VOL. 5

6 } se: 51

DEPARTMENT OF THE ARMY
CORPS OF ENGINEERS

THE

BULLETIN

OF THE

BEACH EROSION BOARD

OFFICE, CHIEF OF ENGINEERS | E
WASHINGTON, D.C. ;

APRIL 1, 1951 NO.2

TABLE OF CONTENTS

Page

Limiting Batter (Slope) Between the Breaking and
Reflection of WaveS ..-...c00e. aware ecatelionen ohare elantene totousrelle ale
Wave Diffraction For Oblique Incidence .ceccovcce DaoR00d 1.3)

Qbservations Wade on Karentes Beach coococccocce voces cace 19

Comparison of Observed Wave Direction With a Refraction
IMPEAM copoos osododanadoooooouodooooooopOorOd ao6o00 6 24

Boca Mae shicin Shywcklos) soy Gagsodoanacoo AAS ancien enti etn 26

Beach srosion Literature cecercsccvccversevccevvcvrov000 31

VOL. 5 5 NO. 2

woo ina

LIMITING BATTER(SLOPE) BETWEEN THE BREAKING AND
REFLECTION OF WAVES

by

Ramon Iribarren Cavanilles
and
Casto Nogales Y Olano
(Highway Engineers)

One of the interesting subjects treated in the XVIIth Inter-
national Congress of Navigation, Lisbon, is the determination of the
limiting conditions between breaking and reflection of waves.

This determination is necessary not only for differentiating
wave breaking from reflecting embankments or Shores, but is also
fundamental for the important project of the wave—breaking slopes
(rampas rompeolas) or dampers (amortiguadoras ) in contrast to re-
flecting walls of piers (muros de muelles) in the zones in which the
plan of wave travel of the interior of the port indicates the ex-
istence of incident waves of some importance and whose successive
reflections might be disruptive to its needed calmness.

In justified cases these slopes can be covered with open pile
piers (muelles en claraboya), but reflecting walls of piers never
should be constructed in these zones of agitation in the interior
of a port, and especially not in its angles.

In connection with this subject, we submit for those interested
in the question, the following quotation from our report, from
Section II, 4th Commnication, of the cited Gongress:

"It is of great interest to determine the limiting batter
between the breaking and reflection of the wave, as a function of
its characteristics. There exists theoretically a limiting batter
Such that both phenomena occur Simultaneously, and that at the
same time separates them in such manner that, for more severe
batters, the wave is reflected, while for gentler batters the wave
breaks, but in practice, there is a range of batters on which the
wave breaks and reflects partially, one or the other phenomenon pre-
dominating until one arrives at a batter of total breaking or of
total reflection. One cannot properly speak of gentle and severe
batters in an absolute sense, but rather this depends om the
characteristics of the wave that impinges."

Based on the quadrilateral of approach, which clarifies so
many previously inexplicable extremes, we determine in the cited

y
ae

* This article has been translated by the Waterways Experiment
Station, Vicksburg, Mississippi, and published as Translation
No. 50-2, 1950

report the expression for slopes (reciprocal of batter) theoretically
limiting between breaking and reflection:

i=4 J/h
T g

the wave period being 2T, its height 2h, and g, the acceleration of

eravity.

In continuation, we are going to examine whether the degree of
approximation, or rather the exactitude, of that simple formula is
really satisfactory .

Using the notation indicated in Figure 1, adopted to facilitate
the comparisons that we are going to make in our notation, the
representatives of the Laboratory of Delft indicated, in one of the
sessions of the Congress, their determination that always if

Sabo
De,

the reflection was complete; and that always if

the breaking was complete.

At the first simple view of these results we estimated then and
here that they could not be acceptable as general expressions for
criteria of breaking or reflection of the waves.

In the first place, it would not be admissible that the con—
ditions of breaking should not be a function of the wave height as
well as the length or rather period of the wave; and neither would
it seem admissible that the relation t/2L defined said conditions of
breaking except in the particular cases in which 2L/H were to be
constant, given that the batter, which accordins to our understand-
ing of the question is the appropriate variable, can be expressed

= t oe 2L

i gc
H 2 i

The horizontal dimension of the batter, t, can only define
the batter, or its inverse the slope, with depth, H, constant.

These logical objections are in accord with our formula

Greph of the Loberotory of Delft
GRAFICO DEL LABORATORIO T.

TALUDES CONTINUOS.
Continues: BarfhICS

figura 4

Sher)
(ratup)

z=
Sa $
Q/ .2
#2
88,
S 2
AS}

y (~)
3 2'§
eae
Ne
© G
¥ §
a:
>

o
>

tS)

&

Figura t!

te ea hi

_—

T g

whose slope, i, or theoretical limiting batter 1/i, is a function of
period 2T, which, for each depth H, fixes the length through the

expression
Tal7 .LxCthy H .
g L

but said limiting slope is likewise a function of the height of the
wave, 2h.

Lately, there came to our attention a report, presented like-
wise by the representatives of the Laboratory of Delft, to the third
meeting in Grenoble of the International Association of Investiga-—
tions for Hydraulic Works, in which were described with great detail
the experiments made and the results obtained.

In the graph shown in Figure 1, which refers to continuous
batters (constant slopes), presented in Grenoble, the ordinates re-
present the relation h)/h between the semi-heights hy of the reflect-
ed wave, and h of the incident, and the abscissas are the relation
+/21, between the horizontal dimension t of the batter and the length
of the wave 2L.

It is emphasized that this second variable is not adequate,
but that what we might call the factor of damping hj/h would depend,
as in our formula, not on the arbitrary relation t/2L, but on the
slope H/t, or its reciprocal the batter t/H.

Thus, logically it comes out that, passing to this new variable,
by means of the simple expression

t = tx 2L
H 2L H

we obtain the graph of Figure 1', in which the aggregation of points
turns out even better grouped than in Figure 1.

Using the data outlined in the cited report presented in
Grenoble, from the extreme observations we obtain the following data
and their corresponding results. Intermediate results would come
from intermediate conditions.

Waves Classified Short

(a) Data.

Semi-period, T = 0.45 sec.; depth, H = 0.3 m.; length, 2L =

1.2 m.; relation, 2L = 4; relation

2m = 15; then, semi-height, h = 1.2
H ;

—ee

L
h 2x1

= 0,04 m.

By application of our formla, we obtain the limiting —

ean [0.04 = 0.568
0.45 V9.81

that is, the batter:

slope:

t = i = nae 0)
h 0.568
(b) Data. °

ny = 0.45 sec.; H = 0.3 M. § 2L = 1.2 m3 2b = 43 L = 203
H h
h eb 52 3 0.03 Moy
2x20
we obtain:

al oS ee 0.03 = 0.492

—_—_———

0.45 9.31
that is, the batter:

t = iL = 2208

il ORLSDD
Waves Classified Long ~
(enmweauar.

T = 0.8 secs; H = 0.25 m3 2L = 203 m3 2b = 9,3; L = 15;
#

7

h s 203 s 0.077 Me
2x NS

we obtain:

that is, the batter;

Gipe 2626
H 0.442

(d) Data

T = 0.8 sec.3;H =.0.25 m; 2L = 2.3 m; 2L = 9.3;
H

h = 2.3 =-0.0575 m.,
2x 2O

we obtain:

= al: = 2.61

The mean value of the limiting batters obtained is then:

t= 1.96 + 2.08 + 2.26 +1261 = 2.165
H 4

As has already been indicated in our report, including ex-
perimental verification, the theoretical limiting batter should
correspond, very approximately, to the mean value between the batter
of complete breaking hj = 0 or of complete reflection hj = 1, that is

h h
ought to logically correspond to the value hy =@ 40 = 6.5:
h 2
With this ordinate, hl = 0.5, and the abscissa obtained, t =
ee H

2.165, we fix the point P in Figure l' which, coinciding with great
exactitude with the center of the collection of experimental points,
demonstrates once again the degree of exactitude of the formula
employed.

Even more interesting are the results obtained by analogous
procedure on the discontinuous batters, whose graph, as is already
apparent in the report presented in Grenoble (Figure 2), is so
disordered and its points so dispersed tmt it is difficult for one
to deduce from it reliable or dependable conclusions.

The principal reason for this dispersion is that the variable
adopted for the abscissas t_, is even less acceptable, in these
cases of discontinuous 2L batters, by virtue of the real
batter of the wave breaking slope being now

GRAFICO DEL LABORATORIO DE DELFT.

TALUDES DISCONTINUOS.
Discontinuous Beatlers

GRAFICO CON ABCISAS correciDAs £ (TaLuD)
TALUDES DISCONTINUOS.

usa peau d

"
Ss
oN
nH

@ eed 4 belle yaaa

>) Galea

Figura 2:

and the relation

eL =2L x
Z a

Nya

is here essentially variable, principally by 2 being so.
H
If, by simple change of variable, we adopt as abscissas the
values of the batter t/z instead of the arbitrary t/2I we obtain the
eraph of Figure 2', as ordered and acceptable as that of Figure 1',

This very interesting result shows once again that the funda—
mental variable for the study of the breaking or reflection of the
waves must be always the batter of the slope t/z, or its inverse the
slope, and not t/2L. The data corresponding to the new experimenta-
tion, done for these discontinuous slopes shovm in the report
presented in Grenoble, are the followinz:

A= Dey; ee = 4 Os a = 0, Sle ime:
H 2h

from which is deduced:

2L = 0.316 x 5 alt = 1.80 Me § h = 1.80 = OnO225 Ms

K =| Cthy Ho = 1166s) 1 = jer ob Kk ="O.58 sees.
L

0a

In the experiments the relation 27H had the values:
“EWA RB) ANS Oevas Amen Olay)
H H H
For z/H = 1 we have the case of continuous batters whose experi-
mentation has been utilized here again, and the corresponding point
P will be approximated much as that obtained previously.
For:
Z = 046; 2 = 0.45 x 0.316 = 0.142 m. >2h = 0.045
H

and we obtain the limiting slope:

see Ge) 8 [0.0225 =1
0.58 V9.81 3

that is, the batter:

ae
Z

10

whose point Q is shown likewise in Figure 2!.
For:

gis 0.273) 2 =,0.27 x 02316 = 0.853 me. Sceh =40.45
H
we obtain the same limiting slope i = 1/3 and the same batter

t/z = 3 by which its representative point Q will be the same as
that we have just obtained.

The mean abscissa corresponding to the point P and to the
points Q, the number of experiments of each of which approximate
those of P, will be:

cine On siren sree
2

Z eo}

which determines for us the point R perfectly centered likewise
in the group of points of Figure 2'.

This shows definitely, we beltieve, that the formula for the
limiting slope:

is very acceptable, and even though its application could be
refined even more, we estimate that what has been done gives a
sufficient approximation for the majority of practical cases.
Perhaps it should be repeated once again that in the methods and
formulas that we recommend for the maritime technician we do not
pretend to utopian theoretical exactitudes, but acceptable
practical approximations.

It should be noted here that the experimentation made for
the discontinuous batters is acceptable and as has been show,
all of it fulfills the condition determined in our report to the
Gongress that the depth in the extreme cf the batter Hy = 2
should be equai or greater than the height 2h of the wave, that
LS fa SA SS Bag

Summarizing, the results of the interesting experimentation

of Delft have confirmed ours, contained in the cited report to the

Congress, according to which the batters of complete reflection
and complete breaking, whose msan is the limiting slope between
both, are practically in the relation of two to one, so that
having obtained the limiting batter, or its inverse the slope,
by means of our formula, it suffices to mitiply it or divide it
by 3/4 or 3/2 to obtain each one of them.

il

For the wave-breaking slopes or wave dampers whose effectiveness
is somewhat increased by the roughness of the rock fill, it generally
suffices to give them the limiting slope i = T/4 + h/g under the
condition H, = 2 > 2h, because if the height of the reflected wave
is reduced to less than half of that of the incident wave its energy
is reduced to much less than 1/4 part.

All the foregoing confirms our criteria, accordinc to which in
order to get acceptable approximate solutions for the complicated
subjects of the maritime technician the enthusiastic and frank
collaboration of theorists, technicians and laboratories is necessary.

12

WAVE DIFFRACTION FOR OBLIQUE INCIDENCE

by
Henri Lacombe
Principal Hydrographic Hngeineer
Comite Central D’Oceanographie et D' Etude Des Cotes

The following article was translated from
the December 1950 Information Bulletin, Comite
Central d'Oceanographie et d'Etudes des Cotes.
It is published here as a means of informing

interested readers of the interest in and pro-
eress being made in foreign countries on ocean
wave problems. Numbers in parentheses refer to
the bibliography following the article.

Status of the Question in June 1950

In the Information Bulletin of Comite Central d'Oceanographie
et d'Etudes des Cotes for June 1949, we discussed the general features
of a study on wave diffraction which had been published in consider-
able detail in the Anmales Hydrographique of 1949 (1). We pro-
posed in that article that diffraction for normal incidence could
be studied by an almost exclusively graphic method which would not
differ greatly from the rigorous solution in the case of semi-
infinite breakwaters. The method was equally applicable to the
diffraction caused by vertical obstacles, with the condition that
the waves approach the obstacles normally.

An American study by Blue and Johnson (5) discussed the the-
oretical solution, which was already known, and extended it by
means of a certain approximation to the case of openings, but sole-
ly for the situation where the waves approached normal to the open-
ing. Photographs obtained in the course of the study on the model
show the existence of areas of agitation analogous to those that
we had noted, and which are encountered also in the emission pattern
of certain submarine acoustic apparatus and radio antenna systems.
This study further had the advantage of showing clearly the effect
of what one my call wave guides, defining the opening and which
are constituted by vertical walls connected to the breakwater and
directed toward the exterior of the port in a direction opposite
to that of propagation. These suides had the effect of making a
little more rigorous the simplified solution of Putnam and Arthur(4)

% Lamb (2) pp. 538-540; Bateman (3) pp. 476-490; Putnam and Arthur (4)

by wave reflected from one part or another of the jetty.

The General Rigorous Solution and Its Practical Complexity

At a meeting of the Section on Fluvial and Maritimes Hydraulics
of the Societe Hydrotechnique de France in June 1950, Mr. Schoemacher
of the Hydraulic Laboratory of the University of Delft, stated that
the rigorous solution of the diffraction problem for any incidence
of wave and for a limited opening in a breakwater had been discovered
for cylindrical waves and reported by P. M. Worse and P. J. Ruben-=
stein (6) in 1938. These latter studies referred to a German study
made in 1902 and it is curious to note that neither Putnam and Arthur
nor Blue and Johnson apparently had any knowledge of these studies.
J. Larras (7) showed in 1942 that the acoustic solution in two
dimension was applicable to ocean waves and it followed that the ex-
pression given by Morse and Rubenstein entirely solved the problem
for swell, at least in theory; but for practical use these authors
referred to tables of Mathieu functions which were established by
themselves

The solution utilizes in effect cylindrical elliptic coordinate
and the lathieu function which occurs in various elliptic or hyper-
bolic problems, noteably those of the vibration of an elliptic
membrane (8) and the propagation of electro-magnetic waves in
elliptical guides (9). Bateman,(3) on pages 429, 430, and 490,
gives supplementary references and adds; "These elliptical
coordinates are useful for the treatment of the problem of vibration
of an elliptic membrane and the diffusion of electro-magnetic waves
by an obstacle having the form of an elliptic or hyperbolic cylinder.
A sereeén containing a rectansular slot of constant width may be con-
sidered as the limit of an obstacle having the form of a hyperbolic
cylinder,” The author did not discuss the solution in detail.

It is a fact that this knowledge which has been unknown
theoretically for almost 50 years has not become of current usage.
Perhaps the reason may bs found in the slight mention made of the
Mathieu functions in classical teaching and in the absence of or
very slight knowledge of the tables of these functions,

Research Toward a Practical _Approximate Solution Based on a

Generalization of Huyghen's Principle, Its Justification Justification and d_ Its
Expression

Despite the slight theoretical interest of a solution founded
on Huyghen's principle, which is necessarily only approxima te, we
have devised a solution of this type whose practical use is simple
and elementary. It is nevertheless necessary to assure oneself
that Huyghen's principle, or an adapted Huyghen's principle, is
applicable to oblique incidence of the wave. Its approximate

14

justification is made in the case of propagation in three dimensions
by classical reasoning due to Kirchoff and Giscussed notably by
Bateman (3), on pages 184-186, as well as by numerous analytical
treatises (10) with respect tothe propagation of waves. It re-
lates the value of the propagation function at a point P to the
values which the function and its normal derivative take on a surface
around the point P. Generally one applies the formula obtained to
diffraction by means of a certain number of hypotheses which are de-
tailed by DeBroglie (9) on page 91 and following. However, for
waves in three dimensions the auxiliary function that must be in-
troduced in addition to the propagation function has a simple ex-
pression, it is rigorously /r, where r is the distance froma
point source of movement P to a point M situated on the surface
surrounding point P, It is not the same in a harmonic movement in
two dimensions, where the expression of movement originating in

a point source involves Bessel functions of zere order of the first
and second types (see Lamb (2) pages 296-297, and 527-529) for
which the asmytotic expression occurs in r~2 and applies only to a
distance from point P which is greater than 2 or 3 wave lengths.
The first term neglected in the alternate series development

of the exact movement is r73/-. However, in employing the exact
expression of the movement starting at a point source P one arrives
easily at an expression analogous to that obtained by Kirchoff,
but which applies to a cylindrical movement and more precisely to
a wave or swell. The expression gives the movement at point P

if one knows the velocity potential and its normal derivative on

a cylinder with normal generatrices in the plane of still water.
One deduces from this formula a Huyghen's principle which applies
to oblique incidence, tut which, containing the asmytotic ex-—
pression of elementary movement and not the exach expression, in-
volves therefore an approximation and is practically applicable
only if the distance PM is sreater than > or 3 wave leneths.

Referring to Figure 1, the letters CC define a wave crest,
let S be the cross section of the cylinder cn which one assumes
the propagation function and its normal derivative are known.

Let oc be the angie between the wave crest and the tangent to

S, @ is the angle of incidence; let @ be the angie between the
normal to the crest and the straight line MP, and r the distance
from the point P wheres one seeks to determine the movenent, to a
very small elementary source. situated on 5S. The depth is supposed
constant throughout.

The transposition of the Kirchoff formla, which transposes
in a fashion the roles of P and M permits the establishment of
a Huyghen's principle of the following form. The movement at
P is obtained always in first approximation by considering that
the movement is the resultant of elementary movements of the
sources, such as M, distributed on the surface S (known movement),

| el
ben

Figure |. Figure 2.

and which causes a2 movement at P of amplitude

cos @ + cose _/

2 Vr Eq. 1

that is to say proportional to the inverse of the square root of
the distance MP multiplied by the term cos@ + Cos , and whose
phase at P is equal to that existing at M retarded by 27 Pr
measuring r in terms of wave length as unity. lor normal incidence

& =0. One then has /+§%?. — which is the mean of the two
solutions that we have considered (1), and which are quite similar
especially in the nsighborhood of the limit of the geometrical
shadow »

The Approximtions Made

To this point the sole approximations that have been mde are
due to the fact that the variation of amplitude in r-2 is only ap-
proximate and that in order to obtain equation 1, one has neglected
a new term in r- 3/2, But how does one apply practically the result
acquired according to Kirchoff, to the diffraction through a very
large opening ina breakwater. It is here that new approximations
mist be made.

Referring to Figure 2, let A and B be the arms of the break-
water and AB the opening, CC is again the crest of a wave. We will
choose a surface S where the velocity potential and its normal
derivative are supposed known. This surface is the contour formed
by the opening AB, the part X'A and BX of the supposed infinitely
long breakwater and a dotted line traced very far from AB toward
the interior of the port and on which the agitation is ccnsidered
to be negligible.

We suppose further that:

1. On X'A and BK the velocity potential is zero (its
normal derivative is necessarily so);

2. Between A and B the movement is the same as though
the jetty did not exist.

16

3. We will suppose that we are able to apply Kirchoff's
reasoning, that is to say Green's formula, to the
surface §, although the velocity potential and its
derivatives are subject to discontinuity in A and B.
There is no reason not to believe that one may avoid
this discontinuity by taking a different path of
integration, avoiding A and B but passing very close
thereto (e.g. the ends of the breakwater rounded).

These are without doubt approximations whose effect is difficult
to appreciate; even more s since it appears that the second
hypotheses is hard to accdpt. In view cf the fact that it states
that diffraction does not exist at all or at least does not alter
the movement at the opening AB, which is certainly not correct, The
sole verification possible consists in comparing the results ob-
tained from the use of this method to the results obtained by the
exact hydrodynamic solution.

The Resultant Solution

One deduces from. Equation 1 the movement in P by an integra-
tion from A to B. This movement is practically poven by the
distance, measured on a spiral analogous te Cornu spirais of
optics, between two points characterized by parameters which are
related simply to the position of point P in relation to the two
arms A and B. The lines of equal value of these parameters are
parabolas whose foci are the points A and B and which may be traced
in advance by supposing that all horizontal lengths are measured
in terms of wave length as the unit of length. In fact if one
compares the results obtained thus to the simplified solution of
Putnam and Arthur (4) one finds an almost perfect coincidence, the
differences not exceeding one hundredth of the amplitude of the
incident wave. This is the verification a posteriori of the validity
of the hypotheses that have been made.

The form of solution obtained, which is Gguite similar to that
of Putnam and Arthur, leds, for the case of a semi-infinite break-
water (i.e. a very large opening} to a distribution of amplitudes
which is in some fashion rigidly bound te the direction of incidence

of the wave. If for example, in normal incidence one obtains at a
distance y behind the jetty (i.e. at a distance y from the wave
coinciding with the breakwater). A certain distribution of move-
ment, then one obtains the distribution in obit ue incidence by
tumning the normal incidence distribution around the breakwater
through an angle equal to the angle of incidence. The homolog of
the line situated at the distance y from the breakwater in n normad
incidence is then the line situated at the distance y ey 1
crest passing the jetty. The line of equal amplitude are
parabolas whose focus is at the breakwater.

Db

=)
A

For a limited opening the distribution of the amplitudes that
one obtains presents "tongues" or "leaves of the agitation analogous
to those noted in normal incidence; and moreover they are more
numerous as the opening is made larger with respect to the length
of the wave. If one examines the area at quite a distance from the
opening the agitation is found to take a very simple form analogous
to that which characterizes the emission from radio antenna systems.
The maximum agitation is found on the axis of the opening extended
parallel to the direction of propagation. If one expresses all the
horizontal distaness in terms of wave Isngth it is equal to the
length of the opening projected on the crests of the incident wave,
Very close to the opening maximum agitation approaches the down
stream arm of the breakwater, i.e. the one reached last by the
incident wave. The maximum amplitude possible is 1.36 times the
amplitude of the incident wave.

According to some fieures presented in the article of Morse
and Rubenstein (6) the exact solution ought to present “tongues”
or areas of secondary mximum agitation. In the actual state of
the question, development is left to reduced scale model experi-
mentation since verification of the results is required.

BIBLIO GRAPHY

H. Lacombe — Note sur la diffraction de la houle én incidence
normale, Annales Hydrographiques de 1949 ou
extrait No. 1363 des ces Annales.

Lamb - Hydrodynamics, 6th edition, Cambridge 1932.

H. Rateman - Partial differential equations of mathematical
physics, New York 1944, Dover publications.

Jo A. Putnam et Re S. Arthur —- Diffraction of Water waves by
breakwaters, Trans. Amer. Geophys. Union, Vol. 29,
No. 4, Aout 1948.

Fo L. Blue st J. N. Johnson - Diffraction of water waves passing
through a breakwater gap. Trans. Amer. Geophys.
Union, Vol 30, No. 5, October 1949.

P. M. Morse st P. J. Rubenstein - The diffraction of waves by
ribbons and slits. Physical Review, Vol. 54,
December 1938, pp. 895-598.

Larras - La deformation ondulatoire des jetsss verticales.
Travaux, Juin 1942.

KH. Mathieu - Cours de physique mathematique. Gauthier-Villars
Paris, 1873, pp. 122-165.

L. de Broglie - Problemes des propagations guidees des ondes
electromagnetiques. Gauthier—Villars, Paris 1941.

P. Levy - Cours dlanalyse de 1'Hcole Polytechnique. Tome II,
1934, p ° 267-268 °
18

OBSERVATIONS MADE ON KARENTES BEAGH

; by
. Professor W. W. Williams & Miss C.AM. King

FO REWORD

D! Oceanographie et ee Des asene S ort fae: pmb

1950, reporting work done by Professor W. W. W:

of Cambridge University and his associate Miss

King in August 1950. The original technical not
been translated and briefed, the ee et omission
being data on the observat bions made with respect AG)
vaves and winds and a daily account of the bshavior
of the beach. The Beach Erosion Board was fortunate
in having the opportunity to be host to Professor

Williams for a period of several w seks during %
late days of World War Il, at which tine Professor

Williams’ interest in offshore bars Hence
stimulating discussions between himself, the

of the Board, and Dr. G. H. Keulegan.

Karentes Beach is situated in the Guif of Lion
Sete. In this region the coast is oriented approximat
southwest, and is constituted by a low beach of fine sand, (median
diameter of 0.30 mm), backed by low dunes situated about 300 feet
from the mean sea level shore line. The amplitude of the tide is
very small and probably does not exceed about + foot. On such a
beach it should be possibile for submarine sand bars
they should be of a simple type since they are fo
plications due to tides and or the abnormal tidal curre
ing in the neighborhood cf headlands or the mouths of r:
Currents parallel to the coast are habitually weak and we hav

ble to observe that at any moment they were uniquely a function of
the predominant wind at the site.

Figure 1 shows the profile of the beach, to a depth of approx—
imately 20 feet, as it existed during the last three summers. It
will be noted that there is a large bar situated between 630 and $30
feet from the water line and that the form of the bar remains

if
markably constant although the bar itself was displaced as much a

Ee
&

j aunbiy

© ° gore “
ts 7 N
VSS / en Np
NX a / “aN
XN o PSS o~! ‘\ Z
ES pee? ae SS = S \.
e \ J ge
9 i ‘, 3 ie
ey ee
\ S / —
O9 uoljo1ab0Xx~a /DIIJ1/ eda NES Bl L i
— — — 409 sn0ysim Vie ae
——- 066] isnbay NN ETE
mnie BK BL an be

ay PS

86) Ajnr

SO/IJOJd YyIDIG

Sa}usIeYy Sd]

/ana7 13040M voapy

S4ajaus 999 00g 004 oog 002

ool

wd
Sf
®
7s
Y
&§
RS
<
Eq
Y
Q
pe Nea Ne
7 \ IG
wer?
ie i
a O

20

50 60 Bo 100 10 120 “NEFELS

Les Karentes
Interior Bar
August 1950

N Water leve/
+ Accretion
- €7rosion

Ce)
A=
-7 ~N
a
@% 4
~ +f7-
8 aN
8
am Ne Ne
~
S
mS)
_
S
= - ee
7 Ts
a N
ee, —
NS ER
2” r~
i+ /\S
meters / j

Figure 2

Za

125 feet in 2 years. It is possible, in fact probable, that this
bar is subjected to displacements which are much larger and more
rapid during the course of violent winter storms. At a point about
1420 feet beyond the bar there is another which is less evident.
Within 300 feet of the water's edge there are found several bars,
which we will call interior bars, which move rapidly as is shown

n Figure 2.

On Figure 1 there has been traced a line which I call the "beach
profile supposed without bars." We will refer to this later.

The interior bars have been studied in detail. They were measured
daily whenever there was any reason to suppose that there had been an
appreciable displacement in the bar position. The velocity and
direction of the wind and the height and period of the wave also were
noted, Available personnel was not numerous and therefore no night
observations were made; which is unfortunate, for judging by the
noiss of the sea at night the height of the waves on certain ni
was certainly much greater than the heights which had been noted
during the day. Up to a distance of 350 feet from shore the beach
profile was determined by leveling with the aid of a stadia board
and the heights given are probably correct within about 1 inch.
Farther seaward a sounding lead was used to determine depths, The
positions of the Soundings on the profile were determined by means
of a plane table.

Previous research a led to six considerations which were to
be confirmed or modified by the observations made on the beach.
These considerations are: :

A. Ears form to the highest elevation at the breaking point of
the wave. At Karentes the exterior bars are formed and
displaced by the storm waves of winter. During calm
weather they are displaced only slightly, however the
interior bar is displaced by very small waves.

B. After breaking the waves transport sand, and by consequence
the bars, toward the sea.

GC. Before breaking the waves transport sand, and by consequence
the bars, toward the beach.

D. If one smooths the profiles of the bars observed on many
different occasions, one will obtain a theoretical profile
which we have called."beach profile supposed without bars’
It appears to be reasonable to assum that the bars are
accumulations of sand which are moved freely on the pro-
file, depending upon the action of the waves and as the
waves vary. Numerous experiments in the laboratory have
shown that there exists a,constant ratio between the
height of the crest of thé bar above the “beach profile

59

Kibo

supposed without bars" and the depth of the water above
the crest of the bar. This ratio is 0.5.

E. ‘The mean ratio between the depths to the crest and to the
trough is 1.5; however, the range of the walues observed
is large and extends from about 1.2 to 1.9.

F. There should exist som relation between the velocity of
displacement of bottom material and therefore of the bars,
and the energy of the waves.

The last three statements may have a considerable importance in
time of war. It was to verify these points on a beach of the simplest
type that we made our observations at Karentes, We take this op-
portunity to express our gratitude to the Comite Central d'Oceanographie
et d'Etude Des Cotes and to Mr. Gougenhsin in particular, for the
authorization to work on the beach and the aid which has been given.

The modifications which occurred on the beach in the course of
the month of August are shown on Figure 2.

Conclusions

The following conclusions are reached with respect to the con-
siderations listed above. By reason of the small range of dimensions
of the waves and the absence of observations during the night our
conclusions may be erroneous, nevertheless it appears to us reason—
able to deduce from our work the following results:

Considerations A, B. and C appear to be verified without dis-
cussion.

Consideration D. It is probable that the bars are displaced
on a relatively easily defined base “profile of a beach supposed
without bars", but the ratio 4 observed in laboratory work may not
be verified universally in nature. For example, it is false for the
deep rudimentary bar found at about 1300 feet from the shore line .
and shown on Figure 1. I+ seems equally clear that when the bars
are in the course of displacement the crests are somewhat augmented
and the depth of the water over them is diminished. Any extended
period of calm weather appears to flatten the bars and make them
disappear.

Consideration HE. More observations are required to reach a
conclusion on this point. It is clear that the ratio depends to a
great extent upon the factors mentioned above in connection with
consideration D.

Consideration F. A study of the use of the energy equation has

given some interesting results but the observations made have been
too few to produce reliable and useful deductions.

22

COMPARISON OF OBSERVED WAVE DIRECTION
WITH A REFRACTION DIAGRAM

by
Donald R. Forrest

During the course of routine wave observations at Mission Bay,
San Diego, California, by the Field Research Group. there occurred
recently a comparatively rare combination of visibility and wave
condition which made possible a comparison of observed wave direction
with a refraction diagram. Observations have been made twice daily
for more than a year and the usual case is that the region on the
sea surface where swell can be clearly defined from the observation
point is limited. On 19 January 1951, however, visibility was ex-
ceptionally good and even though the swell was less than one foot
high it was possible to measure its direction of travel over a
relatively large area.

Directions were measured by the transit-sighting bar method
described in Beach Erosion Board Bulletin, Volume 4, Number 2, 1 April
1950. The 536-foot elevation of the instrument at the observing
point, U. §. Coast and Geodetic Survey triangulation station "Jolla",
permitted direction measurements to be made up to about five miles
distant.

The most seaward of a group of fifteen observations was
plotted on a hydrographic chart and an orthogonal drawn seaward to
deep water. This depth was the 72.fathom contour for the 13-second
wave observed. The deep water azimth thus established was 292°,
a change in direction of two degrees from thatobserved at 38 fathoms.
Other orthogonals were drawn, resulting in the accompanying re-—
fraction diagram. The method of plotting orthogonals was that
developed by Johnson, O'Brien and Isaacs (Hydrographic Office
Publication No. 605). The remaining fourteen observations were then
plotted on the refraction diagram. These are shown as short heavy
lines.

On the whole, agreement is good. However, directions observed
close inshore show a little less curvature than does the refraction
diagram, while those observed at the 15 to 20efathom region tend to
indicate a little more curvature. The largest difference, about
ten degrees, seems to be that of the most northerly observation,
close to shore. The observing technique dictates that this particular
measurement should not be in error by more than about two degrees.
The reason for the difference probably lies in the fact that the
bottom in this area is shallow, very rocky and irregular. This por-
tion of the refraction diagram, based on rather sketchy hydrographic
information, is probably erroneous.

24

BEACH EROSION STUDIES

The principal types of beach erosion control reports of studies
at specific localities are the following:

ae Cooperative studies (authorization by the Chief of
Engineers in accordance with Section 2 River and
Harbor Act approved on 3 July 1930).

b. Preliminary examinations and surveys (Congressional
authorization by reference to locality by name).

ec. Reports on shore line changes which may result from
improvements of the entrances at the mouths of rivers
and inlets (Section 5, Public Law No. 409, 74th Congress).

d. Reports on shore protection of Federal property
(authorization by the Chief of Engineers).

Of these types of studies, cooperative beach erosion studies
are the type most frequently made when a community desires investi-
gation of its particular problem. As these studies have greater
general interest, information concerning studies of specific
localities Rontaaned in these quarterly bulletins will be confined
to cooperative studies. Information about other types of studies
can be obtained upon inquiry to this office.

Cooperative studies of beach erosion are studies made by the
Corps of Engineers in cooperation with appropriate agencies of the
various States by authority of Section 2, of the River and Harbor
Act approved 3 July 1930. By executive ruling the cost of these
studies is divided equally between the United States and the co-
operating agency. Information concerning the initiation of a co-
operative study may be obtained from any District Engineer of the
Corps of Engineers. After a report on a cooperative study has been
transmitted to Congress, a summary thereof is included in the next
issue of this bulletin. A summary of a report transmitted to
Congress and a list of cooperative studies now in progress follow:

SUMMARY OF REPORT TRANSMITTED TO CONGRESS

The area studied is Lowa tec in the Town of Hull, Massachusetts,
12 miles southeast of the City of Boston. It consists of the 1l—mile
length of beach in the publicly owned Metropolitan District Commission
Nantasket Beach Reservation adjacent to the mainland base of a 38-
mile long tombolo which is extensively developed as an amusement

26

area. The reservation consists of a wide beach, a seawall, a highway,
and a public area 125 to 150 feet wide which contains parking space,
pavilions, a hotel, a bathhouse, and sanitary buildings. The beach
is wide and flat and is composed of hard-packed sand in foreshore and
offshore areas, and soft sand and stones in backshore areas, The
beach is extensively used for recreation.

Nantasket Beach has for many years been relatively stable. Where
erosion has occurred, the beach has subsequently been rebuilt by
natural forces. The seawalls have withstood direct wave attack and
adequately serve their purpose. The only beach problem existing at
Nantasket Reach is that caused by stones on the beach which interfere
with its recreational use,

The division engineer considered the desires of the cooperating
agency and studied the character, sources, and movement of beach
material, the existing structures, the changes in the shore lines and
offshore bottom, and the effects of winds and storms. He concluded
that improvement of the composition of the beach is a problem of
periodic beach maintenance which is the responsibility of local
authorities and that this maintenance may be accomplished bys
(1) covering the stones with sand, or (2) burying stones in trenches
or pits dug in the backshors area, or (3) removing stone deposits and
replacing with equal volums of suitable sand. He recommended that
the cooperating agency continue maintaing the beach in a suitable
condition for recreational use, expanding its maintenance methods
as necessary to bury and cover the stone deposits more completely or
to remove the stones and replace them with equal volumes of sand.

The Board carefully considered the report of the division
engineer and concurred in his conclusions and recommendations. In
compliance with existing statutory requirements the Beach Hrosion
Board stated its opinion thats

a. It is not advisable for the United States to adopt a
project for the locality at this times:

b. The recommended maintenance measures are in the public
interest, butsince the policy established by Public Law
727, 79th Goneress does not include a provision for
Federal participation in maintenance costs;

c¢. No share of the expense should be borne by the United
States.

COOPERATIVE BEACH EROSION SfUDIES IN PROGRESS”
NER HAMPSHIRE

HAMPTON BEACH. Cooperative Agency: New Hampshire Bhore and Beach
Preservation and Development Commission.

Problem: To determine the best method of preventing further
erosion and of stabilizing and restoring the beaches,
also to determine the extent of Federal aid in any
proposed plans of protection and improvement.

MASSACHUSETTS

PEMBERTON POINT TO GURNET POINT. Cooperating Agencys Department of
Public Works, Commonwealth of Massachusetts. NE

Problem; To determine the best methods of shore protection
prevention of further erosion and improvement of
beaches, and specifically to develop plans fer pro-
tection of Crascent Beach, The Glades, North Scituate
Beach and Brant Rock.

CONNECTICOT

-STATE OF CONNECTICUT. Cooperating Agency: State of Connecticut
(Acting through the Flood Control and Water Policy
Commission).

Problem: To determine the most suitable methods of stabilizing
and improving the shore line. Sections of the coast
will be studied in order of priority as requested by
the cooperating agency until the entire coast is in-
cluded.

NEW YORK
JONEG BEACH. Cooperating Agency: Long Island State Parks Commission
“ Problem: To datgpmine behavior of the shore during a 12-month
cycle, in¢luding study of littoral drift, wave re-
fraction’ and movement of artificial sand supply between
Fire Island and Jones Inlets.
NEW JERSEY
OCEAN Crry . Cooperating Agency: City of Ocean City.
Problem: To determine the causes of erosion pr accretion am
the effect of previously constructed groins and

structures, and to recommend remedial measures to
prevent further erosion and to restore the beaches.

28

VIRGINIA
VIRGINIA BEACH. Cooperating Agency: Town of Virginia Beach.
Problem: To determine the methods for the improvement and
protection of the beach and existing concrete sea
wall.
SOUTH CAROLINA

STATE OF SOUTH CAROLINA. Cooperating Agency: State Highway Depart—
ment.

Problem: To determine the best method of preventing erosion,
stabilizing and improving the beaches.

FLORIDA ~
PINELLAS COUNTY. Cooperating Agency: Board of County Commissioners,
Problem: To determine the best methods of preventing further
recession of the gulf shore line, stabilizing the
gulf shores of certain passes, and widening certain

beaches within the study area,

LOUISIANA

LAKE PONTCHARTRAIN. Cooperating Agency: Board of Levee Commissioners,

Orleans Levee District.

Problem: To determine the best method of effecting necessary
repairs to the existing sea wall and the desirability
of building an artificial beach to previde protection

to the wall and also to provide additional recreational

beach area.
TEXAS

GALVESTON COUNTY. Cooperating Agency: County Commissioners Court
of Galveston County.

Problem: To determine the best method of providing a permanent
beach and the necessity for further protection or ex—
tending the sea wall within the area bounded by the
Galveston South Jetty and Hight Mile Road.

To determine the most practicable and economical
method of preventing or retarding bank recession on
the shore of Galveston Bay between April Fool Point
and Kemah.

29

CALIFORNIA

STATE OF CALIFORNIA. Cooperating Agency. Division of Beaches and
Parks, State of California.

Problem: To conduct a study of the problems of beach erosion
and shore protection along the entire’ coast of Calif-
ornia. The initial studies are being made in the
Ventura—Port Hueneme area, the Santa Monica Bay area
and the Santa Cruz area.

WISCONSIN
RACINE COUNTY. Cooperating Agency: Racine County.

Problem: To prevent erosion by waves and currents, and to
determine the most suitable methods for protection,
restoration and development of beaches.

KENOSHA, Cooperating Agency, City of Kenosha.

Problem; To determine the best method of shore protection and
beach erosion control.

OHIO

STATE OF OHIO. Cooperating Agency: State of Ohio (Acting through
x the Superintendent of Public Works).

Problem: To determine the best method of preventing further
erosion of and stabilizing existing beaches, of re-
storing and creating new beaches, and appropriate
locations for the development of recreational
facilities by the State along the Lake Erie shore
line.

TERRITORY OF HAWAII

WAIKIKI BEACH:
WAIMEA & HANAPEPE, KAUAI. Cooperating Agency: Board of Harbor
Commissioners, Territory of Hawaii.

‘i Problem: To determine the most suitable «method of prevent—
ing erosion, and of increasing the usable recreational
beach area, and to determine the extent of Federal
aid in effecting the desired improvement.

30

BEACH EROSION LITERATURE

There are listed below a collection of papers presented at the
Institution on Coastal Engineering, University of California, Long
Beach, California. Copies of these papers can be obtained on a
two-week loan by interested official agencies.

Munk, W. H., “Origin and Generation of Waves"

Wiegel, Re L., “Elements of Wave Theory*

Mason, Me A., “Transformation of Waves in Shallow Water'
Shepard, F. P., "Nearshore Circulation"

Vanoni, Vito A., "Harbor Surging"

Arthur, R..S., “Vave Forecasting and Hindcasting"

Studds, R.F.A., "Coast and Geodetic Survey Data - An
Aid to the Coastal Enzineer"

Fleming, R. H., “The Engineering pplication of Sea
, and Swell, Data"

Elliott, D. O., "The Beach Hrosion Board"

Handin, John W., "The Geological Aspects of Coastal
inginsering"

Hinstein, Ho. Ae, “Kstimated Quantities of Sediment
Supplied by Streams to a Coasti

Krumbein, W. C., “Littoral Processes in Lakes"
Baton, R. @., "Littoral Processes on Sandy Coasts”
Blackman, Berkeley, “Dredging at Inlets on Sandy Coasts"

Reilly, G. P., "Dredging at Coastal Inlets - The Sea-
going Hopper Dredge"

Simmons, H. B.,."“Contribution of Hydraulic Models to
Coastal Sedimentation Studies"

Peel, Ke. P., “Location of Harbors"

Rawn, A. M., "Factors Influencing and Limiting the
Location of Sewer Ocean Outfalls®

Kenyon, E. C., Jr., “Case History of Ocean Inlets Los
Angeles County Flood Control District"

Hickson, R. E., “Case History of Columbia River Jetties™
McQuat, H. W., ™ Case History of the Los Angeles Harbor" |’
Wicker, C. F., "Case History of the. New Jersey Coastline"

Hudson, R. Y., “Ihe Hydraulic Model as an Aid in Break-
water Design"

Morison, J. R., "Design of Piling"

Schauffle, H. J., "Erosion and Corrosion on Marine
Structures"

Ayers, J. R., “Seawalls and Breakwaters"

Hickson, Re H., "Design and Construction of Jetties*
Kaplan, "Design of Breakwaters*

Streblow, A. Go, “Breakwater Construction"

Horton, D. Fe, "The Design and Construction of Groins"™

Caldwell, J. M., “By-Passing Sand at South Lake Worth
Inlet™

32