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SS SS ee a SS

AN EXPERIMENTAL STUDY
OF
SUBMARINE SAND BARS

TECHNICAL REPORT NO. 3

BEACH EROSION BOARD
OFFICE OF THE CHIEF OF ENGINEERS

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DEPARTMENT OF THE ARMY

OFFICE OF THE CHIEF OF ENGINEERS

BEACH EROSION BOARD

The Beach Erosion Board of the Department of the Army was
established by section 2 of the River and Harbor Act approved 3 July
1930 (Public, 520, 71st Cong.), to cause investigations and studies to
be made in cooperation with the appropriate agencies of various
States on the Atlantic, Pacific, and Gulf coasts, and on the Great
Lakes, and the Territories, with a view to devising effective means of
preventing erosion of the shores of the coastal and lake waters by waves
and currents. The duties of the Board were modified by an act ap-
proved 31 July 1945 (Public, 166, 79th Cong.), ‘‘to make investigations
with a view to preventing erosion of the shores of the United States
by waves and currents and determining the most suitable methods
for the protection, restoration, and development of beaches; and to
publish from time to time such useful data and information concern-
ing the protection of the beaches as the Board may deem to be of
value to the people of the United States * * *,”

(1)

TECHNICAL REPORTS

Technical Report No. 1—A Study of Progressive Oscillatory Waves
in Water, 1941.

Technical Report No. 2—A Summary: of the Theory of Oscillatory
Waves, 1941.

Technical Report No. 3—An Experimental Study of Submarine
Sand Bars, 1948. |

| Aveust 1948.

This paper was prepared for the Beach Erosion Board during World
War II by Dr. Garbis H. Keulegan of the Hydraulics Laboratory of
the National Bureau of Standards. Dr. Keulegan completed work
for this paper in 1944 while detailed to the Beach Erosion Board as a
consultant in hydrodynamics. While this paper was prepared pri-
marily for departmental use and the opinions and conclusions ex-
pressed are those of the author, it is believed that it will be of material
value to engineering and research agencies engaged in work involving”
submarine sand bars.

(Iv)

CONTENTS

Sicerom. ). Introduction 2 8a5e SRW DEE Qos Me Bett CN SON A ee
IJ. Preliminary Considerations
A. Bar Formation

B. Depth and Form of Bars
III. Equipment and Procedure

2

2

+

5

IV. Results of Laboratory Experiments 9
WoeBreld Observations. 22 522.26 A 4. en ee eter oe a 19
19

20

B. Lake Michigan Bars and Observed Waves
VI. Factors Affecting Mechanism of Bar Formation and Move-

MONA a ee ee ae | ae en, SRT Os ee Se 24
A. Sand Transportation on Smooth Beaches____--____-- 24
B. Sand Transportation and Sand Ripples____________- 29
C. Extreme Water Surface Variations in the Bar Environ-

TC Tae ee, ee te MEL SD BI POOL ee eee ee. mee 33
D. Energy Distribution in the Bar Environment_--__-__-__ 35

MIP BAcknowledements. 42 280) (4 Posrove es) pass 0 eee 39

Figure
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Figure

Figure 8.
Figure 9.
Figure 10.
Figure 11.
Figure 12.
Figure 13.
Figure 14.
Figure 15.

Figure 16.
Figure 17.
Figure 18.
Figure 19.

Figure 20.

NO oR wy

LIST OF ILLUSTRATIONS

. Definition of terms describing the bar environment.

Schematic views of beach and wave apparatus.

. Analysis of sand used in experiments.

Example of graphic data.
Wave height—Length relations in deep and shallow water..
Stages ia formation of a bar.

. Relation between beach slope, wave height and lengths, depths of bar

crest and base.

Configuration of bar on a steep beach.

Configuration of bar on a flat beach.

Relation between wave steepness and bar depths.

Bar profiles, Lake Michigan.

Average configuration of natural bars.

Water surface elevations for initial conditions.

Water surface elevations with bar formed.

Relation between rate of initial sand transportation and fall of water
surface.

Quantity of sand transported in bar environment for initial conditions.

Mechanism of movement of sand over sand ripples.

Crest height— Water depth ratios as a function of wave steepness at
start of deformation and recovery of waves.

Crest height ratios as a function of wave steepness at start of deforma-
tion and recovery of waves.

Crest height—Trough depth ratios as a function of wave steepness at
start of deformation and recovery of waves.

(VI)

LIST OF SYMBOLS

A= Cross-sectional area.
a= Measured wave height in front of wave generator.
a,—Corresponding deep water wave height.
C= Crest of the submarine sand bar.
Y;=Specific weight of sand.
D= Distance traveled by a sand ripple.
dgu= Median diamteer of sand.
Afi= Elevation of the wave crest above the undisturbed water surface.
Afz= Depression of the wave trough below the undisturbed water surface.
AH,= Maximum elevation of the surface above the undisturbed water surface
during the passage of waves.
AH,= Depression of the trough at the start of wave break.
H,= Undisturbed water depth.
H,= Depth of water at the point where wave reformation begins.
Hz= Depth from the undisturbed water surface to the bar base when the bar
becomes relatively stable.
H’,z=Depth from the undisturbed water surface to the bar base when the
bar is in the process of formation.
H,= Depth of the bar crest below the undisturbed water surface.
H,= Depth of the bar trough below the undisturbed water surface.
zt=Slope of the beach.
A= Measured wave length in front of the wave generator.
Ao= Corresponding deep water wave length.
v= Kinematic viscosity.
0= Reference point for measuring and identifying stations.
Q= Rate of sand transportation.
Q,= Weight of sand transported per hour per foot of width.
P,= Density of sand.
P,,= Density of water.
s= Distance between the point where the wave begins to break and the
breaker is completed.
Si= Point of impending wave break.
o,— Logarithmic standard deviation (Krumbein’s notation).
T= Period of the wave.
t;= Time required for the bar to become relatively stable.
y= Depth of any point with respect to the undisturbed water surface.
z=Horizontal distance from a vertical line passing through the center of
the crest of the bar to any other point at depth y.

(VII)

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AN EXPERIMENTAL STUDY OF SUBMARINE SAND BARS
Section I. INTRODUCTION

Submarine sand bars are frequently found as characteristic features
of ocean and lake beaches. The bars may occur singly or in series
and are usually associated with sand beaches and offshore areas.
An individual bar formation consists of a crest or ridge to seaward
and a trough or depression, to shoreward. A series of bars is a number
of related crests and troughs. These bar formations have been
studied by geographers and geologists, and the literature on the
subject has a history of almost three quarters of a century. The
earliest descriptions of these underwater formations were those of
the German investigator Hagen, whose work appeared in 1863
(reference 1) and was followed by that of Otto, Lehmann, Hartnack,
and Evans (references 2, 3, 4, and 5) among others. Although
considerable data on the material aspects (form, dimensions, and
number) of bars is thus available, almost no authoritative information
on the mechanism of bar formation and movement has been obtained.

Admittedly the formation and migration of offshore sand bars are
hydrodynamical phenomena of a complex nature difficult to study
in a natural environment; under these circumstances laboratory
experimentation can prove of considerable value. Economy of time,
the limitations of available apparatus, and the nature of the problem
require that the general case be resolved into components. Each
component can then be considered individually as involving only a
limited number of variables. The present investigation concerns, as
a first step, the metrical aspects of bars; i. e., the shape and disposition
of bars, as they are influenced by the size characteristics of waves.

This paper reports the results of experiments made to determine
the existence of basic relationships governing bar phenomena. Ob-
servations were made of the form, dimensions, and number of bars;
wave characteristics; ripple formation; and nature and volume of
sand movement involved in bar phenomena. During the investigation
certain qualitative observations on some of the factors affecting the
mechanism of bar formation and movement were made. Although
these qualitative observations are limited, they will be discussed
briefly because of their implications for further study.

806229—48-2 (1)

2
Section II. PRELIMINARY CONSIDERATIONS

A. Bar formation.—If an experimental beach be assumed having
initially a smooth sand surface of constant slope and subject to the
action of waves, the part of the beach which lies between the point of
impending wave break and that of reformation following the breaker
is the area of most active change. This region may be called the
bar environment since it is here that the bar is ultimately formed.
As will be discussed more fully later, the breaker is the most important
element of the region and is the genetic cause of the bar itself.

The deformation of the beach surface is in the form of a ridge,
which is relatively flat in the initial stages and moves toward the
shore in an observable manner. In the course of time the ridge is
enlarged, its form is established or stabilized and its motion decreases
to an imperceptible value. When the changes in the position and
shape of the bar become minute, the bar can be considered relatively
stable. Thus, beginning with an initial smooth beach of slope 7 at
time ¢=0, the bar will reach a stable position and assume a stable
shape at time t, such that after that time the changes in the bar
position and the bar shape take place at an imperceptible rate.
Hence, ¢, may be said to be the time required for the bar to become
relatively stable. The first problem in studying bar formation
therefore is to determine the factors which affect the position of the
bar at the time that it becomes relatively stable.

The dimension defining the position of bars in a hydrodynamical
sense must necessarily be selected. Inasmuch as the types of bars
considered here are associated with breaking waves, the relevant
dimension is the depth of the bar with respect to the undisturbed water
surface rather than the distance between the bar and the shore:
The depth of the bar can be represented by the bar base, defined as
the straight line joining the seaward and shoreward toes of the bar
(fig. 1.) The depth of the bar base is measured from the undisturbed —
water surface to the point directly below the crest, C. This depth will
~ be denoted by H’s when the bar is in the process of formation and
by Hz after the bar becomes relatively stable.

Suppose that the variables affecting H’, are the deep water wave
length X, wave height a,, time ¢, the sand size dey, the sand size
distribution coefficient co, (Krumbein notation), the kinematic
viscosity v, the sand and water densities P, and P,,, respectively, and -
the slope of beach 7. The usual considerations of dimensional analysis
then lead to the general relation:

Qo 2 EE Ao, Der T \=
f(a ? ae Oo, 1; Dey Xo =0 (1)

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The dimensionless terms are the transfer parameters relating
phenomena of different scale. The multiplicity of these parameters
makes the possibility of a true model experiment of bar processes
questionable, if not negative. It is possible, however, that approxi-
mate models which ignore the least important terms are sufficient.

The term, 23. may be omitted because the densities are constant.
wD

If it is assumed that hydrodynamic effects on the motion of sand are
independent of the Reynolds number, the parameter containing the
kinematic viscosity also may be eliminated. With these simplifica-
tions,

Qo

do Xo ie 2
vi H’; Xo Gag Oy) 1, 7x )=0 (2)

To obtain the equation for the bar after it is relatively stable, Hz may
replace H’, and the term involving time be omitted. Accordingly,

Qo Qo
hi Hex =!) = Og, i, )= 0 (3)

or omitting o,, which may be a constant for a series of tests.

Hea h (SS Ft) (4)

The last expression suggests that the effect of wave height, a,, and
the bar base depth Hz can be conveniently related by plotting @,/Hz
against the wave steepness ratio @,/No.

B. Depth and form of bars.—The depth of the bar crest below the
water surface (H, in fig. 1) is the highest point of the bar and therefore
is probably a critical feature. It may then be assumed that a relation-
ship exists between the bar depth, the wave characteristics and the
nature of the sand. The general expression

r a
he Sy Ae gee Og, i)=0 | (5)
is obtained from dimensional analysis by assuming that the bar is

relatively stable and the effect of kinematic viscosity is negligible. In
view of equation 3, H./a, may be replaced by H./Hz, giving

H Qo X j
Heh Se aeue #) _
If og is the same in all tests it suffices to write

Ee Gee Nowe F-
Heh Saas) y:

5

Accordingly the depth and form of the bar can be related to wave
characteristics by considering the dependence of H,/Hz upon a@,/Ao.

A more complete description of a bar is obtained from an examina-
tion of variations in the form, considering the bar profile as the form.
The general relation defining the bar profile is obtained as follows.
Any point on the profile may be determined by the parameters y and
2 (see fig. 1) in which y is the depth of the point with respect to the
undisturbed water surface and z is the horizontal distance from a
vertical line passing through the crest, C, of the bar. This distance
will be regarded as positive if the point is shoreward of C. By dimen-
sional reasoning as in Equation 6, we obtain the relationship:

Wemtey 64 Go Nae
Hone Ga do dang at! i) (8)

and again neglecting az,

2 te ARR
H, J (a No dear i) (9)

which is the desired general relation for the form or configuration of
bars.

Section HI. EQUIPMENT AND PROCEDURE

Experiments were made in two wave tanks, one 1.5 feet wide, 2 feet
deep and 30 feet long; the other 14 feet wide, 4 feet deep and 85 feet
long. Schematic views of the small and large tanks are shown in
figure 2A and 1B, respectively. The conditions illustrated are those
for a beach having a slope of 1 on 70. The experimental beach for
each sketch is to the left of the mark 0; the section to the right of the
mark is a transition. In a more suitable tank this transition section
would not have been necessary; instead the beach slope would have
extended to the bottom. The distorted proportions of the approach,
or transition segment, restrict the normal motion of the sand toward
the experimental beach and thus might alter the dimensions of the
bar eventually formed. It is considered that the test results for the
1/70 slope may be affected by these conditions but those for the steeper
slopes are believed to be reliable. The mark 0 of the figure is solely
a reference point for measuring distances and identifying stations.
The depth of water in front of the wave generator is variable, the
depth being adjusted, depending on the height of the waves, to control
the position of the breaking waves on the experimental beach.

Tests in the small tank were made with beach slopes of 1 on 70,
1 on 30, and 1 on 15. Those slopes were established always to the
left of 0, with the transition segment to the right of 0 the same for all

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three slopes. The small tank is provided with a glass wall allowing
observation of the experimental beach. |

The large tank was subdivided into three parallel compartments
with beach slopes of 1 on 30, 1 on 50, and 1 on 70, respectively. The
experimental beach of slope 1 on 70 is shown in figure 2B. The sand
of the experimental beach and of a part of the approach segment is
placed on a platform. The remaining major part of the approach
segment, consisting of a wooden platform, is firmly attached to the
concrete bottom of the tank. In the sketch the experimental beach is
to the left of mark 0, and the approach segment of the right of 0.
In the three compartments the approach conditions were alike.

The sand used for the experiments was commercial Potomac River
sand. Sieve analysis for the distribution of grain size gave the follow-
ing percentage values:

Grain size Percent

(millimeter) larger than
MASI. aces Bae Bee ee See en Se ee a ee Ee ee ea eee ee: ee Sere eee 1.01
ToS none Se Eek SR Re SR ES _ SNe ies eae iaee Soest Sane es SE £2 Meee sf De ee RE Sere. ee 6. 68
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QUOI 8... Uk Se Ee ee | ee oe ee oe eee es eee ee aero ee. ome 99. 30

The defining parameters of the experimental sand in Krumbein’s
notation are, dgy=0.42 millimeter and o,=0.96. (See fig. 3.)

The quantities observed during a test are shown in figure 4, which
is representative of the basic data. The envelopes of the wave crest
and trough, the undisturbed water surface, and the initial beach
profile were marked on the glass tank wall, then transferred to a
graph. The point of impending wave break, point S; in figure 4, was
believed to be significant and was noted. At the point of impending
wave break the wave front in the immediate vicinity of the crest is
almost vertical and the water particles on the crest are moving with
a velocity slightly below the velocity of wave propagation. As motion
continues the crests deform rapidly, then break. It will be shown
later that the transport of sand at the point of impending wave break
is a Maximum.

In the tests made with the small tank, the gradual formation and
stabilizing of the bar was observed through the glass wall. Roughly
speaking, stability was reached in one hour with a slope of 1 on 15,
in two hours with a slope of 1 on 30, and in 4 hours with a slope 1 on
70, the time intervals being the duration of the respective tests made
in the small tank. It was not possible to observe the formation of
the bars during the tests made in the larger tank; therefore, the time
intervals for those tests were determined from the values obtamed for
tests in the smaller tank by comparing wave lengths and using a
transference equation based on the Froude number. The minimum

CUMULATIVE PERCENTAGE COARSER

GRAIN SIZE IN MILLIMETERS

ANALYSIS OF SAND USED IN EXPERIMENTS

FIGuRE 3.

0s

9

test durations adopted for the larger tank on this basis were 4 hours
for the slope 1 on 30, 8 hours for the slope 1 on 50, and 12 hours for
the slope 1 on 70.

The configuration of the bar shown in figure 4 is indicated by a
smooth curve. This is an idealization in that the bar surface and the
adjacent bottom actually were covered with ripple-marks. The
curve shown was derived from master traces of the actual surface;
it and similar data are the average of two or more runs.

It is desirable to define the waves reaching the experimental beach
in terms of their deep-water characteristics but the waves generated
in both the small and large tanks were shallow water waves. Since
the period is constant, the wave length, d, and the wave height, a,
may be related to the corresponding deep water characteristics \, and

a, by the following approximate expressions: M2 tank, ie:

No »
a, f2cHy . , IaH 27H\ | 2a
eo AT Be coshi / cosh —— (10)

where \ and a are measured at a point having the undisturbed water
depth H (reference 7). The elimination of H between equations 1
and 2 gives an expression for the dependence of a,/a upon d/\,. This
expression is shown graphically in figure 5 and is the basis of the
desired reductions. Taking the observed value of the wave period,
T, the deep water wave length \, may be computed from

A=gT?/2r (11)

The value a,/a corresponding to the ratio \/A, where d is the measured
wave length in front of the wave generator, may be read from the
curve in figure 5. Knowing the value of the ratio a,/a, the deep water
wave height a,, is readily obtained since the wave height a@ measured
in front of the wave generator is known. These computations were
made for all the tests.

Section IV. RESULTS OF LABORATORY EXPERIMENTS

The initial changes in the form and movement of the bar are il-
lustrated in figure 6 which shows a typical bar development. The
form and movement changes do not progress uniformly with time,
but oscillate about mean values. The inset on figure 6 is a smooth
curve representing the average position of the bar with time. The
initial assumption as set forth in section Ila, to the effect that the
motion of the bar is decreased to an imperceptible value is thereby
shown to be substantiated by the laboratory observations.

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The factors controlling the position of the bar at the time it becomes

relatively stable are stated in equation 4. The results of the tests to

. evaluate the.terms in equation 4 are plotted on figure 7. Figure 7A
shows the relation between a,/Hz i. e., the ratio of wave height to the
depth of bar base; and the wave steepness, @,\o.

It appears that for the same value of wave steepness, @,/Ao, the
initial slope 7, of the beaches did not affect the ratio a,/H,. If an
effect is present, it is of the order of the error of observation.

Since the points are plotted without differentiating between the

-values of the ratio a and do not scatter to any great extent, it may
- Gem

be concluded that the effect of this ratio, if present, also cannot be
larger than the observation error. For example, considering the
following pair of data taken from the records:

Ao=48 ft, ao/M=0.06, a,/Hp=0.50
Ay=18 ft., Go/My=0.06, a,/Hzp=0.45

a 300 percent variation is indicated in the at values, yet the variation
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in the values of the ratio a,/H, is only 10 percent.

Interpretation of the curve in figure 7A leads to the conclusion that
if the water depth remains constant, the depth of bar base and con-
sequently, the position of the bar, formed by a single system of waves
is a function of the wave height, a,, and wave steepness @,/A». If the
water depth and wave steepness are held constant, an increase in the
wave height will move the bar seaward. Furthermore, if the water
depth and wave height are held constant, an increase in the wave

_....steepness will move the bar shoreward. Further, if the wave height
and wave length are held constant, Hz, will likewise remain constant,
and any increase in the depth of water moves the bar shoreward.

In figure 7B the data from the beaches of different slopes are plotted
individually. No distinction is made, however, between the tests
made in the large and small tank. Similarly, no differentiation is
made between points associated with the variable sand parameter,
Xo/dew. It should be noted that the ratio of the depth of bar crest
to the depth of bar base H,/Hsz, is practically independent of the wave
steepness ratio a,/d, and of the slope 7. Since the data covers a wide
variation in the values of the ratio \o/dear, it may also be inferred that
the ratio H,/Hz is likewise independent of the ratio \o/d¢xr within the
order of the error of the observations. The data as plotted in figure
7B give a constant value

H./Hz=0.58 (12)

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the basis of equation 9 to determine -the importance of a,/Xo, it is
desirable to replace the bar base depth Hz with the depth of the initial
beach surface Hz measured at the point just below the crest. See
figure 1. This substitution is useful because Hz is more accurately
determined than Hz and is permissible because the ratio H,/Hz
approximates a constant quantity, which is a function of the slope 7.

Figure 8 gives the results of data obtained in the small tank for a
slope of 1/15. The values of \/Hz as plotted in figure 8 were grouped
and averaged for certain intervals of the wave steepness ratio. No
systematic variation in the configuration of the bars with wave steep-
ness is discernible. Therefore a single curve passing through the points
may be considered as defining the form of the bar for this slope
(t=1/15). The results indicate that the form of the bar is apparently
independent of the size of the generating waves. The bar base and
the initial beach surface also are shown in the same figure. It is seen
that here H;/H;—1.2. This ratio differs from unity due to the fact
that the slope of the initial beach surface is steep and the bar troughs
have descended relatively far below the initial surface. A similar
analysis of the bar form developed in tests made on beaches with a 1/30
slope leads to the same conclusion; i. e., the form of the bar is inde-
pendent of the wave steepness ratio. The ratio H,/Hz in this case
has the value of 1.09, indicating that although the troughs of the bar
descend below the initial surface, the effect is not so pronounced as
that obtained with the steeper slope, =1/15.

The results of the test with 1/70 beach slopes are shown in figure
10. In this case the effect of wave steepness is noticeable. Bars
formed by waves of small steepness ratio have wider crests and longer
bar bases. With an increasing steepness ratio the bars become slender
and pointed.

An examination of figures 8 and 9 shows that in the bar environ-
ment of beaches having steep slopes the troughs descend markedly
below the level of the initial undisturbed beach. In the bar environ-
ment of beaches having flat slopes the troughs hardly descend below
the level of the undisturbed beaches. As a result of this behavior,
the trough depth below the undisturbed water surface manifests a
remarkably uniform relationship with the depth of water over the
bar crest.

Let H, denote the depth of the trough below still water level. The
dependence of H,,/H, on a,/d, is indicated in figure 10. To facilitate
the comparison the data for the different slopes are shown separately.
It is seen, making due allowance for errors of observation, that H/,/H,
is practically independent of wave steepness and of beach slope. The
ratio is found to have an average value of 1.69.

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19
Section V. FIELD OBSERVATIONS

A. Form and dimensions of natural bars.—The bars of the Pomeran-
ian coast have been sounded by Otto and Hartnack (references2 and 4),
and the latter’s paper includes records of soundings by Lehmann and
others. Evans (reference 5) measured the bars of Lake Michigan.
The data from these investigations will. be examined here to ascertain
whether any similarity exists between the form of natural bars and
those developed in laboratory tests.

Consider the profiles of Lake Michigan bars shown in figure 11, from
which two groups of bars are selected; one consists of the bars in a
first zone of the four profiles, i. e., the bars closest to the shore and
the other, those in a second zone the next seaward. The bar base
depth H; of each bar was determined, and the distance y and z of pro-
file points of a given bar obtained, from which the space parameters
y/Hz and z/H, are formed. The averaged and smoothed values of
these parameters for the bars of the two groups are shown in figure 12.
The values are plotted as rectangles for the first zone and circles for
the second zone. Since the distribution of points for the two zones is
approximately the same, a mean curve can be drawn as shown in the
figure. The curve may be representative of the bars of Lake Michigan.
In preparing the Pomeranian coast data, the average for the given
locality was taken irrespective of the zones of bars. These data are
represented by dots, triangles and crosses and the mean curve through
the points is representative of the bars of the Pomeranian coast. The
lowest curve represents. the average value for the bar determined by
the experiments based on the data shown in figure 9.

The form of the experimental bars varies considerably from that of
the natural bars. The natural bars are flatter and longer than the
experimental bars; significantly, however, the depth of the crests ex-
_ pressed as a fraction of the depths of bar base have practically the
same value for the natural and the experimental bars.

No adequate explanation can be offered now as to the cause of the
difference in the forms of the natural and experimental bars, except
to suggest that the waves producing the natural bars are of a mixed
type, and the order and extent of turbulence is of a different scale
resulting in a difference in the suspension and motion of particles.
It is believed that such factors would tend to produce flatter bars.

A more detailed comparison of the crest and trough depths of the
natural bars is given in table 1. The sequence of bars defines the
zones in which the bars occur, the bar nearest shore lying in the first
zone.

20

TABLE I.—Crest and trough depths of natural bars

: : Zone
Locality and investigator sony Factor -
I II III IV

Pomeranian Coast (Otto)_-_______-_ 0.013" |= Aa/ Gt) == | 3.8 6b ae
7 013 | Hc/Hp--------|_--------- 46 OG LEE ee E

O135 | PAG |e eee 1.79 1956: |=-ee ee

Pomeranian Coast (Lehmann) ______ -017 | Hs/(ft.)------- 3. 12 5. 74 10. 40 14.8
017 | H/cHs-------- . 54 55 57 58
017 | H,)He--------- 1.71 1. 50 1. 66 1. 56

Pomeranian Coast (Otto) _---_-______ S014”) SA Gt 2 Se ee 7.02 12. 20 20. 8
014 | He/Hz_---.---|--.------- 46 52 52

QUA: | DER ET Gy os | Bae ee 1. 87 1.77 1. 81

Pomeranian Coast (Hartnack) ______ .017 | He (ft.)___--_- 2. 22 6. 56 9. 88 13.1
017 | He/Hs--.----- 42 49 55 62

017 | Ai/He--------- 1. 86 W77: 22-22 | eee

‘Lake Michigan (Evans) __-__--_____ BOD 7a |eedian abs) | ee 7.12 13. 37 20. 6
: OU ae bape aa } 55 62 62
O17 eG ics a ee 1. 45 1.42 1. 55

The dimensions shown are average values derived from different
profiles for natural bars found in similar zones. The bar nearest the
shore is designated as being in the first zone, but at times it was
difficult to recognize that zone. Therefore, in some cases the zone
designation may not be correct. The data in table I, as would be
expected, indicate that the depth of the bar base increases with the
order of zones. The ratio of the crest depth to the bar base depth
tends to increase as the distance the bars lie from the shore increases.
This may indicate that the bars that are farther away from the shore
have lost material from their surface. It is possible, also, that the
larger waves do not operate long enough in this area for the complete
formation of bars. The mean value. of H;/Hz when computed from
the table is 0.54 which compares favorably with the corresponding
ratio, H;/Hz of 0.58, obtained from the laboratory tests. For the —
different determinations the ratio of trough depth to the bar crest
depth varies between 1.42 and 1.86. The average value H,He¢ of 1.66
compares well with the average of the laboratory determination;
H,/He of 1.69.

B. Lake Michigan bars and observed waves —The profiles of the bars
of Sylvan Beach, near the White Lake piers on the eastern shore of
Lake Michigan, were measured by Evans in 1939 (reference 5). Some
of these measurements are reproduced in figure 11. The essential
bars in a given profile are three in number, ignoring the indefinite
forms near the shore. An application of the results shown in figure 7A
might be to determine the magnitude of the waves necessary to’ pro-
duce those bars. The underlying supposition is, of course, that each
bar in a given profile is created under the action of a singular system

DEPTH IN FEET BELOW WATER LEVEL

DISTANCE FROM SHORELINE IN FEET
400 600 6800 1000 1200

“—SYLVAN NO.6

BAR PROFILES, LAKE MICHIGAN

Figure 11.

22

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23

of waves, and the respective wave systems are different. It is recog-
nized that natural conditions may not approximate this supposition.
The bar base depths of the bars in the different zones are defined by
lines joining the troughs of the bars. According to the definition, the
segment of the line under the body of a given bar becomes the bar
base. From figure 11 the average value of the bar base depth, Hz for
the bars in the first zone of the profiles is 7.2 feet; in the second zone,
13.2 feet; and in the third zone, 19.5 feet. If it be assumed, for
example, that the bars are formed under waves of steepness ratio
@o/\o= 0.03, the required value a,/H, from figure7A is0.75. Accordingly,
the bar of the first zone required 5.4-foot waves for its formation, that
of the second zone required 10.5-foot waves, and the third zone 14.5-
foot waves. H. A. Montgomery observed wave heights in Lake
Michigan, near Milwaukee, from 10 April 1931 to 28 September 1932
(reference 7). The maximum wave heights recorded over the first 12
months of the investigation were:

Number of
Mazimum wave height days on which
in feet recorded
fe UP REET OE Ore Owe ae 2 Se oa aoe ont aa eae a seen oso nae eee ee ae se eee eee 1
ici ae arenes. SPER SE CE s408 RW Jie ESS eek De Ee DE pee oe 7 eee ee 0
TIPE con co SE ee ne ee ee ee ee ee ee eee eee oe 0
Sill) Ee ene ee ee eR een oo Seas eee neat See ae Bae eee teas ee eee 1
il (kt eee SR RRO EE EL Sule top ty sales ess ONS. Ao tT So nye 2 ee Neer. See EN eee eee ee 2
S| () Ee nietes Says ee te ae Se ee ee ee 7
RO ERR Reena RM ee NSS ORs 2 ee ee ee a aoe a ee ee 6
Pore eee Meese ThE ee 0h eer teeter e ee ith eee ie) eee eee 5
Qo ee ee, Se ee ee a eS ee eS See 5 ee Se ee = 22
BReccccete teers ed wept. RIE cr le 8 ARI ntl epee a ee abate are rope fen Sp ec 32
RD oe Ly Bn oe Te Oy Oe Od a Ce ee ee a Ee eee eee eee ese See ee ee 8 | 48
Tein DN RE See 2 eee ener een eae 5 are Mey Men See Ae AER Sen, Ae ei, te 71

The average period of the waves of lowest height was 4.3 seconds,
with a minimum period of 3.5 and a maximum of 5.0 seconds. Since
the water at the location of the measurements was quite deep, the
inference is that the low waves were about 95 feet long. The wave
steepness ratio was about 0.03. The above measurements were
made on the western shore of the Lake. If they apply to the eastern
shore near White Lake, the waves are of the order of magnitude
expected to create bars of the dimensions observed in the first and
second zones.

The above comparison of laboratory experiments and natural
processes would be of greater value were it possible to discuss simul-
taneously the effect of a spectrum of waves operating over a beach.
Actually it may be surmised that waves in nature are far from a singu-
lar system,

24

Section VI. FACTORS AFFECTING MECHANISM OF BAR FORMATION
AND MOVEMENT

As stated in the introduction, certain observations of the mecha-
_nism of bar formation and movement were made during the course of
the experiments. While these observations are somewhat limited
it is believed that they are of sufficient value to justify some discussion.

A. Sand transportation on smooth beaches.—A qualitative measure
of the bed motion of sand in the initial and the final stages of bar
formation was obtained from an experiment in the small tank with
a 1 on 70 slope, wave height of 0.32 foot, and wave period of 1.4
seconds. The undisturbed water depth H,, at the point where waves
began to deform was 0.58 feet. In figure 13 are shown the initial
condition of the beach, and the envelopes of the wave crests and
troughs. The breaker was of the spilling type and was completed
at station 4. In figure 14 are shown the last stages of the bar formation
and the corresponding envelopes of crests and troughs. Very marked
changes are noticeable both in the beach surface and the water surface.
Ripple marks in the area in front of the bar are in evidence. The
general slope of this area is almost horizontal. The breaker, now of the
plunging types, has moved toward the shore. It is important to note
that the bar was formed, not where the breaker was initially present,
but appreciably nearer the shore.

Determinations of the rate of transportation of sand were made for
the initial and the final conditions of the beach. Initially measure-
ments were made with galvanized rectangular traps, one-half inch —
wide, one inch deep, and extending across the tank. Four of these
traps were set level with the sand surface at one-foot intervals to
collect the sand in motion along the beach. Samples were collected
for selected intervals and the locations of the traps were changed
periodically so as to obtain the rate of sand transportation over a.
stretch of 13 feet between the stations 0 and 13. The data thus
obtained are shown graphically in figure 15 on which the rate of sand
transportation, Q, is plotted against the station distances. The rate
is expressed as pounds per hour per foot of width. On the graph are
shown the location of the breaker as it initially occurred and the final |
position of the bar. Mf

The maximum rate of transportation of sand on the initial smooth
beach in the area investigated occurred at the point of impending
wave break. Considerable movement of sand was observed along
the beach surface where the waves were breaking. The breaker was
of the spilling type, and the distinctive discontinuity of the plunging
type of breaker was lacking. The bar is eventually formed at a
point where the rate of sand transportation is nearly a minimum.
Movement of sand occurs on the beach up to the limit of wave action.

25

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Obviously, the non-uniform transportation of sand is effected by the
total displacement of the water surface at a given point during the
passage of waves. If AH, is the maximum elevation of the surface
above the undisturbed water level during the passage of the waves
and AH, the maximum depression, the total displacement is given
by AH=AH,+AH,. The relation between the rate of initial sand
transportation and the fall of the water surface is shown in figure 15.
It is apparent that correlation exists between the rate of sand trans-
portation and the total displacement of the water surface during the
wave passage. A clear picture of the correlation of surface displace-
ment with the rate of sand transport is obtained by considering the
dependence of the rate of sand transportation Q, upon AH expressed
in terms of dimensionless quantities. Three distinct regions must
be kept in mind in dealing with the problem of sand transportation
under wave action. The most important region has been referred
to previously as the bar environment, which is bounded seaward by
the point of impending wave break and shoreward by the point of
reformation of waves beyond the breaker; the limits of this region are
readily obtained in laboratory experiments. The remaining two
regions are: the one extending indefinitely seaward from the point of
impending wave break; the other extending to the shore line from the
point of reformation of waves. The laws of transportation of sand in
these regions are expected to be different.

In this investigation only the movement of material in the region
of bar environment under initial conditions will be considered. The
assumption is made that the rate of sand transportation Q depends on
the total surface displacement AH, the depth of water H, at the point of
impending wave break; the period of the wave 7’; the characteristic
sand grain size dg, the kinematic viscosity v; the densities of water
and sand p, and p;, respectively; the gravity constant g; the slope 2;
and the sand dispersion coefficientc,. The ordinary arguments of
dimensional analysis result in the general relationship

=F (AHH, pole VES, Hig, doulBhs i, 66) (18)
PsGl1, v

defining the law of sand transportation. In the equation the dimen-
sions of the initial deep water waves are missing, since the wave
_ characteristics are determined by J and AH. The last six quantities
may be considered constants for a given test hence it suffices to write

2a J OHI) (14)

28

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Q eS so ve) + Ce)

18

16

RELATION BETWEEN RATE OF INITIAL SAND TRANSPORTATION

b
=
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wi
re

uz
C)
uw
)
2
=

On

~a
z
°
Ee
=

2H

+ a ° o ) + ca 2
(14 Y3d YH Y3d “SB81) NOILVLYOdSNVYL GNYS JO 3LVu

o

16

AND FALL OF WATER SURFACE

Figure 15

29

Figure 16 is plotted in accordance with the latter equation from
figure 15. The product p,g, being the specific weight of sand, is taken
to equal 137 pounds (weight) per cubic foot. . The quantities Q and
AH are read from the curves in figure 15. Since Q was expressed as
pounds (weight) per foot width per hour and T is given in seconds, the
numerical value of QT involves the ratios of two different units of time.
Interpretation of the curve reveals that the rate of sand transportation
in the bar environment is controlled, among other things, by the total
displacement of the water surface, AH and that there is a critical dis-
placement value below which no appreciable transportation of sand is
possible.

Further conclusions cannot be drawn on the basis of the meager
data available, but the problem is important enough to warrant a
separate and a more complete study. In an extended study the
effects of all the parameters entering into the right hand side of equa-
tion 14, should be examined.

B. Sand transportation and sand ripples —The discussion of sand
transportation given above refers to the initial conditions when the
beach is smooth and the slope of the surface is constant. Certain
changes in the beach surface occur as transportation continues. See
figure 14. Sand accumulates in the process of bar development and
sand ripples appear seaward of the bar. The effects of these two
changes are quite significant. With the formation of the bar the
breaker characteristics are changed, the total displacement of water
surface in the area between the breaker and the point of impending
break of the waves becoming nearly constant everywhere in this area.
The sand ripples signify a new mode of sand movement end a very
considerable reduction in the rate of sand transport, as will be shown.

Sand ripples begin to form in the area just under the breaking wave,
apparently from two causes. One is the secondary undulations, which
manifest themselves at the surface of the breaker and are transmitted
to the bottom to form corrugations. The other is an accidental
unevenness of the bottom surface due to nonuniformity of resistance
of the sand or some other discontinuity of the bottom geometry.
Once the sand ripples have been started, in one way or another, a
continuous series of them results under favorable conditions. Their
final dimensions are controlled by the depth of water, the period of
the waves, the total displacement of the water surface during the
passage of waves, and the size and dispersion coefficient of the sand.

Movement of sand in the bar environment after sand ripples have
covered the entire beach surface appears to take place in the following
manner. At the instant that a crest is moving over a sand ripple, the
motion of water particles just above the ripple is toward the shore.
At this moment the sand of the ripple surface is moved toward the

30

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crest of the ripple and then downward into the depression of the
next ripple. Sce figure 17. Simultaneously an eddy is formed in the
depression and is loaded with small sand particles picked up from the
depression. The hydrodynamic reaction between the rotating eddy
and the sand surface ejects the eddy together with its sand particles
upward some distance. With the trough of the wave following, the
movement of water particles near the sand ripple surface is reversed.
This moving mass of water carries with it toward the sea the susj:ended
sand particles of the previous eddy. In this reversed motion the
major portion of the suspended particles is deposited on the surface
of the sand ripple from which it initially derived and a minor amount
is taken over the next ripple on the seaward side. The cumulative
effect of this cyclic process is that a ripple moves gradually by small
steps toward the shore. In the process the larger particles of sand of
the environment are taken to the shore and the lighter particles,
through suspension, are taken to sea. Thus the sand ripple, rather
than its motion, becomes the sorting agent.

SS.

eddy formed

TROUGH PASSING

MECHANISM OF MOVEMENT OF SAND OVER SAND RIPPLES

Figure 17.

o2

The net rate of sand transportation in the form of ripples can be
‘ascertained once the distance traveled by the ripple and the area of
the vertical section of the ripple is known. Let Q, denote the weight
of sand transported per hour per foot of width. Denoting the dis-
tance traveled by a sand ripple per hour by D and the sectional area
by A,

Q,=7;DA (15)

‘where 7; is the specific weight of sand. For the sand ripples in test 51
A=0.54hl (16)

‘where A is the height of the ripple mark and / the length of the base,
both quantities being expressed in feet. Accordingly,

Q,=0.54hIDy, (17)

Table 2 lists the dimensions of some thirteen sand ripples of test
51 together with their rate of travel. The first ripple mark was
located at station 2, and the remainder covered the distance between
the bar and that station. See figure 14. It can be seen from the
table that with a few exceptions the ripples have nearly equal sizes
and travel with nearly equal velocities. This can be explained on
the basis that the depth of water over the ripples is constant and the
displacement of the surface uniform during the passage of waves.
On the average the rate of transportation of each is Q,=1.21 pounds
‘per hour per foot of width. Comparing this with the data of figure
15 it is obvious that sand transportation in the form of ripples is
‘much smaller than the sand movement under initial conditions.

Taste II1.—Transportation of Sand by Ripples

No 1 h D Q
Ft. Ft. Ft.fhr. | Lbs./hr./ft.
Tete eth iS oat that en nas A ie ee 0.31 0.06 0. 75 0.97
CL ee ie lee Ba ais Tay ah Ayal ee bey tin Ade 6 ay S05) | wae ia5e 1.93
SE ANN AWE a Sate A eg el» An En beeee .34 .06 .90 1.25
See ae ae atee: Cher: emer nee ie NON T Ae ere 135 .06 .60 .87
(oe ap ae eae BUN RI SaaS ON Se Seat WO ee Be .34 .07 45 74
[er Oe SET Eero Ween meee 2 34 .06 .60 83
1 bits are, etraRe ee eit eae ees Dae 33 .06 .90 1. 28
see a en LN 2 eben ewe us 87, SC ie ereiaie 36 06 .90 1. 34
Gi 6 Sc ec ta We ae On OS ee Chee 32 .07 .90 1.41
POS Le As a a eee 37 05 .90 1.10
AT pies Lemme Pon eS NCTE SP OCTE a ese ca Se 36 05 1.35 1.68
AD eek AUS NG Mal Mea Do is ie | Seas 40 04 1.50 1. 62
“Ee eRe on er nee attr Sele rea Ute enw LCA int Ga 28 .02 2, 40 75

33

The general expression for the law of sand transportation in the form
of sand ripples remains to be obtained. As it is assumed on the basis:
of table 2 that in the bar environment sand transportation is inde-
pendent of the position of sand ripples, the desired law can be
written immediately from equation 15 with the term AH/H, omitted.
Accordingly,

ia Hd, Hd /
paren te( _/Py Ve aie ate rn Tt vs)

(18)

In the test under study the parameters on the right hand side are
constants and from the data for the test we obtain

ieee 039 (19)

The data used to establish the relation are: Q,—1.21 pounds (weight)
per foot per hour, 7=1.4 seconds, P,g=137 pounds (weight) per
cubic foot and H,=0.58 foot. As an application to a natural con-
dition, let us suppose that H,=6 feet, 7=38 seconds and other condi-
tions are similar to those of the test. The rate of transportation
of sand would be: Q,=0.039X 1376 6/3, or 64 pounds per foot
width per hour.

The problem of sand transportation in this form of ripples on sea:
beds is an important one and can be treated with sufficient detail only
by an independent investigation. The investigation should be made
for the bar environment, and for the seaward region adjacent to the bar
environment. The law probably assumes different forms in the two
regions.

C. Extreme water surface variations in the bar environment.—The bar
environment, as was mentioned previously, is limited seaward by
waves which are beginning to break and shoreward by waves which
reform following the breaker. Variations in the water surface eleva-
tions at these points determine the flow of energy into and out of the
bar environment. In studying breaking of waves, the first question to
consider is the determination of the locality where breaking is im-
pending. The upper curve in figure 18 shows the dependence of the
ratio AH,/H,, upon the wave steepness, @,/Xo. Here H, is the depth of
water at the locality where the waves are beginning to break, and
AH, is the elevation of the crests at the point of impending wave
break, measured from the level of the undisturbed water. Thus, if
AH, is known the curve just given will enable us to evaluate A.
Obviously, AH, is a function of a, and a,Xo in the form

AH, /a.= J (Go/Xo) ; (20)

34

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35

if the effect of the slope, 7, can be neglected. This relationship is
shown in the upper curve of figure 19, and it is noted that the effect
of slope can be neglected within the order of experimental error.

The second question to consider is the depression of the trough
AH, at the start of wave break. The desired information can be ob-
tained conveniently if we consider the ratio AH,/AH, as a function of
@\o. The upper curve in figure 20 approximates the relationship.

Evaluation of the energy flowing into the bar environment can be
made on the basis of the quantities H,, AH,, and AH,/AH,. Un-
fortunately, the theory of shallow water waves is not complete enough
for an appropriate analysis, there being available no general energy
formula covering the case of breaking waves.

It is necessary to justify the absence of a, and a,/dp in the general
and transportation formulas in equations 13 and 18. It was established
that a,/H,=f(a,/). By multiplying both sides of the equation by
A/a, it can be shown that there is a one to one correspondence

2
between a,/X) and \)/H,;. Inasmuch as ro ,it may be stated that

Q,/h) and H,/gT? are uniquely related and the latter may replace the
former. Because of these relations the quantities H, and H,/g7? are
sufficient to characterize, in a functional manner, the magnitude and
the shape of waves entering the bar environment.

The movement of sand from the shoreward limit of the bar environ-
ment to the shore is controlled by the energy content of waves at the
instant of reformation. For the evaluation of the energy the quan-
tities H., Af,, and Af,/Af, are fundamental. H is the depth of water
at the point where wave reformation begins, Aj, is the crest elevation
above the undisturbed level and Af, the depression of the trough
below that same level. The relationship between these quantities
and the wave steepness ratio a,/Ay is given in figures 18, 19, and 20.
‘These relationships should be studied further.

D. Energy distribution in the bar environment.—The ease with
which the mathematical expressions for the form and position of bars
is established results from the simplicity of the experimental pro-
cedure. The tests represent conditions under a singular system of
waves and in the absence of tidal flow. Each test began with a smooth
sloping surface and a bar was allowed to form and attain a relatively
fixed position in association with the breaker; then, the quantities
having a bearing on the form of the bar were measured. The resulting
data is of the most elementary nature and really contains little infor-
mation on the beach processes involved in the production of the bar.
During the tests, however, attempts were made to observe these
processes qualitatively.

Sand movement along the bar appears to take place in the following

36

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manner. Initially, in the absence of the bar, a certain distance exists
between the points where the wave begins to break and the breaker is
completed; this distance may be referred to as the breaker distance
and is denoted by s in figure 1. On flat beaches the breaker initially
is of the spilling type. As the bar is formed it changes into the plung-
ing type and the distance, s, increases. The magnitude of s is a func-
tion of the slope of the beach and of the depth of water at the point
of impending wave break. The functional dependence may be

expressed as |
8/H,= f (d/o, 1) (21),
or

which probably assumes a different form when no bar is present.
When bars are present, it is found that, to a rough approximation.
s/H,=5, 8, 9, and 11 when 1=1/15, 1/30, 1/50, and 1/70, respectively..

It is the interpretation of the writer that the two changes mentioned.
above have a bearing on the energy distribution in the bar environ-
ment. This may be explained by reference to the two curves in figure
4 which represent the maximum and minimum elevation of the water
surface with reference to the still water level during the passage of
waves. In a sense these curves represent the paths of travel of the
crests and troughs. In tracing the travel path of the crests, it is.
seen that the crest deforms rapidly and develops a central curl some-
where between the point where the wave starts to break and the
plunge point. The position of the fully developed curl is shown in
figure 14. The curl, or rotating body of water, causes a greater
internal dissipation of energy than otherwise would be possible.
After the appearance of the central curl when the water elevation is.
maximum at the bar crest, a strong current moves parallel to the
seaward slope of the bar over the crest and is directed shoreward.
The current in passing over the crest develops a curl and falls into.
the trough of the bar. At the instant of the fall of the curl its linear
momentum is imparted to the waters ahead and the rotary momen--
tum is consumed locally. The impact of the curl reforms the wave.
In following the path of the wave troughs seaward it is seen that the:
current attains a maximum at the crest of the bar. The trough path.
at this point is a concave surface just above the seaside face of the bar,.
as shown in figure 14; here the hydraulic gradient is quite steep. This.
together with the fact that depths are small indicate the existence of a.
strong seaward current. The net result is that in all probability the.
main role of the bar is to reduce the energy of the incoming waves
causing them to impart a lesser amount of energy to the reforming:

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39

waves than would be the case if the bar were absent. It is believed
that the longer breaker distance signifies a greater dissipation of
energy over the sand ripples as a result of increased turbulence. In
such cases the rotary motions of the central curl and plunging curl
are locally damped. Maximum currents over the bar are associated
with loss of energy in the boundary layers over the sand.

The currents naturally affect the movement of sand in the bar
environment. When the water depth over the crest is greatest the
strong currents directed toward the shore move the layers of sand
from that part of the seaside slope of the bar surface and the crest
area into the trough of the bar. Subsequently, when the plunging
curl is falling in the bar trough, the rotating water picks up sand from
the trough and suspends it in the main body of water. At the instant
that the depth of water over the crest is minimum, the seaward current
transports the suspended sand particles and at the same time induces
a movement of sand over the bar surface toward the seaside foot of
the bar. In fact, the cyclic movement of sand over the bar is quite
similar to that over a sand ripple, except that the advance of the main
body of the bar is imperceptible. In the case of bars, the sand trans-
ported to the seaside foot of the bar by sand ripples is collected at the
shore side of the bar by the recovering and reforming waves and is
transported to the shore.

The formation of the main body of the bar is the final step in the
sorting of sand. The coarser sand reaches the shore through the
cyclic creeping movement on the surface of the sand ripples and the
bar; the finer particles return to the sea through the turbulent action
of the eddies of the sand ripples and the bar. This interpretation is
based on incomplete data and a detailed investigation on this particular
subject as an independent problem is necessary.

Section VII. ACKNOWLEDGMENT

The author acknowledges gratefully the numerous suggestions

made by Dr. M. A. Mason, Chief, Engineering and Research

Branch of the Beach Erosion Board, during the conduct of the tests.
The importance and significance of bar structures were brought to the
author’s attention by Prof. W. W. Williams of Cambridge University,
and his general discussions of different problems relating to the
subject have been very valuable. The work of Messrs. W. H. Vesper
and D. G. Dumm, who operated the models and assisted in the com-
putations, was invaluable.

a

REFERENCES

. Hagen, Gotthilf: Handbuch der Wasserbaukunst, 3 Bde. 1863.
. Otto, Theodore: Der Darss und Zingst. Jahresber. Geogr. Ges. Greisswald,

XIII, 1911-12.

: fginmecn F. W. Paul: ie Kustengebiet Hinterpommers. Zschr. d. Ges. f.

Erdk. Berlin 19, p. 391, 1884.

. Hartnack, Wilhelm: Uber Sandriffe, Jahresber. Geogr. Ges. Greisswald,

XL-XLII, 1924.

. Evans, O. F.: The Low and Ball of the Eastern Shore of Lake Rel a The

Journal of Geology, Vol. XLVIII, p. 476, 1940.

. Burnside, W.: On the Modification of a Train of Waves as it advances into

Shallow Water, Proc. London Math. Soe., Vol. 14, p. 131, 1915.

. Montgomery, Lt. H. A.: Lake Michigan Wave Measurements at Milwaukee,

U. S. Engineers, Milwaukee Office, Nov. 4, 1933.
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U.S. GOVERNMENT PRINTING OFFICE: 1948