CHAPTER 66
EQUILIBRIUM CONDITIONS IN BEACH WAVE INTERACTION
Dr. H. Raman, and John J. Barattupuzha
Hydraulic Engineering Laboratory
Indian Institute of Technology
Madras, India
ABSTRACT
Laboratory studies were conducted in an attempt to find
out a relationship between beach and wave characteristics
when equilibrium conditions are reached in beach wave interaction for the simple case of regular waves acting normal to
the beach. Experimental results indicate the existence of
stable points on beach profiles where the coordinates of the
profile do not change with time when waves of constant
characteristics act on the beach. Erperical relationship
between the wave and beach properties are proposed. A new
criterion for classification of beach profiles is indicated.
INTRODUCTION
In beach-wave interaction and the resultant process of
mutual modification there is a tendency to attain equilibrium
conditions under which beach and wave no longer cause any
further change on each other. In nature however, the everchanging meteorological factors which cause rapid changes in
waves, tides and currents seldom permit this interaction to
reach equilibrium conditions. Nevertheless* certain dominant
tendencies in the wave pattern may persist over a season or
part of a season thereby enabling us to delineate a few
significant characteristics of waves to work out their
influence on beach modification problems. This in turn
offers us an opportunity for a meaningful determination of
beach response to wave action, provided necessary numerical
relationships are available for this purpose. Accurate
formulae for determination of the nature and extent of changes
on beach profiles for given wave conditions, though of great
importance in a large number of shore problems, are not available at present. It appears that the evolution of theoretical
relationships, which can be used on practical problems with a
reasonable degree of accuracy, may have to wait till better
tools are available to deal with problems of sediment transport in an unstable and oscillating wave velocity field. This
is discussed in detail by Silvester (1,2). Baperical relationship can however be derived for certain conditions ot profile
1237
1238
COASTAL ENGINEERING
changes. la this paper an attempt la made to derive
emperical relationships for the determination of the type
and quantum of beach profile changes in the simple case
of regular waves acting on a beach of uniform initial
slope aligned normal to wave action.
STABLB POINTS OH BBACH PROFILES
Available experimental and field data on beach profiles
indicate that usually there is a stable point on beach
profiles near the breaker sons which does not undergo any
considerable change during the process of profile modification due to incident waves of essentially constant characteristics. In the case of storm profiles a second stable
point may exist in the offshore zone depending on the steepness of the incident waves (See ?ig.1). The stable point
(or the second stable point in case two stable points exist)
acts as a fulcrum about which the waves try to keep the
material distribution in balance - erosion onshore and deposition offshore - within the active sone of the profile.
Tor normal profiles the stable point does not exist in the
strict sense, but there is one point on the modified profile
which seem to serve the same function as above (See Fig.2).
In this case there is erosion offshore and deposition onshore.
Heutral points on beach profiles where, material of a
given size will be in oscillating equilibrium, with no net
movement, have been discussed in a few previous works.
Cornaglla (1898) was perhaps the first to make this observation^). Inman*s(4; observations on sediment sorting and
studies of Miller and Zeigler (5) showed that there are
zones of oscillating equilibrium for each sediment size for
the given wave conditions. Bagleson et al (6) in their
theoretical analysis of the problem of sediment transport
on sloping beaches under waves equated the forces acting on
a bed pactiefeand arrived at a criterion for oscillatory
equilibrium. Wells (?) defined a neutral point on the beach
profile such that the skewness of the probability distribution of horizontal water velocity of second order gravity
waves will be zero at the point.
Che neutral positions of particle motion described by
these authors Indicate that particles of the same size will
move in opposite directions when placed on either side of
the neutral point. (A sediment particle on the seaward side
will move seaward while one on the shoreward side will move
shoreward of the neutral point for the same sediment size).
The stable points mentioned In this paper are those which
act as a fulcrum about which the profile swings while
material is moved from one side of the point to the other.
The net sediment motion is unidirectional, either towards
shore or away from it, for appreciable distances on either
side of the stable point, A directional variation is expected
EQUILIBRIUM CONDITIONS
1239
only due to changes la Telocity field imposed toy reflection or due to changes in material sice* The influence
of the former is comparatively small and can he easily
identified by the formation of nearly stationary humps
and hollows on the bed whose spacing will he closely
related to the wave length. This is easily understood
from the profiles shown in figures 3 to 8 and from the
profiles given in the references cited.
Pig.3 shows the stable point on a set of natural
beach profiles taken at a shore which is in equilibrium.
Pig.14.22 of Bef (8) shows 12 sets of profiles on an
actual beach. Bach set contains 5 profiles taken during
the course of two to three months at a single station. It
is seen that in most cases the profiles remain stationary
at a depth of 5 ft. (1.52 m).
Figures 4 to 7 show the stable points on experimental
beach profiles of the storm type and Figure 8 shows the
modified version of the stable point on a normal beach. In
the experimental beach profile shown in Pig.14 of Ref(7)
the stable point is visible at 1?.5 en depth. At this point
there is no change for the original profile irrespective
of the time of propagation of waves, although adjacent
areas are getting altered with time of wave action.
EXPERIMENTAL SET-UP AND PROCEDURE
The experiments were conducted in a wave flume 31 »
long 90 cm wide and 90 cm deep. The plunger type wave
maker powered by a variable speed motor is fixed at one
end for wave generation. At the other end of the flume,
model beaches are laid to uniform initial slope using sand
of required size. All tests in the present series were
done under a constant water depth of 35 cm at the toe of
the beach.
Waves of constant characteristics were generated and
allowed to work on the beach till the profile showed no
significant change in time. In order to locate the stable
points and to find out the pattern of modifying process
the beach profile was measured at 5 to 10 hour intervals.
Normally it took 40 to 50 hours for the profile to reach
equilibrium conditions. Wave characteristics were measured
by resistance type wave gauges and a recorder unit.
EXPERIMENTAL RESULTS AND DISCUSSION
Even though experiments were done with sand of different
sizes and coal powder and with different initial slopes, the
results of tests on 0.3 am sand alone is reported in this
1240
COASTAL ENGINEERING
paper* Other profiles also show the same pattern. But
detailed analysis with respect to stable point is done
only for 0.3 ma sand for different wave conditions and
initial slopes and hence this is discussed in detail.
typical storm type profiles for slopes 1/8, 1/10,
1/12 and 1/15 are shown in Figures 4,5,6 and ? indicating the progressive development of profiles towards
equilibrium. Similarly Fig.8 shows the development of a
normal type profile on a 1/10 slope.
TYPES OF PS0F1LB CHARGES
It is found that mainly there are three types of
profile changes:
(1) Storm type profiles with erosion on the shore
and deposition in deeperwa&ers.Srosion takes place shoreward of the second stable point when it exists or shoreward of the first stable point when the second one is
absent. Long shore bar is present.
(ii) Normal profiles with deposition taking place on
the shore from the material dug out by waves from offshore
of the breaker by waves. Longshore bar is absent.
(iii) Storm type profile with longshore bar, but
deposition takes place on the shore. Shoreward of the
second stable point erosion takes place and seaward of it
deposition is seen.
The last mentioned of these is an interesting phenomenon
in that it is customary to think of storm profiles as
indicative of shore erosion. Examination of this type
of profiles show that onshore transport in storm profiles
takes place when the distance between the two stable points
is more than one local wave length . It appears that in
such a case more than one set of mass transport circulation
cells will be formed with possibility of a reversal in the
direction of transport. It may be recalled that LonguetHiggins (9) showed that mass transport circulation cells
may be formed in pairs at spacings equal to half wave
length for standing waves. In the present case there is
reflection from the beach and hence there will be a partial
standing wave, this incidentally explains why flat beaches
usually show accretion on the shore while steep beaches
erode for the same wave and bed material characteristics.
It is interesting to note that the second stable point being
the offshore limit of erosion of the original beach profile,
its location is controlled by the local wave characteristics,
which is primarily dependent on the depth below still water
level, and bed material characteristics. The first stable
EQUILIBRIUM CONDITIONS
point is * product of the breaker at the longshore bar
and hence is controlled by the elope of the beach also,
therefore the distance between these points is a function of the slope also. As the slope becomes flatter,
the distance increases, increasing the possibility of
creating more than one mass transport circulation cell.
This helps in reversing the direction of transport at
some point between the offshore bar and the second stable
point, thereby permitting deposition on the shore.
SISNIFICAirCB OF STABLE POIKTS
A stable point can exist only on a beach which is
in equilibrium - ie. a beach with a shore line which may
oscillate between certain shoreward and seaward positions
due to seasonal ohanges in waves while maintaining its
mean position without significant change when considered
over a period of one or two decades. This means that
there should not be a net loss or gain of material to the
beach in its active zone either due to onshore-offshore
transport or longshore transport mechanism. The presence
of a stable point on a beach profile can be taken as an
indication that the beach is in equilibrium.
She presence of stable points on natural beaches
Fig.3 and Hef.(8) indicate that small changes in wave
characteristics as may occur within a season for short
periods do not annihilate the stable point or the
corresponding alongshore stable zone.
BQ0ILIBBIUM PBOBTLB ATO STABLE POINTS
If the local wave steepness B/L» where H is the local
wave height and L the local wave length calculated from
the linear theory with Rayleigh's assumptions, is plotted
against the ratio of x/h for a straight uniformly sloping
beach, a straight line graph will be obtained for h/Lo
ratios less than nearly 0.15. Here x is the horizontal
distance from a suitable origin to be chosen on the still
water line (SWX) to any desired point on the bed profile
and h is the depth of that point below SWL. (See Fig.1 for
definition). The slope of this line is controlled by the
location of the origin on the still waterline as can be
seen from I4g.9. Fig.9 gives plots of H/L Vs x/h for the
case of uniform beach slope of 1/10 when a wave of deep
water wave height, Ho • 11.28 cm and wave period, T • 1 sec
acts on the beach for different locations of origins. The
straight vertical line is for origin at the intersection
of the SWL with beach slope. This is the point of zero
chainage also. The plot becomes flatter and flatter as
the origin is shifted further and further offshore.
1241
1242
COASTAL ENGINEERING
Available experimental results indicate that a plot
of H/L Vs x/h prepared for the final equilibrium profile,
even though not falling strictly into a straight line,
can be approximated to one for portions of profile offshore of the longshore bar* The dashed line in Fig.9 is
the line of best fit drawn in this fashion for the final
profile resulting from the action of the above wave on
the 1 on 10 initial beach slope. In this case the projection of the 1st stable point on SWI< at ch.104 cm is taken
as the origin. This line will then correspond to a
straight line which represents the mean profile in the
offshore part.
Fig.10 gives plots of H/L Vs x/a for the final
equilibrium profiles resulting from waves of different
characteristics acting on the 1 on 10 initial beach slope.
In all cases the origin is ohosen as the first stable point.
The lines marked 1 to 4 represent storm type profiles and
5 and 6 represent normal profiles. It will be noted that
the lines for the same type of profile, either storm or
normal are nearly parallel to each other. Also the distances between these lines along the ordinate are nearly
equal to the differences in deepwater wave steepness in
the case of profiles of the same type. This perhaps,
indicates the possibility of obtaining the mean profile
resulting from any incident wave if H/L Vs x/h relationship for one wave condition is known for the same initial
beach slope.
LOCATIOH Of STABLE POIHTS
Prom the available experimental results concerning
an initially uniform slope of 1/10 with material having a
median dia of 0.3 mm it is found that the depth at the
first stable point is a function of Ho and X for the given
slope. The depth at the stable point is found to increase
linearly with the nondimensional parameter (Ho g T)/jJ
where g is the acceleration due to gravity and JJ is the
kinematic viscosity of water. Shis is shown in Pig.11.
The second stable point, when it exists, corresponds
to the point of intersection of the plots H/L Vs x/h. for
the initial and final profiles calculated with the origin
at the first stable point for both cases. Pig.12 shows
H/L Vs x/h relationships for initial and final conditions,
for the four profiles shown in figures 4 to 7* These four
profiles have four different initial slopes. The wave
characteristics are given in the drawing. It can be easily
seen that the intersection of the initial and final plots
of H/L Vs x/h represents the second stable point.
EQUILIBRIUM CONDITIONS
1243
CLASSIFICATION OF BEACH PH0KLB8
A new criterion for Johnson*s(11) classification of
beach profiles as 'atom and normal* is briefly examined.
If the formation of a normal profile la examined it will
be seen that transport of material is onshore everywhere
along the profile. This would suggest that the fluid
particle movement at bottom, at least predominantly if
not always is towards shore. A type of wave which satisfies this condition is the solitary wave which in its
ideal case has only forward movement everywhere) across
its depth. In shoaling water it is possible that wave
condition may approximate to solitary wave. If the point
of threshold movement of bed material is onshore of the
point at which the wave becomes nearly solitary, only
onshore movement of sediments can take place.
Taking Bagnold's (10) criterion T - lirv! to
find the depth h at which a progressive wave may be
approximated to a solitary wave for a given period T
and finding the depth a. at which threshold movement
starts by equating the maximum instantaneous horizontal
velocity from linear theory to the threshold velocity for
material movement as given in Hef.(11), the criterion for
classification of profiles can be written as:
—= !» 1
*1
h.
—- «£ 1
''T
normal profile
storm profile
Here, the quantity ah is given as a function of the
ratio of amplitude n to depth below SWL, h by Bagnold (10)
for easy extraction of the same for calculation purposes.
Available experimental and field data generally confirm
the validity of this test. A detailed description of the
same is not attempted in this paper.
CONCLUSIONS
Available experimental and field data Indicate the
existence of stable points on beach profiles of equilibrium
beaches. On a normal beach two stable points may be present,
one near the breaker sons and one at the offshore limit of
erosion of original profile seaward of the longshore bar.
The variation of the local wave steepness H/L with the
ratio of coordinates x/n with the projection of stable point
on SWL as the origin is nearly linear for the final equilibrium
1244
COASTAL ENGINEERING
profile. For the same initial slope and material characteristics, different wave characteristics give nearly
parallel plots of H/L Vs x/ht the distance between these
lines along the H/L axis being equal to the deepwater
wave steepness* This indicates the possibility of getting the mean final equilibrium profile if the depth at
the first stable point for the profile is known along
with one plot of H/L Vs x/h for the same initial slope
and bed material and any other wave characteristics.
Experimente on a 1/10 initial beach slope using
0.3 mm dia sand show that the depth at the stable point
is proportional to the dimensionless parameter (Ho g T)/JJ
She second stable point can be fixed if the first
one is determined and the plot of H/L Vs x/h plot can be
drawn as already described.
ACKNOWLEDGMENTS
The authors are greatly indebted to Professor
V. Sethuraman, Professor of Hydraulic Engineering, I.I.T.
Madras for his constant encouragement in this work. The
authors also thank the Kerala Engineering Research Institute for permission to use the field data given in Pig.?
and Mr. Balakrishnan for his help in carrying out part
of the experimental work.
REFERENCES
1. Silvester, R., "Sediment movement beyond the breaker
zone , Civil Engineering Transactions, April 1970,
PP 6J-71
2. Silvester,„H., "Beach Profiles and Littoral drift
assessment , La Houille Blanche/No.6-1969, pp 615-622
3. Reference by James C. Ingle Jr., "Movement of Beach
Sand P.91, Developments in Sedimentology 5, Elsevier,
New Tork 1$66
4. Inman, C.L., "Areal and Seasonal variations in beach
and near shore sediment at La Jo11a, California ,
U.S. Army Corps of Engineers, B.E.B. Tech. Memo No.39,
March 1953
5. Miller, Robert R.L. and John M. Zeigler, "A model
relating dynamics and sediment pattern in equilibrium
in the region of shoaling waves, breaker zone and
foreshore , Journal of Geology, 66, 4 (July 1958)
pp 17-41
EQUILIBRIUM CONDITIONS
6. Bagleson, P.S., B. Glenne and J.A. Dracup Equilibrium
characteristics of sand beaches in the offshore zone
U.S. Army Corps of Engrs., B.E.B., Tech. Memo No.126,
July 1961.
7« Wells,„D.B., "Beach Equilibrium and Second Order Wave
Theory , Journal of Geop. Hes. Vol.72, Ho.2, Jan 15,
1967, PP 497-50*
8. Wlegel, R.I., HOceanographic&l Engineering", p.365,
Prentice-Hall Inc., 1964.
ranepo
9« Longuet-Higgins M.S., "Mass transport
in water waves",
Phil. Trans. Hoy. Soc. A 245 (90?)
(903) 535-581
10. Technical Report No.4, Coastal Eng. Res. Centre, tT.S.
Army Corps of Engrs., 1966.
11. Johnson J.W., "Scale Effects in Hydraulic Models
involving wave motion Trans. Amer. Geophys. Union,
30, 4 (1949), 517-25.
1245
1246
COASTAL ENGINEERING
-ORIGIN OF CO-ORDINATES
X
NOTE:
<D,® INDICATE SUCCESSIVE
STAGES IN PROFILE MODIFICATION
STABLE POINTS
FIG.1.. PROFILE MODIFICATION FOR STORM
BEACH PROFILE
STABLE POINT ON
MODIFIED PROFILE
FI6.2..PR0FILE MODIFICATION FOR NORMAL
BEACH PROFILE
EQUILIBRIUM CONDITIONS
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