CETA 805
W H I
DOCUMENT
'LECTION
Interpretation of Wave Energy Spectra
by
Edward F. Thompson
COASTAL ENGINEERING TECHNICAL AID NO. 805
JULY 1980
Approved for public release;
distribution unlimited.
330
lA*.
20 S'
U.S. ARMY, CORPS OF ENGINEERS
COASTAL ENGINEERING
RESEARCH CENTER
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Fort Belvoir, Va. 22060
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Department of the Army position unless so designated by other
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BEFORE COMPLETING FORM
t. REPORT NUMBER
CETA 805
2. GOVT ACCESSION NO.
3. RECIPIENT'S CATALOG NUMBER
Â«. TITLE (and Subtitle)
INTERPRETATION OF WAVE ENERGY SPECTRA
S. TYPE OF REPORT & PERIOD COVERED
Coastal Engineering
Technical Aid
6. PERFORMING ORG. REPORT NUMBER
7. AUTHORfa;
Edward F. Thompson
8. CONTRACT OR GRANT NUMBERfs;
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Department of the Army
Coastal Engineering Research Center (CERRECO)
Kingman Building, Fort Belvoir, Virginia 22060
A31463
11. CONTROLLING OFFICE NAME AND ADDRESS
Department of the Army
Coastal Engineering Research Center
Kingman Building, Fort Belvoir, Virginia 22060
12. REPORT DATE
July 1980
13. NUMBER OF PAGES
21
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15. SECURITY CLASS, (of th'm report)
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18. SUPPLEMENTARY NOTES
19. KEY WORDS fConfinuB I
cesaary and identify by block number)
Spectral analysis
Wave energy spectra
Wave gages
Wave height
Wave hindcast
Wave period
20. ABSTRACT fCazTtÂ£auc ox iwera* Â»Me ff n^ce^seaey and Identify by block number)
Guidelines for interpreting nondirectional wave energy spectra are presented.
A simple method is given for using the spectrum to estimate a significant height
and period for each major wave train in most sea states. The method allows a
more detailed and accurate description of ocean surface waves than that given by
a single significant height and period, yet it eliminates much of the formidable
detail of a full spectrum. An example problem illustrating application of the
method is presented. Spectral analysis and display techniques, and the natural
variation of spectra in space and time, are discussed.
DD.'
1473
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SECURITY CLASSIFICATIOK OF THIS PAGE (Whan Data Entered)
PREFACE
This report presents guidelines for interpreting nondirectional wave energy
spectra. The guidelines apply to spectra derived from both wave gage measure
ments and from numerical wave hindcasting models. A method is provided for
using the spectrum to estimate a significant height and period for each major
wave train in a sea state, except major wave trains with nearly the same period
and different directions cannot be distinguished. The method has undergone
limited testing and has been applied to 7 stationyears of gage data, but fur
ther testing in well documented field situations is needed. The guidelines and
method are consistent with but are more practical and explicit than the material
in the Shore Protection Manual (SPM) . The work was done under the wave measure
ment program of the U.S. Army Coastal Engineering Research Center (CERC) .
This report was prepared by Edward F. Thompson, Hydraulic Engineer, under
the general supervision of Dr. C.L. Vincent, Chief, Coastal Oceanography Branch.
Helpful reviews by Dr. C.L. Vincent, Dr. D.L. Harris, P. Knutson, and P. Vitale
are acknowledged. Dr. D. Esteva provided the data for Figure 6.
Comments on this publication are invited.
Approved for publication in accordance with Public Law 1966, 79th Congress,
approved 31 July 1945, as supplemented by Public Law 1972, 88th Congress,
approved 7 November 1963.
TED E. BISHOP
Colonel, Corps of Engineers
Commander and Director
CONTENTS
Page
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) 5
SYMBOLS AND DEFINITIONS 6
I INTRODUCTION 7
II PRIMARY GOAL OF SPECTRAL ANALYSIS 7
III PRACTICAL LIMITATIONS OF SPECTRAL ANALYSIS 9
1. Calculation Procedures 9
2. Display Formats 9
3. Natural Variability 11
IV INTERPRETATION OF SPECTRA 14
1. Highest Spectral Peak 14
2. Major Secondary Spectral Peaks 14
3. Example Problem 17
V INTERPRETATION OF SPECTRA FOR APPLICATIONS SENSITIVE
TO SPECIFIC FREQUENCIES 19
VI SUMMARY 20
LITERATURE CITED 21
TABLE
Spectrum for Huntington Beach, California 18
FIGURES
1 Spectrum for Wrightsville Beach, North Carolina 8
2 Energy spectra for wave record at 0400 e.s.t., 17 March 1968 10
3 Energy spectra for record composed of three superimposed
sinusoidal waves 10
4 Five formats frequently used in displaying wave energy spectra 11
5 Wave energy spectra from bottommounted pressure gages at Channel
Islands Harbor, California 12
6 Wave energy spectra from piermounted continuous wire gages
at the CERC Field Research Facility 13
7 Directional spectrum obtained in the Atlantic Ocean 68 miles
east of Jacksonville, Florida 15
8 Spectrum for Huntington Beach, California 18
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT
U.S. customary units of measurement used in this report can be converted to
metric (SI) units as follows:
Multiply
by
To obtain
inches
25.4
millimeters
2.54
centimeters
square inches
6.452
square centimisters
cubic inches
16.39
cubic centimeters
feet
30.48
centimeters
0.3048
meters
square feet
0.0929
square meters
cubic feet
0.0283
cubic meters
yards
0.9144
meters
square yards
0.836
square meters
cubic yards
0.7646
cubic meters
miles
1.6093
kilometers
square miles
259.0
hectares
knots
1.852
kilometers per hour
acres
0.4047
hectares
footpounds
1.3558
newton meters
millibars
1.0197 X 10"3
kilograms per square centimeter
ounces
28.35
grEuns
pounds
453.6
grams
0.4536
kilograms
ton, long
1.0160
metric tons
ton, short
0.9072
metric tons
degrees (angle)
0.01745
radians
Fahrenheit degrees
5/9
Celsius degrees or Kelvins^
^To obtain Celsius (C) temperature readings from Fahrenheit (F) readings,
use formula: C = (5/9) (F 32).
To obtain Kelvin (K) readings, use formula: K = (5/9) (F 32) + 273.15.
SYMBOLS AND DEFINITIONS
a^ wave component amplitude
d water depth
E sum of all Ej
Ej, E(fj) spectral energy density values
^max energy density for the highest spectral peak
fj frequency of spectral component in hertz
Â£p frequency corresponding to the highest spectral peak
Â£p^ frequency corresponding to the ith highest spectral peak
g acceleration due to gravity
Hg significant wave height corresponding to the full spectrum
Hg^ significant wave height corresponding to the ith highest spectral
peak
^ ' 2 indices representing the lower and upper bounds of the highest
spectral peak
Kii, ^2i indices representing the lower and upper bounds of the ith highest
spectral peak
N total number of spectral frequency components
S^, SCfj) spectral energy values
Sp^ energy contained in ith highest spectral peak
t time
Tp period corresponding to the highest spectral peak
Tp^ period corresponding to the ith highest spectral peak
Af, (Af), frequency bandwidth represented by each spectral energy density
(in hertz)
n seasurface elevation referenced to local mean water level
(J)^ phase of spectral component
(lij frequency of spectral component (in radians per second)
INTERPRETATION OF WAVE ENERGY SPECTRA
by
Edward F. Thompson
I INTRODUCTION
The ocean usually has more than one independent train of waves propagating
along its surface in U.S. coastal areas. The common practice of using a single
significant height and period for a sea state can be misleading because no indi
cation is given to the existence or characteristics of other trains. On the
other hand, an estimate of the wave energy spectrum provides more information
than is generally used in coastal engineering. The spectrum can be reduced to
estimates of significant wave height and period for all major wave trains pres
ent. A knowledge of these characteristics for major wave trains is often im
portant to coastal engineers.
Spectra are becoming widely available through various field wave measurement
programs, laboratory tests with programable wave generators, and numerical wave
hindcasting projects. Because of the availability and applications of spectra,
practicing coastal engineers should become familiar with spectra and their
interpretation.
II. PRIMARY GOAL OF SPECTRAL ANALYSIS
A fundamental parameter for characterizing a wave field is some measure of
the periodicity of the waves. For many years a significant period, which could
be subjectively estimated in various ways, was used. However, the ocean surface
often has waves characterized by several distinct periods occurring simultane
ously. A record of the variation of seasurface elevation with time, commonly
called a time series, frequently appears confusing and is difficult to interpret.
Developments in computer technology and in mathematical analysis of time
series have provided a practical approach to an objective, more comprehensive
analysis of periodicity in wave records. The approach is to express the time
series as a sum of periodic functions with different frequency, amplitude, and
phase. The simplest functions to apply are the trigonometric sine and cosine
functions. Thus, the time series of seasurface deviations from the mean sur
face, n(t), is expressed by equation (311) in the Shore Protection Manual
(SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977)
as
n
Ti(t) = ^ aj cos (a)jt(f)j) jj^.
where
Sij  amplitude
(jOj = frequency in radians per second
<f)j = phase
t = time
Frequency is often expressed in terms of hertz units where one hertz is equal
to one cycle per second. One hertz is also equivalent to 2tt radians per
second. If the symbol fj denotes frequency in hertz, then Zirf â€¢ = o).
The amplitudes, aj, computed for a time series, give an indication of im
portance of each frequency, fj. The sum of the squared amplitudes is related
to the variance of seasurface elevations in the original time series (eq. 312
in the SPM) and hence to the potential energy contained in the wavy sea surface.
Because of this relationship, the distribution of squared amplitudes as a func
tion of frequency can be used to estimate the distribution of wave energy as a
function of frequency. This distribution is called the energy spectrum and is
often expressed as
(Ej) (Af)
a^
 ^  S
(2)
where Ej = E(fj) is the energy density in jth component of energy spectrum,
(Af)j the frequency bandwidth in hertz (difference between successive fj) ,
and Sj = S(fj) energy in jth component of energy spectrum. Equation (2) is
similar to equation (315) in the SPM.
An energy spectrum computed from an ocean wave record is plotted in Fig
ure 1. Frequencies associated with large values of energy density (or large
values of a^/[2(Af)]; see eq. 2) represent dominant periodicities in the orig
inal time series. Frequencies associated with small values of aW[2(Af) â€¢] are
usually unimportant. It is common for ocean wave spectra to show two or more
dominant periodicities (Fig. 1).
25,000
20,000
Ji 15,000
Q 10,000
5,000
I I Region used to Compute Spi
^^ Region used to Compute Spz
Peak 1
Figure 1
fp2 .p
Spectrum for Wrightsville Beach, North Carolina, 0700 e.s.t.,
12 February 1972; Hg = 4.2 feet (128 centimeters), Af = 0.01074
hertz, and depth = 17.7 feet (5.4 meters).
The primary goal of spectral analysis is to obgeGtively identify all im
portant frequencies in a wave record. Since wave period in seconds is equal
to the reciprocal of frequency in hertz, important wave periods are also
identified.
III. PRACTICAL LIMITATIONS OF SPECTRAL ANALYSIS
1. Calculation Procedures .
The appearance of a spectrum can be noticeably influenced by the methods
used for calculation and display, neither of which is standardized in coastal
and ocean engineering activities.
Spectra are computed from both digital wave records and analog records for
which an assortment of analog spectral analysis devices exists. A spectrum is
computed from a digital wave record by either (a) computing the fourier trans
form of the autocovariance function of the record, or (b) computing the fourier
transform of the record directly from the record using the fast fourier trans
form (FFT) approach (see Harris, 1974, for further detail). Both of these
algorithms are used in conjunction with wave records subjected to various fil
ters and smoothing functions before analysis. Further, some form of smoothing,
averaging, or summing is often applied to the computed spectral components.
Different methods for calculating a spectrum will produce slightly different
estimates of the spectrum when applied to a particular wave record. Major dif
ferences in the height of the spectral peak were shown by Wilson, Chakrabarti,
and Snider (1974) when different approximations to the autocovariance function
were used and different smoothing functions were applied to the spectrum of a
field wave record (Fig. 2). Major differences in the height of the spectral
peak and energy levels between peaks were noted by Harris (1974) when a time
series composed of three superimposed sinusoidal waves was analyzed by several
accepted methods (Fig. 3) .
Hindcast wave energy spectra are computed by estimating atmospheric input
of energy to the sea surface and redistribution of energy within a spectrum.
The estimates are based on a series of equations derived from the physics of
airsea interaction and waves. The quality and characteristics of hindcast
spectra are a function of the model used to perform the calculations (Resio
and Vincent, 1979) as well as the accuracy of the input wind field.
Spectra obtained from either measurements or hindcasts are also limited by
the resolution of the computation technique. The energy density and frequency
at a spectral peak can be noticeably distorted if the frequency bandwidths,
(Af)j, are not small enough to permit clear definition of major peaks.
2. Display Formats .
The appearance of a spectrum can be strongly affected by the display format
used. Harris (1972) showed five often used formats for plotting spectra (Fig.
4). Each format alters the appearance of the spectrum. Format E shifts the
relative magnitudes of spectral peaks enough that the second highest peak in
formats A, B, C, and D becomes the highest peak in format E. A and C are the
two most frequently used formats.
800 
600 
200 
0.05 0.10 0.15
Frequency ( Hz )
0.20
Figure 2. Energy spectra for wave record at 0400 e.s.t., 17 March
1968, from a weather ship in the North Atlantic, computed
with different approximations to the autocovariance func
tion and different spectral smoothing functions (after
Wilson, Chakrabarti, and Snider, 1974).
10
Q lO"
â€¢^
tl ill
m
FFT used with 1,024 second lime
series smoothed by cosine bell
Avg. of spectrol volues computed with
FFT lor eight 128second time series
Autocovorionce procedure
ih
\l\
JA I ^f 1
1/64 1/8 1/4 3/8 1/2 5/8 3/4 7/8
Frequency ( Hz)
Figure 3. Energy spectra for record composed of three superimposed
sinusoidal waves. Simulated frequencies indicated by
vertical lines from top of graph (after Harris 1974) .
0.5
Frequency ( Hz)
Frequency (Hz)
lao
1
LOO

f\
c
; â€¢' Â°
J
\
h\
.0 1

\
.0 01
0.
1
h
!)l
0.1
1.
F
eq
uency
(H
z)
0.5
Frequency (Hz)
10.0 20.0 30.0
Period ( s)
Figure 4. Five formats frequently used in displaying wave energy spectra. The
actual spectrum is identical in all five graphs. The frequency band
width, (Af)j, is constant for all j so that energy values and energy
density values differ by a constant factor (see eq. 2). The plots
would look the same if energy were replaced by energy density, but
the vertical scale would change (from Harris, 1972).
3. Natural Variability .
Wave energy spectra are naturally variable simply because they are based on
a finite length record of a wave field which varies in time and space. Spectra
computed for successive records of a relatively stationary wave field are never
identical and often differ noticeably. The magnitude of spectral variation in
time is illustrated by spectra derived at 2hour intervals from two pressure
gages along the southern California coast (Fig. 5). The significant wave height
is nearly constant in the figure.
Spatial variation of the spectrum over short alongshore distances in shallow
water is also shown in Figure 5. Each spectrum in the top row of the figure can
be compared to the spectrum immediately below it to see variations between spec
tra from two gages 80 feet (24 meters) apart. In this figure, spatial variations
are smaller than temporal variations. Spatial variations would be expected to
be greater if the gages were farther apart or the water depth varied between
measurement points. Variations between spectra from gages situated along a line
perpendicular to shore are shown in Figure 6. The spatial variations are more
prominent in this figure than in Figure 5.
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12
13 Jon, 1979
1058 hr e.s.t.
Hs= 5.5 ft (168 cm)
Hs = 4.5ff (136 cm)
13 Jon. 1979
1258 hr e.s.t.
Hs:5.5ff (168 cm)
Hs = 4.6 ft (139 cm)
0.2 0.3
Frequency (Hz)
0.2 03
Frequency ( Hz]
II Sept. 1978
llOOhr e.s.t.
Hs = 3.3 ft (100 cm]
.Hs = 2.2 ft (67cm)
.Hs= 1.8ft (54 cm]
0.2 0.3
Frequency ( Hz)
13 Sept. 1978
0000 hr est.
Hs= 3,5 ft (108 cm )
H. 2.2 ft (67cm)
Hc=l.8ft (54cm)
0.2 0.3
Frequency ( Hz )
Figure 6. Wave energy spectra from piermounted continuous wire gages
at the CERC Field Research Facility (FRF] near Duck, North
Carolina, showing variation along a line perpendicular to
shore. Solid lines represent a gage at the seaward pier end
(depth 29 feet or 8.8 meters); dashlines represent a gage 480
feet (146 meters) from the seaward pier end (depth 22 feet
or 6.6 meters); dotdash lines represent a gage 840 feet (256
meters) from the pier end (depth 17 feet or 5.1 meters).
IV. INTERPRETATION OF SPECTRA
Because of natural var^ahility in the spectrum and ar'tificial variability
induced by analysis and display techniques j the spectrum should never be in
terpreted as an exact representation of energy density versus frequency for a
wave field. However, certain major features of the spectrum are consistent
and meaningful .
1. Highest Spectral Peak .
a. Frequency and Period . Frequency corresponding to the highest spectral
peak, fp, is usually a reliable measure of the dominant wave frequency; fp
is shown in Figure 1. Period corresponding to the highest spectral peak, Tp,
is equal to the reciprocal of fp and is usually a good estimate of the domi
nant wave period.
b. Energy and Significant Wave Height. Energy contained in the highest
peak, Sp2j is defined as the total energy in the vicinity of the highest peak.
Si = I E^(Af)j (3)
j=Ki+l
where K;^ and K2 are indices representing the lower and upper bounds of the
main peak. The upper and lower bounds sometimes represent a broad range of
frequencies (see Fig. 1). Spl is relatively consistent, and is less influenced
than the magnitude of the highest peak by data collection procedures, by analy
sis and summarization procedures, and by temporal and spatial variation.
Some spectral analysis procedures are designed so that (^f)j = ^f is
constant for all j, which leads to
Si = (Af) I Ej
j=Ki+l
(3a)
Significant wave height corresponding to highest spectral peak Hgi, is an
estimate of the significant height for the wave train represented by the highest
spectral peak. It is computed by the relationship
Hsi = ^yf^i (4)
Energy density at the highest spectral peak, ^max, can be an indicator of
how well focused the wave energy is in frequency. Although this parameter is
variable, major differences in ^rmx (on the order of 50 percent) between
spectra analyzed by the same method can be meaningful, ^max is shovm in
Figure 1.
2 . Major Secondary Spectral Peaks.
a. Identification of Major Secondary Peaks . Major secondary spectral peaks
are often indicative of independent secondary wave trains characterized by dif
ferent heights, periods, and directions than the train represented by the main
peak (examples are given in McClenan and Harris, 1975). Identification of major
secondary spectral peaks involves some subjective judgment, but an objective
14
test for major secondary peaks has been developed and used at CERC. The test
is applied to the difference in energy density between a spectral peak and the
preceding and following spectral valleys. If that difference exceeds 3 percent
of the total of all spectral energy density values, E, then the peak is con
sidered major. Details of the procedure with a computerized version are given
in Thompson (1980). The procedure was applied to the spectrum in Figure 1, com
puted from an ocean wave record, and two major peaks were identified. The pro
cedure has been applied to 7 stationyears of shallowwater spectra by Thompson