft.
H3B significant height of pressure amplitudesmeasured at the recorder (in this case onthe bottom)
H3S significant height of the surface waves
PCT Pacific Coast Time date/hours/minutes
2var variance ft.
N degrees of freedom
ACKNOWLEDGEMENTS
The author wishes to express appreciation for the advice and
encouragement of Professor J« B. Wickham under whose direction this
study was carried out.
In addition, the author is indebted to Lieutenant Commander G. M.
Griswold, USN, and Lieutenant No L. Perkins, USN, of the Fleet
Numerical Weather Facility staff, for assistance in programming and
for suggestions which were incorporated into this study.
Appreciation is also expressed to Captain P. M. Wolff, USN,
Officer in Charge, Fleet Numerical Weather Facility, who made available
the CDC 6500 computer used in part for this study.
I. Objectives
The randomness of the sea has puzzled investigators for centuries.
Before forecasts of the behavior of the sea can be made, there must
exist a more complete knowledge of the many-faceted energy relationships
between the air and the sea. Most modern authorities feel that the
true behavior of the sea can be best approximated by considering the
behavior of each of its spectral components."
In addition, with the increasing use of major structures situated
on the sea floor, it becomes more important to know about the possibil-
ity of pulsating water motions, with particular frequencies, and changes
in the magnitude of the pulsations. The knowledge of bottom pressure
activity is of extreme importance in mine warfare. In view of these
and other needs, this paper has the following objectives;
A. To display the spectral patterns of energy density of waves
and associated bottom pressures versus time for the analogue
wave records recorded at Pt. Sur, California, during the
periods:
1. November, December, and January, 1947-48, (winter), and,
2. June, July, and August, 1949, (summer),,
The spectra are to be obtained by digitizing the analogue data
and performing a spectrum analysis on the resulting time series
B. To compare the results from the summer and winter periods;
C. To provide background data for other studies, such as;
1. Consideration of effects of shoaling and refraction in
obtaining a deep water spectrum;
2. Wave energy attenuation studies;
3. Locating of sources of wave activity reaching the Pt. Sur
area; and,
11
4. Examining the analysis for peculiarities such as favored
frequencies that might indicate the influence of local
topography
.
12
II. Related Work
Munk, Snodgrass, and Miller (1) displayed contours of equal power
density, E(f,t) on a frequency time plot and used the spectral peaks'
arrival times to estimate distances to the areas of wave generation.
They confirmed the relationship by determining direction and distance
in a separate analysis-
The use of a digitizer to convert the analogue data of the wave
records is discussed by Moskowitz, Pierson, and Mehr (2). Their
records were part of the data used in ""the problem of developing
numerical wave forecasting procedures for the North Atlantic Ocean",
now functioning as a part of ASWEPS.
13
III. Data Used in This Study
The wave pressure records used for this study were recorded by a
University of California Mark III bottom pressure recorder, anchored
on the bottom in sixty-eight feet of water, measured from MLLW. A
comprehensive discussion of bottom pressure recorders and the manner
in which they act as a low pass filter was presented by Munk (3).
The major advantage of choosing these somewhat dated records is that
they provide a long and nearly continuous history from which patterns
of wave activity may be shown. Johnson (4) has also published the re-
sults of another kind of analysis of these same records.
14
IV. Spectral Analysis
A time series of pressure variations is analyzed using the fast
Fourier transform, a technique developed by Cooley and Tukey (5), in-
corporated in the IBM computer subroutine HARM to obtain the amplitude
spectrum. The Fourier amplitudes are then used to find the power spectrum
which is smoothed by averaging over several frequency bands. The entire
procedure and several associated processes have been incorporated into a
computer program written in FORTRAN IV.
A basic step in the analysis is selection of a suitable sample inter-
val. For this study the primary interest was in wind waves; and, both
theory and observation indicate that, due to filtering, very little
activity at depths of sixty-eight feet is found in waves of frequencies
-1greater than 0.25 sec. . The appropriate sampling interval for this
frequency is A t = 2 seconds, given by
1(1)max 2 At
The fast Fourier transform requires 2 (n a positive integer)
discrete data points for each analysis. With a 2-second sampling inter-
val, the largest n possible was 9 (512 data points) equivalent to 17
minutes and 4 seconds,
As a check on the spectral analysis, an independent calculation
of the total energy was made directly from the pressure amplitudes, by
N 2
E = 2 var(Ak
) = 2/N^ Afc
(2)
where, A, = pressure amplitude, and,
M = 512 is the number of observations of A, .
The Fourier series analysis, HARM, produces a complex amplitude
spectrum which represents the points of the time series by the sum of
15
the same number of discrete frequencies. The complex amplitudes are
given by
512 -2irf.tA(f .) = 2 A(t, )e J
, (3)J k=l
k
where, denotes Fourier transformation,
f . = j A f is frequency, and,
t, = k At is the time of the pressure data points
The energy in each band of width A f is proportional to
2B. = AE. = 2J J
A.J
(4)
where the factor of two enters because of the symmetry of A. about
j = 256, and the total energy is proportional to
s= 2512
A.
j=l J
2?S6
= 1 B • (5)
Because the sea surface disturbance is a random variable, the
Aspectral values A(f .) are also random variables. Thus equation (3)
holds only for averaging over many realizations. The energy is also
a random variable and equations (4) and (5) are only estimates of the
respective energies. The confidence limits for these estimates can
be determined in the manner presented by Pierson and Marks (6) and
discussed here below.
The energy in each elementary band width has been determined from
only two independent bits of information, and the estimates are distri-
buted according to the ~Y' distribution with two degrees of freedom.
By averaging a number, say m, of elementary bands of width A f (see
figure 1) , one obtains an estimate of the energy which is more stable
than the estimates of the contributing narrow bands.
16
B r
T
.—.*—-— * .. ,i l ! I .mi.I —i—
X
— ... .... 1ii.» i,- M»l
BjH-1 Bj+3 Bjt5 Bj+7 B^ t ^B^-f Bj+2 Bj+4- Bjt£ E^t 8
j^+m/2 m 3 m 5 m1/m ^ B 2' 2 ' 2 ' '"
2j-l-m/2+1 J
.M- 5
Figure 1. B. , representing A E in band width m Af
2These estimates have a Xi distribution with 2m degrees of freedom.
Confidence limits may be assigned utilizing this distribution of B .
Band averaging was done over 10 and 16 B. The total energy per band
of width m A f was computed by
AE(fx)
j-l+m/21/m 2 B;
j-l-m/2+1m 5"j (6)
where, f. is considered the center frequency of the band, and,
m - number of bands.
The cumulative energy was obtained by summing over m bands,
M - m/i M - MA
E - ]>AE(f ) - ^B . (7)
The average energy density in the pressure variation in the band
width m A f was defined as
S^) a AECf^/m Af, (8)
The average energy density at the surface was then determined by
8§(£
1) - SCf^/r, (9)
where K is the well known pressure response factor,
1K
cosh 2 XT d/L ' (10)
17
where, d = depth of water (meter assumed on sea floor),
and,
L = wave length at the depth d.
18
V. Performing the Analysis
The wave records used in this project were analogue records of
pressure variations recorded on Esterline Angus Co., Inc. chart 4313,
shown in figure 2, which has curvilinear co-ordinates (t,R9). The
records were digitized using the Calma Co. model 480 digitizer, which
recorded incremental pairs, in the order, Ax and Ay (of the cartesian
co-ordinates of the pressure curves), every .01 inch of stylus travel.
This resulted in the wave form being approximated by a step - function,
as shown in figure 3, with a secondary wave form including very high
frequency energy being superimposed on the i ecorded wave form. This
is a source of error and will be discussed further in section VI.
Figure 3. Steplike approximation to the recorded trace.
The digitizer recorded incremental values and it was therefore
necessary to form the cartesian co-ordinates by summation within the
computer. These cartesian co-ordinates were then converted to the ap-
proximately correct t,R9 co-ordinates by employing geometric relation-
ships.
Computation of the correct value of t. (the time axis) was critical
to the conversion. On the time scale, 0.1 inch of record was equivalent
to the sample interval of 2_seconds. The summation of Ax, x , repre-i
senting the horizontal travel, was converted to the t,R0 co-ordinates
-L2.
kDE nusa the Esterline-Angus Co., Inc.,
CO
FIGURE 2. ESTERLINE ANGUS CO., INC. CHART 4313
20
by,
t . = x . + R - V R2
- y2
i i *Jx
(11)
where, x. is the time scale in equally spaced increments of
At on the cartesian scale,
y. is the ordinate on the cartesian scale,1
t. is the time in t,R9 co-ordinates, and,1
R is the constant radius for arcs of constant t .
These are shown in figure 4.
• Y = f(RG)
0. = tan"1
yj \jR2
- y2
(12)
Figure 4. Geometric relations for conversion to t,R0co-ordinates
Because 2-second intervals read by the digitizer on the rectangular
co-ordinates do not in general correspond to the same interval in curvi-
linear co-ordinates, the following scheme was used to approximate even
time steps. The cartesian amplitudes used to form the time series were
extracted from the record when the value of t exceeded or closely ap-l
proximated a separately incremented test value which represented the
true position in time; i.e.,
21
if t - test or if test - t - .001, then
y. = y.
where, i is the position in the record, and,
j is the position in the time series.
The cartesian amplitudes were then converted to arc length measured
in scale units; by
y = R9 (13)
The value of R used in these calculations was determined by graphi-
cal construction. By repeated measurement, with the digitizer, the
radius was determined to be 4.36 + .02 inches or approximately 490 scale
units.
22
VI. Discussion of Results
One feature of the surface spectra was the characteristic "rapid
increase in the high frequencies due to the presence of noise ..."
as described by Moskowitz, et. al. (2). As indicated in section V,
the digitizing procedure superimposed a very high frequency waveform
on the recorded trace, the energy being aliased eventually into the
frequencies used in the analysis.
An effort was made to estimate the contribution of this to the
total energy. The maximum reading error of the digitizer was .01
inch on the recording or .0167 feet in the pressure amplitude. As-
suming that the errors have a uniform distribution (where each event
is equally likely) between zero and 0.0167 and that the errors in ampli-
tude are not correlated with the amplitudes themselves, the contribution
of noise due to digitizing appears to be less than one per cent, usually
much less. However, even a small amount of noise is critical at the
higher frequencies, since the energy there is also very small. Moreover,
there was considerable additional noise not associated with digitizing
as spectral results of non-digitized data also displayed the rise in
high frequencies.
In the absence of more precise information about the total noise
contribution, it was assumed that most of the energy appearing in the
spectra at frequencies greater than 0.18 sec. was noise. Much of
this was then arbitrarily removed in a manner analogous to that employed
by Moskowitz et. al. (2), wherein it was assumed that after aliasing
and smoothing over a number of spectral bands, the noise was white. The
resultant determination of noise was subtracted from all spectral bands
thus giving spectra with reasonable energy in the higher frequencies.
23
All values above 0.25 sec.' indicated on computer printouts are beyond
the range of the sampling interval and are spurious.
The total energy computed by the sum of the Fourier amplitudes
agreed well with that found from the variance of the pressure amplitudes.
The results of band averaging over 16 and 10 elementary band widths were
essentially the same (in part due to the smoothing to eliminate noise);
however, the greater detail expected from averaging over fewer bands was
not entirely lost as can be seen from the selected samples in Appendices
I and II.
The corrected values of bottom pressure spectra and inferred surface
spectra were used to calculate the respective significant heights. The
significant heights at the surface, for the month of January, 1948, are
compared with values obtained by Johnson (4) in figure 5. Figures 6
and 7 show values of significant height on the bottom and at the surface
plotted versus time for the selected periods January, 1948, and August,
1949. A tabular display of significant heights for all periods of this
investigation are presented in Appendix III.
An attempt was made to determine distance to the generating area of
the swell by observing the time lags between spectral peaks of increas-
ing frequency, as done by Munk et. al. (1). Figures 8 and 9 are contours
of equal power density, drawn for the characteristic of the logarithm
of the spectral component of energy density (N = 32) at the surface.
The very steep slopes are indicative of a relatively close storm area.
This would be expected for the winter months. The summer months show
no significant patterns either, perhaps due to the smoothing. In Munk's
investigation a much more sensitive pressure recorder was used (vibra-
tron developed by F. E. Snodgrass) . The great amount of correlated
24
?5
5 10 (5 20
TIME IN DAYS
FIGURE 6 H3S AND H3B VS. T I ME F0 R JANUARY, |94$
26
HE
I
GHT
N
FEET
15 20TIME IN DAYS
25
FIGURE 7. H3SANDH3B VS. TIME FOR AUGUST, 1949
27
ooooo
22 22 2 2
18 20 25JULY
JO 04 09 14
AUGUST24 29 31
FIGURE 8. FREQUENCY (CPS) VS, TIMECDAYS) WITH CONTOURS OF EQUAL POWER DENSITY
NUMBERS ON CONTOURS ARE CHARACTERISTICS OF LN |Q S(f^
?R
.21 I
,227j~
.24 2 i-
-1 -1°-i?0
3 32 23 3 3 33 3332 2
22 24 24
DECEMBER03 08 3 I 8
JANUARY23 28
FiGURE 9. FREQUENCY (£PS) VS. HMECDAYS) WIT H CONTOURS OF EQUAL POWER DENS|
i y
NUT/3ERS ON CONTOURS ARE CHARACTERISTICS OF LN)0 S(f.).
20
detail upon which Munk's hypothesis was based occurred between 0.40
and 0.80 sec. " and therefore is beyond the range of this analysis.
The ambiguity might be cleared up by more frequent analyses, say 12
or 6 hours apart vice 24.
30
VII. Conclusions and Recommendations
Intensity peaks in the summer and the winter occur about a week
apart. The surface values of significant height in winter are about
twice those of the summer period. The bottom ratios are much smaller.
This difference is due to the different distribution of the energy with
respect to frequency between winter and summer. There seems to be a
preferred band of frequencies about 0.08 sec. for the winter. No
similar characteristic appears for the summer months.
Further investigations to determine distance to the generating
area should be conducted with a more sensitive digitizing instrument
located at a shallower depth with analyses more often than daily. These
procedures would tend to eliminate the high frequency noise.
31
BIBLIOGRAPHY
1. . Ocean Wave Spectra . Englewood Cliffs, New Jersey,:
Prentice - Hall, Inc., 1963, pp. 158 - 160.
2. Moskowitz, L. , W. J. Pierson, Jr., and E. Mehr. Wave SpectraEstimated from Wave Records Obtained by the OWS WEATHER EXPLORERand the OWS WEATHER REPORTER (I) and (II) . New York UniversityTechnical Report under Contr. N61206 - 1042, School of Engineer-ing and Science, Research Division, 1962, 1963, 1965.
3. Munk, W. H. Ocean Waves as a Meteorological Tool, Compendium of
Meteorology , 1952, pp. 1090 - 1100.
4. Johnson, J. W. Wave Records of the Pacific Coast. Institute ofEngineering Research , University of California , Series 14, Issue
4, September, 1950.
5. Cooley, J. W. , and J. W. Tukey. An Algorithm for the MachineCalculation of Complex Fourier Series, Mathematics of
Computations , Vol. 19, p. 297, April, 1965.
6. Pierson, J. W. , Jr.. and W. Marks. The Power Spectrum Analysisof Ocean -Wave Records. Transactions , American Geophysical Union ,
vol. 33, No. 6, December, 1952.
7. Tukey, J. W. , and R. W. Hamming. Measuring Noise Color 1. MurrayHill, New Jersey: Bell Telephone Laboratories, 1949.
8. Jenkins, G. M. General Considerations in the Analysis of Spectra.Technometrics , Vol. 3, No. 2, pp. 133 - 166, 1961.
9. Weigel, R. L. Oceanographic Engineering . Englewood Cliffs, NewJersey: Prentice Hall, 1964.
10. Kinsman, B. Wind Waves . Englewood Cliffs, New Jersey: PrenticeHall, 1965.
11. Blackman, R. B. , and J. W. Tukey. The Measurement of Power Spectra .
New York: Dover Publications, 1958.
32
APPENDIX I
TABULAR DISPLAY OF SPECTRAL COMPONENTS FOR SELECTED INTERVALS
33
_ o m ao •o * r> m 00 rr> rv p- o cp oo >o —i o o pi •-<
u o <o o m *-H e-l r> rg -4 i—) p- p- cp cp rg (M p- rg cp r—
t
LU c m <r a fSJ <f> in m —
*
>t f> .—
<
a fO cp m a * m U'
«/) o •c fr r- (VJ m t-4 m •* 00 rg eg a p- r- o p» -j •* ow o •o <r m rg <o •£> (^ cp <£ m in cp o f\ cp 0D p- cp f\l
o o cp or <NJ m <* m <VJ m cp •a —4 >* f~* rg P- m in 00
c o <C IT f\j Pll <c ou in in p- o in r-4 r- •* *4 00 -0 >r (Vj
c00 (M in oo -* 1-4 cr ao t- *o >o in m «» * J" m m fi m
a eg >» fSI r-t --> ^4 <
OLL,
l/l
ol/l in*- au. 0^1
-* a
i*> >o in
m rg m cp
3 •-" O
1—
»
•o o «_ vOvf r-i -t o |\j
o f—l in p- piCP P- m c Or- <» m inrg •V CO rr> o
en rg in rg rg P-o P- P- e~i ao p-
e> o P- O rg m—
<
in o O ao p-
.-g O rg in O *ao P~ «c in in m
o
O •-< rg m cp eg o
IT. t-i CM-» CP ^m rd CI00 pg in
<r> pi -1-
O pg o-
in in rr,
u1-1
a
o -< —i rg ^ ro o
w p- CO
o o>- i-o a>- p- ^i~ *4 O
o -< m
rri ^^ p-N »-t
p- rg >r ao rg m cr
<f t-4 c. a rg «* -0
-c o CC rg p- o in
^ >0 -* gs »H ao oB> ^ o pi CO rg oPi in fs, t—
i
o o oO o O o o o <-J
o o o o o
o
0- p- g3 pg o o o- g? o *-*
-0 o rri ro p- p- p- >J- p-l Uim m <o •—i in m in p- a £p- «—
i
PA J- >t >»• -i- «« » Ca o> & CP o- CP a> o> tp O-
o c o o o o o o o Oa o c- o o o o o o o
oIS)
oo rg oa> m «
rn _4
<f ~C
O co <_
l^ o
<. ^-" co
— LI^ .—I .—
I
fp, m o> rO of—i ^-t «*• * o<r rg o O C)e o o o oc o o u wo o O o o
oo
— rg
o o
m m in m
rg rr> in
in in in mp- rg p- rg p- rg p- rg p- rg r~
p-l o co ^H rri O oo wl p-i -c co^f o -p f^ CP in —
'
00 ~»- o -Oa >* CT in O ~o rg p- p-i CP <t
<r c P~ a rH rg >* in p- 00 org rg rg rg rg rg m
O O O O O O O
34
ft —t -4
— OD <o m —
.
o r» CM —
<
^4u (T 43 a J"in <r O e'- ~*•_ a- 4) er h-
o> 43 IT ma r- cm a CM
* 00 o a-
a. a <o j rv
*4 m -
r\j f^ pH am CM cc •c
gl- «o IT <tee co o o-<••• co f- a_>
IT. m o h-r» %o c> r-
in a; h43
o43
o
•J
o43o43
o
a j p-c -<
n m »<J- —i eg t-a
H Ifl Nu\ m inN (D K\ N
o p» r r- f\j o of—
<
c. <_> <c w* •o oM o- »4 •* 0- 00 Irf-i r- .—
<
•* U"> <o .—
<
m cm ,_ f- a IT vf
C •© lf> in r«- —
i
43
0" f- IP co <•* © CC
- 43 o -«CM m* N
•-4 in43 en
ro r-
om inro CM
cm co in ro c\i o(M ,-4 ,-4 -4 •-> •-!
in in m
a. cm cm cm in s *-< .—
i
<r- r» r-- <f _ CM 4 >! o o CM r- 00 in t>- r- h- r>,
O CM r- CO -0 r» CD CO t-> m O r- o -0 co o «o ro
,_< 43 CM •* o- co -o CM CO o CM CO •» .—
<
•c r- in OIT CM or •4" ec p-4 CO O p-4 oo a or. o CO CM —
*
r» 43
a CO cc r- «-4 fM •c CO o- o CM r- <* r» m CO a rvr-s CM •43 •* f-* o- r- CO CM CO o •o o ** o> co oc C-
o <J o o o e.- --> #-4 •o CM o o •o m CO m 4 <Nj
p4 m Sin o r-j
JD 00 <f
m -c CL>
Cj tr or
O fM O or
O oo CM -o
4 K oOOC»»-tf\l«-HO'*>»-«fr, 0000
•-« *^ OD
1
oK c- u\
U. cr co*- «—i c
rr •*
V m -J
1— o or•— CM O
ao >h irt -H ^e CO cr r- H CM r- cn CM o cr r^ o CM CO <*• r- O 4 •-<
a r- CM co ro 00 ~t m O <t in oc o 4 r- o co oc f- a•t 0- .» -* o rM o r~ m CO O —
'
<o 0> o CO oo CO w4 r>m -o r- -0 f-l J- 41 r- ao CO -c vO r- m «t oc CO •* o oCO CM i— CO (M O CM C^i CM •c —
!
•4- in CM o ~-t l_l c o oO o c_ c o o O U o o o O o o o o o o o c
^-1 X ~t r-
—i r-. 0> CO
o o o oo o o o
l_>
_ — ^
in in in
i-i in ro r^* o- f" in
h- <o h in co
O >o in cm in co cm in
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57
APPENDIX II
GRAPHIC DISPLAY OF SPECTRAL COMPONENTS FOR SELECTED INTERVALS
58
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APPENDIX III
SUMMARIES OF H3B AND H3S
79
SUMMARY OF H3B AND H3S FOR NOVEMBER, 1947
Date Time (PCT)
2 1138 - 1156
5 1124 - 1142
6 1147 - 1205
7 1143 - 1201
8 1143 - 1201
9 1142 - 1200
10 1150 - 1208
11 1145 - 1203
12 1142 - 1200
13 1201 - 1219
14 1800 - 1818
15 1202 - 1220
16 1153 - 1211
17 1805 - 1823
18 0854 - 091219 2103 - 211220 1459 - 1517
21 1456 - 151422 1519 - 1537
23 1617 - 163524 1605 - 1623
H3B (ft.) H3S (ft.)
1.299 *12.528
0.436 9.9690.621 7.4080.413 5.0080.417 2.7661.075 6.5310.995 6.2240.575 2.687
0.754 8.1250.597 4.9220.853 7.1861.022 5.0420.971 8.0360.820 5.2551.121 10.1500.768 13.5331.148 6.6931.382 5.5971.036 4.3550.727 1.7380.588 2.917
80
SUMMARY OF H3B AND H3S FOR DECEMBER, 1947
Date Time (PCT) H3B (ft.) H3S (ft.)
12 1323 - 1341 1.718 5.81113 1340 - 1349* 1.105 4.21014 1305 - 1323 1.326 3.59115 1820 - 1838 1.827 6.08516 1804 - 1822 1.337 5.053
22 1102 - 1119 2.306 9.34123 1107 - 1116* 2.021 14.192
1323 - 1341
1340 - 1349*
1305 - 13231820 - 18381804 - 1822
1102 - 11191107 - 1116*
2009 - 20270804 - 08221421 - 14391418 - 1436
1421 - 14392022 - 20401420 - 14381410 - 1428
24 2009 - 2027 2.491 10.41925 0804 - 0822 1.119 7.21526 1421 - 1439 0.794 2.90227 1418 - 1436 1.100 1.93628 1421 - 1439 0.875 3.48229 2022 - 2040 1.920 10.27630 1420 - 1438 1.119 3.28331 1410 - 1428 0.473 1.679
* - Record doubled
81
Date
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
3031
SUMMARY OF H3B AND H3S FOR JANUARY, 1948
Time (PCT) H3B (ft.)
0.2781.1593.2132.7951.8400.9511.5721.8042.4581.4921.2741.4590.7850.5890.6530.5851.147
1.0791.208
0.8300.7211.2711.1351.4891.2080.8221.4561.1551.8091.3690.958
0818 - 08360814 - 08321428 - 1446
1422 - 1440
0820 - 08381432 - 1450
1610 - 1628
1005 - 1023
0943 - 1001
1634 - 16521610 - 1628
1042 - 11001056 - 1114
1043 - 11011642 - 17001641 - 1659
2243 - 2251
0446 - 05040442 - 05001042 - 1100
1115 - 1133
1111 - 1129
1110 - 1128
0510 - 05281105 - 11231103 - 1121
1105 - 1123
1102 - 1120
1043 - 1101
1046 - 11041043 - 1101
H3S (ft.)
1.2066.309
11.57313.4345.1404.8117.3846.3879.8055.5675.0893.9571.607
0.8612.0272.2963.8464.3462.5032.9912.5382.3984.3447.0896.7373.2652.8974.8693.9825.7093.364
82
SUMMARY OF H3B AND H3S FOR JJNE, 1949
late Time (PCT)
5 1345 - 1403
6 1404 - 1422
7 0808 - 08268 1411 - 1429
9 1547 - 160510 1400 - 1418
11 0800 - 081812 0600 - 061813 0600 - 061814 1442 - 1500
15 1448 - 1506
16 1453 - 1511
17 1734 - 1752
18 1123 - 1141
19 1148 - 1206
20 1155 - 1213
21 2208 - 2226
22 1005 - 1023
23 1010 - 1028
24 1010 - 1028
25 1731 - 174926 1134 - 115227 1145 - 120328 1223 - 1241
29 1106 - 1124
30 1136 - 1154
H3B (ft.) H3S (ft.)
0.741 1.8731.025 4.1870.467 2.2410.753 1.4481.004 2.6412.062 6.6470.907 2.4881.205 3.1001.005 4.1091.045 5.8621.506 5.7261.312 9.0321.262 9.9790.865 3.9730.839 3.7530.878 3.2820.702 1.5110.486 1.3741.127 5.1630.646 3.3641.465 8.3571.023 8.6720.745 4.1460.822 3.3190.719 1.8570.861 4.735
83
iate Time (PCT)
1 1948 - 2006
2 1400 - 1418
3 1355 - 14134 1407 - 1425
5 1355 - 1413
6 1824 - 1842
7 1223 - 1241
9 0246 - 0304
18 1352 - 1410
19 1715 - 1733
20 1122 - 1140
21 1125 - 1143
22 1540 - 1558
23 1545 - 1603
24 1547 - 1605
25 0949 - 1007
26 1322 - 1340
27 1457 - 151528 1505 - 1523
29 1506 - 1524
30 1827 - 1845
31 1230 - 1248
SUMMARY OF H3B AND H3S FOR JULY, 1949
H3B (ft.) H3S (ft.)
0.660 1.5090.470 1.2070.866 2.1430.798 2.6950.644 2.0170.915 2.3230.319 3.498
0.860 3.923
0.494 3.4190.130 6.4591.062 2.1720.733 3.3431.098 5.1640.753 2.5860.635 2.3920.877 6.0800.730 4.6860.983 4.5611.049 4.5950.957 2.5360.932 2.6550.681 2.153
84
SUMMARY OF H3B AND H3S FOR AUGUST, 1949
late Time (PCT)
1 1242 - 1300
2 1457 - 1515
3 1436 - 1454
4 1530 - 1548
5 1528 - 1546
6 1504 - 1522
7 1505 - 1523
8 1503 - 1521
9 1522 - 1540
10 1521 - 1539
11 1520 - 1538
12 1529 - 1547
13 1530 - 1548
14 1528 - 1546
15 1529 - 1547
16 1541 - 1559
17 1541 - 1559
18 1541 - 1559
19 0942 - 1000
20 1211 - 1229
21 1210 - 1228
22 1208 - 1226
23 1131 - 1149
24 1130 - 1148
25 1131 - 1149
26 1129 - 1147
27 1137 - 1155
28 1139 - 1157
29 1140 - 1158
30 1201 - 1219
31 1141 - 1159
H3B (ft.) H3S (ft.)
0.876 3.2140.652 1.7130.733 1.4740.698 3.5531.147 3.5580.696 1.2991.089 6.1071.000 5.5390.960 2.8600.484 4.6620.419 2.2130.565 1.867
0.769 3.6410.887 3.1900.524 0.8760.324 1.2070.574 1.5550.515 2.1280.627 1.162
0.567 1.4800.240 1.9720.281 2.1280.288 1.5860.313 1.0880.500 2.9430.721 2.4450.824 2.3210.941 3.3850.590 2.1580.663 1.7110.506 2.891
85
INITIAL DISTRIBUTION LIST
No. Copies
1. Defense Documentation Center 20
Cameron StationAlexandria, Virginia 22314
2. Library 2
Naval Postgraduate SchoolMonterey, California 93940
3. Prof. Jacob B. Wickham 5
Department of Meteorology and OceanographyNaval Postgraduate SchoolMonterey, California 93940
4. LT Boyden T. Steele, USN 5
Naval AcademyAnnapolis, Maryland 21402
5. Department of Meteorology and Oceanography 3
Naval Postgraduate SchoolMonterey, California 93940
6. Naval Weather Service Command 1
Washington Navy YardWashington, D. C. 20390
7
.
Officer in Charge 3
Fleet Numerical Weather FacilityNaval Postgraduate SchoolMonterey, California 93940
8. AFCRL - Research Library 1
L. G. Hanscom FieldAttn: Nancy Davis/Stop 29
Bedford, Massachusetts 01730
9. Superintendent 1
Naval AcademyAnnapolis, Maryland 21402
10. Oceanographer of the Navy 1
The Madison Building732 N. Washington StreetAlexandria, Virginia 22314
11. Naval Oceanographic Office 1
Attn: LibraryWashington, D. C. 20390
86
No. Copies
12. National Oceanographic Data Center 1
Washington, D. C. 20390
13. Director, Coast and Geodetic Survey 1
Department of CommerceAttn: Office of OceanographyWashington, D. C. .
• 20235
14. Mission Bay Research Foundation 1
7730 Herschel AvenueLa Jolla, California 92038
15. Director, Maury Center for Ocean Sciences 1
Naval Research LaboratoryWashington, D. C. 20390
16. Office of Naval Research 1
Attn: Undersea Warfare (Code 466)Department of NavyWashington, D. C. 20360
17. Office of Naval Research 1
Attn: Director, Surface and Amphibious Programs(Code 463)Navy DepartmentWashington, D. C. 20360
18. Commanding Officer and Director 1
Naval Civil Engineering LaboratoryPort Hueneme, California 93041
87
UnclassifiedSecurity Classification
DOCUMENT CONTROL DATA -R&Dinlv Classification ol title, body ol abstract and indexing annotation must be entered when tlte overall report is classified)
Originating ACTIVITY (Corporate author)
Naval Postgraduate School
Monterey, California 93940
2a. REPORT SECURITY CLASSIFICATION
Unclassified2h. GROUP
1 REPOR T TITLE
Spectral Analyses of Waves and Associated Bottom Pressures Off Point Sur, California
pfscriptivE NOTES (Type ol report and. inclusive dates)
Thesis5 AUTHORiSI fFirif name, middle initial, last name)
Boyden Tracy Steele
6 REPORT DATE
June 1968
7a. TOTAL NO. OF PAGES
8676. NO. OF RE FS
11
la. CONTRACT OR GRANT NO.
b. PROJEC T NO
9a. ORIGINATOR'S REPORT NUMBER(S)
9b. OTHER REPORT N O (S) (Any other numbers that may be as signedthis report)
10. DISTRIBUTION STATEMENTMammmimmmm^mim m^mf^m^mmmmmmmmmmm^mmmmm
vflAa1 - C .-taMMM
1*1 1I P
immmII. SUPPLEMENTARY NOTES 12. SPONSO RING Ml LI T AR Y ACTIVITY
Naval Postgraduate School
Monterey, California 93940
13 ABSTRACT
A time series of pressure variations was obtained by digitizing analoguerecords of bottom pressure, recorded at Pt. Sur, California, during the
periods November, December, and January, 1947-1948, and June, July, and
August, 1949. The power spectrum of the pressure variations was found via the
amplitude spectra obtained by means of the fast Fourier transform. Thisspectrum was then smoothed by averaging over several frequency bands andconfidence limits were assigned.
Surface wave power spectra were inferred from the pressure power spectraand significant wave heights were estimated from these, after applying a
procedure designed to remove noise.
Values of significant height on the bottom and at the surface are
displayed in tabular form and plotted versus time for the selected periodsJanuary, 1948, and August, 1949. The spectral patterns of wave activityshow the transient nature of the energy due to storm swell and local winds.
DD ,
F
°o1M.,1473 ,PAGE "
:0RMNO V 65
S/N 01 01 -807-681 1
89Unclassified
Security ClassificationA-31408
UnclassifiedSecurity Classification
KFY WORDSLINK A LINK B
Wave Pressure Records
Bottom Pressures
Significant Height, Graphic Display
Fast Fourier Transform
Digital Analysis
DD .'•?.,1473 (back)Unclassified
i i- 90 Security Classification
