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DATA LIBRARY |
REFERENCE COLLECTION H.o. Got
SH QODS HOLE OCEANOGRAPHIC INSTITUTIC’s

WIND WAVES AND SWELL

PRINCIPLES IN FORECASTING

Prepared for the

Hydrographic Office, U. S. Navy
by

The Scripps Institution of Oceanography
University of California
La Jolla, California

3 |

BI if |
WS |

H.O. Misc. 11,275|

ce entaeoo TOeE0

MUNN, TW

1OHM/ 1814

WIND WAVES AND SWELL
PRINCIPLES IN FORECASTING

Table of Contents

INTRODUCTION
SURFACE WAVES IN WATER

General discussion

Waves of very small] height

Deep-water waves of moderate and great height
Interference of waves; short-crested waves; white caps

EMPTRICAL KNOWLEDGE OF WIND WAVES AND SWELL

Measurements of waves and swell
Comparison of measured and computed values
Empirical relationships between wind and waves

GROWTH Of WIND WAVES
DECAY OF WAVES

Waves advancing into regions cf calm

Effect of following or opposing winds

Distance from which observed swell comes; travel time;
velecity of wind which produced the swell

TH STATE OF THE SEA
FPORECASTING OF WAVES FCR SHORT FETCHES
FCRECASTING OF SWELL 9 6

Determination of wind, fetch, and duration
Wind direction
Wind velocity
fetch
Duration of wind
Determination of highest wince waves
Determination of the swell
Waves advancing througk regions of caim
Following or opvosing winds
affect of following or cpposing winds
Remarks on forecast
Example

FORECASTING Of THE STATE OF TH SEA

APPENDIX: WAVES ENTERING SHALLOW WATER: BREAKERS AND SURF

WIND WAVES AND SWELL

PRINCIPLES IN FORECASTING

Prevared for the Hydrogravhic Office, U. S. Navy
by
The Scripps Institution of Oceanography
University of California

INTRODUCTION

Study of the problem of forecasting sea and swell was started
at the request of the Army Air Forces and is being continued under
the direction of the Hydrogravhic Office, U. S. Navy.

Four vroblems of forecasting are involved: (1) forecasting
the length and height of the swell in the open sea, (2) forecast-
ing the swell reaching exposed or partially exposed anchorages,
(3) forecasting the height of breakers and the amount of surf on
any given beach, and (4) forecasting the state of the sea in any
civen ocean area. The first problem involves two steps: (a) de-
termination of height and period of the waves which emerge from
any given wind area and which may arrive as swell on a distant
coast, (b) determination of the travel time and the decrease of
height of the waves as they proceed from the wind area. For the
second and third problems an additional factor is involved, namely,
the determination of the transformation of the waves as they enter
into shallow water and wash the beach. The fourth problem involves
two steps: (a) determination of the highest waves found under giv-

en wind conditions and (b) establishment of the relation of these

waves to the state of the sea as described by a scale such as the
Doweilas Sea Seale.

This manual deals with the generation of waves by wind and
with the travel of waves in deeo water after they have left the
POSLOMS OF SoicoOde viekals5 Wiswlaiocls ecuee Ceseirillsscl stoic Cloweieiul ies mye
the characteristics of wind waves by means of.data from adequate,
consecutive synoptic weather maos and for forecasting swell off
coasts.

Relationships between waves and the three important variables,
wind at the sea surface, fetch (the stretch of water over which the
wind blows), and duration (the length of time the wind nas blown)
are discussed. Verifications and interovretations of the empirical
laws developed by various observers of waves are given, together
with gravhs for use in forecasting wind waves and swell.

In order to use the granhs most effectively their »hysical
significance and limitations must be clearly understood. Forecasts
should therefore not be atteupted until the forecaster has studied
the first part of the paper which describes the processes leading
to the growth and decay of waves.

Tests of the wethod made to date indicate that swell forecasts
can be made with about the saine certainty as that of Most meteor-
OGQLOLLCGLL WOCSCAS OSS LicOLMOS wIC Cleiess lies MOQ Mijooiwweias wOrwr wide
forecasting of swell because considerable tiie elapses between tre
generation of waves in distant storm areas and their arrival at the
coast. Thus, after exverience has been gained, it 1s possible to
forecast swell several days in advance. Forecasts of the state of

the sea, on thevother hand), must (be sbasied) ina param onmord@ =m osrpnc

weather maps und cannot be prepared for periods longer than those
for which these maps can be considered valid.

It is contemplated that a more comprehensive edition of this
manual will be issued in the near future. This will contain meth-
ods for determining the transformation of waves in shallow water
and for forecasting surf from synoptic weather data or from observa-

tions of waves offshore.

SURFACE WAVES IN WATER

General Discussion

AV Waves) described by ats Venebhs is ije) the horizontal
distance from crest to crest or trough to trough (see fig. 14),
and bi iu smc ohne hy ic. bnew ver tically dustance som atacoued
to crest. A wave is furthermore characterized by its period, I,
i.e. the tiine interval between the appearance of two consecutive

crests ab a given position.

Ener pote aaa

WATER LEVEL

Figure 1. Surface waves. A. Profile of wave.
B. Advance of wave, showing the wave profile at the
EINES % 6 O, % = Wik, Elael lo W/Aa il wey bi Wy
the wave has advanced one half wave length, L/2.

A wave may be standing or progressive, but this discussion
deals with vrogressive waves only. In a progressive wave, if
the length and energy are constant, the wave height is the same
at all localities and the wave crest appears to advance with a
certain velocity (fig. 1B). During one wave period, T, the wave
crest advances one wave length, L, and the velocity of the wave,

C, is therefore defined as
zal
Hl

The motion of the water particles depends on the wave length
and the depth of the water. In general, it can be stated that
the advance of the wave form is caused by convergences and diver-
sences of the horizontal motion. In front of the crest the motion
is converging and the surface is rising, but behind the crest the
motion is diverging and the surface is sinking.

By energy of the wave is always understood the average energy
over one wave length. The energy is in part potential, Ey. asso-
ciated with the displacement of the water particles abone ok below
the level of equilibrium, and in part it is kinetic, E,, associated
with the motion of the particles. In surface waves half the energy
is present as kinetic and half as potential. The total average
energy per square foot is E = 1/8 2 pH’, where g is the accelera-
tion of gravity and pis the density of the water. For a 10-foot
high wave the total average energy is 800 foot-pounds per square
foot. Since g and p can be considered constant the energy per unit

area in a wave is vroportional only to the square of the wave height.

For the total energy per unit width along a wave length it is

necessary to multiply the energy per unit area by the wave length.

Waves of Very Small Height

By waves of very small height are understood waves for
WALGlA WAG weeLO C1 M@iedlrs wo LeVian a6) W/KOO (oie Wess Bas
simplest wave theory deals with such waves, the form of which
can be represented by a sine curve (see fig. 3). In water of

constant depth, d, such waves travel with the velocity

c= /g_L_ tanh 279
270 L
where g is the acceleration due to gravity.

If d/L is large, that is, if the wave length is small com-
pared to the depth, tanh 27 d/L approaches unity and one obtains

x) * tm

These waves are called deep-water waves.

If d/L is small, that is, if the wave length is large com-

pared to the depth, tanh 271d/L approaches 27d/L and one obtains

—

These waves are called shallow-water waves.

In general, waves have the character of deev-water waves when
the depth to the bottom is greater than one half the wave length
(d=>L/2). However, for shallow-water waves the depth must be less

than one twenty-fifth of the wave length (d<L/25).

In a low deep-water wave the water particles move in circles.
At any depth, z, below the surface the radius of the circular path

followed by a particle is

because the particles complete one revolution in the time T (see
i> 2))o

A water varticle at the sea surface remains at the surface
throughout its orbit. A water particle at a given ‘averace wWdepth
below the sea surface is farthest from the surface when it moves
in the direction of wave progress.

In a low shallow-water wave the vertical motion of the
particles is negligible and the horizontal motion is independent
of depth. The particles move back and forth, following nearly
straight lines.

In a deep-water wave only half of the energy advances with
wave velocity, whereas in a shallow-water wave all the energy
advances with wave velocity. The reason for this difference is
that in a deep-water wave only the potential energy varies period-
ically and advances with the wave form, but in a shallow-water
wave both potential and kinetic energy vary veriodically and both
advance with the wave form. These laws can also be stated by say-

ing that the energy advances at a rate which, in a deep-water wave,

equals half the product of energy and wave velocity, whereas in
a Shallow-water wave it equals the product of energy and wave

velocity.

DIRECTION OF PROGRESS

TT) Paar eee!
| OO
d
|
|
|

Figure 2. Movement of water particles in a deep-
water wave of very small height. The circles show
the paths in which the water particles tiove. The wave
profiles and the positions of a series of water parti-
cles are shown at two instants which are one quarter
of a period apart. The full-drawn, nearly vertical
lines indicate the relative vosSitions of water par-
ticles) which Iie exactly on-vertical lines when the
CES Ole WINE wicOUiela Oi WINS WEHYS IOVS) Eval Wis) Glelsiase|
lines show the relative positions of the same particles
one quarter of a period later.

Deep-water Waves of Moderate and Great Height

By waves of moderate and great height are understood waves for

which the ratio of height to length (H/L) is from 1/100 to 1/25

and aeGasomenly/ 215 itiously/a/ arlene Citpiaviclaya meu Cit @rcMeOtL hie Sem wanes
Can not be represented by a sine curve. For waves of moder-
ate height the form closely approaches the trochoid, that is’,
Ae GWuAVS Wiaslela WS ClesSCGiceilloacl iy Cl jOOLmMG OM 2 CGISeE Winwela wos
below a flat surface (fig. 3). Waves of great height deviate
from the trochoid; the troughs are wider and flatter and the
crests narrower and steeper. The wave form becomes unstable

when the ratio H/L equals 1/7.

LINE ALONG WHICH DISC ROLLS

Figure 3. Profile of a trochoidal wave (full-drawn
lines) and of a sine wave (dashed lines).

The wave velocity increases with increasing steepness (in-
creasing values of H/L), but the increase of velocity never
ExC@oas 2 weie O©Smlo>

IMAI) uses S Het CASA WO ed NP NEI EN I TOONS) ee
radii of which decrease rapidly with depth. The particle
velocity is not uniform but is greatest when the varticles
GOOG) WSUS ME) WOO Oi Wl Creloaim (siony aia Wisi wine GClaweSoe wali Ox
wave progress), with the result that the particles upon com-

pletion of each nearly circular motion have advanced a short

distance in the direction of progress of the wave (fig. 4). Con-
sequently, there is a mass transport in the direction of progress
of the wave. The mass transport velocity (u') at the sea surface

is expressed by the formula,

Figure 4. Orbital motion during two wave periods
of a water particle in a deep-water wave of iioderate
or great height. In two wave periods the forward dis-
placement equals 2u'T.

The velocity is appreciable for high, steep waves but is
very small for low waves of long period. Mass transport in
waves has received little attention in previous work because
in most practical applications it is sufficient to consider
the water particles as ioving in circles regardless of the
wave height. In order to understand the growth of waves
through wind action, however, it is necessary to take the

mass transport velocity into account.

Interference of Waves; Short-crested Waves; White Caps

When waves of different heights and lengths are vresent simul-
taneously the appearance of the free surface becomes very compli-
cated. At Some voints tie waves are opposite in phase and there-
fore tend to eliminate each other, whereas at other points they
coincide in phase and reinforce each other.

As a Simple case, consider two trains of waves which have the
same height and nearly the same velocity of progress. Owing to
interference, groups of waves are formed with wave heights rough-
ly twice those in the component wave trains, and between the wave
groups are regions in which the waves nearly disappear (fig. 5A).
Analysis shows that these groups advance with a velocity which is
nearly equal to one half of the average velocity of the two trains.

AS another example; consider the simultaneous presence of long,
low swell and short but high wind waves. The resultant pattern is
illustrated in Figure 5B from which it is evident that the short,
high waves dominate to such an extent that the presence of the
SWE sl SmoODsicumedr

So far, the discussion has dealt only with long-crested waves,
that is, waves with very long straight crests and troughs. Waves
can, however, also have short, irregular crests and troughs. In
the presence of such short-crested waves the free surface shows a
series of alternating "highs" and "lows", as indicated in Figure 6.
This figure illustrates the topography of the sea surface, "highs"

being shown with full-drawn lines and "lows" with dashed lines.

Figure 5. Wave vatterns resulting from interfer-
ence. A. Interference of two waves of equal height
and nearly equal length, forming wave groups. B. In-
terference between short wind waves and long swell.

Figure 6. Short-crested waves. L = wave length,
IY & ©wesw IeiMaeul.

; White caps are formed by the breaking of relatively short
waves which often appear as "riders" on longer waves (fig. 5B).
Such short waves may grow so rapidly that their steeoness reaches
the critical value H/l = 1/7 and they break. If interference

occurs long waves may attain this steepness and break.

EMPIRICAL KNOWLEDGE OF WIND WAVES AND SWELL

Measurements of Waves and Swell

Wind waves are defined as waves which are growing in height
under the influence of the wind.

Swell consists of wind-generated waves which have advanced in-
to regions of weaker winds or calms and are decreasing in height.

So far, the discussion of surface weves has dealt mainly with
waves which appear as rhythmic and regular deformations of the
surface. Because of interference, the formation of snoreeerested
waves, and the breaking of waves there is, however, little
regularity in the appearance of the sea surface, particularly when
a strong wind blows. Although individual waves can be recognized
and their heights, periods,-lengths, and velocities measured, such
measurements are extremely difficult and comparatively inaccurate.
The lengths of most waves and the heights of low waves are likely
to be underestimated, while the heights of large waves are general-
ly overestimated. Wave heights above 55 feet are extremely rare,
yet the literature contains many reports of waves exceeding 80 feet
in height. Such errors are vrobably due to the complexity of the
sea surface and the movement of the ships from which measurements

are made.

Reliable measurements of wave height, H, are so diffi-
CMbtunabe eA Cnemoll tic me poOmbtied = Valwes) se pimresient) crude
estimates. The height of a large wave is estimated as the
eye height of the observer above the water line when the
Shlpens. on even keel in the trough of the wave, provided
that the observer sees the crest of the wave coincide with
the horizon. The height of a small wave is estimated dir-
ectly, using the dimensions of the ship for comparison.

On board a siiall ship the height of waves which are more
than twice as long as the ship can be recorded by a micro-
barograph.

Mas WWE SwsLOG, W, SLi WE MOASUIPSC Iiy awSCOIwGlAMsZ wins
time interval between successive appearances (on a wave
crest) of a well-defined patch of foam at a considerable
distance from the ship. In order to obtain a reliable
value, observations should be made for several minutes and
averaged.

The wave length, L, can be estimated by comparing the
ship's length with the distance between two successive
GPESUS 5 Wans preocsdwuice Leads CO WaAGSieiweiiiad wesvllas., ladiomwevere-,
because it is often difficult to locate both crests relative
to the ship and because of disturbance caused by the move-
ment of the ship.

he velocitylon aheswame, ©) Can pe found by cecording
the time needed for the wave to run a weasured distance
along the side of the ship and by applying a correction for

the ship's sveed.

Comparison of Measured and Computed Values

Theory indicated that velocity, length, and period for deep-

water waves are interrelated by the formulae

= Be wie = pele ° Surette z= 2, -/2£7 =) 2h
C= a z/6 a= Sg3 8 nae mo Te "ape oa

Wawa G ain Isaiows, Ih sind wesw, ial WY iia Sooo

G= 1.34. /E = 3.03 T
= 00555 C= 5.12 1
tS 0.42Q/L = On23 6

Thus, if one characteristic is measured the other two can be com-
puted, and if two or three are measured the correctness of the
theory as applied to ocean waves can be checked. Comparisons of
measured and computed values have given satisfactory results, in-
dicating that wind waves and swell in deep water do have the char-
acteristics described above. In general, the conclusion that the
ratio H/L always remains less than 1/7 is also confirmed by obser-
vations, as waves of this or greater steepness are very rarely

reported.

Empirical Relationships between Wind and Waves

Observations of waves have not Ted to clear-cut conclusions
about the empirical relationships between the wind and waves. The
following nine approximate relationships have been proposed by

various workers:

1. Maximum wave height and fetch. For a given wind velocity

the wave height becomes greater the longer the stretch of water
(fetch) over which the wind has blown. Even with a very strong
wind the wave height for a given fetch does not exceed a certain

maximum value. For fetches larger than 10 nautical miles it has

eee eo IL [F

where caine represents the maximum probable wave height in feet

been observed that

with very strong winds and F is the fetch in nautical isiles.
2. Wave velocity and fetch. At a given wind velocity the
wave velocity increases witn increasing fetch.
36

Wave height and wind velocit The height in feet of the

ereatest waves with high wind velocities has been observed tc be
about 0.& of the wind velocity in knots. If the entire range of

wind velocities is considered, the observed data conform to
H = 0.026 UM

where U revresents the wind velocity in knots.

4}. Wave velocity and wind velocity. Although the ratio of
wave velocity to wind velocity has been observed to vary from
less than 0.1 to nearly 2.0, the average maximum wave velocity
apvarently slightly exceeds the wind velocity when the latter is
less than about 25 knots, and is somewhat less than the wind
velccity at higher wind speeds.

5. Wave height and duraticn cf wind. The time required to

develop waves of maximum height corresponding to a given wind

increases with increasing wind velocity. Observations show
that with strong winds high waves will develoo in less than
LZ MOwes ,

6. Wave velocity and duration of wind. Although observa-~
bional data are inadequate, it is Known that for a given fetch
and wind velccity, the wave velocity increases rapidly with time.

7. Wave steepness. No well established relationship exists
between wind velocity and wave steepness, that is, the ratio of
wave height to Length. This ais orobably due to the fact that
wave steepness is not directly related to the wince velocity, but
depends upon the stage of development of the wave. The stage of
development), Om age of (the wave, can be conventently expressed
by the ratio of wave velocity to wind velocity (C/U), because
during the early stages of their formation the waves are short
and travel with a velocity much less than that of the wind, while
at later stages the wave velocity may exceed the wind velocity.
In order to establish the probable relation between wave steep-
ness and wave age all wave observations were examined which
appeared to be consistent with certain basic requirements and
HO Wintela WeULMes Cie Il, ih (one ( Cie BE), Euscl W Weise ieeeord sel.
The corresponding values of H/L and C/U were plotted in a dia-
eram (fig. 7). The scattering of the values is no greater than
MROWLLG| INE Ehcorxowmeol, COOMSilClSigiias while) EACSee Giciceies| Cl MSASpECSiMSinws -
There apvears to be a definite relationship between the steep-
ness and the age of the wave. This relationship, shown by the
curve in Figure 7, plays an important part in the theoretical

discussion.

x|-

WAVE STEEPNESS,4, IN PERCENT
uw

1.0 Ma 1.2 1.3 1.4 5 1.6 1.7 1.8

ie) al 2 BS) 4 5 6 7 8 K:)
WAVE AGE, <

Figure 7. Relation between wave steepness as
expressed by the ratio wave height to wave length,
H/L, and wave age as expressed by the ratio wave vel-
ocity to wind velocity, C/U. Observed values shown
In? OL1eCLOS ¢

Se WecreascmoOtmlelehbOmeswelel. | Lhe hea this som ssweliide=

creases as the swell advances. Roughly, the waves lose one-third
of their height each time they travel a distance in miles equal to
their length in feet.

9. Increase of period of swell. Some authors claim that
the veriod of tne swell remains unaltered when the swell ad-
vances from the generating area, wiereas others claim that
the veriod increases. The greater amount of evidence at the
present time indicates that the veriod of the sweil increases

as the swell advances.

GROWTH OF WIND WAVES

A knowledge of the height, velocity, and direction of progress
of wind waves is necessary if their arrival as swell at a distant
coast is to be vredicted. Direct observations of these wind waves
are rarely available, but their height and veriod can be determined
from consecutive synoptic weather maps if the relationship between
wind and waves is known.

In the area of wave Pomme ttton the highest waves present at any
time depend upon the wind velocity, the stretch of water over which
the wind has blown (the fetch), the length of time the wind has
been blowing cver the fetch (the duration of the wind), and the
waves which were vresent when the wind started blowing (the state
cf the sea). These four factors can all be determined if a sequence
of weather maps is available showing the meteorological conditions
OVE wa OGSEIAS CMs sdeomyels Oi, Seay LZ Cre Zh Inewics, WHSese@ mej
must be based on a sufficient number of ships' observations to make
possible the plotting of fairly accurate isobars from which winds
may be deteriined. In the tropics wind observations imust be avail-
able from ships or exposed stations on islands. In middle and
higher latitudes direct wind observations on ships will serve as
checks on wind estimates from the isobars.

Thus, with adequate weather maps at one's disposal, an estimate
of the wind waves can be made if accurate relationships between wave
height and wind velocity, fetch, and duration are known. Such
accurate relationships have not been developed in the past because

of the inadequacy of observational data on waves, but they can be

determined theoretically from a consideration of the wind energy
available for wave forination if tre fundamental assumption is
made that the velocity (pericd) of a wave always increases with
time.

The area in which waves are formed is called the generating
area. In such an area waves receive energy from the wind by two
orocesses, by the push of the wind against the wave crests and
by the pull or drag of the wind on the water.

ihe eneney transite by push depends upon the diftterence
between wind velocity and wave velocity. If the waves advance
with a speed much less than that of the wind the vush is great,
but if the two velocities are equal no energy is transferred.

If the waves travel faster than the wind they receive no snergy

by push but on the contrary they meet an air resistance comoarable
to the air resistance against a traveling automobile. The effect
of the push of the wind or of the air resistance against the wave
depends on the wave form. There enters, therefore, a fundamental
coefficient which is related to the degree to which the wave is
streamlined and which is called the "sheltering coefficient." The
determination of this coefficient is necessary for an exact evalua-
tion of energy transfer by push.

The pulling force of the wind always acts in the direction of
the wind. It is the same at the wave crest and the wave trough
but the effect differs. Energy is transferred from the air to the
water (the movement of the surface layer is speeded uv) if the sur-

face water moves in the direction of the wind, but energy is given

WY)

off from the water to the air (the movement of the surface water
is slowed down) if the surface water moves against the wind. If
wind and waves move in the same direction the water particles

move in the direction of the wind drag while at the crest, but
against the drag when in the trough (see fig. 2). In the absence
of a mass transport velocity the particle velocities at the Bee
and the trough are equal but in opposite directions, so that the
effect of the pulling force of the wind at the wave crest is ex-
actly balanced by the effect at the wave trough. In the presence
of a mass transport velocity, however, the forward motion at the
crest is greater than the backward motion in the trough (sestjeey dh)
and a net amount of energy is transferred to the water. No satis-
factory explanation of the growth of waves can be given without
assuming a transfer of energy due to the wind pulling at the water
particles; and this fact is the best ergument for the presence of
a mass transport velocity in ocean waves.

Since the pulling force cf the wind over the ocean is known,
the energy transfer from the air to the water by wind drag can be
computed with considerable accuracy from the theoretical values
for uiass transport velocity given on page 9. iven when the wave
velocity exceeds the wind velocity, the effect of the wind drag
remains nearly the same because it depends unon the difference
between wind velocity and varticle velocity in the water, and in
eeneral the water varticles jwove imuch more slowly than the wind
even when the wave fori: moves much faster. If the wind can not

transfer -aergy to the water by pulling at the water particles,

no satisfactory exolanation can be given of the fact that waves
frequently have a higher velocity than the wind which produces
them.

Energy is dissipated by viscosity but the viscosity of the
water is so slight that this process can be neglected. There is
no evidence that energy is dissipated by turbulent motion in the
wave. The chief processes which can alter the wave height or
the wave velocity in desp water are therefore the push of the
wind, which becomes an air resistance if the wave travels faster
than the wind, and the drag or vull of the wind on the sea
surface.

Knowing the rate of energy transfer from the wind and the
rate at which the wave energy advances (page 6) it is possible
to establish a differential equation from which the relationships
between the waves and wind velocity, fetch, and duration are
obtained as special solutions. The equation contains three
numerical constants (including the "sheltering coefficient")
which have to be determined in such a manner that all the nine
empirical relationships are satisfied. This can be accomplished,
and at the same time discrepancies between existing empirical re-
lationships can be accounted for.

The growth of waves as determined in this manner is illus-
trated in Figures 8 and 9 which are constructed on the assumption
that a wind of a constant velocity of 30 knots started to blow
over an undisturbed water surface extending for 600 or more nau-
tical miles from a coast line. Figure 8 shows the height and

period of the waves as functions of the distance from the coast

WIND VELOCITY 30 KNOTS

5
fe <
Ww m
Ww

me
ce) m
z zy
ee fo)
Be go
bE a
ae
2 2
Wi
sf 2
uJ
> (2)
aq [o}

Zz
= iz)

FETCH, F, IN NAUT, MILES

Figure 8. +- Wave height and wave period as func-
tions of distance from coast line at 51 to 35h after
a wind of 30 knots started to blow over an undisturbed
water surface.

for every fifth hour after the wind started. First, small waves
are formed, probably by eddies striking the sea surface. At the
coast the waves remain low, but off the coast they travel with
the wind and grow as they receive energy by push and oull. When
the wind has blown for 5 hours one finds that with increasing
distance from the coast the waves increase rapidly in height and
veriod out to a distance of 35 miles. There the waves are 8.4
feet high with a veriod of 4.7 seconds. Beyond 35 miles similar
waves are present but there exists a striking difference between

conditions inside and beyond the 35-mile point. Inside of 35 miles

a stscady state has been reached, that is at any given point the
waves do not change, no matter how long the wind lasts, but beyond
35 wiles the waves continue to grow for a length of time which
depends upon the distance from the coast. After 10 hours a steady
state has been established to a distance of $5 miles, after 15
hous) ToOluamdustance on MOOlmilles: Jand Soons sin Bicure Cithe ful d—
drawn and dashed curves show the steady state. Parts of the curves
and the horizontal lines represent wave height and period as func-
tions of the distance from the coast at 5 to 35 hours after the con-
stant wind of 30 knots started to blow.

The fetch shown in Figure 8 can be limited either by the pre-
sence of a coast line or by the characteristics of a wind system
Over the epen ocean. it may be seen from the fisure that forva
given wind velocity the time needed to establish a steady state
depends only upon the length of the fetch. For a given fetch this
time depends, however, on the wind velocity and is longer for weak
winds than for strong winds. This time is called the minimum dur-
ation and is measured in hours. Plate I shows the minimum duration
as function of wind velocity and fetch.

Plates II and III show wave heights and periods as functions of
fetch and wind velocity when the duration is longer than the iinimun.

If the time is shorter than the minimum duration, the waves at
the end of the fetch depend on the wind velocity and the duration in
a wanner similar to that shown for a 30 knot wind in Figure 8. For
practical use Plates IV and V show wave heizhts and periods as func-

tions of wind velocity and duration.

£3

When using Plates II to V it should be borne in mind that the
curves are constructed on the assumption that a constant wind sud-
denly starts to blow over an undisturbed water surface. If the
wind velocity changes gradually, an average velocity has to be
introduced according to rules which are discussed when dealing
with the prectical applications. Also, allowances must be made
for-waves that are oresent when the wind starts blowing.

Some other characteristics of the growing waves are shown in
Figure 9. In the upper curve the wave steepness as expressed by
the rato H/lis) plotted azainst the tebceh tor la wandwotm SO isnots.
The curve shows the steady state and the horizontal lines show
the stage or development after 10, 20, and 30 hours. Before a
Steady state has been reached, that is, when the duration is
Shorter than the minimum duration, the steepness decreases with
time, and when a steady state has been established it decreases
Wath hebieh.

In the lower curve of Figure 9 the wave age as expressed by
the ratio, wave velocity to wind velocity, C/U, is plotted ageinst
fetch. The wave age increases with duration before the minimum

value is reached and with fetch after the establishment of a steady

If the corresponding values of H/L and C/U are plotted in a
gravh with wave steepness, H/L, and wave age, C/U, as coordinates
Wideny sabe lll” Eodeloielhy Cia qin Clave) slat WaeqbkeS 7/. WiaalGla IWSjoreesSemgs wlae
empirical data. Actually, this curve has been used for determin-

ing the constants needed for carrying out all computations. By

ineans of the curves in Plates II to V it can be ascertained that

the ewoirical relationships 1 to 6 are satisfied.

oats

wre)

ic:

we NAUT. MILES
a

$ 400

9
TS

(e)
ine)

WAVE AGE, ¢-

FETGH, F IN NAUT. MILES f

200 300 400

Figure 9. Wave steepness (upper graph), expressed
by the ratio H/L, as function of distance from coast
line at Loe. 20h, and 30h after a wind of 30 knots
started to blow over an undisturbed water surface, and
corresvonding representation of wave age (lower graph)
exoressed by the ratio of wave velocity to wind vel-
OGY, G/U.

According to Figure 9, with a 30-knot wind the wave veloc-
ity remains lower than the wind velocity at fetches of 600 miles
or shorter. With increasing fetch the wave velocity would, how-
ever, exceed the wind velocity and the waves would continue to
grow in height but decrease in steepness.

If the wave velocity exceeds the wind velocity the waves

can no longer receive energy by push but will lose energy because

of the air resistance they meet. They will however continue to
receive energy by the pulling force of the wind and will grow in
height until this gain is compensated by the loss due to air re-
sistance, which occurs when the ratio C/U equals 1.45. The fetch
and duration needed for reaching this stage increase rapidly with
increasing wind velocity, as shown by the values in Table 1. If
the fetch and the duration are longer than those listed in the
table the highest possible waves will be present regardless of

how much longer the wind blows.

Table I

Highest Possible Waves Produced by Different Wind Velocities,
and Corresponding Fetches and Durations.

(Ratio wave velocity to wind velocity equals 1.45,
ratio wave height to wave length equals 1/45)

Wind Highest waves Fetch Duration
velocity Height Period (naut. m. ) (hours)
(knots) (feet) (seconds)
10 26 L.8 260 25
20 OR 9.6 1040 50
30 Zou 14.4 2340 (D>
LO I 55) 19.2 4150 100
50 66n2 24.0 * 6500 1.25

Waves of the character shown in Table I may be present in
the trade wind regions and may be approached in the westerlies
of the southern oceans. In the middle and higher latitudes of
the Northern Heisphere the fetches are so short that with strong
winds the wave velocity always remains less than the wind veloc-

WWW 6

Plates II to V show only the highest waves present. These
waves have traveled the entire distance from the beginning of
the fetch. However, the wind can raise new waves anywhere in
the fetch, and some of these may grow slowly and reach heights
corresponding to the distances they travel, while others may
srow Gapidly and break. These eontribute to the broken ap-
pearance of the sea surface which is described as the "state of
the sea." The relationship between the wind and the state of

the sea is discussed later.

DECAY OF WAVES

Waves Advancing into Regions of Calm

When waves spread out frou a generating area into a region
of calm only half of the energy of the wave advances with wave
velocity. The consequence of this characteristic can be recog-
nized by. examining a Simple example. Assume that a series of
waves is foriied by rhythmical strokes of a wave machine which
at each stroke adds the energy E/2 in a given locality. The
first stroke creates a wave of energy E/2. In the time interval
between the first and the second stroke one half of this energy,
E/4, advances one wave length and one half, E/4, is left behind.
The second stroke adds E/2 to the part of the energy which was
left behind. On completion of the second stroke two waves are
present, one close to the wave machine with an energy 3E/4 and
one which has advanced one wave length with energy E/4. By re-

peating this reasoning, Table II has been vrevared, showing the

distribution of energy in the waves after sach of the first five
SipeOktes. | ANG) Slain Tid wine esi ibiime O18 wai wellolle e Cleit tinwe
pattern has already developed after five strokes; the waves which
have traveled the greatest distance have very little energy, the
wave which has traveled half way has an energy B/2, and each of
the waves closest to the machine has an energy which approaches
the full amount HH. When a large number of strokes have been
completed these gradations are much clearer and the distribution
of energy can be represented schematically by the curve in Figure
10, which shows that the energy advances with a definite "front."
At the front the wave height increases from nearly zero to nearly

its full value in a distance corresponding to a small number of

wave lengths, and this front advances with half the wave velocity.

Table II

Advance of Waves from a Wave Machine into Still Water

Number of Relative energy of advancing waves
strokes

iL L/2

z 3/h ih

3 Ws ys Uys

Ih IS/lS IU/ML F/G — L/L6

y) BIB) 26/32 MS /32 O/ 32 1/32

When applying the above reasoning to the behavior of wind
waves which advance into regions of calm it is necessary to con-

sider also the following facts: (1) the wave loses energy because

of the air resistance against the wave form, (2) the wave velocity

(period) increases continuously.

"FRONT " OF ADVANCING ENERGY

Fisure 10. Advance of wave energy in time t from
a Source INTO Sitti waiter. A viery smalieanounitn om
the energy has advanced the distance Ct. The region
of ravid increase, “the front," has advanced the dis-
tance C t.
ZR

When the problem is treated analytically it is not necess-
ary to introduce any new constants. The travel time of the waves
and the decrease in wave height can be obtained as special solu-
tions of the fundaiental equation which was discussed in the sec-
tion on the growth of waves.

The results cf this analysis are presented in Plate VI. The
coordinates are the wave period at the end of the fetch, Tp and
the distance of decay, D, that is, the distance which the waved
travel through areas of calm. The main part of the graph contains
two sets of curves. One set gives the factor by which the wave
height at the end of the fetch, Ha must be multipvlied in order to
fine the heisht o* the swell at Hie end of the distance of decay,

Hp. The other set zives the travel ti.ue, Up» (in hours) for the

distance, D. Inset I sives the length and velocity of a deep-water
wave for which the period is known. Inset Il gives the factor by

which the veriod at the end of the fetch, Tp must be multiplied in

OLeCGSie CO sinc aie josielOGcl Gig wine Giacl Of wae, ChiswemneSs Oi CleCeyy/, Th:

Hats factor cdejoeiacls only upon the reduction factor for the wave

nedela 5 il et The use of the diagrams will be described when dis-

DE dd

cussing the forecasting of swell.

Effect of Following or Opposing Winds

The effect of a following or an opposing wind on the decrease
of the heisht of the swell is also found from a special solution of
the fundamental equation of the "energy budget" of the wave.

LG is assumed that the increase in wave velocity over the distance
of decay is not influenced by following or opposing winds. Although
this assumption has little basis in either theory or observation it
MeO Moly MEAS cO BwppicomiimMewoly COieeSOw iceESuilos, lid tine Case OF A
following wind the computed wave heights may be somewhat too high
and the wave periods somewhat too low, whereas in the case of an
opvosing wind the heights may be too low and the periods too high.
Consistent differences between values computed on this basis
and observed values may later be used to improve the theoretical
aporoach.

The following or ovvosing wind may blow over only a part of
the distance of decay. The vroblem is to determine how much more
or how imtuch less the wave height decreases in any given distance

as compared to its decrease in the absence of any wind. This

problem can ber solved by ticans of Plate Vil, the use of which will

be explained when discussing the practical procedure.

Distance from which Observed Swell Comes;

Travel Time; Velocity of Wind which Produced the Swell

If the height and the period of the swell are observed it is
possible to find approximate values of the distance to the end of
the generating area from which the swell came, of the travel time
of the swell, and of the wind velocity in the generating area. In
Plate VIII the coordinates are the height of the swell (in feet)
and the veriod of the swell (in seconds).- The plate contains three
families of curves: full-drawn curves giving the distance to the
generating area in nautical miles, light dashed amienes giving the
travel time from the generating area in hours, and heavy dashed
curves giving the wind velocity in the generating area in knots.

The values which can be derived from the plate are only
apvroximate because the height and period of the swell depend also
upon the ratio between wave velocity and wind velocity (C/U) at
Ghe end ofthe: feten. Lhe sraph us constructed for C/U = O16.
corresponding to average conditions, and gives too high values if
C/U is smaller and too low values if C/U is larger. However,
variations ia 6/0 between 0.7 and O.9 will not introduce errors
exceeding 10 per cent, but errors will also arise from inaccuracies
in the observations of height and period of the swell and from lack
of knowledge as to changes caused by following or opposing winds.
The values read off from the graphs may therefore be 25 ver cent

iid, GICICOIe

THE STATE OF THE SEA

The preceding discussion has dealt only with long-crested
waves, that is, waves with very long crests and troughs. Waves
may also have wave-shaped crests and troughs. In the presence
of such short-crested waves the free water surface shows a ser-
168 Of alwcrmeGing Varelas” eiacl Vhows” (mace 10) 5 Mwrcineimnere ,
attention has been paid only to the waves which accumulate the
largest amount of energy and attain the greatest heights and
longest periods. In addition a these waves, and superimposed
upon them, a large variety of shorter and lower waves will also
be peesent. AL wind vellocities exceeding Beaurort 3 many of
these shorter waves imerease So rapidly in heleht thaiv they breaks
forming white caps. It appears that at -low wind velocities a
great amount of energy goes into the formation of regular long-
erested waves, while at high wind velocities a large part is
used in the generation of small and short-crested waves.

After the waves leave the generating area, the small waves

and the short-crested waves die out quickly because they contain

little energy, and the long-crested waves of maximum height, which
MEWS ISSA ClSelLiy Walwla alin wine jueoCSClibas CML YSIS, EIS wespoMmsiilole

for the emerging swell. In the generating area, however, the brok-
en appearance of the sea surface is chiefly determined by the pres-
ence of the small, short-crested waves and is déeSscribed- by the term
"state of the sea."

For the state of the sea there have been proposed several

scales of which the Douglas Sea Scale is the most widely used.

There exist, however, discrepancies between definitions of the
term "state of the sea" and cisagreeiients as to the wave heights
to be assigned to the descriptive terins. The following discus-
sion appears in "Instructions to Marine Meteorological Observers."

(U. S. Weather Bureau. Circulur M, 6th ed., 1938). nages 53-55:

"Ordinary waves which are moving with the wind
constitute the 'sea' while a relatively low, undulat-
ing sea surface, with motion in a direction different
from the local wind, is the 'swell.'

"These definitions are not entirely satisfactory.
Usually, the ocean surface is disturbed by both forms
of wave motion, with the swell from distant winds
crossing the local sea. The combined effect is the
"sea,' while the well-defined ridges of waves sioving
in a different direction from the local wind are the
4 SHE ILILSS 5 Y

Mating 565 Seale? (Maile inl, Golkuiiins I, 2s etal 3)/
SHOW d ben UScd sin wCKaASsciityincs tines Cheracter some
sea disturbance. In recording observations in accord-
ance with this scals, 'sea' may be considered to be com-
posed of swells, combined with waves produced by the
winds at the place of observation.

"The scale of sea disturbance is approximate, based
roughly on the observer's judgment as to the height of
waves."

On the other hand, the "Admiralty Weather Manual," 1938, pages

5O=5llL, Sweisese

"The state of the sea should be reported according
to the Douglas Sea Scale (Code XIII), which is here re-
produced with = table of heights of waves corresponding
to the code figures" /Table III, columns 1, 2, and 4/
"... Careful distinction skould be made between sea and
Swell, sea being the waves caused by the wind at the
place and time of observation, while swell is wave motion
due to past wind or wind at a distance. The direction
from which the swell comes should be noted to nearest
colpass point."

On the Meteor Expedition wave heights were measured from
stereophotogrammetric pictures and these wave heights were
compared to Simultaneous estimates of the state of the sea
made by the ships' cfficers. A comparison of the two sets ae
observations led to the assignient of the wave heights which
are given in Table III, column 5.

In view of the discrepancies between different systems for
describing sea state, only a tentative assignment of wave heights
(in feet) to the different terms of the Douglas Sea Scale is
eiven in Table 2121, columm 6. It Should be noted that this
assignment intends to relate the wave heights as obtained from
Plates fk and Vi vol the terns of the Sea Scale. Gensiderabilke
Weight has been given to the Meteor data and to the fact that
for low waves the observed wave heights are in general too
low. L& the letter Leasure as taken a nito acicount there rius mo
great discrepancy between the values of the Admiralty Weather
Manual (Table III, column 4) and the values introduced here
(column 6). The validity of the tentative assignment can be
WSSCSC [di COmipesling iweSjooOIeGs Cie wae See oO wie Se wO WellwSes
derived from wind fetches and durations as determined by weans
of weather maps.

In columns 7 and 8 are stated the corresponding wind vel-
ocities based on fetches of 500 wiles and durations of 24 hours.

The frequency and direction of different states of sea in
certain parts of the oceans, as well as the frequency and
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Way

FORECASTING OF WAVES FOR SHORT FETCHES

In areas such as the Méditsrranean and other partially enclosed
bodies of water, it is often necessary to forecast waves senerated
over short fetcnes wnich are dstermined entirely by coast lines and
Wind direction. In this case the problem of forecasting becomes
primarily a meteorological problen of forecasting the direction and
velocity of the wind. If this cam be done, the wave height and waive
WSieLod aAiweS Foumc irom PleawSs MLL Af wae Fewci LS Sinorweie wide LOO
MEMEO ilies, feliael wie Ieee INE iw aig as Womecre, US <vaPeteLOmM ws
rarely limiting if the feteh is less than 200 nautical miles, but

should such be the case Plate IV or V must be used.

Example

A strait running north-south has a width of 30 miles. At 1299
it is forecast that a northwest wind will reach the strait, will
attain a velocity of 30 knots at 299% and will continue to blow
with that velocity nom i2 hours. What weiviels\ can belexpecied omits
the eastern shore of the strait at %6¢@@ the next morning?

The pertinent values are:

Fetch 43 naut. m.

Wind velocity 30 knots

Duration, og to G69 10 hours

Minismm duration 6 hours (Plate I)

Since the duration is longer than the minimum duration Plate II is

used, from which

= 9.5 feet, T = 5.0 seconds

The next noon the wind velocity decreases and it is forecast
that from 189% on it will be nearly calm. What waves can be ex-
pected to reach the eastern shore on the following morning at $699?

Assuming that the wind suddenly died at 1590 the following

wave heights and periods are determined fron Plate III:

Distance from Distance to Wave height Wave period
lee shore windward shore (feet) (seconds)
(SAE ithe }) (naut. m.)
5 38 207 3.0
15 28 6.90 4.0
30 13 8.0 6
The time interval under consideration is 15 hours. com Plate

VI it is evident that the waves listed above would travel 50 nauti-
cal miles or more in 15 hours. Therefore, the waves have diea out
before J69% on the following morning.

The procedure indicated in this example can be modified accord-
ing to the nature of the vroblem. The forecaster should attempt to

gain local experience and modify his use of the graphs accordingly.

FORECASTING OF SWELL

Forecasts of swell can be made with considerable accuracy if
adequate consecutive weather maps are available from which (1) wind
direction, (2) wind velocity, (3) fetch, and (4) duration can be
determined for the wave-generating areas. The details of the prc-
eedure will depend upon the character of the weather maps but some
general principles can be outlined. In Tables IV, V, and VI are
listed the fundamental. and the auxiliary quantities which are used

when preparing a forecast. The quantities summarized in Table IV

Dl ‘

always have to be computed, but when some experience has been
gained the computations indicated in Table VI need not be carried
out but an estimate of the final values can be made directly. In
the following discussion the numbers in oairantheees ELSS GO COr=

responding terms in Tables IV and VL.

Determination of Wind, Fetch, and Duration

1. .Wind direction. Outside of the tropics the wind direc-
tion over the ocean is obtained ICOM, Glas COURSE Oi wide AsSoOlbews ,
applying the rule that the wind deviates 80 degrees to the right
of the pressure gradient in the Northern Hemisphere and 80 degrees
to the left in the Southern Hemisphere. Where the isobars are
nearly straight (fig. 11, A and B) the winds to be considered in
forecasting swell are those with directions within 30° Of a lime
joining the generating area and the locality for which forecasts
are to be made. Where the isobars are curved (fig. 11C) the winds
to be considered are those with directions within 45° of a line
jOining the generating area _and the locality for which forecasts
Bugs TO INS lel, Wale QSmleiceliaine eueSas TO ie COMisslCleieel cace iin wel
by these restrictions. The reasons for these rules are that the
course of the isobars is not exactly known and that the swell prob-
ably spreads out somewhat when entering areas of calm. The spread-
ing. out will be greater from a region with curved -isobars.

In the tropics the wind direction must be obtained from

Glo Se VvEIoems) Omi OCKiecl KBaLps Cre Clu Sx OoOsed asileing SiwEewioMs

Table IV

Summary of Quantities to be Determined when Forecasting Swell
in the Absence of Following or Opposing Winds

Term

Generating area, mean distance
between isobars drawn at inter-
vals of 5 mb

Mean latitude
Curvature of isobars
Geostrophic wind

Wind velocity at sea surface

Average wind velocity

Observed wind at sea surface
Fetch

Duration of wind

Minimum duration

Wave height at end of fetch

Wave period at end of fetch

- Distance of decay

Reduction factor for wave height Hp/Hp

Wave height near coast
Factor of period increase
Period near coast

Travel time

Wave length

Wave velocity

Degrees
of
latitude

Degrees

Knots

Beaufort
Naut. m.

Hours

Feet

Seconds

Naut. m.

Feet

Seconds
Hours
Feet

Knots

Symbol Units

}

Source

Synoptic chart

Synoptic chart
Synoptic chart
Tables or graph

From (4),
of isobars.

considering curvature
Multiply by:
0.60 Great cycl.

curv.

0.63 Small cycl. curv.
0.65 Straight isobars

0.67 Small anticycl.
0.70 Great anticycl.

curv.
curv.

Current and preceding synoptic

charts
Synoptic chart

Synoptic chart’

Current and preceding synoptic

charts
Plate I

IDileloe) IIL Cpe ILILIL -
(is) alae a St,
using and

Synoptic charts
Plate VI, using
(11) times (14)
Plate VI, Inset
(12) times (16)
Plate VI, using
Plate VI, Inset

Plate VI, Inset

mig} ae ta =<t

using (6) and
Pilate: LV or Vi,

min
(12) and (13)
Tat, msatrys ((i/h))
(12) and (13)

ib, Wisaliayes (17)

I, using (17)

Table V

Scheme of Nomenclature Used for Forecasting in the Presence of a Second Wind System

Locality for which forecast is desired

Yee
oe
Sx
ox
aces
ox
Sx
ates
SX
Sx
ace
ne
x

2
o,

|
| ¥No

Second

Wind S50
SKK KKK XS es
RR

>
C.0,0,0,0-0,6-0,0. 0,000.0,
[a a SOOO

S52
O

| | Wave Period

| |

[eee [ie ik Sr AN ne a
(diate ener alee!
| Special D=0 | Tp =7
Waray ae Boe i Hi
a : Hp'y'sHp y', D'= 'D
Sa er aig ape

Keio Pesci Nanna sey cline

ves
rene

Second Wind Area

Special e ‘
| Case | D'=D | Hp"=Hp o'=Tof
é Hp! u'=Hp.u'

[Sra atl een Sa

—— ore rn Se \/ KEK KKM KOKO
[ : fr a al SOR KR KK
| Special | ‘ | It T OOOR
Case | D=0 niay ve | Moyes 5
Nscieg! | |
|i oer ier | oe ll ee See a SE QOS
Definitions:
D Distance from end of generating area to locality for which forecast is desired.
D' Distance from end of generating area to beginning of second wind area.
D" Distance from end of generating area to end of second wind area.
Hp! Wave height at the beginning of second wind area.

Hp"'y' Wave height at the end of second wind area

Hp" Wave height at the end of second wind area, if secondary wind U' were zero.

(The same system of notations applies to wave periods, except that Tp" u! always equals Tp)

# Note: Hp',Hp",Tp', Tp", are wave heights and periods in area of decay at distances D'and D’ from generating

area. (No second wind area ).

Te)

Bil

Table VI

Summary of Quantities to be Determined when Forecasting Swell
in the Presence of Following or Opposing Winds

Term

Distance to beginning of second
wind area

Distance to end of second wind
area

Wind velocity in second wind
area (D" - D')

Sign to be applied to U'

Reduction factor for wave
height for distance D' (U' = 0)

Wave height at distance D'
(U' = 0)

Factor of period increase for
distance D!

Period at distance D'

Reduction factor for wave height
for distance D" (U' = 0)

Wave height at distance D"
(U' = 0)

Factor of period increase for
distance D"

Period at distance D"

Average period for distance
(D" Ee D')

Ratio wind velocity to wave
period in second wind area

Ratio of wave heights in second
wind area (U' = 0)

Correction factor to (35)

Wave height at end of second
wind area

Reduction factor of wave
height for distance (D - D")

Wave height near coast

Symbol Units

D' Naut. m.
Dy Naut. m.
Uy Knots
Be
Hp: /Hp
Hp: Feet
Tp. /Tp
Tp: Seconds
Hon /Hp
Hp
Thy /Tp
Tow Seconds
oe Seconds
Ui Knots/sec
|.
pao
Hn ur Feet
Hp yt/Epw ys
Feet

Hp ur

ral

Source

Current and prognostic synoptic
charts

Estimated from current and
prognostic synoptic charts

Following or opposing wind

Plate VI, using (12) from
Table IV and (21)

(11), Table IV, times (25)

Plate VI, Inset II, using (25)

(12), Table IV, times (27)

Plate VI, using
IV, and (22)

(12), Table
(11), Table IV, times (29)
Plate VI, Inset II, using (29)

(12), Table IV,
Average of (28)
(23), considering (24), divided
by (33)

(30) divided by (26)

Plate VIII, using (34) and (35)

(30) times (36)

Plate VI, using (32) and (D - D")

(37) times (38)

2. Wind velocity (5). Outside the tropics the wind vellocity,
over the generating area is obtained from the pressure distribution.

Instead of computing

i=)

the gradient wind it is sufficient to compute
the geostrophic wind (4) and to multiply the value so obtained by a
LEOGuUcCETOMGaAchOm whieh tpealce sm niiOmmac COMMIT mul c uc Usa cil Uie Cs Omembiae
LS OMEUCS o

The following factors appear to be sufficiently accurate to

dispose of the somewhat uncertain computation of the gradient wind:

Great cyclonic curvature of isobars 0.60
Small cyclonic curvature of isobars 0.63
Straight isobars 0.65

Siaall anticyclonic curvature of isobars 0.67
Great anticyclonic curvature of isobars 0.70

The computations may have to be carried out for different
parts of the fetch in order to obtain the average wind velocity
in the generating area. Ships' observations should be used as a
check on the computed value. A difference of not more than one
on the Beaufort Scale between computed and observed velocity is
A SeoLSLACuory ClmSCk<s

The wind velocity obtained in this manner applies to the
current weather map and may differ from the wind velocity over
the same area according to the preceding map. A constant wind
velocity was assumed in the preparation of Plates II to V which
are used to determine the wave height, and it is therefore
necessary to introduce an average wind velocity (6) which can
be considered applicable to the entire time interval between
the two maps. Although the manner in which the velocity has changed
is not known, the fact that strong winds raise waves more rapidly

permits the application of the following crude procedure:

Find the component of the wind which on the preceding map
blew in the direction of the wind on the current map. Subtract
one-fourth of the difference between these two velocities from
the, ereater velocity. Lhe wresult 1s considered the average
velocity during the time interval between the maps.

If the wind is decreasing this rule should be applied only
if the velocity remains above 15 knots. If the velocity drops
below 15 knots the effect of a following wind should be ex-
amined.

This procedure may have to be modified according to the
experience of the forecaster.

In the tropics the wind velocities have to be obtained from
observations on board ships or at exposed stations on islands.

3, Weve (8), MWae 2S 1S Bae lemeinin Ox wie SeSinereio—
ing area in the direction of the wind, that is, the stretch between
the rear and the front boundaries of this area. In general, the
boundaries are determined by coast lines or by one of the following:
(a) fanning out of isobars, (b) meteorological fronts, or (c) cur-
vature of isobars, as shown schematically in Figure 11. When the
boundaries have been decided upon the fetch is measured on the map.

hen the isobars have a great curvature two fetches should be meas-
ured, as shown in Figure 11C. Computations of wave height and wave
period at the end of both fetches should be carried out, since
inspection alone will not indicate which fetch should be used.
In making the forecast consideration should be given the higher

values.

)

- CURVATURE

LOW

Fisure 11. Boundaries of the fetch for different
byOOS Oi USOloeuws

Ii) WibbeeiG Lora Tone yatincl (9) 5 ull) Chibi Oia tent iolaley WyiatinGl ss
determined from a comparison of current and preceding weather
mMaos. The duration of the average wind velocity equals the
time interval between the last two maps plus a correction
determined from the height of the waves present at the beginning
of that titze interval. These waves should be known from the ex-
amination of the preceding map. Only the waves which travel
at an angle less than 45° from the average wind direction
SINOUILG! “Ss EHEwIAiaSCl 4 ~~ Wie COwPSCulom LS, wowidGl wWicoim WilhawSs IOV
by the following procedure:

finter the graph with the average wind velocity and follow
& horizontal line to the curve which gives the wave height on
the preceding day. The corresponding duration, as read off
from the top or bottom scale, represents the correction to be

added to the time interval between the maps.

Example

Ao Wimc welo@iinay wien Cube Wwe 5 45) 46 Be iaalogs}
foo Waitacl yelloorimy Zh Inouies; Gelellaleie 6 ¢ 5 20 Ikinous
@. Average wind velocity for last 12 owes
a-b 2)

(a = =) Semin (ican Mommie an Geese) ie eo MOS
d. Maximum wave height 24 hours earlier

(from preceding map). . SERS ORT Celt.
e. Time needed by 29-knot wind to raise

lO-root weaves (eileise IN) 5) 2 ate hs 74 hours
f. Duration of 29-knot wind (24 hours a) ) | 32 Ia@wies

Determination of Highest Wind Waves (11,12)

When wind velocity, fetch, and duration have been determined
the minimum duration (10) is read off from Plate 1. If the dura-
wilOMm 1S LOmeeie winein wae maladie duration the wave height and wave
DSLILOG Be Wa Sul Oi WINE WEweln eles ClyweaLMSel iwigei Iles) IIL Ore

Ti, aif ae as Sla@mser, aticom Pilewes IV ow W.
Example

Wind velocity 29 knots, fetch 800 nautical miles, duration 32 hours.

Wicoa, IPLenGe: Ie minimum duration, 43 hours.
From Plate IV: wave height, 18.0 feet,
wave period, 9.0 seconds.

Determination of the Swell (13-20)

1. Waves advancing through regions of calm. The distance of
decay (the distance from the end of the fetch to the locality for

which the forecast is made) is measured on the map. Entering Plate

Vi With bie tdistance of decay, D, and the perreodvausthe vend of whe

fetch, T,, the reduction factor to be applied to the wave height

at the end of the fetch and the travel time (in hours) are read off.

Example

At end of fetch:

ENS plaveabedane oi Mg a 8G) TS feet

PE TNO Wie Miata pak, elena aga 9 seconds
Di SoeiNee CH CECE co « 4° 6 OOO TANG 6 10
Reduction factor to be

applied to wave height. . 0.47
IACyeis, Mitel epihe y em eee ye lO hours
FICHE Gea Ostee SWiiulllrs eet Mi call retire So) lekeels

Meow IMse~ IIL sia IleoS Wil we weaCGwoe AS w])UiAGl ly WidwlEla, wine

wave period at the end of the fetch, T must be multiplied in

ype

order to find the period of the swell at the end of the distance

Ox GeCAay o

Example

Wave period at end of fetch . . 9Q.O seconds
Reduction factor to be applied
to wave height . 0 5 : 0. Och?
Factor of period increase (from
IDM Gee NAGE, AAtINSVeN Ces IGIE) rn io a URUR oe eo tilled a7
PerilOc Or Swell +5 56 -o 6 oo» 6 Lilo seconds

From Inset I in Plate VI are found velocity and length corres-

ponding to any given wave period. Exact values are obtained by
2

USLMI Wa sowunbass iG (aid, Kaos)! & S503) Ww, 1b (alia weer) SS S52 UW.
Example
Period Velocity Length
(seconds) (kniotis)) “(eet)
Waves at end of fetch 9.0 ZH 5) 15
Swell at end of distance

of decay IES 35.0 690

2. Following or opposing winds. In general, swell should

be forecast on the basis of preceding and current weather maps,
assuming that it travels through regions of calm. However, if a
prognosis of the weather situation or if a subsequent weather map
shows that the waves travel through regions where the wind has a
COMMA jEUCEILIVSIL WO wlats} CliwGO wal Oi Oak joGOQZAG SSS 5 wiles) seh SOEISc
should be modified by taking into account the effect of a follow-
ing or an opposing wind.

The region of a following or an opposing wind has to be
considered as a second wind area, the boundaries of which have
to be selected as the boundaries of the region in which the
component, U', of the wind parallel to the direction of progress
of the swell exceeds 6 knots. The reasons for this limitation
CLO) TIMES CIGLOSS WaliaGls ElceS “MO wW OOiadsalCleoeeyol iGO) sine I Vieiaiole)” jelae
swell and that the effect of very weak winds is negligible. MThe
wind velocity in the second wind area is obtained by estimate if
a prognosis is made or in the manner described above if a subse-
quent weather map is used.

30 IMTS wW Cit WOILOWALINE Oie OMIOSLioes Walls > ~ WalS GwieSow Ort
the following or opposing wind on the wave height only has to be
determined, because it is assumed that the wave period is not in-
fluenced by these winds and that, consequently, the travel time
remains unaltered. Travel time and wave period at the end of
the distance of decay are therefore found by means of Plate VI
in the manner described above.

In order to determine the wave height at the end of the dis-

tanee of decay, H the wax lrary quanbinnes Tasted in Fe ple vi

DU?

have somber mound.) lines values Oil, Hp: and Hp are obtained from
Plate VI by entering the graph with the period at the end of the
fetch, Tp and the partial distances of decay, D'; and D" (see
Table V). The corresponding veriods, Tp and Tp are obradliacd
from Inset II to Plate VI in the manner described above. Having
determined these quantities, Boe oh is obtained from Plate VII
in the following manner:

The average value of Tp: and Ton is computed and called
T, The ratio between the wind velocity in the second wind

area, U', and the average veriod, T, is found and is taken as

vositive for a following wind and negative for an opposing wind.

From Plate VII which is entered with the ratio U'/T and the ratio
Hpn/Ep: a correction factor, Hn yr /Hpw> is read off. Multiply-
ing Side) Pacer by How the value of he a is found.

Finally, Hp ut is obtained from Plate VI by entering this
gravh with the period Tp and the distance (D - D").

If the second wind area extends over the entire distance of

decay or if there is only one region of calm (see Table V) the

procedure is shortened, as evident from the following examples.

Example 1 (Table V, special case a)
Wave height at end of fetch, H 18 feet

Wave period at end of fetch, ie BEY fo GO mesic Gongs
Distance of decay, D. ; 600 naut. m.

It is estimated that a following wind of 10 knots will blow
over the entire distance of decay. The computation of the wave

height at the end of the distance of decay, Hp ur is carried out

as follows, using the symbols in Table VI:

is}

Number Syinbol Numerical Value

21 D! 0)

22 p" 600 maw. ma, (iD = 1)

DD 23) U! 10 knots

29 Hp/Hp On a7 (Hp. = Hp)

30 Hy 8.5 feet

31 T)/Ty Wa27y (Thx = Ty)

32 Ty TLL 6 de seconds

33 ie Ove seconds

34 oo / Ww 0.98

36 i 5 (Ml Loh3 fusiae (29) aac (BL) 7
Dt! OD -

39 Hp ut W252 feet

TMS, Wine COMMOECHOC Iieisinng Ol wine Swell as L252 sews Mine
Peclod ass Vly seconds. sas) in vee vEeceding examples, and) ume
travel time, 40 hours. It is probable that the method gives wave

heights that are somewhat too great and periods that are too short.

Example 2 (Table V, sveciai case b)
Wave height at end of fetch, H amet 18 feet
Wave period at end of fetch, ae oo CEOmSecomads
Distance of decay, D. Ader 5 (00) meibnes im-
On the basis of the subsequent weather map it is estimated
that the swell will meet an opposing wind of 30 knots over the
last 200 nautical miles of the distance of decay. Again using

the symbols in Table VI:

Number Syibol Numerical value

Ad ID) LOO MEG 5 ils
22 iD" 600 WAG. Wi, (ID 5S 1D)
23) 5 Gl. (Ui -30 knots

25 Hy. /Hp 0.58

26 Hp: LO ch wee

27] Tp: /Tq gz

28 Ths 10.8 seconds
29 a ae Ook? (Bon = Hy)
30 Hp 8.5 feet

Bil Wy) oe Loe7 (Mn = Ty)
32 Ty 11.4 seconds

33 ae lies seconds
3h Wy AE =2 07

35 - Hp/Hp, 0.82

36 Hy yt /Ap 0.62

39 Hy ys Jos wOSt

Thus, the corrected wave height is 5.3 feeb, bub period and
travel time remain unchanged.
Remarks on Forecasts

Estimates of the probable decrease and increase of the swell
have to be based in part upon a prognosis of weather conditions.

Usually the forecaster need not construct a prognostic map but can

base his estimate on the conditions he anticipates from his examin-
ation of the weather maps. The following or the opposing winds can
be estimated in a Similar manner.

In order to arrive at an estimate of the rapidity with which
Swell nay de out at Ts advisable to split the fetch anvo several
parts and compute the swell from each.

In middle latitudes a sequence of low-pressure systems, that
iS, a sequence of generating areas, often travels across the oceans.
It is recommended that the swell which is forecast from each genera-
ting area be plotted on graph paper, using height of swell and time
OmecmciniclwasmCOOrdiMicnes|.sObServcd values) Should bemcniteiced monte
Same graph in order to test the accuracy of the forecasts.

lin Carrying out the forecasting it may be found that several
wave trains arrive at approximately the same time; in this case the
resulting swell will be complicated because of interference. The
greatest wave heights may eoual the sum of the heights in the indiv-
idual wave trains but the average height will be that characteristic
of the train having the highest waves. it apovears probable that
with experience the complexity of the expected swell can be forecast.

The general procedure which has been outlined should be modified
according to the type of weather maps which are available and accord-

ing to the experience of the forecaster. However, it should be em-

phasized that the continuity of the processes must be borne in mind.

Example

Forecast of swell for Casablanca and vicinity,
Northwest coast of Africa, November 7, 1931.

Dal

The forecast is based on the weather map for the North
ame or Nowenlosre 7 ey IAC, ELM. (Pie, 12) eiaél om joPeced=
ing maps. The weather mav of November 6 showed an elongated
low-pressure area to the south of Greenland from which a cold
front extended south in longitude 32° W, bending toward SW in
latitude 40° N. Behind the cold front the wind was WNW with an
average sneed of about 30) knots. Elo the casi, of they mcontee co—
ward the coast of Spain, the wind was nearly W and the average
speed about 20 knots. |

On November 7 the low-pressure area and the cold front had
advanced toward SSE and a well-defined generating area was pre-
sent to the northeast of the Azores (fig. 12). The isobars,
drawn at intervals of 5 mb, were nearly straight and in 40° N
they were 1.6 degrees of latitude apart. The corresvonding geo-
strophic wind was 50 knots and, with a reduction factor of 0.65,
the wind at the sea surface was 32.5 knots. Ships revorted wind
velocities of 8 Beaufort (30-35 knots according to the scale ad-
opted by the International Meteorological Committee). The aver-
aze wind velocity during the past 24 hours is found to be 29
knots, according to the rule given when the determination of the
wind velocity was discussed.

In selecting the boundaries of the generating area the front
boundary was placed somewhat behind the cold front because of the
curving of the isobars, and the rear boundary was placed where
the isobars fanned out. This selection gave a fetch of 800 naut-

LOCUL fit LES 5

AREA OF DECAY LAND
aes

NOV. 7, 1931
ee 1300 GMT
AREA OF GENERATION * TETCH [Al 7]

Figure 12. Isobars over the North Atlantic on
Nov. 7, 1931, at 139% G.M.T. taken from the meteor-
ological charts of the Northern Hemisphere. Original
observations are omitted, except a number of ships'
observations of wind. A generating area and the dis-
tance of decay for swell traveling toward northwest
Africa are indicated.

The duration was determined in the following manner: On
November 6 a wind of 20 knots had blown over the generating
area and had been preceded by stronger winds. The waves present
on November 6, therefore, were the highest possible at that wind
WELOGIGy 5, WINENS WS, ElOCOwehlide qe) Iilenes) IML foe I) Wleieyy, viene 10) ieosiw
high. A wind velocity of 29 knots would need 7 to & hours to
raise these waves (Plate IV) and the duration of the wind was

therefore 32 hours.

With tnese values one obtains from Plate IL:

Minimum duration, CA = /h3) laoibaes}

SuLMOS WlalS GlbseEHErLCin als} Slniopeioese elatetal 7h5) Invowboss!, Ilene BOY as) wisiecl .

from which one obtains:
He = 18.0 feet, Tp = 9.0 seconds.

The distance of decay was 600 miles. Entering Plate VI with
a period of 9.0 seconds and a distance of decay of 600 miles, a
travel time of 40 hours and a reduction factor of 0.47 are read
off. Consequently, the swell should arrive at the northwest coast
of Morocco in 40 hours, that is, on November 9 at 959%, G.M.T.,
with a height of 8.5 feet. From Inset II in Plate VI one finds a
factor of 1.27 for the period increase, that is, the swell should
arrive with a period of about 11.4 seconds.

The calculations can be tabulated as follows, using the sym-

IOS iin Welloile INVES

Number Symbols Numerical value
iL G 1.6 degrees of latitude
2 ) L0° N.
3 - straight
L Ug 50 knots
5 Uy 32.5 knots (factor 0.65)
6 U 29 knots (adopted average)
qT (U) 8 Beaufort
8 F 800 naut. m.
9 tg 32 hours

Nuiber Syiubols Numerical value

ake) erie 43 hours

TAL Hy 18.0 feet
LZ Ty 9.0 seconds
13 D 600 naut. m.
ih. H)/Hp 0.47

15 Hp 8.5 feet

16 T)/Tp Lo 27

1L7/ Ty 11.4 seconds
18 ty L0 hours

19 Ly 690 feet

20 Ch 35 knots

When vreparing a forecast on the basis of this analysis it
must be considered that the winds over the distance of decay can be
exovected to continue to blow in the direction of progress of the
swell so that the decrease in height will be less than that obtained
from Plate VI. From this prognosis of the weather conditions it is
estiiated that a following wind of 10 knots will be present over the
entire distance of decay. According to the procedure outlined above
(example 1) the swell should then arrive with a height of 12.2 feet
and the veriod of the swell should remain unchanged, but this wave
height may be soimewhat high and the period may be too short. Fur-
thermore, the wind system causing the swell will vorobably continue
to advance towards the east so that the height of the swell can be
exvected to increase for some time as the distance of decay shortens.

The following forecast should therefore be issued;

Casablanca and vicinity: On November 9 between 24

and @8@@: Swell from NW, height 8 to 12 feet, veriod

ll.4 to i) seconds. Swell increasing during the day.

This forecast did not need modification on November & because
the weather map of the 8th showed the estimate of the following wind
to be nearly correct.

The following values were observed on the morning of November 9:

Locality Approx. height Period Swell
(feet) (seconds) from
Mehedya 7, 15 NW
Rabat g ILS) NW
Casablanca 12 15 NW
Safi 6 12 W
Mogador W WZ NW

The observations at Safi give consistently lower values than
those at neighboring stations, possibly because the locality at
which observations were made is less exposed. For the other sta-
tions the forecast height of the swell was nearly correct, but at

the northern stations the forecast veriod was too short.

FORECASTING OF THH STATE OF THE SEA

A forecast of the state of the sea must be based on the con-
clusions as to the state of the sea drawn from preceding and cur-
rent weather maps and upon a prognostic weather map. The procedure

in using the prognostic weather map is exactly the same as that

which applies to the current mav. When winds, fetches, and dura-
tions have been estimated the wave heights and periods in the
generating areas are found in the manner described when discussing
the forecasting of swell. If desired, the state of the sea may be
described by a term on the Douglas Scale, according to Table III.
Although the method has not yet been tested extensively, it
is believed that the accuracy of the forecast will correspond to
the accuracy of the prognostic map. It must again be emvhasized
that success can be expected only if the continuity of the proc-

esses are borne in mind.

Appnendix
WAVES ENTERING SHALLOW WATER: BREAKERS AND SURF.

A manual on forecasting breakers and surf is in vreparation.
For temporary guidance the transformations of waves that enter
shallow water are briefly discussed here.

Consider a wave which approaches a straight coast off which
the depth to the bottom increases regularly and slowly, and as-
sume that in deeo water the wave crest is parallel to the coast
line. At a distance from a coast at which the depth to the bottom,
ad, 18 Bloowle 1/2 the wave length transformation from a deep-water
wave to a shallow-water wave begins to be perceptible. The veloc-
ity of progress decreases but the period remains unaltered so that
the decrease in velocity avpears as a decrease in wave length. If

the wave lengths in deep water, Lo» and in shallow water, Ls, are

known, the depth to the bottom is obtained from the equation:

tanh 27 G_ = _S
Ls
Oo

Where the depth is less than 1/25 of in the equation is reduced to

ely eas)
em Ly

These equations have been used to determine the bottom topography
from aerial photographs of waves.

The wave height remains constant until a depth is reached
which equals about 1/25 of the wave length in deep water. This is

explained by the fact that if the effect of friction is disregarded

changes in wave height depend uvon changes in the rate at which
energy advances. In deep water the amount of energy which
advances through a cross section of the wave is 1/2 CEG: where
C Alte Land E is the ttean energy of the wave per unit

O Cm fo) fo)
Surface area. In shallow water the corresvonding amount is Cle
where oF m/ed- hit ia) Gigacey as) MOS lony loo wml TIPU CelOMN ES ielaS

wave advances toward shore, 1/2 COB o = CE. Where Bo =: one

has 1/2 Cy = C, oney/2 Ly = L.- The corresvonding devth is
= ho te RS
Gar = 25 Weo5

Therefore, the wave height, which is provortional to the
square root of the wave energy, is the same in deep water and in
shallow water where the devth is avvroximately L,/25- The wave
height however does appear higher. The steeoness of the wave has
been doubled because the wave length has decreased one half.

As the depth becomes less than a/25 the wave height increases
rapidly and the wave length continues to decrease. When long and
low swell approaches a gently sloping beach, narrow, steep crests,
SSOWeIcAwSGl loy/ Lome, wilaw wie Oulelasi, EoOSEIe GO WISS El Siao@ies CaS wosideS
Igo, wie DSACll, AiuGl wMeSS C1esSus SOOM MOSCOMmEe SOQ SwSSo wise wae
break. It is the narrowness, however, and not the height of the
crests which makes them plainly visible. The breaker height, Hy

and the depth of breaking, qd, depend unon a number of factors:
the steepness and direction of the waves in deep water, the slope
and regularity of.the bottom, the strength and direction of local

winds, and the number of wave trains present. As yet no general

rules can be given, but the ratio H,/H, appears to lie between one

and two with the smaller value referring to steep waves on
Sembly Slopilias DeAcMes, Whe wWelioi© d,/H, varies between one
and three, the smaller value referring to a gently sloning
beach.

Where a wave train apvroaches the coast at an angle the
CUIRSCULOM Oi DiKOLKPSSS Cliaises Aas one waves enter shallow water.
Snimicie! the velocity ws Mess! in shallow waiters ithe part of the
wave which first reaches shallow water vrogresses at a slower
rate than the part which is Still in deeo water and consequent—
ly the wave front turns gradually until it becomes narallel to
the beach. The height of the waves will be less than thab of
waves which advance directly against the coast as the energy
must be distributed over a greater length of beach.

As a simole example consider a straight coast off whicn -
Hae Gswigln COidwowne IdiMSs) Cucee joeeeiWILSil, Gali, Gly Smeicsy Cit
the waves in deep water Eo Lew a, 0G wae ginyeile yWidwteia wae
wave crest in deep water forms with the coast line, and let

a, be the angle with the coast line where d = L/25. Where

- a.),

d = yf 25 the energy of the wave equals Eo cos ( a, _

and the wave height is

because the wave height is vroportional to the square root
Of ithe enereys) “huss the areduiciihontein. inecntehiGs sswsimcwel: be-

cause even for ( a= a.) = 5°

If the bottom topography is not too complicated and a good
chart is available the bending of the waves can be computed, but
such computations should be checked by measurements or aerial
nhotographs. Methods for computations will be dealt with in

the forthcoming manual of forecasting breakers and surf.

ait
‘.
Lee

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SLONYH NI‘N *ALIDOT3A ANIM

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yOS ys? yor yor

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Plate II

suOTZOUN]T Se POTZed SsABM pues YUSTSy sAeM

“SoABIN

SFTW “LAVN NI 3 ‘HOL3S4S

OMS a jw NATOOTOA jorsriew go

UTM JO WAMory

ata YMG

OO€! 00d! OOl| ooo! 006 008 OOL 009 OOS OOV OO€e 002 [oJe)}
4 |
T 1 al i r el i) if il
| = +} DaS NI ‘L ‘GOId3ad JAVM TvNDda JO SsaNi1—— - 4 lac r
i ry laa 133453 Ni ‘H “CLHOISH SAVM WVNDS 3O SAND T St +
4 | | t 4 | 4 4 + ste =t JL. = +
rie
= | a ee ee Oe Se Te aes = ay =P sf AS b z
= | = = eal = = —
= [a al Ge a oe = ale eal eee yop 0 ss
a == — 9+ — SSEeIe = izes
Se Se ee le et im San = 2 Ty
ae a | eee eT Sans i 5 j : ISIN
—_ + — | — —_— s6- — i— Ol i | ~~ 8
—— — se > | :
Gc $s = al — oe : = ol =i \
= = : => vl | IESE za aa
= =| 91 = Ey
ae es a s ae =: = ; $I vA
s/7 :8||—> - S64 I 9I |
pees peters = “7102 sal Ni
tS ~ = ™~ | | , eon | BI f
a ES She Jt 1 — _ } : Oz
= sees 4 SS (ie le I J S SS
— — SS ; Sj ~ . | Ne |
| = i oc|—— ~ Sn | GeX
sé! = ¥ —= = JL Ss | a Sei \ 1
> ieee (Si
qi SE t ir pel
— ™~ | |
-: SS 4 ; | : Tae T TS
~~ | I
sb x Ov ja eS r
ss — ~ eet ==
— = BY, aeta | 3a
~ i= a ne ~ ail
= OG aisetsel at
SS T + — + =F
— —= — sS/ = _ = s¢s ——- je +=
eee AS 35 | = re = sf =
<A _ Bo] = a5 it 0 es a So:
— “ SS
OO€l 002! OOll 000! 006 008 OOL 009

‘ALIDOT3SA GNIM

SLON NI

we) iil

eb fel

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Plate V

DURATION,t,IN HOURS

Wave height and-wave period as functions

Growth of wind waves.
of wind velocity and short duration of wind.

Plate V.

Plate VI

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