Effects of currents on waves

U.S .Army
Gre. Res Cte

ee
CETA 81-14

Effects of Currents on Waves

by

WHO!

DOCUMENT
COLLECTION

Barry E. Herchenroder

COASTAL ENGINEERING TECHNICAL AID NO. 81-14

OCTOBER 1981

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Approved for public release;
distribution unlimited.

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COASTAL ENGINEERING

Te
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au30

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Technical Aid

6. PERFORMING ORG. REPORT NUMBER

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EFFECTS OF CURRENTS ON WAVES

7. AUTHOR(s)

Barry E. Herchenroder

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Horizontal currents Wave characteristics
Surface gravity Wave measurements

ABSTRACT (Continue on reverse side if necesaary and identify by block number)

This report presents ways in which a horizontal current influences surface
gravity waves and their measuremente Relatively simple hand-calculation meth-
ods are described which provide a means to estimate (a) the wavelength modifi-
cation due to a current, (b) whether a current can prevent waves from reaching
a particular location, (c) the correction needed to compensate for a current
when observed bottom pressure fluctuations are used to estimate wave heights,
and (d) the range of periods (if any) where the effects of currents can be
neglected when wave heights are estimated from bottom pressure fluctuations.

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PREFACE

This report presents some ways in which a horizontal current influences
surface gravity waves and their measurement. Relatively simple hand-
calculation methods are described which provide ways to estimate how much
currents affect the wavelength and measured height of waves. These methods
are the initial results of a research effort designed to furnish practical
guidance on how to account for the effects of currents on wavese Such guid-
ance fills an information gap in the Shore Protection Manual (SPM). The work
was carried out under the waves on currents program of the U.S. Army Coastal
Engineering Research Center (CERC).

The report was prepared by Dr. Barry Ee. Herchenroder, Oceanographer, under
the general supervision of Dr. C.L.e Vincent, Chief, Coastal Oceanography

Branche

Comments on this publication are invited.

Approved for publication in accordance with Public Law 166, 79th Congress,
approved 31 July 1945, as supplemented by Public Law 172, 88th Congress,
approved 7 November 1963.

TED Ee ‘BLSHOP
Colonel, Corps of Engineers
Commander and Director

CONTENTS

Page
CONVERSION FACTORS, UeS. CUSTOMARY TO METRIC (SL1)ccccccccccccecccrces 5

SYMBOLS AND DEFINITIONSe coccccccceccccecccccececceescesseccccsccececs 6
MNTRODU GION ererecereelesercieveveretelelelelelelevelcilesciciovercteleheeiele ene lel oleleiereicieie.e eee s el sve elefele 9
WAVELENGTH MODIFICATION AND THE STOPPING OF WAVESccccccvcccceccseeeee 10

CURRENT-INDUCED CORRECTLONS WHEN BOTTOM PRESSURE MEASUREMENTS
ARE USED TO DETERMINE SURFACE WAVE HEIGHTS. .ccccccvcccccceesessessees 14

EXAMP INE PROB IGE Mistereiercisl eisieheictereseeleneleterelclelclctetel evelielis) cic veie cle cisisicl siecle! el je!si elec ctars/ mG
SUMMARY cctet vere tshevetstelatel olelelcleteletercleleialokeletertstekeieieteelelcloteicheteieeie/ et eleie/e/e clelelsie slicieleleie 20
TABLE

(La/dp) anda AM “kore vardlous) wallies ots Ntectelereieleierslele clelcic cteielsicie/eis/clele/s/eiefemminl ©

FIGURES

Relationship between horizontal current velocity, V, horizontal wave
vector, K, wave orthogonal, wave crest, wave ray, and angle, Oeeee. ial

Contours of dimensionless wavelength factor, Ryeeccececceesrceeeseeeee Il
Contours of dimensionless wave height factor, Ryeseeceececeeseeeceeeee 15

Schematic representation of how SA and %D are related to FL
and FS for the example problem in Section [Veccccccscccccccccccessee 1/

CONVERSION FACTORS, UeS.e 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 2544 millimeters
254 centimeters
square inches 626452 Square centimeters
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 0.336 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
foot-pounds 1.3558 newton meters
Pa Phare 1.0197 x 1073 kilograms per square centimeter
ounces 28.35 grams
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!

1To 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.

FL

FS

Si

SYMBOLS AND DEFINITIONS

time-averaged water depth

largest time-averaged water depth expected
smallest time-averaged water depth expected
effective Froude number (eq. 3)

minimum effective Froude number for which waves with dimensionless
frequency can propagate (Table)

largest F that can be expected (eqe 15)

smallest F that can be expected (eq. 16)

acceleration of gravity = 32.2 ft (9.8 m)/s?

height of a monochromatic wave

height of monochromatic wave when current is absent (eqe 10)

height of monochromatic surface gravity wave when current is present
(eqse 9 and 12)

horizontal wave vector
wavelength of monochromatic wave

wavelength of monochromatic wave when current is absent (solution of
eq. 1)

wavelength of monochromatic wave when current is present (eq.e 7 and
solution of eq. 2)

bottom pressure fluctuation magnitude accompanying monochromatic surface
gravity wave

dimensionless wavelength factor = Ly/ Ly
dimensionless wave height factor = Hy /Hy (eqe 11)
largest value of Ry expected

smallest value of Ry expected

period of monochromatic wave

smallest wave period of interest

smallest wave period for which currents can be neglected when observed
bottom pressures are used to estimate wave heights (eq 1/7)

horizontal current speed

horizontal current velocity vector

VL

VS

6L

6S

SYMBOLS AND DEFINITIONS--Continued
largest value of V_ expected
smallest value of V_ expected
“stopping velocity” of a wave (eq 6)
3214159.6.
angle between V and K
largest value of 6 expected
smallest value of 986 expected
water density = 2.00 slugs/ft3 (1,031 kg/m?)
dimensionless frequency (eq. 4)
defined in equation (19)
largest value of & that can be expected (eq.

defined in equation (18)

hyew
Mie

v

: fee

ij

iu
fy

EFFECTS OF CURRENTS ON WAVES

by
Barry E. Herchenroder

I. INTRODUCTION

A horizontal current can modify surface gravity waves in several ways.
One modification is to the wavelength. A current can locally stretch or
shrink features in a wave train. This wave-train distortion produces a "Dop-
pler shift" in the wave period. As a result, a stationary observer measures a
different wave period than an observer moving with the local current velocity.
To a good approximation, the period of a monochromatic wave is fixed in the
stationary coordinate system. To an observer moving with the local current
velocity, the measured period of this wave changes in time and at each point
in response to the changing currente

The wave orthogonal, crest, and ray directions are also modified. An
orthogonal is a line perpendicular to the local wave crest direction. A ray
is a line parallel to the local group velocity vector, i.e», tangent to the
local direction of wave energy flow. In the absence of a current, rays are
parallel to orthogonals. When a current is present, orthogonals (by defini-
tion) are still perpendicular to the local crest orientation but rays are not
parallel to the orthogonals unless the wave is propagating in the same direc—
tion as the current.

Another way a current modifies surface waves is to change the wave energy
by causing an exchange of energy between wave and currente Energy per unit
length of wave crest is no longer approximately conserved between rays, but
rather a quantity called "wave action" is conserved. This new quantity, a
generalization of wave energy, is given by the energy per unit length of wave
crest divided by the “intrinsic angular frequency" of the wave, the latter
being the Doppler-shifted angular frequency seen by an observer moving with
the local current velocity. Since wave action rather than energy is conserved
between rays, enough wave energy is gained from the current or lost to the
current to keep the wave action the same as the wave propagates.

A modification also occurs in the pressure field accompanying the wave.
This change can be an appreciable source of error in measuring wave character-
istics if an existing current is not accounted fore In particular, a signifi-
cant error can sometimes arise if bottom pressure measurements are used to
determine surface wave heights and lengths.

Prediction of current-modified wave energy, heights, directions, and
pressures usually involves the use of complex numerical models and computer
programs. Such models and programs are under development and will be avail-
able over the next few years. However, relatively easy but useful calcula-
tions can be done without a computer. This report presents some of these
simpler calculations, such as methods to determine (a) the wavelength modifi-
cation due to a current, (b) whether a current can stop a wave from reaching
a particular location, (c) a correction for the presence of a current when
bottom pressure measurements are used to determine wave heights, and (d) the
range of periods (if any) for which currents can be neglected when these
pressure measurements are used to determine wave heights.

Il. WAVELENGTH MODIFICATION AND THE STOPPING OF WAVES

When a current is absent, Peregrine (1976)+ and the Shore Protection Man-
ual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research Center,
1977)% indicate that the wavelength, L, of a monochromatic wave is given by
Lea Uasaunee,” lx 7s computed by solving the “dispersion relation"

27 1 2ady
ae eee are (1)
T*g
and A A
tanh = the hyperbolic tangent
I = the wave period
g = the acceleration of gravity 32.2 feet (9.8 meters) per second
squared
dy = the time-averaged (over enough wave periods to filter out waves)
water depth
T = the factor 3.14159...

When a current is present, Peregrine (1976)? indicates that L is no
longer equal to Ly, but to Ly, where Ly is given by solving the equation

= es 4
GD) (V cos 8/Ly) | l 21d»
> ———— = Fan
8 Ly Ly

(2)

where cos is the cosine, V the horizontal current speed, and 98 the angle
between the horizontal current vector, V, and the horizonal wave vector, K.
The angle, 68, is taken to be greater than or equal to zero and less than or
equal to 180° (nm radians). The vector K has a magnitude of 2n/Ly and
points along the wave orthogonal in the direction of wave crest and trough
propagation. The relationship between horizontal current velocity, V, hori-
zontal wave vector, K, wave orthogonal, wave crest, wave ray, and angle, 8,
is shown in Figure 1. The figure shows that when a current is present, wave
rays are not parallel to wave orthogonals and hence not perpendicular to wave
crestSe

Figure 2 shows a set of curves which represents the dimensionless solution

of equation (2). The F axis represents various values of the effective
Froude number

1PEREGRINE, D.H., “Interaction of Water Waves and Currents," Advances in
Applted Mechanics, Vol. 16, Academic Press, New York, 1976, pp. 9-117.

2U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore
Proteetton Manual, 3d ed., Vols. I, II, and III, Stock No. 008-022-00113-1,
U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp.

3PEREGRINE, DeH., ope cit.

10

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. (V cos 8)

STEARNS ela aT ATIOTE 6}
(ede) s/ (3)

where the & axis represents values of dimensionless frequency

(2m) dp 1/2
Q = = oa (4)

Each curve is a fixed value of wavelength factor, R,, where

(5)

L ?
V \_ wavelength with current
wavelength without current

Negative values of F correspond to waves with a component of phase velocity
in the direction opposite to that of the current. The phase velocity is the
velocity that a point on the crest or trough is moving. The curve Ri = 1.0 is
the case where a current is absent; ieee, where L = UN fulfilling the
usual dispersion relationship (eq. 1). ‘

The curve F = FM in Figure 2 represents the limiting effective Froude num-
bers for which waves can propagates If F is less than FM for a particular
@, then waves cannot propagate against the current for the particular 2 and
F values. In other words, the current acts as a filter under certain condt-
ttons and prevents waves from ever reaching a given point. The limiting value
of “effective current velocity,” V cos 9, for which waves of a given period,
T, in a mean depth of water, dis can propagate is the “stopping velocity,”
VST, given by

VST = (FM) (gd) 1/2 (6)

VST is negative (since FM is negative) and depends on dimensionless fre-
quency, & (since FM, as shown in Fige 2, depends on &), and mean depth,
dy (through the (gd_)}/2 factor). If a wave is propagating against the cur-
rent (iee., cos 8 is less than zero), then the wave cannot reach any area
where the current speed V is greater than the local value of VST. Such an
unreachable area in physical space or on a plot like Figure 2 is called the
“forbidden region.” Waves traveling with the current (iee., with a cos 0
greater than or equal to zero) have no forbidden region.

The Table gives the minimum effective Froude number, FM, and the ratio

ES

_A _ wavelength without current
7 average (in time) depth
i

for various values of dimensionless frequency, For a particular 2
value, this table (used in conjunction with Fig. 2 and linear interpolation)
allows the following to be found: (a) whether a wave can propagate against

the current (i-e., is F less than FM), and (b) if a wave can propagate, the
value of its current modified wavelength, Ly. Using the value of (L,/dy)
from the table and Ry, from Figure 2, Ly is computed from

12

Q

0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0. 30
Onrsi2
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
0.50
0.52
0.54
0.56
0.58
0.60
0.62
0.64
0.66
0.68
0.70
OnyL 2
0.74
0.76
0.78
0.80
0.82
0.84
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.02
1.04
1.06
1.08
1.10

Table. (La/dy) and FM for various values of 2.

La/dry

62.73
D229
44.73
3:95.10
34.72
SMe ei 2il
28.33
PEDO) o}
23689
Lore lS
20.63
19.30
18.12
17.08
16.14
15.29
14.52
36 82
13.18
122.59
12.04
11.54
MEALS (O)7/
10.63
NOG 472
09.84
09.48
09.14
08.83
O85:2
08.24
07.97
O7es 1
07-47
07.24
07.01
06.80
06-60
06-40
06.22
06-04
05.86
05.70
05-54
05.39
05.24
05.09
04.96
04.82
04.69
04.57

FM

-0.787
-0.714
= Oils 617.3
=0...653
=0'3 6:3)
=O. 617
-0.600
-0.584
-0.568
-0.538
-0.524
= 0). 5:10
-0.497
-0.472
-0.448
=0)..4:25
-0.404
=. 366
=0.397
-0.349
-0.340
= Ole 3S OV2
-0.310
= (153103
= ker 2916
-0.289
=(Oie283
-0.266
-0.260
= Oier2510)
-0.245
-0.240
-0.236
=O eu2 Si
= Ole 22:7

1S)

Q

We 2
Tes 1A
1.16
1.18
6.20
We22
1.24
1.26
ee 238)
1.30
E32
1.34
ES 210)
1.38
1.40
1.42
1.44
1.46
1.48
1.50
eZ
1.54
1.56
1.58
1.60
1.62
1.64
1.66
1.68
1.70
W572
1.74
1.76
1.78
1.80
1.82
1.84
1.86
1.88
1.90
re 92:
1.94
1.96
1.98
2.00
LOW
2.04
2rei0i6
2-08
PR NAO)

4.45
4.33
4.22
4.11
4.00
5\6.9'0
3.80
3.70
3-61
4p
3.43
3.34
3-26
3.18
3.10
3.02
Zire DD
2.8/7
2.80
2.74
2.6/7
2-61
ZieeD1D
249
2243
Zeal
PDS?)
2.26
PAA
2616
2-11
Die Of
2-02
1.98
Us Sis!
1.89
1.85
1.81
re
1.74
1.70
IEA 7
1.63
1.60
e57
1.54
ayeyal
1.48
1.45
1.42

FM

=OVer212)3
= Orere2 9
-0. 216
(0) 72d
-0.205
-0.202
Ore Led
=(ie 195
=O 92
-0.187
= Ole 184
-0.181
—Olenlyi 9,
-0.174
-0.167
-0.164
= (ie Lo
=O lS
-0.156
= lOven Lesics
=e lis 2
=O.6 151
Oks 147
-0.145
-0.144
-0.142
-0.140
—Or 139
-0.137
—Ole Los
=0.130
—Olonl 29
=O 126
=Ore tl 2)5
-0.124
=O. P21
=O 120
= Ore laleg

A =-
Ly = (8) (— \G@,) (7)

III. CURRENT-INDUCED CORRECTIONS WHEN BOTTOM PRESSURE
MEASUREMENTS ARE USED TO DETERMINE SURFACE WAVE HEIGHTS

Peregrine (1976)* and Jonsson, Skovgaard, and Wang (1970)° indicate the
errors involved in not correcting for the current when using bottom pressure
measurements to determine wave heights. They give numerical results which
cover some of the conditions encountered in practicee A more extensive set of
correction curves is presented in this section. These curves allow an esti-
mate of current corrections to be made for the range of circumstances which
usually occur in the field.

Peregrine (1976)© indicates that if a monochromatic surface wave height,
H, has a bottom pressure fluctuation magnitude, P, then H can be esti-
mated from P_ by

H= Hy pe)

where

dnd,

(9)

Hy = 2}cosh L ae
V W

with cosh the hyperbolic cosine, Py the water density (2.00 slugs per cubic
foot or 1,031 kilograms per cubic meter); g, dp, and Ly are the same as
previously defined. Equation (9) takes the current into account since the
wavelength Ly for a nonzero current rather than La for no current is
used. Usually the current is not taken into account when estimating H from
P. Rather, the equation analogous to (9) but with L instead of Ly is
used to estimate H from P; i.ee., H = H, is usually used where

2nd
Hy = 2}/cosh a s! ae (10)
A Pye
The wave height factor, Ry,» where
H cosh( 21d jt )
gee ee oN Gal

ty cosh( 2nd, /L)

4PEREGRINE, DeHe, ope cite, pe 10.

SJONSSON, I.G., SKOVGAARD, C., and WANG, J.D., “Interaction Between Waves
and Currents,” Proceedings of the 12th Conference on Coastal Engineering,
American Society of Civil Engineers, Vol. 1, 1970, pp. 489-509.

SPEREGRINE, DeHe, op. cit.

14

allows the "correct" wave height value, H, to be determined from the uncor-
rected height, Has by using the simple equation

HN= (Rg) Hy (12)

Contours of R are shown in Figure 3. The F and & axes are the same as
in Figure 2. The boundary of the forbidden region is again indicated by the
curve labeled F = FM. Figure 3 shows that Ry, is always greater than 1.0
for F less than zero (for “adverse” currents where waves propagate against
the current) and Ry is less than 1.0 for F greater than zero (for "“follow-
ing" currents where waves propagate with the current). Figure 3 also shows
that for a given value of F, adverse currents have a larger effect (i.e.,
give Ry values further removed from 1.0) than following currents.

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0.2
0.!

-08 -06 -04 -02 0 02 O04 O06 O8 41.0
F[F=(veos8)/(9 ay)" |

t
°

Figure 3. Contours of dimensionless wave height factor,
Ry» given by equation (11). Waves cannot prop-
agate for values of F and %& which lie in the
forbidden region (boundary line F = FM).

15

Knowledge of Ry, can also be used to estimate whether current effects are
great enough to warrant being taken into account when using an observed bottom
pressure fluctuation magnitude, P, to get a wave height, H. For the range
of current speeds, average depths, angles between the current vector and wave
orthogonal, and wave periods, the largest and smallest values of Ry (denoted
by RHYL and Ry,S;> respectively) are estimated. If both are within the
acceptable range, then the current correction can be neglected and the wave
height estimated from

at =, (13)

where Hy is given by equation QHO)ie = Lt edithexr Ry, Has Ry, g or both are
out of range, then equation (12) must be used to compute H. Practical limits
on the acceptable range for both Ry eal and Ry.s are 1.15 and’’0.855" 4vecn rye
the wave current interaction produces Ry values which differ from 1.0 by
+0.15 or less, the effect of the current can be neglected. Figure 3 shows
that Ry py is the Ry value corresponding to the smallest F and largest
Q that is expected. Analogously, Ry S is the Ry value corresponding to
the largest F and largest Q that is expected. The largest expected 2,

QL, is given approximately by
27 dr T 1/2
a, = Sef SE) 14)
TSN otea/

where TS is the smallest wave period of interest and dt »L, thee dargest
time-averaged water depth expected (i-e., largest value of dy expected).
The largest F, FL, and the smallest F, FS, that can be expected are
given approximately by

FL ree cre 1f. 20° =< 65 7< "90° (0 rad < 6S < > rad) (15a)
Vpe2
,s)
ine ee (VS)_cos(@S) if 90° <) 6S) < 180°( = 0S < 7 rad) @5b)
(gd yes
a
and
Fs acest if 90° < @L < 180°(> rad < OL < rad) (16a)
(8dy 5)
FS = Se! if 0° < eL < 90° (0 rad < OL < 5 rad) (16b)
(2d, ,)

where (VL, VS) are the (largest, smallest) current speeds expected, (6L, 6S)
are the (largest, smallest) values of the angle 6 expected, dr, gis the
smallest time-averaged water depth expected (i.e., smallest value of dy
expected), and dy, L has already been defined. If FS is less than the
FM value corresponding to 2 = QL, then FS is reset equal to FM.

In computing the wave height elevation, H, from the fluctuating bottom

pressure magnitude, P, there may be a range of wave periods over which a
current can be neglected (H is given approximately by Hy where H, is

16

defined by eq. 10) and another range over which the current must be accounted
for (H is given by (Ry) Hy where R, is defined by eq. 11). Figure 3 and
the values of FL and FS, once computed, can be used to determine the
smallest wave period beyond which currents can be neglected, TS'. For all
wave periods greater than or equal to TS', the current can be neglected when
computing H from P. TS' is given by

d IW A72
27 Teles
tir r= 5) 1 4
TS qs7 5 for 2S 0) (17a)
TS' = © (iee., currents are never important) for 2S' = 0 (17b)
where
QS' = MIN(QA, QB, RC, QD) (18)

MINC ) means take the minimum value of the numbers in parentheses, and the
factors MA, NB, LC, and MD are estimated from Figure 3 by using the
following definitions:

QA = the value of & which corresponds to Ry = 0-85 when F = FL
(QA = 0 if FL is less than or equal to 0)

QB = the value of & which corresponds to Ry = 1.15 when F = FL

(2B = 0 if FL is greater than or equal to 0) Gay
QC = the value of ® which corresponds to Ry= 0.85 when F = FS

(2c = 0 if FS is less than or equal to 0)

QD = the value of which corrresponds to Ry = 1.15 when F FS

(QD = 0 if FS is greater than or equal to 0)

Figure 4 gives a schematic representation of how fA and QD are related
to FL and FS for the example problem presented in the next section (in the
problem, 2B = NC = OQ).

Q

FORBIDDEN
REGION

Figure 4. Schematic representation of how fA and QD are related to FL and FS
for the example problem in Section IV. For the problem, 2B = RC = 0.

IV. EXAMPLE PROBLEM

This example illustrates the method used to estimate the maximum and mini-—
mum wave height factors, Ry and Rus whether the current can _ be
neglected completely in computing wave height, H, from the fluctuating bot-
tom pressure magnitude, P, and if not what range (if any) the current can be
neglected.

GIVEN: A bottom-mounted wave pressure gage is located near an inlet at an
elevation of -19.68 feet (-6 meters) referenced to the 1929 National Geo-
detic Vertical Datum (NGVD). The tide elevation ranges from -3.28 to 3.6
feet (-l1 to 1.1 meters) NGVD. Current speeds from 0 to 4.92 feet (1.5
meters) per second are known to occur at the location of the gage. The wave
climatology indicates that the dominant waves have periods between 6 and 15
seconds and can travel at any angle with respect to the current.

FIND:

(a) The largest and smallest values of Ry (Ry L and Ry 3) for the
conditions expected and the range of wave periods of interést (periods
between 6 and 15 seconds). :

(b) Whether the currents can be neglected for the entire range of wave
periods of interest when bottom pressure observations are used to determine
wave heights.

(c) The range of wave periods over which currents can and cannot be
neglected if they cannot be neglected over the entire range of interest.

SOLUTION:

Step l. Set 9S and 9L. Since the angle between the direction of wave
travel and the current can take on any value,

6S = 0; OL = 180° = m rad

Step 2.2 Set VL, VS, and TS. From the given values of the problen,

Vin =] 4.92 "ft (1.5 m/s
vs = 0
TS = 658

Step 3. Compute dy S and dp L* From the tide elevation and _ bottom
elevation information Ziven, ‘

dps = 19.68 -3.28 = 16.4 ft (5 m)

19.68 + 3.60 = 23.28 ft.(7.1 m)

ae
Step 4. Compute QL using equation (14).

2(3.14159) ( re Cy ap ae \

6 32.2 ft/s2 = 0.89 (dimensionless)
Ss ° EVs

18

Step 5. Compute FL using equation (15) and FS using equation (16).
Since 06S is zero, equation (15a) is used for FL, giving

(4.92 ft/s) (1.0)

L. = —<—_—_$_$_$______—____________. = 0.214. (dimensionless)
[(32.2 ft/s?) 16.40 fr] }/2

Since @L is 180° (mw rad), equation (l6a) is used for FS, giving

~(4.92 £t/s) (1.0)
7S = SS = = 0 DY (daimensatonilles's))
[(32.2 ft/s*) 16.40 £r])/2

Step 6. In Figure 3, find the value of Ry corresponding to FL and
QL. This value of Ry, is Ry g given approximately by
3

Rus = 0.7/7 (dimensionless)

Step 7. In Figure 3, find the vwalue of Ry corresponding to FS _ and
QL. This value of R, is Rabe given approximately by

Ruy = 1.75 (dimensionless)

Steps 1 through 7 solve part 1 under "Find."

Step 8. To answer part 2 under "Find," note that Ryo is less than
0.85 and Ry py is greater than 1.15. As a result, the current cannot be
neglected for all wave periods between 6 and 15 seconds.

Step 9. To solve part 3 under "Find,” Figure 3 is first used to deter-
Mine the factors MA, QB, QC, and 2D. Invoking the definitions for
these parameters given in equation (19), it is found that approximately (see

Fig. 4)
QA = 0.80; QB = NC = 0; QD = 0.59 (all dimensionless)
Step 10. Compute &S' using equation (18). The result is

QS' = MIN(O.80, 0.0, 0.0, 0.59) = 0.59 (dimensionless)

Step ll. Compute TS' using equation (17) and the value of
found in step 3. The result is

dy 1
[263 1SO)) 7 23e2e te Vis
Ts! = ————_______ ( —______- = 9.1 s
0.59 32.2 ft/s?

This value of TS'- shows that only for waves between 9.1 and 15 seconds
can the current be neglected in computing H from P. Waves between 6 and
9.1 seconds must have the influence of the current taken into account when
using bottom pressure measurements to compute wave heights.

19

Ve. SUMMARY

Some of the ways in which a horizontal current influences surface gravity
waves and their measurement have been presented. Relatively simple hand-
calculation methods have been described which provide a means to estimate
(a) the wavelength modification due to a current, (b) whether a current can
prevent waves from reaching a particular location, (c) the correction needed
to compensate for a current when observed bottom pressure fluctuations are
used to estimate surface gravity wave heights, and (d) the range of periods
(if any) where the effects of currents can be neglected when surface wave
heights are estimated from bottom pressure fluctuations.

20

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