US Army Corps of Engineers
0.5 ALONGSHORE DISTANCE (x/L)
(7/8) NOTLISCd 3NIT3YOHS
TECHNICAL REPORT CERC-87-15
ANALYTICAL SOLUTIONS OF THE ONE-LINE MODEL OF SHORELINE CHANGE
by Magnus Larson, Hans Hanson
Department of Water Resources Engineering Institute of Science and Technology University of Lund Box 118, Lund, Sweden S-221-00
and
Nicholas C. Kraus Coastal Engineering Research Center DEPARTMENT OF THE ARMY
Waterways Experiment Station, Corps of Engineers PO Box 631, Vicksburg, Mississippi 39180-0631
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October 1987 Final Report
Approved For Public Release, Distribution Unlimited
Prepared for DEPARTMENT OF THE ARMY US Army Corps of Engineers Washington, DC 20314-1000
Under Coastal Sediment Transport Processes Work Unit 324-1
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11. TITLE (Include Security Classification) Analytical Solutions of the One-Line Model of Shoreline Change
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17. COSAT!I CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number) BIECD SUBCROUE Coast changes (LC) Mathematical models (LC)
Beach erosion (LC) Shore-lines (LC)
19. ABSTRACT (Continue on reverse if necessary and identify by block number)
This report presents more than 25 closed-form solutions of the shoreline change mathematical model for simulating the evolution of sandy beaches. The governing equation is developed in a general form, and the assumptions and techniques used to arrive at tractable closed-form solutions are described. Previous solutions are reviewed, and many new solutions are derived. Solutions for beach evolution with and without the influence of coastal structures are given that cover situations involving beach fill of almost arbitrary initial shapes, sand mining, river discharges, groins and jetties, detached breakwaters, and seawalls. Techniques for combining and extending the solutions are discussed. Appendixes provide details of mathematical techniques used and complete derivations of selected new solutions. Such analytical solutions can provide a simple and economical means to make a quick qualitative evaluation of shoreline response under a wide range of environmental and engineering conditions.
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PREFACE
The investigation described in this report was authorized as a part of the Civil Works Research and Development Program by the Office, Chief of Engi- neers (OCE), US Army Corps of Engineers. The work was performed under the Coastal Sediment Transport Processes Work Unit 324-1, Shore Protection and Restoration Program, at the Coastal Engineering Research Center (CERC) of the US Army Engineer Waterways Experiment Station (WES). Messrs. John H, Lock- hart, Jr., and John G. Housley were the OCE Technical Monitors.
The study was conducted from 1 July 1986 through 31 December 1986 by Dr. Nicholas C. Kraus, Research Physical Scientist and Principal Investigator, Coastal Sediment Transport Processes Work Unit, Research Division (CR), CERC, in conjunction with related engineering studies by Messrs. Magnus Larson and Hans Hanson of the University of Lund, Sweden. This report presents the over- all results of these efforts. The CERC portion of the study was conducted under general supervision of Dr. James R. Houston, Chief, CERC; Mr. Charles C. Calhoun, Jr., Assistant Chief, CERC; and Dr. Charles L. Vincent, Program Manager, Shore Protection and Restoration Program, CERC; and under direct supervision of Mr. H. Lee Butler, Chief, CR, CERC. Work at the University of Lund was performed under general supervision of Dr. Gunnar Lindh, Head, Department of Water Resources Engineering, Institute of Science and Technol- ogy. Mr. Bruce A. Ebersole provided technical review. This report was edited by Ms. Shirley A. J. Hanshaw, Information Technology Laboratory, Information Products Division, WES.
COL Dwayne G. Lee, CE, was Commander and Director of WES during publica-
tion of this report. Dr. Robert W. Whalin was Technical Director.
CONTENTS
PREPACE tierce cic clclec SSOOCIOOIOIOICOICIC CII HOCIOIOIC Cece ecececcvesecccceene. coe LIST OF FIGURES..... SSICIICISIOISIC OOO COO i IOC cece c cece ccs ccc ccccces PART I: DNTRODUCION Neretsite creletelelotelsireilciaretelsrslaare AOISIOIOIOIC ICICI ICICI
BACK ETOUNcteteletatetetetelalelelcleletellsteletelciers OMENS WNAIAo paDbODDDCdDOaDDD00N0E B0000000 CD00 000000NDO000G00 Overview of Previous Analytical Work........
General Approach in the Present Work
SOLUTIONS FOR SHORELINE EVOLUTION WITHOUT
BAR Temlelie COASTAL STRUCTURES...
General Formal Solution............
Finite Rectangular Beach Fill........
Triangular-Shaped Beach.... Trapezoidal-Shaped Beach...
Nah AclWBle WEEE 6 qg50000000000
Semiicincullar=ShapedPBeachi\amtercrelelelsteleleleveleleleleleleleleielaretetenelevets Semicircular Cut in a Beach......
eoececeee eee ee
Semi-Infinite Rectangular Beach Fill............seee0. Rectangular Cut in a Beach....
eoceoeceveeeee eee eee o
eececeeoeve
eeccece
Sand Discharge from a River Acting as a Point Source..........-
Sand Discharge from a River Mouth of Finite Length.. SOLUTIONS FOR SHORELINE EVOLUTION INVOLVING
PART III: COASTAL STRUCTURES....
Shoreline Change Initially Filled Shoreline Change Shoreline Change Shoreline Change
REIEREN GES rreteteteretenetelsnoletonsrers APPENDIX A:
APPENDIX B:
APPENDIX C: APPENDIX D:
WHERE FLANKING OCCURS.........
APPENDIX E:
A SHORT INTRODUCTION TO THE LAPLACE TRANSFORM TECHNIQUE.........
SHORELINE EVOLUTION DOWNDRIFT OF A GROIN WITH BYPASSING REPRESENTED BY AN EXPONENTIAL FUNCTION.
SHORELINE EVOLUTION BEHIND A DETACHED BREAKWATER........ SHORELINE EVOLUTION IN THE VICINITY OF A SEAWALL
at Groins and Jetties......... Groin System.. at a Detached Breakwater. at aS Sawailalyyerererelelicvelclevene cverenelerciareleversiciele ate at a Jetty, Including Diffraction.............
SHORELINE EVOLUTION DOWNDRIFT OF A JETTY IF AN
ARBITRARY NUMBER OF SOLUTION AREAS IS USED TO
MODEL DIFFRACTION...
SHORELINE EVOLUTION BEHIND A JETTY FOR LINEARLY
APPENDIX F:
VARYING BREAKING WAVE ANGLE....... APPENDIX G:
VARYING BREAKING WAVE ANGLE. APPEND DXS Hs NOTATION yeveleretletelsiere
SHORELINE EVOLUTION BEHIND A JETTY FOR EXPONENTIALLY
eoceoeee
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Al
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12
13
14
15
16
17
18
LIST OF FIGURES
Schematic illustration of a hypothetical equilibrium beach PRORIMNEG 550000000 CoodDdDOOOODDDDD0DDNE SoadaooCD0DDKDD0DN0N 6090000000
Definition sketch for geometric properties at a specific locattonwasmrelated@tonshorelbinerchangelrererercjencielersicieteletleleresercio crcl
Comparison between experimental and theoretical shoreline GHOIMETOING od5000 D0ODDDDADDDDDDOUDGDDODNDDDORODDDNOONN oo000000000
Definwtionasketchwromithe stwor-lenentheoryelpeleieters chererelchercnelehelelererele Lwo-ikine theory solutelon) LOTrmane oun Sy SiEeMeroeleleielerelcleleleieielieleleiereienete
Shoreline evolution between two groins initially filled with
SATU dccisras oy sealers roan neers oneliskevarens Mevekelenccheneiens FODDOGCOOOOD OOOO DON OOe auetielic
Shoreline evolution of an initially rectangular beach fill gqoseal TO WANES Eipeabialiays soYoramell (ee) SIME ogoqgq00000000000000000
Percentage of sand volume lost from a rectangular fill as a function of dimensionless time............ Jo0DKbaDDD0UD0ND00RDNDN
Shoreline evolution when sand is supplied at x = 0 to maintain a specific beach width Vo tort t tees e eee eee eee goo0Kd
Shoreline evolution of an initially semi-infinite meCieaneil arb ea Chtreverdertererareleicleraisnencneleleionslekcickelekenenene 900000000 9000
Shoreline evolution of a rectangular cut in an infinite beach of width Vo Tite t eee t cece cece reece cece eneeeeeceneccees
Shoreline evolution of an initially triangular beach...........¢.
Comparison between analytical solution with the linearized transport equation and numerical solution with the original transport equation for a triangular beach fill (for height-to- Width rattosvor lOlvaird OS) ccdia srenerercneitotatsse lave teuoveraelteneneiremevieireaseniencatie
Shoreline evolution of an initially trapezoidal beach form.......
Shoreline of arbitrary shape approximated by N_ straight
AN TAM 'S Fates atfastatvon etre tisycosie rellwnakiolie we veite ever HOFer oh ee) eievene' tacoliwel alah elok ex erarroweunvewen euebane eoaeatone
Semicircular-shaped beach approximated by a polygon......cccescee Shoreline evolution of an initially semicircular beach...........
Definition sketch for a circular segment-shaped beach.......cccce
10
13
14
15
16
19
22
23
24
2S)
27
28
29
30
Sil
32
33
20
21
22
23
24
7235)
26
27
28
29
30
Sil
372
33
Shoreline evolution of an initially circular segment-shaped
beach (a = 45 deg).
Comparison between analytical and numerical solutions for the case of a circular segment-shaped beach.............
Shoreline evolution of an initially semicircular cut in a
Shoreline evolution of an initially circular segment cut in
a beach (a = 45 deg).
eoeeeeee eee eoeoeeeee
Shoreline evolution of an initially cosine-shaped beach (a distance of one beach cusp height added to the shoreline POSSHELCM)) oo00000000000000 aia’ ciewekenegenshe
Shoreline evolution in the vicinity of a river discharging
sand and acting as a point SOUTCE.......ceeeceees
Shoreline evolution in the vicinity of a river discharging sand
with a periodic variation in strength as a function of time
ne OD. §eO FRIIS). O25) aoosonsbasadbospcc:
Shoreline evolution in the vicinity of a sand-discharging river
mouth ok ehamitelwaldithiaciclehereickerelelerers
Maximum delta growth from a sand-discharging river mouth of
finite length......
Definition sketch for the case of a groin..........-. 9
Shoreline evolution updrift of a groin which is totally
blocking the transport of sand alongshore........eccceeees
Shoreline evolution downdrift of a groin with bypassing described by Q,(1 - erly @L/O SS 0s/ 5 OG. = Osh aad 5
2 B B *‘o fo) Mb US © 2) oobanoooo0odcooo0c so00006
Comparison between analytical and numerical solutions of shoreline evolution updrift of a groin with incident breaking
waviel ang dlem 2 OMdelocreneretsicielenerchelcneieleleiens
Comparison between analytical and numerical solutions of shoreline evolution updrift of a groin with incident breaking
wave angle 45 deg.
eooeeececeeee eee ec ee eee eee oe oe eo ee oO ee 8
Shoreline evolution between two groins initially filled with
sand (L/W = 0.33,
Qa oO
= 0.25 rad).
35
35
36
37
39
40
43
44
47
48
51
2
>2.
54
No. Page
34 Bypassing sand transport rate at the downdrift end of a groin x = W as a function of time...... pODDOD DOR ODDOO DOO DSDOOODDOONDNN 55
35 Definition sketch for the problem of shoreline change in the vicinity of a detached breakwater..... DoodCoDDDDDOODODDOOD DOORN 56
36 Initial shoreline evolution in the vicinity of a shore-parallel
detached breakwater (6 = 0.5 , aT = 0.4 rad , Cm) = ON eerie re 59 37 Final shoreline position in the vicinity of a shore-parallel
detached breakwater (6 = 0.5 , Ona OR4uradis oa O) Rae 60 38 Definition sketch for a semi-infinite seawall for which no
erosion occurs behind the seawall........... o600d0000000000 5000 61
39 Definition sketch for a semi-infinite seawall for which erosion occurs behind the seawall.........cccccecce Sle Mie ei Weems saiehereiewe 61
40 Shoreline evolution in the vicinity of a seawall where erosion and flanking may occur behind it (a =02 radu, Ge" Old. rade Oo =i OLO) een yatrn aches ah Ra Nie Tee MOS 02 41 Definition sketch for shoreline evolution downdrift of a jetty for which a finite number of solution areas is used to model diffraction........ apacoe6e atelelisloloielckehohsrssicuclehevehensncliens By onelichehens 66
42 Shoreline evolution in the vicinity of a groin for variable sand transport rate conditions (two solution areas; 6 = 0.5 , Oo -0.1 rad , Cages =—OeAvrad) ari wearcielericiin erste acs SICA Oto oe oe 67 43 Shoreline evolution behind a jetty with linear variation in breaking wave angle in the shadow zone (a_ = -0.1 rad , Oy, = (SAUD een Pe ge RUN Mircg On ye Re o00000D0OOEDOON 69
44 Shoreline evolution behind a jetty with exponential variation in breaking wave angle (o. SV Oca sna! is Viva \isrerusceie emcnave texensrebeusueneke 70
ANALYTICAL SOLUTIONS OF THE ONE-LINE MODEL OF SHORELINE CHANGE
PART I: INTRODUCTION
Background
1. Mathematical modeling of shoreline change has proven to be a useful engineering technique for understanding and predicting the evolution of the plan shape of sandy beaches. In particular, mathematical models provide a concise, quantitative means of describing systematic trends in shoreline evo- lution commonly observed at groins, jetties, and detached breakwaters and produced by coastal engineering activities such as beach nourishment and sand mining.
2. Qualitative and quantitative understanding of idealized shoreline response to the governing processes is necessary in investigations of beach behavior. By developing analytical or closed-form solutions originating from mathematical models which describe the basic physics involved to a satisfac- tory level of accuracy, essential features of beach response may be derived, isolated, and more readily comprehended than in complex approaches such as numerical and physical modeling. Also, with an analytical solution as a starting point, it is possible to estimate, rapidly and economically, charac- teristic quantities associated with the phenomenon, such as the time elapsed before bypassing of a groin occurs, percentage of volume lost from a beach fill, and growth of a salient (emerging tombolo) behind a detached breakwater. Another useful property is the capability to obtain equilibrium conditions from asymptotic solutions. Closed-form solutions for shoreline change can also be used as a teaching aid. However, the complexity of beach change implies that results obtained from a model should be interpreted with care and with awareness of the underlying assumptions. Closed-form mathematical models cannot be expected to provide quantitatively accurate solutions to problems involving complex boundary conditions and wave inputs. In engineering design, a numerical model of shoreline evolution would be more appropriate.
3. The equations describing shoreline evolution fast become excessively
complicated to permit analytical treatment if too many phenomena are described
in one formulation. Therefore, to obtain a closed-form solution to shoreline change, a simple mathematical formulation has to be used, but one which still preserves the important mechanisms involved. The one-line (denoting the shoreline) theory was introduced by Pelnard-Considere (1956), and it has been demonstrated to be adequate in this respect. Considerable numerical modeling of long-term shoreline evolution (time-scale on the order of years) has been done on the basis of this work. However, not many analytical approaches have been published, probably because of their limited applicability for describing the finer details of shoreline change. Contributors in this field include Bakker and Edelman (1965), Bakker (1969), Bakker, Klein-Breteler, and Roos (1971), Bakker (1970), Grijm (1961, 1965), Le Méhauté and Brebner (1961),
Le Méhauté and Soldate (1977, 1978, 1979), and Walton and Chiu (1979).
One-Line Theory
4. The aim of the one-line theory is to describe long-term variations in shoreline position. Short-term variations (e.g., changes caused by storms or by rip currents) are regarded as negligible perturbations superimposed on the main trend of shoreline evolution. In the one-line theory, the beach pro- file is assumed to maintain an equilibrium shape, implying that all bottom contours are parallel. Consequently, under this assumption it is sufficient to consider the movement of one line in studying the shoreline change, and that line is conveniently taken to be the shoreline, which is easily observed (Figure 1).
5. In the model, longshore sand transport is assumed to occur uniformly over the whole beach profile down to a certain critical depth D called the depth of closure. No sand is presumed to move alongshore in the region sea- ward of this depth. If the beach profile moves only parallel to itself (maintaining its shape), a change in shoreline position Ay at a certain point is related to the change in cross-sectional area AA at the same
point according to Equation 1:
AA = AyD (1)
where
> 3S fl
; . 2 change in cross-sectional beach area (m )
Ay = change in shoreline position (m)
oO Il
maximum depth for sand motion (depth of closure) (m)
6. The principle of mass conservation must apply to the system at all times. By considering a control volume of sand and formulating a mass balance during an infinitesimal interval of time, the following differential equation
is obtained (see Figure 1): s+ St =0 (2)
where Q = longshore sand transport rate Gio yieee) = cross-sectional area of the beach Ge)
x = space coordinate along the axis parallel to the trend of the shoreline (m)
t = time (sec)
Weak A Q (egal ce uae Figure 1. Schematic illustration of a hypothetical equilibrium beach profile
7. Equation 2 states that the longshore variation in the sand transport rate is balanced by changes in the shoreline position. If, in addition to longshore transport, a line source or sink of sand at the shoreline is con-
sidered, Equation 2 takes the following form:
eee il Para (3)
where q denotes the source or sink of sand per unit length of beach ae m2) 6 The minus sign denotes a sink (loss of sand), and the plus sign denotes a source.
8. In order to solve Equation 2, it is necessary to specify an expres- sion for the longshore sand transport rate. Longshore sand transport on an open coast is believed to bear a close relation to the longshore current which is generated by waves obliquely incident to the shoreline. A general expres-
sion for the longshore transport rate is
Q= Q, sin 20, (4) where OF = amplitude of longshore sand transport rate (Paes) Cae angle between breaking wave crests and shoreline
In the generally accepted formula for longshore current, the speed of the cur- rent is proportional to sin 204, (Longuet-Higgins 1970a,b). 9. The angle between the breaking wave crests and the shoreline
(Figure 2) may be expressed as
= Es oy on a arc tan (2) (5)
in which
a
e angle of breaking wave crests relative to an axis set parallel
to the trend of the shoreline
dy/dx = local shoreline orientation
10. A wide range of expressions exists for the amplitude of the long- shore sand transport rate, mainly based on empirical results. For example,
the Shore Protection Manual (SPM) (1984) gives the following equation:
= 28 42 ee eS 16 EoD C8, (0. - p)rA Me)
where o = density of water (kg/m) g = acceleration of gravity (aeece) H = significant breaking wave height (m) Cg, = wave group velocity at breaking point (m/sec) K = nondimensional empirical constant 0. = density of sand (keyme)
s \ = porosity of sand
xX
Figure 2. Definition sketch for geometric properties at a specific location as related to shoreline change 11. If Equation 5 is substituted into Equation 4, the sand transport
rate can be written: = if _— ax) Q Qe sin a arc tan (3 (7)
12. For beaches with mild slopes, it can safely be assumed that the breaking wave angle relative to the shoreline and the shoreline orientation are small. The consequences and validity of this assumption, which linearizes Equation 7, are discussed further in this report. Under the assumption of
small angles, to first order in a Taylor series,
10
AS =o DY Ove a, (22, 2 ox (8)
13. If the amplitude of the longshore sand transport rate and the inci- dent breaking wave angle are constant (independent of x and t) the follow-
ing equation may be derived from Equations 1, 2, and 8:
2 LOnyaPS OY SINE (9) ox where 2Q 6 = = (10)
14. Equation 9 is formally identical to the one-dimensional equation describing conduction of heat in solids or the diffusion equation. Thus, many analytical solutions can be found by applying the proper analogies between initial and boundary conditions for shoreline evolution and the processes of heat conduction and diffusion. The coefficient e , having the dimensions of length squared over time, is interpreted as a diffusion coefficient expressing the time scale of shoreline change following a disturbance (wave action). A high amplitude of the longshore sand transport rate produces a rapid shoreline response to achieve a new state of equilibrium with the incident waves. Fur- thermore, a larger depth of closure indicates that a larger part of the beach profile participates in the sand movement, leading to a slower shoreline response.
15. If the amplitude of the longshore sand transport rate is a function of x , the governing differential equation for the shoreline position will
take a different form: Ooh Gs Oy — oy Elmaneteae Gee eos as + (il)
where it is assumed that the depth of closure is constant. Equation 11 makes it possible, in a simplified way, to account for diffraction behind a groin,
where the wave height varies with distance alongshore. However, the
11
expression describing the variation in oh in a diffraction zone must be simple enough to allow an analytical solution. Otherwise, a numerical solution technique must be employed (Kraus and Harikai 1983, Kraus 1983, and Hanson and Kraus 1986). If the incident breaking wave angle OL, is also a function of the distance x , another term, eda, /dx » must be added to the right side of Equation ll.
16. In summary, the assumptions which comprise the one-line model, in which breaking waves are the dominant sand-moving process, are as follows:
a. The beach profile moves parallel to itself (assumption of equilibrium of the beach profile).
b. Longshore sand transport takes place uniformly over the beach profile down to a depth D (depth of closure).
Details of the nearshore circulation are neglected.
| [0
The longshore sand transport rate is proportional to the angle of incidence of breaking wave crests to the shoreline.
17. In addition, the following assumptions will be used to derive analytical (closed-form) solutions of the one-line model (Equation 9):
a. The angle between the breaking wave crests and the shoreline is small (small-angle approximation).
b. The angle of the shoreline with respect to the x-axis is small.
18. In arriving at all solutions, it is tacitly assumed that sand is always available for transport unless explicitly restricted by boundary and/or
initial conditions. Overview of Previous Analytical Work
19. Pelnard-Considére (1956) was the first to employ mathematical modeling as a method of describing shoreline evolution. He introduced the one-line theory and verified its applicability with laboratory experiments. Figure 3 shows a comparison between experimental results and the analytical solution for the case of an updrift groin, as obtained by Pelnard-Considére. Pelnard-Considére derived analytical solutions of Equation 9, the linearized shoreline change equation, for three different boundary conditions: shoreline evolution updrift of a groin (with and without bypassing) and release of an instantaneous plane source of sand on the beach.
20. Grijm (1961) studied delta formation from rivers discharging sand.
In the transport equation discussed in his article, the sand transport rate
72
Initial Shoreline
SCALE LEGEND Physical model (0) Im [MA ED es ——-—— Analytical model
Figure 3. Comparison between experimental and theoretical shoreline evolution (after Pelnard-Considére 1956)
is set to be proportional to twice the incident breaking wave angle to the shoreline. Only solutions which were similar in shape during the course of time are discussed. Two different analytical solutions are presented: one for which the incident breaking wave angle and the shoreline orientation angle are small and one for which the wave angle is small in comparison with the shoreline orientation. The governing equations (sand transport and mass con- servation) are expressed in polar coordinates and solved numerically. Grijm (1965) further develops this technique and presents a wide range of delta for- mations. Komar (1973) also presents numerically obtained solutions of delta growth under highly simplified conditions.
21. Le Méhauté and Brebner (1961) discuss solutions for shoreline change at groins, with and without bypassing of sand, and the effect of sudden dumping of material at a given point. Most of the solutions were previously derived by Pelnard-Considére (1956), but they are more thoroughly presented in Le Méhauté and Brebner's work, especially regarding geometric aspects of the shoreline change. The decay of an undulating shoreline and the equilibrium shape of the shoreline between two headlands are treated.
22. Bakker and Edelman (1965) modify the longshore sand transport rate equation to allow for an analytical treatment without linearization. The sand
transport rate is divided into two different cases:
13
Q= Qk tan a (Sesh Ch S528! (12)
Qn 0 cannon 1.23 < tan a, (13)
where K, and Ky are constants. From these equations as a starting point, the growth of river deltas was studied.
23. Bakker (1969) extends the one-line theory to include two lines to describe beach planform change. The beach profile is divided into two parts, one relating to shoreline movement and one to movement of an offshore contour
(see Figure 4). The two-line theory provides a better description of sand
Figure 4. Definition sketch for the two- line theory (after Bakker 1968)
movement downdrift of a long groin since it describes representative changes in the contours seaward of the groin head. Near structures such as groins, offshore contours may have a different shape from the shoreline. The two lines in the model are represented by a system of two differential equations which are coupled through a term describing cross-shore transport. According to Bakker (1969), the cross-shore transport rate depends on the steepness of the beach profile; a steep profile implies offshore sand transport; and gently sloping profile implies onshore sand transport. Analytical solutions of the two-line theory are not included in the present report. However, an example of a two-line theory solution for a groin system is shown in Figure 5. The solution describes the stationary form of the shoreline for various groin spacings given in multiples of a nondimensional groin length Lo 5
24. The two-line theory is further developed in Bakker, Klein-Breteler, and Roos (1971) in which diffraction behind a groin is treated. In this case, it became necessary to numerically solve the governing equations. Expressions
for the coastal constant (diffusion coefficient €) for the one- and two-line
14
Distance Between Groins: 6Ly
Distance Between Groins: 10 Lo
* Distance Between Groins: © y
x Figure 5. Two-line theory solution for a groin system (after Bakker 1968) theories are also presented. Bakker (1970) developed a phenomenological dif- fraction routine for one-line theory but numerically solved the problem.
25. Le Mehauté and Soldate (1977) present a brief literature survey on the subject of mathematical modeling of shoreline evolution. Analytical solu- tions of the linearized shoreline change equation are discussed together with the spread of a rectangular beach fill. In Le Méhauté and Soldate (1978, 1979) a numerical model is derived which includes variation in sea level, wave refraction and diffraction, rip currents, and the effects of coastal struc- tures in connection with long-term shoreline evolution.
26. Until recently, the most complete summary of analytical solutions to the sand transport equation has been made by Walton and Chiu (1979). Two derivations of the linearized shoreline change equation are presented together with another approach resulting in a nonlinear model. The difference between the two approaches, which both arrive at the diffusion equation, is that one uses the Coastal Engineering Research Center (CERC) formula (SPM 1984, Chap-
ter 4) for describing the longshore sand transport rate by wave action and the
15
other a formula derived by Dean (1973) based on the assumption that the major sand transport occurs as suspended load. Most analytical solutions then appearing in the literature were presented by Walton and Chiu (1979). Addi- tional solutions mainly concern beach nourishment in connection with various shoreline shapes. The new solutions derived by Walton and Chiu (1979) treat beach fill in a triangular shape, a rectangular gap in a beach, and a semi- infinite rectangular fill. Some data on the coastal constant are also pre- sented in the paper.
27. Analytical solutions can be used conveniently to describe the be- havior of beach fill, as mentioned above. Dean (1984) gives a brief survey of some solutions applicable to beach nourishment calculations, especially in the form of characteristic quantities describing loss percentages. One solution describes the shoreline change between two groins initially filled with sand.
The resultant shoreline evolution with time is shown in Figure 6.
Figure 6. Shoreline evolution between two groins initially filled with sand (after Dean 1984)
General Approach in the Present Work
28. The simplified or linearized shoreline change equation (Equation 9) is a linear partial differential equation which is identical to the equation describing one-dimensional conduction of heat in a solid or to the diffusion equation. By specifying boundary and initial conditions in these areas which represent conditions prevailing in a specific shoreline evolution situation,
the corresponding analytical solutions are directly applicable. Carslaw and
16
Jaeger provide many solutions of the heat conduction equation, and Crank (1975) gives solutions to the diffusion equation.
29. The following paragraphs present a review of previously obtained solutions together with new solutions. The new solutions have been derived either from analogies with heat conduction or through the Laplace transform technique, a short outline for which is given in Appendix A. Carslaw and Jaeger (1959) provide a more comprehensive treatment. In order to present the solutions in an efficient and general format dimensionless variables have been used to a large extent although physical understanding may be obscured by the absence of dimensional quantities. Also, in many cases for which the solution is symmetric with respect to a coordinate axis, the solution for only one side of the symmetry line is displayed. The solutions have been divided into two groups based on the physical properties of the initial and boundary condi- tions, not on their mathematical properties, because the object of the report is to present solutions and not to describe details of their derivation. The first group of solutions describes shoreline change situations without coastal structures. Solutions describing shoreline evolution in these cases are applicable both to natural and artificial beach forms (nourished beaches) if similar types of wave conditions prevail. Also, several solutions describing river delta growth are presented covering the cases of a river discharging sand as a point source and a river mouth of finite length.
30. The other group of solutions comprises configurations involving various types of coastal structures such as groins, jetties, detached break- waters, and seawalls. Since the equations quickly become complicated, the influence of coastal structures on shoreline evolution has to be idealized to a considerable extent. However, the essential features of the situation may still be preserved if this idealization is carried out in a physically reason- able manner. Some simple models to account for diffraction downdrift of a groin are shown also.
31. Most of the analytical solutions are presented in the main text without derivation. Reference is made to the appropriate literature in case the reader is interested in deriving the solutions. Also, in Appendixes B-G,
derivations are given for selected new solutions.
17
PART II: SOLUTIONS FOR SHORELINE EVOLUTION WITHOUT COASTAL STRUCTURES
General Formal Solution
32. The basic differential equation to solve is Equation 9, together with the associated initial and boundary conditions. An infinitely long beach is assumed to be exposed to waves of constant height and period with wave crests parallel to the x-axis (parallel to the trend of the shoreline). The shoreline will adjust to reach an equilibrium state in which the longshore sand transport rate is equal at every point along the shoreline. Since the wave crests are parallel to the x-axis, the equilibrium sand transport rate is zero. An initially straight beach is thus the stable shoreline form in this case. If the shoreline shape at time t = 0 is described by a function f(x) , the solution of Equation 9 is given by the following integral (Carslaw and Jaeger 1959, p. 53):
co
2 wee) o fro ep a ee Siac (14)
—oo
ie) a] iu) ct
imope ie Sy 0) eynal =e < Ke K EG
The shoreline position is denoted by y and is a function of x and t
The quantity & is a dummy integration variable. Consequently, the change in both natural and manipulated beach forms can be determined if Equation 14 is evaluated. Equation 14 may be interpreted as a superposition of an infinite number of plane sources instantaneously released at t = 0. The source located at point ¢€ contributes an amount f(é)dé to the system. Infinitely far away from such a single source no effect on the shoreline position is assumed (boundary condition). Equation 14 is used to derive most of the solu-
tions dealing with various shoreline configurations in the following text.
18
Finite Rectangular Beach Fill
33. The solution to this problem in connection with shoreline change is first mentioned by Le Méhauté and Soldate (1977). At time t = 0 , the shore-
line has a rectangular shape of finite length 2a described by Equation 15
(see Figure 7):
Vig |x| <a ACA) S tC) (15) 0 |x| >a The solution is y(x,t) = 5 y_lerf le = ,) i eye | GE *) (16) g 2vet 2vet
0.6
0.4
SHORELINE POSITION (y/y,)
0.2
i) 0.5 1 1.5 2 ALONGSHORE DISTANCE (x/a)
Figure 7. Shoreline evolution of an initially rectangular beach fill exposed to waves arriving normal to shore
19
The symbol erf denotes the error function which is defined as
y, 2 erf z = ae dé (17) vn 0
The error function is tabulated in standard mathematical reference books (e.g., Abramowitz and Stegun 1965). It is convenient to introduce the fol-
lowing dimensionless quantities:
VW y' = (18) Yo We eget x = (19) 0 = Se t' = 5 (20) al
The quantity used to normalize the time variable expresses half the time elapsed before a square beach fill of length a would completely erode at the constant transport rate OF . If the solution is expressed in dimensionless quantities, the resultant shoreline evolution can be displayed in compact form. Figure 7 illustrates how a rectangular fill spreads or diminishes with time according to Equation 16. It should be noted that the vertical scale of this and the following figures has been distorted for the sake of clarity.
34. Dean (1984) discusses how the sand from two different beach nour-
ishment projects spreads with time. The time for a certain percentage
ie 12 P to be lost from the original rectangular beach fill is compared with the
corresponding time toy for different conditions: a 2 € 2 1 tae Sat —} — (21) P2 Pl (2) Ey
35. This formula is obtained by noting that the same percentage of
beach volume is lost during the same dimensionless time. Consequently, a
20
rectangular beach fill which is twice as long maintains its volume four times as long if exposed to the same wave conditions. It is possible to calculate the time it will take for a certain percentage P to be lost from the initial rectangular fill. The following expression is obtained by integrating Equa- tion 16 and comparing the resulting volume at a specific time to the original
fill volume:
p= ver (ee - ierfc | (22) VT ve"
where ierfc denotes the integral of the complementary error function erfc
ierfc z = f ext E dé (23)
erfc z= 1-erf z (24)
Figure 8 shows the percentage of sand volume lost as a function of time. 36. It is possible to determine the rate of sand to be supplied to the fill in order to maintain the original shape. The boundary condition for this
case is that the end of the rectangular fill is kept at the initial position: y(0,t) = y (25)
Note in this case that the x-axis originates from the corner of the fill instead of from the middle of the fill as in Equation 16. The solution de- scribing the resultant shoreline evolution is (Carslaw and Jaeger 1959,
p. 60):
y(x,t) = y_ erfe x ) (26) S =
for t >0O and x20
21
80 iJ (a) Cc im <=
© 60 oO fe) oa wo wo oO
= 40 LJ = J =! (=) >
20
0
0 2 4 8 10
6 TIME let/a’)
Figure 8. Percentage of sand volume lost from a rectangular fill as a function of dimensionless time
Sand has to be added to the corner of the fill at the following rate:
Q (27)
The spread of the moving shoreline front (Equation 26) is illustrated in Figure 9.
37. It is advisable to use the analytical expressions describing shore- line evolution for a rectangular fill with great care, even for rough estima- tions, because the linearization procedure (Equation 8) is based on small shoreline orientation angles, a condition which is violated on the sides of the rectangle. In fact, the linearized transport equation implies an infi- nitely large initial sand transport rate at the edges of the fill. However, the original transport equation (Equation 7) gives a zero transport rate at the corners; thus, a rectangular beach form is stable to parallel incident
waves. In reality, sand transport occurs at the corners because of
22
SHORELINE POSITION (y/y,)
1 ALONGSHORE DISTANCE (x/y,)
Figure 9. Shoreline evolution when sand is supplied at x = 0 to maintain a specific beach width yA
diffraction and refraction, but this realistic situation is not described by the linearized equation. Consequently, the linearization procedure artifi- cially increases the erosion of the fill, implying that the analytical solu- tion overestimates the speed of erosion. The error is, therefore, on the con- servative side. This problem is only an apparent one since it is a practical
impossibility to create a perfectly rectangular fill in the field. Semi-Infinite Rectangular Beach Fill 38. The initial conditions for a semi-infinite rectangular beach fill are
y(x,0) = (28)
Walton and Chiu (1979) give the following solution:
23
ie |
y(x,t) = 5 y, exfe ( x ) (29) Qvet
iene ie S> (0) ehoval Gd & se SCD
The solution is antisymmetric about the y-axis, taking the constant value y,/2 at x =0 . If the shape of the shoreline for x 2 0 is approximated by a triangle having height y,/2 so as to conserve mass, the speed of prop- agation of the triangle's front is inversely proportional to the square root of elapsed time. This relationship is also valid for Equation 26. Figure 10 illustrates the solution of Equation 29. The right side of Equation 29 for
x > 0 equals half the solution of Equation 26.
0.6
SHORELINE POSITION (y/y,)
0.2
-2 -1 1 2
Lt} ALONGSHORE DISTANCE (x/y,)
Figure 10. Shoreline evolution of an initially semi-infinite rectangular beach
Rectangular Cut in a Beach
39. The initial conditions for rectangular cut in a beach are formu-
lated as
24
y(x,0) = (30)
These conditions may represent an excavation or a natural embayment of rec-
tangular shape. Walton and Chiu (1979) present the following solution:
y(x,t) = 5 y,|erfe 2 = + erfc (=) (31) QVet Wet
OG ts 0 land) =o < 5x lo
This: solution may be obtained by superimposing Equation 16 with a negative sign on a beach of width sae In general, with due regard to the boundary and initial conditions, it is possible to derive new solutions simply by superimposing existing solutions since the governing differential equation (Equation 9) is linear. Equation 31 is symmetric with respect to the y-axis,
and only half of the solution region is illustrated in Figure 11.
0.8 0.6
0.4
SHORELINE POSITION (y/y,)
0.2
0 0.5 1 ALONGSHORE DISTANCE (x/a)
Figure 11. Shoreline evolution of a rectangular cut in an infinite beach of width Ve
25
40. Since the present situation is the inverse problem of the rectangu- lar beach fill, Figure 8 can be used to evaluate the rate of infilling of a
certain volumetric percentage of sand. Triangular-Shaped Beach 41. The triangular-shaped solution is also mentioned by Walton and Chiu
(1979). The original beach has the shape of a triangle according to the
initial conditions as follows:
a-x ye ( = ) O<x<a atx y(x,0) = ye ( = ) Al @ 5 < (0) (32) 0 es] = & In this case the solution takes the following form: Yo a= xX ap > x WIESE) = OF (a - x) aati "55 (a + x) erf Ce - 2x erf — 2vet Qet 2Qvet et #2) fast =n) fA t x Magt +2,/£-Je +e SF es DE (33) T
toe (© s @ ema a5 <2 o
A nondimensional illustration of the shoreline evolution from an initially triangular beach is shown in Figure 12.
42. Depending upon the height-to-width ratio of the triangle, lineari- zation of the transport equation may reduce accuracy of the analytical solu- tion. However, even though the assumptions forming the basis for the lineari- zation procedure appear to be extremely limiting (particularly in requiring small wave angles), in practice the analytical solution is found to be appli- cable for angles as large as about 45 deg between the shoreline and the break-—
ing waves. In order to estimate the effect of the linearization, a comparison
26
SHORELINE POSITION (y/y,)
tt} 0.5 1.5 2
1 ALONGSHORE DISTANCE (x/a)
Figure 12. Shoreline evolution of an initially triangular beach
was made between the analytical solution and a numerical solution with the original sand transport equation (Equation 7). Figure 13 shows the result as a function of the height-to-width ratio and elapsed time.
43. It is quite clear that the analytical solution produces a higher rate of shoreline change by overestimating the longshore sand transport rate (since a> sina). Thus, if the analytical solution is used to estimate the time scale involved in beach nourishment problems, a higher rate of attenua-
tion of the fill will always be obtained than is expected to actually occur. Trapezoidal-Shaped Beach
44, A trapezoidal beach form is described by the following initial
conditions: Voy 7 Yl YO) Ee? a se 3 Sp SAS x) Xa Xo oy) i 2 1 y(x,0) = (34) 0 x <X) > X > Xy
27
Here yy and y, denote shoreline positions corresponding to the longshore
locations xy and Xo - The solution is
d (= = =) et (9) Jeet ~(ep%)// +e ae
oie je = 0) final DK KR KO,
Analytical Soln:
Numerical Solns —-—-——————————-
2et
t’ a2
SHORELINE POSITION (y/y,)
0 0.5 1.5
1
ALONGSHORE DISTANCE (x/a)
Figure 13. Comparison between analytical solution with the
linearized transport equation and numerical solution with
the original transport equation for a triangular beach fill
(for height-to-width ratios 1.0 and 0.5)
The solution for the triangular beach form (Equation 33) can be obtained by superimposing two trapezoidal beach shapes which reduce to triangles. In the same way, in principle, the analytical solution for any arbitrary shoreline
shape may be obtained by approximating the shoreline with a series of straight
lines. Even though the sand transport at each boundary of the trapezoids in
28
such a case is overestimated (because of the large incident wave angle) super- imposition of the solutions eliminates these effects. In Figure 14 the solu- tion for a single trapezoidal beach form is shown. A representative length L
has been chosen to normalize the shoreline position and the alongshore
distance.
SHORELINE POSITION (y/L)
i 1.5 2 2.5 3 ALONGSHORE DISTANCE (x/L)
Figure 14. Shoreline evolution of an initially trapezoidal beach form
0 0.5
45. If an arbitrary-shaped shoreline is studied, it is most convenient to approximate it with a series of straight lines and then to superimpose the respective solutions. Consider a shoreline (see Figure 15) divided into N reaches, with each length described by a straight line connecting two neighboring points denoted by (x, ; y,) and (X54) ; Ya+p) for a certain
reach (the fee reach).
29
x; Xi+]
Figure 15. Shoreline of arbitrary shape approximated by N straight lines
46. The shoreline position can be written, accordingly:
7 ay ox Ge
for t >0O and -~ <x<o, Semicircular-Shaped Beach
47. In order to find an analytical solution for a beach formed in a half circle between -a < x < a , the circle is approximated by a polygon with a finite number of corners (Figure 16).
48. The solution can be obtained using Equation 36 with proper expres-
sions for the line segments. The following quantities are defined:
xt = a cos [Ss22] (37) = = a cos (= =| (38)
Lig F in yaa a sin (5 = ) (39)
tan aieal
The integer N is the number of corners in the polygon approximating the semicircle. For example, if N = 3 then a triangular beach form is obtained.
The solution can be written with the previously defined quantities:
N-1 R L
x, -x Ke =) x
y (x,t) -5 ») (Kx) + yy - Kx) erf NI (ee erfilo= fea 2vet 2vet
2 2
R I A -(«$-) 4et (x=) 4et BO. y= e =we (41)
FOI {2 sv@) => eon ce co ¢
Figure 16. Semicircular-shaped beach approximated by a polygon
31
In the limit N+” the polygon coincides with a semicircle. The solution (N = 101) is illustrated in Figure 17 which shows the shoreline evolution as a
function of time for an initially semicircular-shaped beach.
t!=0
SHORELINE POSITION (y/a)
0 0.5 1 1.5 2 ALONGSHORE DISTANCE (x/a)
Figure 17. Shoreline evolution of an initially semicircular beach 49. If the beach is formed as a circular segment, the solution may be
derived by superimposing Equation 41 with the appropriate summation limits and Equation 16 with reversed sign. In Figure 18 a definition sketch is shown.
If the pitch height is denoted by p , then the width of the circle segment becomes 2¥p(2a — p) . Furthermore, the height of the rectangular fill is
a- p., and the angle a (see Figure 18) is arc sin (1 - p/a) . Conse- quently, the summation of the solutions for the polygon stretches should start
at angle a in the semicircle and end at angle t - a. The solution is
32
N-n-1 R by
x, - x x, - x y(x,t) = ; > yy + K,(xi - ) erf Be erf el ae
F 2vet 2vet
i=mt+1
2 2 [ex (xf )/set -(x{- \/vee + 2k, — |e -e i T
Y¥p(2a =p) = :) “dare ae =0p) at =) (42)
Qvet
1 - = (a - p)]| erf 2 ( ovet
for t >0QO and -- <x<o,
Figure 18. Definition sketch for a circular segment-shaped beach
The quantity N is, as before, the number of corners in the polygon, and m represents the number of corners minus one contained in the angle a. Fig- ure 19 illustrates the transformation of an initially circular segment-shaped shoreline.
50. Since the tangent of the shoreline orientation (see Equation 5) is
infinite at the corners of the semicircle (x = +a), the condition of small
333}
SHORELINE POSITION (y/a)
0 0.5
1 1.5 2 ALONGSHORE DISTANCE (x/a)
Figure 19. Shoreline evolution of an initially circular segment-shaped beach (a = 45 deg)
angles is violated. This condition implies, as previously discussed, that the sand transport is overestimated, leading to a faster dispersion process of the shoreline toward the stable condition (a beach parallel to the wave crests). An analytical solution for a circular segment-shaped beach, however, will show better agreement with the numerical solution of the original sand transport formula if the angle of shoreline orientation is small at the edges. A com- parison between an analytical and a numerical solution for a circular segment beach is illustrated in Figure 20. In this case the linearization approxi-
mates the transport equation well; thus, the solution is accurate. Semicircular Cut in a Beach 51. The situation of a semicircular cut in a beach is the antisymmetric analog of the case described in the previous section. A solution is obtained
by superimposing Equation 41 with opposite sign for a beach of width a. The
solution is displayed in Figure 21.
34
0.3
t'=0
Aral y tical SOV Ms ees
Numerical Solns: ------—--—-——-——- »_ St Tran
SHORELINE POSITION (y/a)
it} 0.5
1 ALONGSHORE DISTANCE (x/a)
Figure 20. Comparison between analytical and numerical solu- tions for the case of a circular segment-shaped beach
1 =s0)
0.8
0.6 y,,
SHORELINE POSITION (y/a)
0.4 0.30 eit Ser y a 0.2 0 0 0.5 1S 2
1 ALONGSHORE DISTANCE (x/a)
Figure 21. Shoreline evolution of an initially semicircular cut in a beach
35
52. In the same way, shoreline evolution of a bay formed in a circular segment may be calculated. Equation 42 is superimposed with opposite sign on
a beach of width p (pitch height). Figure 22 shows the solution.
0.3
t'=0
SHORELINE POSITION (y/a)
0.0
0 0.5 1 1.5 2 ALONGSHORE DISTANCE (x/a)
Figure 22. Shoreline evolution of an initially circular segment cut in a beach (a = 45 deg)
Rhythmic Beach
53. A beach with a rhythmic shoreline in the form of a cosine wave at- tenuates with time but maintains its rhythmic character. The initial condi-
tion is
y(x,0) = A cos ox (43) where A represents the amplitude of the rhythmic form such as cusps along the beach, and o denotes the wave number of the shoreline oscillation or
cusp. The quantity o can be expressed also as 21/L , where L is the
beach cusp wave length. The solution to this case is found to be
36
2 y(x,t) = A cos ox en. Be (44)
toe {6 S 0) Ehowel ays se 6 GG
Le Mehauté and Brebner (1961) and Bakker (1969) give this solution. A non- dimensional diagram of the shoreline evolution of an initially cosine-shaped
beach is shown in Figure 23. t' 0.0
0.01
0.02
0.03
0.04 0.05
SHORELINE POSITION (y/A)
0 0.5 1 1.5 2 ALONGSHORE DISTANCE (x/L)
Figure 23. Shoreline evolution of an initially cosine-shaped beach (a distance of one beach cusp height added to the shoreline position)
Sand Discharge from a River Acting as a Point Source
54. If a river mouth is small in comparison to the area into which it is discharging sand, the discharge may be approximated by a point source. The sand discharge from the river or the strength of the point source is denoted
as and is a function of time. (The units of dp are at Jace.) A solu-
q R tion may be obtained by considering the continuous sand discharge from the
river to be the sum of discretely released quantities of sand at consecutive
37
times. If a certain volume of sand V is instantaneously released at a point
> at time cs » the solution can be written
S 2 —(x-x 4e(t-t_) y (x,t) = an ee ( .)/ rs (45) 2DvTe(t —- t.)
ROG ee and -~> <x < om,
Equation 45 has been discussed by Le Méhaute and Brebner (1961) and by Le Méhauté and Soldate (1977). Accordingly, a superposition of an infinite number of such released quantities can be used to represent the sand discharge
from a river. According to Carslaw and Jaeger (1959, p. 262), the solution
for a point source with a continuous time variable sand discharge dp may be expressed as 1 Y -(x-x,)"/4e(t-) dt y(x,t) = dp(&) e SS (46) 2Dvre J viene
for t >0O and -~ < x<om,
If qp is constant and equal to q,° the solution is
= — 2 Get = q t (x, x) € ae x 25 3 > 2S —e - (47) TE D €
y(x,t) =
for t >0QO and -~ <x<o,
Equation 47 is identical to the solution describing a constant flux q,/2 on the boundary (x = 0) for a beach of semi-infinite extent. Figure 24 illu- strates the solution where L is used as a normalizing length, and the point source is located at Tt L . The nondimensional quantity containing the shoreline position is formed as the ratio between the amplitude of the sand
transport rate and the sand discharge from the river.
38
0.3
1.0 0.8 »_ €t ae gg 0.6 Ss S 0.2 0.4 z = = on © 0.2 WW z = Ohi GJ oe S x wo 0.0 0 1 2 3 ALONGSHORE DISTANCE (x/L) Figure 24. Shoreline evolution in the vicinity of a river dis- charging sand and acting as a point source 55. If the sand discharge has a periodic behavior, the function dp could take the following form: dp (t) =i qe + q, sin (wt + ) (48) where a xe steady sand discharge from river q_ = amplitude of periodic sand discharge
; = angular frequency = 2n/T
T = period of oscillation of sand discharge from river
» = phase angle of periodic variation The solution consists of two parts, namely Equation 47 describing the shore- line evolution from a steady point source and the following solution which
accounts for the periodic component:
39
vt q, ”) ms [bee wisps) Sl —= fm [uct = (5 )) ar » | e dé (49)
Dvet
The shoreline behavior is composed of one contribution that evolves roughly proportional to the square root of elapsed time and another contribution which is a periodic oscillation that damps out along the x-axis with a decay factor Yw/2e (both in the negative and positive directions). Consequently, beyond a certain distance from the discharge the periodic effect of Equation 49 can be neglected, implying that the solution may be approximated by Equation 47 only. Because of the periodic variation in the discharge, sand waves are generated from the river mouth. These sand waves propagate with a speed V2ew along the x-axis, and the time lag between the oscillation in sand discharge at the river mouth and a specific location is 1/4 + xV¥w/2e . In Figure 25 the shoreline evolution at specific locations in the vicinity of a point source of
sand discharge with a periodic variation in strength is shown as a function of
0.5
0.4 = S a» Ss 0.3 LOCATION (x/L) — = 0.0 _— o Oo a 1.0 rr] re (ibed — a 2.0 a =} = w
0.1
0.0
0 1 5 6
3 TIME (et/L?)
Figure 25. Shoreline evolution in the vicinity of a river discharging sand with a periodic variation in strength as a function of time
(wh? /e SA OS O 4 q,/Q, = 46/9, = 0.5)
40
time. The quantities in the figure are dimensionless, with the sand discharge from the river normalized by the amplitude of the sand transport rate Qe and the angular frequency of the oscillation normalized by =i a baeune eZ
clearly shows how the superimposed sinusoidal-shaped variation damps out with
distance from the source along the x-axis. Sand Discharge from a River Mouth of Finite Length
56. If the river mouth has a finite width in comparison to the area into which it is discharging sand, an approximation by a point source is no longer accurate. Instead of supplying sand to the system via the boundary or initial conditions, the mass conservation equation in the full form of Equa-
tion 3 is applied. The sand discharge from the river is considered a
q R continuous function of x , varying along the river mouth. The river mouth is assigned a length 2a , and the sand discharge is measured per unit width.
Mathematically, the situation is expressed as
2 oy SR My e—r tas OFS }xisia (50) ox ay, V5 e Ta Tae Sd Be) (51) ox y, (x50) = y5(x,0) = (0 (52) ox ox arse ay) aa x = 0 (53) Mal = V9) x= a
41
57. The problem consists of two coupled partial differential equations with appropriate boundary and initial conditions. Since the configuration is symmetric with respect to the center of the river mouth (if qp is constant), only half of the problem domain has to be treated. The boundary conditions are no sand transport through the center of the river (symmetry), and mass conservation should apply between the two solution areas. Also, the beach must be continuous at all times over this boundary. Furthermore, the shore- line is unaffected by the river sand discharge as x approaches infinity.
According to Carslaw and Jaeger (1959, p. 80) the solution is
I
qpt & y, (st) = — ilo 2” eeie G | — 2392 oe | (55)
epee = (0) Evol (0) SS Sx
IA rt)
2q,t Yo (x5 t) = x 6° eete fe — ) ete sala“ z | (56)
ioe jc S 0) Ehol Se S El
58. The function ierfc is defined in Equation 23 and the superscript 2 denotes a double integration. An exponent n represents n integrations
of the complementary error function. The following recurrence relation holds
ore il = I 9 -2
Qn i» erfc x = do erfc x - 2x jet erfc x (57)
In Figure 26 the solution to Equations 55 and 56 is illustrated.
42
0.4
eC) a WN S 2.5 5, BR re) oes NOS — 2.0 & 3 1.5 oO — | mn 0.2 ros) 1.0 (o, Oj Zz Ln I =) OJ oS Pio w
0.0
i) 1 3 4
2 ALONGSHORE DISTANCE (x/a)
Figure 26. Shoreline evolution in the vicinity of a sand- discharging river mouth of finite width 59. A nondimensional quantity describing shoreline change is defined
according to
y(x,t)eD
su (Ge! 5 fe) a (58)
4qpa
The quantity used to normalize Equation 58 can be written by using Equation 10
to arrive at
2aqp Q
oO
a (59)
This quantity can be interpreted as a ratio between sand discharge from the river and the amplitude of the sand transport rate produced by the waves. The solutions given by Equations 47, 49, 55, and 56 are also valid for the place- ment of sand (beach nourishment), provided the placement is made under the
same conditions. Solutions with an opposite sign consequently represent
43
mining of sand. Equations 55 and 56 describe only the general features of delta growth since the river flow conditions within the delta formation are neglected in the present treatment. The time required for the delta to reach a certain distance ye from the original shoreline position is calculated
from the following relationship
t y_(t) = —]1- hee erfc ( g ) (60) e 2Vet
i] j=)
fom ot) >) 0) and) x
Equation 60 is illustrated in the nondimensional diagram of Figure 27. For a
specific wave climate, the above relation implies that an increase in the sand
2.5 oN oa ra a N S
(e] 2 oO “ > ww
= 1.5 (=) Le | ee — wo j=) a
[ea] 1 a _— —| (7e o& (=) w
0.5
0
0 1 4 5
2 3 TIME (et/a’)
Figure 27. Maximum delta growth from a sand-discharging river mouth of finite length
discharge from the river has a proportional effect on the growth of the delta
according to the following relation:
44
me han (61)
Here the indices 1 and 2 refer to two different sand discharge conditions
experiencing the same wave climate.
45
PART III: SOLUTIONS FOR SHORELINE EVOLUTION INVOLVING COASTAL STRUCTURES
60. In the previous chapter, the incident wave crests were restricted to be parallel to the x-axis. In such a case, an initially straight beach will always remain straight, unless material is supplied in an irregular way. If the waves arrive at the same angle to the shoreline everywhere, the beach will also be stable if it is initially straight. However, if an obstacle on the beach disturbs the equilibrium transport conditions, a change in shoreline position occurs in order to achieve a new steady-state configuration. Exam- ples of such obstacles are groins, jetties, detached breakwaters, and sea- walls. In order to treat such complex cases analytically, the situation has to be idealized to a large degree. Properties which generally vary continu- ously along the shoreline (breaking wave angle, amplitude of the sand trans- port rate, etc.) usually must be approximated by means of a series of coupled solutions of simpler problems.. Within each solution area the properties are
held constant but are allowed to vary from one area to another. Shoreline Change at Groins and Jetties
61. The analytical solution for beach change at a groin or any thin shore-normal structure which blocks alongshore sand transport was first ob- tained by Pelnard-Considere (1956). Initially, the beach is in equilibrium (parallel to the x-axis) with the same breaking wave angle existing every- where, thus leading to a uniform sand transport rate along the beach. At time t = 0 a thin groin is instantaneously placed at x =0 , blocking all trans- port. Mathematically, this boundary condition can be formulated as (see
Equation 7)
OY = tan a x = 0 (62) x fo)
This equation states that the shoreline at the groin is at every instant parallel to the wave crests. The wave crests make an angle ay with the x-axis according to Figure 28, giving rise to longshore sand transport in the
negative x-direction.
46
GROIN
Figure 28. Definition sketch for the case of a groin
62. A groin interrupts the transport of sand alongshore, causing an accumulation at the updrift side and erosion at the downdrift side. The solu-
tion describing the accumulation part is
Cee) SY een ty vet ierfe ( =) (63) QVvet
coe te > O zal sx S 0
The solution can also be written as follows:
et 5 here x e
vist) = 2) tan! — - = erfc ( z ) (64) oO TT 2 Wet
This expression is obtained by integrating the function ierfc by parts. A nondimensional plot of the shoreline evolution updrift of a groin is shown in Figure 29.
63. The shoreline position has been normalized with a characteristic
length (the groin length) and the tangent of the incident breaking wave angle.
47
gas p= &t 3 L2 fe fo) » N > 1.0
0.9 z oO 0.8 = fH eC wo 0.6 ©
0.6 ¥ 0.4 = =; i 5 a 0.2 wo 0.3
ie} 0 0.5 1 1.5 2
~ ALONGSHORE DISTANCE (x/L)
Figure 29. Shoreline evolution updrift of a groin which is totally blocking the transport of sand alongshore
For a specified amplitude of the sand transport rate and the depth of closure, the ratio of shoreline positions at a given point for two different incident breaking wave angles is proportional to the following ratio of respective
tangents of the angles: Se Oe (65)
64. Equation 64 is valid only until the shoreline has reached the tip of the groin, after which time bypassing of sand is assumed to take place. This bypassing happens when y = L (length of the groin) at x = 0 , which
occurs at time t% 8
ST L t_ = z 7) (66)
48
The above relationship for a fixed wave climate reveals that if the groin length is doubled, the time required for the shoreline to reach the end of the groin will increase fourfold.
65. If bypassing of a groin occurs, the boundary condition at x = 0 changes into y=L. A correct solution to this situation should fulfill this boundary condition and use as an initial condition the shoreline shape just before bypassing occurred, according to Equation 64. An approximate solution was presented by Pelnard-Considere (1956) who used the solution for a shoreline with fixed position ye at x = 0 (see Equation 26) and matched it against Equation 64 by equating sand volumes. With this criterion, the following relationship between the time elapsed before bypassing occurs to (in Equation 64) and the actual time in the matching solution ty » which
makes the sand volumes equal, is obtained: == (67)
66. Thus, in the case of bypassing, it is possible to use Equation 26, iiasthewtime | tags ereplaced sby aut es=sti— (ila 1” /16)t, for t > to . The rate of sand bypassing the groin for t >t is calculated according to
G Equation 8 to produce the following relationship:
(68)
Here 2Q 095 is the sand transport rate at equilibrium (straight beach) under imposed incident breaking wave angle - andthe asthe modititedetime in the matching solution using Equation 26.
67. Formally, the solution downdrift of a groin is the same as that in Equation 64 but with opposite sign. However, if the groin or jetty extends far outside the wave breaker line, diffraction will occur behind the groin
altering the breaking wave height and angle; thus the transport capacity
49
(Equation 9) does not provide a complete description of the shoreline evolu- tion if diffraction is significant.
68. Bypassing may occur immediately after construction of a groin and not start just at the time when the groin is completely filled. If the by- passing sand transport rate grows exponentially to a limiting value Q, the
boundary condition at the groin will be Q 2 al sie i) x = 0 (69)
69. In Appendix B a derivation is given. The quantity y is a rate coefficient describing the speed at which the bypassing sand discharge grows toward the limiting value Q . The solution downdrift of a groin may be
written (for an initially straight beach) as
Q 12 iL 3 et -x /4et x x y(x,t) - -2f 32) —e - Forte ( ) Co) 2 oy T 2 oVet
| ele al
-yt i gee He a eye! IES Wale (70) [o}
fore ff 20) Amal xs SO ,
Employing the two dimensionless parameters, Q,/Q, and yl? /e » the solution is illustrated in Figure 30.
70. The parameter YL? /e describes the rate at which the sand bypassing increases in comparison to the size of the coastal constant (€). In Equa- tion 70 the second term is a transient which decays with elapsed time. Ac- cordingly, after sufficient elapsed time, Equation 70 will be identical to the solution given by Equation 64 with a modified incident breaking wave angle ae Se = 0) Cee ae ee vo Equation 70 may be used also to describe shoreline change updrift of a groin (with reversed sign) if bypassing occurs immediately
after construction of the groin. If, in Equation 70, Q,/9, = ae » the
50
0.6
0.3 = 1 o©) a Zz Se 1.4 = _ wo oO a yg = -0.9 1.8 —) te 5 ye Bt wm L?
-1.2
Soe)
0 0. 1.5 2
1 ALONGSHORE DISTANCE (x/L) Figure 30. Shoreline evolution downdrift of a groin with bypassing described by Q,(1 - a Vy @jO, = Oi 5
a, = 0.4 rad , le fe 2 ®)
bypassing sand discharge will equal the transport rate alongshore behind the groin at equilibrium conditions. Consequently, the initially eroded area downdrift of the groin will fill when the bypassing sand rate reaches its maximum, and the beach will become straight again.
71. In order to investigate the effects of the linearization of the governing equation (Equation 9) on the solution for a groin, numerical simula- tions were carried out with the original sand transport equation (Equation 7). Selected results are displayed in Figures 31 and 32. From the two figures it is seen that the linearization procedure degrades the solution if the incident breaking wave angle is about 30 deg. However, the analytical solution has
surprising accuracy, considering the approximations made.
51
0.8
Diral y tical S01 nn
Shonen) SOB. SaaS ar
0.6 ts &t L?
SHORELINE POSITION (y/L) o
0.2
0.0
0 0.5 1.5 2
1 ALONGSHORE DISTANCE (x/L)
Figure 31. Comparison between analytical and numerical solutions of shoreline evolution updrift of a groin with incident breaking wave angle 20 deg
0.8 Analytical Solns ——_—_———
Numerical Solns -—--—-—-—-—---———
Da eas te
SHORELINE POSITION (y/L)
0 0.5 1.5 2
1 ALONGSHORE DISTANCE (x/L)
Figure 32. Comparison between analytical and numerical solutions of shoreline evolution updrift of a groin with incident breaking wave angle 45 deg
52
Initially Filled Groin System
72. Dean (1984) presents an analytical solution for shoreline evolution between two identical groins which define a compartment initially filled with sand. The distance between the groins is denoted by W , and the groin length is L. At time t =O , the shoreline is exposed to the action of waves
breaking with angle aos The solution is
2 tan oe = 2 y(x,t) =L- w(i - x) tan ao BS 2 ane + ee
(71)
<6 (Ome) “7 CRIA E + 22] e cos ra OW
core © > © emul 0) Soe Sy
The boundary conditions for this configuration are no sand transport at x = 0 (ay/ox = tan a) and a constant shoreline position of y=L at x=W. Consequently, bypassing occurs at the boundary x = W , whereas no sand enters the system at x = 0 . This occurrence means that the solution is unsuitable for application to a groin system of more than one compartment. Otherwise, bypassing must be accounted for in the boundary conditions at the updrift groin (left) in each compartment leading to a coupled problem. The last term in Equation 71 approaches zero as t + ~ and causes a shoreline parallel to the wave crests to be created between the groins. In Figure 33 the analytical solution is presented in dimensionless form. All distances have been nor-
malized with the compartment width W.
73. The final percentage loss of sand from the groin compartment is
W L tan a (72)
5)3)
SHORELINE POSITION (y/W)
0.4 0.6 0.8 1 -ALONGSHORE DISTANCE (x/W)
Figure 33. Shoreline evolution between two groins initially filled with sand (L/W = 0.33 , ON 0.25 rad) From Equation 71, the sand bypassed (discharge rate) at x = W can be obtained. The sand transport rate as a function of time can be written (if it
is assumed that tan a =a)
- 2 2 2 ue) He, D GU eee ie -
+ 1)T n=0
for t > 0O and x=W.
In Equation 73, the quantity 205% is the sand transport rate along a straight beach exposed to the incident breaking wave angle ao (This is the transport initially existing when the groin compartment is completely filled.) If Q in Equation 73 is normalized with this quantity, the bypassing sand discharge at the downdrift end groin is conveniently displayed in dimension-
less form. Figure 34 shows such a curve.
54
SAND TRANSPORT RATE (0/20,o,)
i} 1.2 1.5
0.6 0. TIME (et/W?)
Figure 34. Bypassing sand transport rate at the downdrift end of a groin x =W asa function of time
Shoreline Change at a Detached Breakwater
74. A detached breakwater reduces the wave height behind it and pro- duces a circular wave pattern at each tip, thus decreasing the longshore sand transport rate. The actual effects are quite complex to describe and involve diffraction and the current field resulting from spatial changes in wave height and direction. However, it is possible to find an analytical solution if the situation is idealized.
75. It is assumed that the incident breaking wave crests are parallel to the x-axis and to the detached breakwater. When the waves reach the break- water, they are assumed to be diffracted at a constant angle behind the break- water (shadowed region) and remain parallel to the x-axis outside of the breakwater (the illuminated region). The diffraction behind the breakwater is symmetric about the center of the breakwater and, accordingly, only half of the problem domain needs to be considered. In Figure 35, a definition sketch
is shown.
a)
DETACHED BREAKWATER
-2L =L N50 Figure 35. Definition sketch for the problem of shoreline change in the vicinity of a detached breakwater 76. Since the incident breaking wave angles and the amplitudes of the sand transport rates Qo1 and Q02 » respectively, are different in the shadowed and illuminated regions, a coupled problem arises. The boundary conditions for this case are as follows:
a. No sand should be transported across the line of symmetry behind the breakwater.
b. The sand transport rate out of the area on the right side of the breakwater should be equal to that into the area behind the breakwater.
c. The shoreline is continuous over the boundary between the two areas.
Furthermore, the shoreline should be undisturbed (y = 0) far from the struc-— ture. With yy denoting the shoreline position in solution area number 1 (shadow region) and Yo denoting the shoreline position in solution area num- ber 2 (the illuminated region), the mathematical formulation of the situation is
56
2 r) yy ay
ox ay, V5 E5 5) = ae x >0 (75) ox y, (x,0) = Y (x,0) = 0 (76) oY) aie = tan on x = -L (77)
— —ta x = 0 (78) ox ox oo ol Ym xs FO
77. The derivation of this solution is presented in Appendix C. The
quantities Orr and v9 are the amplitudes of the longshore sand transport rate in the respective areas, and OO1 is the diffracted breaking wave angle behind the breakwater. The angle a) is zero since the wave crests in this
area are parallel to the x-axis throughout time. The solution is, with
Bo |-S5 | (80)
57
6a) — _ y, (x,t) = - 2ve,t ierfc l 6 + 1 i (se)
n + tan or » (; = r) 2ve,t ierfc (n+ DL +x n=0 2ve,t
n+l + (; = +) 2ve,t ierfc (2n + DL =x
S : 2ve,t
n (3 1) 2ve st ierfc Aone ln ies
n=0 2ve,t aINGL MG = Who = + (; t) 27 ene ierfe | ———————_— (81) 2ve,t Fore 2 SO ema Gh Sx < @ ei 6 Yo (x,t) Tie SEESI 2ve,t ierfc x 2ve,t 540 zs . n -2— 5 ( - 7) 2ve,t ierfc | “$+ 204 DE (CRE) ee, 2ve,t 6 tan a = n + 2 a > (; = t) 2ve,t jerfce | SX + Cn + DL (82) n=0 2ve,t
fore 2 = @ amd x S © ,
58
78. The distance L is half the length of the detached breakwater. If Equations 81 and 82 are plotted, the following behavior will be noticed. When the breakwater is placed in front of the initially straight shoreline at time t = 0 , erosion of the shoreline starts at points in line with the corners of the breakwater. Simultaneously, the shoreline grows to form a salient about the line of symmetry behind the breakwater. Because of the gradient of the shoreline outside the shadow of the breakwater, material is transported toward the breakwater in order to achieve a state of equilibrium with the waves. The shoreline behind the breakwater also approaches an equilibrium configuration which is parallel to the wave crests diffracted at the angle The final shoreline will be inclined at an angle a behind the
@l ~ ol breakwater and be straight outside the breakwater. However, the straight
a
portion of the shoreline will at all times be displaced landward a small distance, controlled by the volume of sand that has accumulated behind the breakwater. Figure 36 illustrates the solution in dimensionless form for
0.20
0.05
SHORELINE POSITION (y/L)
0.00
-0.05
“1 -0.5 1.5 2
0 0.5 1 ALONGSHORE DISTANCE (x/L)
Figure 36. Initial shoreline evolution in the vicinity of a shore-parallel detached breakwater (6 = 0.5 , a1 7 0.4 rad , a, = 0) o2
short elapsed times, and Figure 37 shows the features of the solution after a long elapsed time. The length of the salient behind the breakwater increases
in time toward a maximum value of
59
L tan a, (83)
1 The elapsed time is normalized by the quantity te li . Although mass is conserved across the boundary between the two solution areas, the gradient of
the shoreline is not continuous at this point.
ae]
_ &t 0 44 Wwe L2 tia KO
a
>
- sd
z
oO
_—
ol
4
wo
2B aA
a
GJ
Zz
ere
im
tre 0.1
oO
x
wo
-0.5 1.5 2
Le} 0.5 1 ALONGSHORE DISTANCE (x/L)
Figure 37. Final shoreline position in the vicinity of a shore- parallel detached breakwater (6 = 0.5, a = 0.4 rad ,
al ol 262. 8)
Shoreline Change at a Seawall
79. The function of a seawall is to prevent the shoreline from retreat— ing along a specific coastal reach. If the shoreline remains well seaward of the seawall, there will be no influence of the seawall on the shoreline evolu- tion. If the shoreline retreats to the seawall, the location of the seawall determines the minimum allowable shoreline position. If erosion takes place beside a seawall (flanking), various changes in the shoreline position might occur depending on the characteristics of the seawall and the incident waves. If flanking of the seawall is not possible (see Figure 38), the solution for the plan shape of an eroded shoreline will be the same as for erosion downdrift of a groin (Equation 64, with opposite sign). In this case, the
seawall is functioning as a semi-infinite structure.
60
SEAWALL
on fon
YR C8 55
AO" ~ SHORELINE
Figure 38. Definition sketch for a semi-infinite seawall for which no erosion occurs behind the seawall 80. Figure 39 illustrates the case of erosion at the side and behind a seawall, i.e., flanking of the seawall. This must be solved as a coupled problem. The incident breaking wave angle is a) outside the seawall and OO behind it. Wave energy is transported behind the seawall by the process
of diffraction.
SEAWALL
oS Sa
‘SHORELINE
- sian
Figure 39. Definition sketch for a semi-infinite seawall for which erosion occurs behind the seawall
61
81. The ratio between the amplitudes of the longshore sand transport
2 rate in the two solution areas will be denoted as 6 (= Q51/9%52)* Mathemati-
cally, the situation is formulated as
2 Oy e 7 oe - ! ox 2 Pe) Yo 975 € == > 2 9x2 ot y, (*,0) = y,(x,0) = 0 ails Felli 1 oY 4 ox ol 52 02 62 ox Tal Yo x = 0 Sa = 0 x > © Vg = 0 x > ©
It is assumed that the border between the two solution areas at
me) alg
stationary in time, although it moves somewhat in the x-direction as time
evolves. The solution is (for details, see Appendix D)
ol 2 od e,t -x’ fact y, (x,t) = Ser ee 2 TS. + x erfc
for t > 0O and x<0O.
62
(84)
(85)
(86)
(87)
(88)
(89)
Grol eo? e,t -8x"/ue et ae yy (x,t) ae Tagua 2 ae - 6x erfc | —— (91) 2ve_t
roe (© = 0) Eml og 2 4
The quantity aol represents a mean diffracted wave angle behind the seawall. The solution in nondimensional form is presented in Figure 40 (expressed in
terms of the coastal constant E,)-
0.0
t'=0.4 — -0.2 q 0.8 n =z z 1.2 a = 1B ® -0.4 oO 2.0 GJ =z _— —| te > t’= et wo -0.6 L?
<H} 15 -1 -0.5 i) 0.5 1 1.5 2 ALONGSHORE DISTANCE (x/L)
Figure 40. Shoreline evolution in the vicinity of a seawall where erosion and flanking may occur behind it (a = 0.2 rad , ol O = 0.4 rad , 6 = 0.6) 02 82. A characteristic length L is chosen to normalize the shoreline position. In Figure 40 the time has been normalized by use of the quantity
2 L/e, - Shoreline Change at a Jetty, Including Diffraction
83. In the shadow zone of a long groin or jetty, it may be an
63
oversimplification to neglect the process of wave diffraction. Consequently, although Equation 64 (with reversed sign) may give a satisfactory description of shoreline evolution at some distance downdrift of a jetty, in the vicinity of the jetty this solution does not represent what is commonly observed. Ero- sion just behind the jetty will be overestimated if diffraction is neglected since the wave height is assumed to be constant alongshore. Accordingly, by allowing a variation in wave height (and thus in the amplitude of the sand transport rate) in the shadow zone, a more realistic description of shoreline change will be obtained.
84. There are a number of ways to account for a varying amplitude in the longshore sand transport rate (resulting from varying wave height). One way is to assume that, outside the shadow zone, the incident breaking wave angle and the amplitude of the sand transport rate are not influenced by the jetty. In the vicinity of the jetty, Equation 11 may be used to account for a variation in the amplitude of the sand transport rate. An alternative way is to divide the shadow region into distinct solution areas, each having a con- stant amplitude of the sand transport rate. The incident breaking wave angle may also be varied from one solution area to another. With this procedure, a coupled system of equations is obtained which involves intensive calculations for even a small number of solution areas. If the simple case of two solution areas (one inside the shadow zone and one outside) is considered, the mathe- matical formulation is the same as for a detached breakwater. However, the incident breaking wave angle outside the shadow region is not zero (in which case no sand transport would occur) but has a finite value. Therefore, the boundary condition on continuity in sand transport across the border between
the two solution areas takes the following form:
(92)
where 8 is the ratio between the amplitudes of the sand transport rate in- side and outside the shadow region. The analytical solution to this problem is formally identical to Equations 81 and 82, except that certain constants are different. The following substitutions should be made in order to apply Equation 81 and Equation 82 to the diffracting jetty case:
64
7 Sel me (sea) ) a0 “5 +4 ol ol o2 Se ec SCE (94) (6 + 1) (oe)
So Ie is zero, the expressions on the right side reduce to those
o2 on the left side. As can be seen from Equations 81 and 82, even though the description involves only two solution areas, the governing equation is already quite complex. Generalization to an arbitrary number of solution areas is straightforward, in which case the situation is mathematically ex-
pressed for the en area as follows (see Figure 41):
2 r) Yay ay, Eee) Bes Oa Sitiny Wee edeel (5) ox oy oy. i 1 i+l ae eos! BD “ost 72 BSR ae *i41 (96) oy 6 i oy oy. ttl 1 1 i cs mm “oiot 9 Soa ” a tee a Cu) 6. 6. i-l i-l Yi-1 7 V4 xr ris Yi ~ Vidi SF PES e) where Q. Gs — (99) oitl
65
_
SOLUTION
Figure 41. Definition sketch for shoreline evolution
downdrift of a jetty for which a finite number of
solution areas is used to model diffraction For the first and last solution areas, other conditions prevail on the outer boundaries, such as no sand transport at the jetty, and y=0O as x+t>, 86. Extremely complex algebraic manipulations are associated with the analytical solution of coupled systems with several solution areas. In Fig- ure 42 the solution is presented for two areas, with a = -0.1 rad ,
= -0.4 rad , and 6 =0.5.
ol a
02
87. The solution for an arbitrary number of distinct areas is outlined in Appendix E. In Figure 42 are plotted shoreline positions normalized with the length of the shadow region. The length of the geometric shadow region is B = L tan (a5) » where L is the jetty length and a is the incident breaking wave angle in the illuminated region.
88. If the amplitude of the longshore sand transport rate is considered to be a continuous function of x in the shadow zone, Equation 11 is appli- cable. However, this equation is quite complex, and it is difficult to find analytical solutions even if very simple functions are employed. The related
case, in which the incident breaking wave angle is a continuous function of
66
t'=0.4
-0.2 rea) ~ =) 0.8 5 -0.4 C= iS 1.2 wn oO oa LJ 2 -0.6 1.6 ne = te S 2.0 a5 wn -0.8 t’= et B? -1 -1 -0.5 1.5 2
Lt} 0.5 1 ALONGSHORE DISTANCE (x/B)
Figure 42. Shoreline evolution in the vicinity of a groin for variable sand transport rate conditions (two solution AReAss © S205 5 @ = -0.l rad , a = -0.4 rad) ol o2 x , is easier to treat analytically and provides interesting solutions. Under
these circumstances, Equation 11 will take the following form:
dey Wey oe ToS Oe Ge (ae) ox
in which oo is a function of x only. This is formally the same equation as that describing heat conduction in a solid containing a finite source. Consequently, if oe grows linearly with x (e. = xa_/B) the situation will be identical to the one describing a river mouth of finite length which dis- charges sand at a constant rate. Equations 55 and 56 are the solutions to this case, with reversed sign and qp replaced by a /B . The solution is presented in Figure 26 in dimensionless form.
89. If an is different from zero at the jetty, but still grows lin- early along the x-axis in the shadow zone, the variation in breaking wave
angle will be
x a = oT + (x - =) 3 (101)
in which a is the incident breaking wave angle at the jetty, and Oh is the angle in the illuminated region. The mathematical description for this case is almost the same as for a river mouth of finite length which discharges sand but with a modified source term. This is a coupled problem containing
two solution areas but with a boundary condition at the jetty given by
ia = iCan on (102)
The analytical solution to this problem is (see Appendix F)
(a. - a jet y, (x,t) = ee 212 erte (2 = ) +) 2 q? erfc le a _ - 1 2Vet 2Vet
2 - tana 2 fl ens ee - x erfc ( x ) (103) Vv T 2Vet for t >0O and OS x<S<B (a. - a )et
Yo (x,t) egal "7 x 2i erfc e a “| - 2 i erfce (5 — >)
2Vet Qvet
) (104)
for t >0O and x?B.
The quantity B is the geometric length of the shadow zone as before. In Figure 43, the dimensionless shoreline evolution is presented for the specific case of o = -0.1 rad and hy = 0.4 rad . Shoreline position has been normalized by the length of the shadow region.
90. Another case that allows a fairly easy analytical solution is ob- tained by assuming that the incident breaking wave angle varies exponentially
with distance from the jetty according to
68
a. = a (1 2 oe) (105)
Here, the quantity Y is a coefficient describing the rate at which the
breaking wave angle approaches the undisturbed value Oh along the x-axis.
0.0
Og t'=0.4 a SS > =z iS} -0.2 0.8 ee — ip) i=) oO ta 1.2 —or—OeS — —! fe = 1.6 pe St 7p) B?
-0.4
2.0 0.5 0 0.5 2.5 3
ALONGSHORE DISTANCE (x/8) Figure 43. Shoreline evolution behind a jetty with linear variation in breaking wave angle in the shadow zone
(c, = -0.1l rad , Oy = 0.4 rad) The derivation of the analytical solution is presented in Appendix G. The
solution is
ay 2 yoo) «B= NEE Ms 2 § onto (A) Y T (Sie ar ay ks pencil erfe( = - ies) Y 2Vet - 4 ales tae erfc ( x + fe) + al ies ( a! et) (106) Y avet for t >0O and x20.
69
If a dimensionless quantity yL is introduced, the solution may be displayed efficiently in dimensionless form (Figure 44). For large values of y , Equa- tion 106 approaches Equation 64, which is valid for a jetty and constant
oblique breaking wave angle.
0.0 t'= 0.4 0.1 a ~ hy 0.8 S 0.2 _— Ext 1.2 wo oO oa WW 1.6 Z -0.3 Le | — Ld oe 2.0 x= »_ St se sip? 0.4 0.5 i) 0.5 1.5 2
ALONGSHORE DISTANCE (x/B) Figure 44. Shoreline evolution behind a jetty with exponential variation in breaking wave angle (a, = 0.4 rad , yL = 1)
91. The solution obtained for a variable breaking wave angle over- estimates the rate of erosion behind the jetty since it is assumed that the amplitude of the longshore sand transport rate is everywhere the same (and thus that the wave height, in principle, is constant). In reality, diffrac- tion reduces the wave height in the shadow region and, accordingly, the ampli- tude of the longshore sand transport rate there. Despite this reduction, Equations 103 and 104 provide a better description of the actual situation than the commonly used solution (Equation 64) for which maximum erosion will
always appear immediately adjacent to the jetty or long groin.
70
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1984. CRC Handbook of Coastal Processes and Erosion, Komar, PRD ee ditor.CRCmeEressmince a bOCamRaton mbar
Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. 1954. "Tables of Integral Transforms," Vol 1, McGraw-Hill, New York, NY, 391 pp.
Grijm, W. 1961. "Theoretical Forms of Shoreline," Proceedings of the 7th Coastal Engineering Conference, American Society of Civil Engineers, pp 197-202.
. 1965. "Theoretical Forms of Shoreline," Proceedings of the
9th Coastal Engineering Conference, American Society of Civil Engineers, pp 219-235.
Hanson, H., and Kraus, N. C. 1986. "Seawall Boundary Condition in Numerical Models of Shoreline Evolution," Technical Report CERC-86-3, US Army Engineer Waterways Experiment Station, Vicksburg, Miss.
Komar, P. D. 1973. "Computer Models of Delta Growth Due to Sediment Input
from Waves and Longshore Transport," Geological Society of America Bulletin, Vol 84, pp 2217-2226.
71
Kraus, N. C. 1983. "Applications of a Shoreline Prediction Model," Proceed-
ings of Coastal Structures '83, American Society of Civil Engineers, pp 632-645.
Kraus, N. C., and Harikai, S. 1983. "Numerical Model of the Shoreline Change at Oarai Beach," Coastal Engineering, Vol 7, No. 1, pp 1-28.
Le Mehauté, B., and Brebner, A. 1961. "An Introduction to Coastal Morphology and Littoral Processes," Report No. 14, Civil Engineering Department, Queens University at Kingston, Ontario, Canada.
Le Mehauté, B., and Soldate, M. 1977. "Mathematical Modeling of Shoreline Evolution,'' CERC Miscellaneous Report No. 77-10, US Army Engineer Waterways Experiment Station, Vicksburg, Miss.
1978. "Mathematical Modeling of Shoreline Evolution," Report No. TC-831, Tetra Tech, Inc., Pasadena, Calif.
1979. "Mathematical Modeling of Shoreline Evolution," Proceed-
ings of the 16th Coastal Engineering Conference, American Society of Civil Engineers, pp 1163-1179.
Longuet-Higgins, M. S. 1970a. + "Longshore Currents Generated by Obliquely Incident Sea Waves, 1," Journal of Geophysical Research, Vol 75, No. 33, pp 6778-6789.
. 1970b. "Longshore Currents Generated by Obliquely Incident Sea Waves, 2,"" Journal of Geophysical Research, Vol 75, No. 33, pp 6790-6801.
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Shore Protection Manual. 1984. 4th ed., 2 vols, US Army Engineer Waterways Experiment Station, Coastal Engineering Research Center, US Government Print- ing Office, Washington, DC.
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the Sand Transport Equation and Some Simplified Solutions," Proceedings of Coastal Structures '79, American Society of Civil Engineers, pp 809-837.
72
APPENDIX A: A SHORT INTRODUCTION TO THE LAPLACE TRANSFORM TECHNIQUE
1. The Laplace transform is a powerful technique for solving linear partial differential equations. This technique allows the target partial dif- ferential equation to be converted to an ordinary linear differential equation in the transformed plane for solving one-dimensional problems in space. The Laplace transform of a function y is denoted as L{y} and is defined by the
operation:
L{y} = y - [veo ely usdit (Al)
The over bar denotes the transformed function. The transform of a derivative
of a function with respect to time is
oy Se eu = sy - y(x,0) (A2)
This relationship may be derived by performing a partial integration of Equa- tion Al. The term y(x,0) represents the initial conditions for the system. Accordingly, the transform of the diffusion equation may be written (if, with the convention y(x,0) = 0 , that is, a shoreline which is initially parallel
to the x-axis):
ioe
ae ye 0 (A3) x
O10
The general solution of this homogeneous linear differential equation is
y Ss Ase! & ha (A4)
where
Al
2. The coefficients A and B are determined by the transformed boundary conditions and are, in general, functions of the parameter s . To obtain a solution in the time domain, Equation A4 has to be inverse trans- formed. This can be accomplished using tables of known transforms (see, for example, Erdelyi et al. (1954) and Abramowitz and Stegun (1965))* or the
Fourier inversion theorem which states
Gate
jo Rll Situ y= oer i enn y (s)) ds (A5) Bale
The integration is performed as a line integral in the complex plane, for which ¢ is taken sufficiently large to have all singularities of the func- tion y(s) lying to the left. Equation A5 is normally evaluated by means of the residue calculus. If several solution areas are used, the solution within each area is of the form of Equation A4. The solutions are dependent upon each other through their common boundaries (as an example see Appendix E) by the prevailing boundary conditions.
3. Table Al presents a short summary of selected applicable transforms
useful for solving the diffusion equation.
* References cited in the Appendix can be found in the References at the end of the main text.
A2
Table Al Short Table of Laplace Transforms of Functions Often
Encountered in Solving the Diffusion Equation
y(s) y(t) qs ae 1/2 af fle q Tt £ erfc ( zs 2 2vet aw 9 fea asf fhe x x erfec as u 2vet s 2Vvet m2eQ5 il, 2 -qx Wf 2e ane : _ ice) a /4et heelxteth fe ( x a WES 2vet ent 2 ea x het yy lth Bore x qs (q + h) h \t h2 Vet
h is an unrestricted constant
A3
ae i ee a neti
aa eAthena PE tha 5 hey bis Linea | isi wer Pee Semi
tir THe ri i eet) Ba? weal Bane iW i ay ie ii vials Vite ye a : 1 ll ‘pont te ad Pe ad ' a i a nike Py vie : ve ii eto hy (tebe 7 , . , 4 ——re et : me ae roth caer vA “ren gaan na mle ? | : om fr we lel ae bl sl Wn wee = it ig ; : : ae vailiderenuntinee i Giveks itt tibial | nel
; a Ae 4 ge an ‘ 7 cen a , 4 y ; t f j War re ia ! mA H , r s appr nee ‘4 a sh re q it, my Rx aya i Oe eT hei
‘sy a ge pant bor i tan t ; +e | |
APPENDIX B: SHORELINE EVOLUTION DOWNDRIFT OF A GROIN WITH BYPASSING REPRESENTED BY AN EXPONENTIAL FUNCTION
1. Sand is transported past the groin according to the following
relationship: Q=Q(1-e%) (B1)
Here Q denotes the maximum bypassing sand transport rate which occurs at the groin, and y is a rate coefficient describing the rate at which the limiting value Q is approached in time. Using Equation 8, the boundary
condition at the groin is written: ) x = 0 (B2)
Consequently, the mathematical statement of this case is, together with the
above boundary condition:
2 Suter Oy: e— == (B3) are ee y(x,t) = 0 X + © (B4) y(x,0) = 0 (B5)
2. By using the Laplace transform technique, an ordinary linear differ-
ential equation is obtained:
<y-2y¥=0 (B6)
where y denotes the transformed function of y . The transformed boundary
condition is
Bl
a.
ee ee dx Ss 2 ve Ss GaP 47
Solving Equation B6 together with Equations B4 and B7 yields
Q Q —qx B\e ih 1 “B e (B8)
a l y = _ Qa —_——_—_—_ — —— 2 Os qs 2 OR q(s + y)
ie]
where a = =
3. The inverse Laplace transform of the first term in Equation B8 is
found to be (Appendix A)
Q 2 o So Ge s = 2 Ae ent [Eee x erfe — (B9) ) w 2vet
The second term is evaluated by applying Duhamel's theorem (Carslaw and Jaeger
1959, p. 301) which reads t L i £ (1), (t = dtp = Lif, (e)} L{e,(€)3 (B10)
oO
in which L{} represents the Laplace transform operation. The second term of
Equation B8 yields, after some rearranging, Dijae 2 als xX /4eé dé (B11)
2p oe IE aaa Y/ Q 7
Accordingly, the complete solution is
Sy
B2
Vt Q 22: D) ie) Ne sidaiafost” “x /4ek dé (B12) [@) (e)
4. The last term on the right side of Equation Bl2 describes a tran- sient which disappears with time. After the effect of the transient term has vanished, the solution for shoreline change downdrift of a groin will be the same as the solution obtained without bypassing but with a modified breaking
wave angle. If Q& < 25% erosion will take place; whereas if Q > 205%
there will be accretion.
B3
f / i ps r - = > = &, may TN Oe % \ 4 ‘ a . fi ; oP 6
\ a. \ ; hi) ou iw) a yy : ve @ ci ba qarpt Sr hell BAe atl ite ree teak wht 7 a Peay aie Yes mils “nee? Wat a) +e ii netitn i) a ee ov" #, Re 4 ebrbereeity nner ik sudo f met wo hctul dew wt? hind sen Ftd iat ‘. wiciine-a ren a oabel will aden i fram: i tits prac res . tt in i bhocuiten wil. eal ete 5 ; aw sieieoerah : why ‘bs- | ie. - olaytn *
me
ings mo
APPENDIX C: SHORELINE EVOLUTION BEHIND A DETACHED BREAKWATER
1. In Figure 35 (in the main text) a definition sketch is shown for the case of a detached breakwater and normal incident waves. The shoreline evolution is symmetric about the centerline of the detached breakwater; thus, only half of the problem domain needs to be considered. Since the amplitude of the sand transport rate Qe and the incident breaking wave angle a, are different behind the breakwater and outside the breakwater, two solution areas are required. Mathematically, the shoreline evolution is described by Equa- tions 74-79. After the Laplace transform technique is applied, the following
system of ordinary linear differential equations is obtained:
2-— d yy -—y, =0 -L<x<0O (C1) ask ey 4 ay, s= pate oe a = 0 x > 0 (C2) dx 2 dy tan a jee ol je "a See ee Y5 = 0 x > 0 (C4) Vy 2 Va x = 0 (C5) dy Q dy (o} 1 o2 2 ol (C6)
Cl
in which yy and Y5 denote the transformed shoreline position corresponding to the regions behind and outside the breakwater, respectively, and L is half the length of the breakwater. Solving the system of equations subject to
the boundary conditions yields
q,x - So) e ; i aie Sl a5 ar ° cosh (a,x) — sinh (a, )| @ a - ol sd) ol 6 + 1 q,s(6 sinh q,L + cosh qb) i e-2 0 (7) 6a -q,L -q,x -q,x a ol 1 2 2g. Ste oe hie ee OM eas) sy) § +1 qs q,s(6 sinh q,L + cosh q,2) where Q 2 1 2 2 6 Sieg att a q5 => (C9) 02 al 2.
2. The inverse transform of Equations C7 and C8 may be obtained by use of the Fourier inversion theorem (Appendix A) or by expanding the denominator in a Taylor series and finding the inverse transform of each term in the series. The latter method will be used here. The denominator may be
rewritten as
1 ane Sel wae q,8(6 sinh q,L + cosh q,1) =7 4,5 e (6 + 1)]1 - (: = *) e (C10)
The last term in Equation C10 is expanded in a Taylor series according to
ihe @ - L ae Re YS a - aT a be Gis § +1) ° 5 +1
c2
3. Only the inverse transform of Equation C8 will be obtained here to illustrate the procedure. The inverse transform of the first term in
Equation C8 is (noting that ao = 6q,)
1 OE RE 6x Vo = 2 Soa Z ierfc (C12)
in which the function ierfce is defined according to Equation 23. The second
part of Equation C8 is rewritten by using Equation Cll:
6a -q,L sete a -2q,nL Soe = (ea - oF ors! Spek hae 2 (Ss ae i (C13) 1 é +1 “ol 6 + 1 Rearranging Equation C13 by moving terms inside the summation gives SERA a, co -q, [L(2nt1)+6x] er ip) ae V9 Ties eae q,s © -q, [2L(nt+1)+6x] 5 ae ow 2.9) See See eee (8 + ae ® ap il q\s
n=0
This expression is inverse transformed term by term (Appendix A). The solu-
tion is
6 tan os n v5 = 2 ———— = € — t) 2ve,t ierfc Sesae Wein a IW)
Sa re n 269) wats » (3 S t) 2ve,8 terfe 6x + 2L(n + 1) (C15) (Gone a car, 2ve,t
c3
The complete solution to Equation C8 is written as
6a n 2) 1 - » (s z +) ave,t Temae ox + 2L(n + 1) (C16) (Si Wa ae 2ve,t
In the same way, Equation C7 may be inverse transformed, resulting in
Equation 81 (main text).
C4
APPENDIX D: SHORELINE EVOLUTION IN THE VICINITY OF A SEAWALL WHERE FLANKING OCCURS
1. Two solution areas are employed to describe flanking of a semi- infinite seawall, one area behind the seawall and the other away from the sea- wall. The amplitudes of the sand transport rate are denoted as Ol and Qo9 in the respective solution areas, and the corresponding incident breaking wave angles are denoted as aT and Qo ° The incident breaking wave angle oy behind the seawall (solution area 1) should be interpreted as a repre- sentative mean value related to the sand transport rate. Equations 84-89 (main text) constitute the mathematical formulation of shoreline evolution in the vicinity of a seawall subject to flanking. The Laplace transformed system
of equations and the boundary conditions are
dy A ~- 23, =0 x <0 (D1) dx eI ay, Si OF ee 0 xe = (0) (D2) dx 2 i > 0 x > -@ (D3) Y5 = 0 xX > © (D4) Iq 2 Yo x= 0 (D5) dy dy een | 2 1 1 u Ge Re x Li (s0 i: ee “) s eT Y (D6)
2 Solving the system of ordinary linear differential equations subject
to the boundary conditions yields
= bol Tyee) coe ale y, = x <0 (D7) : ee aie 5 a x = 6 afi! Lol Ai oils y = SEE x > 0) (D8) ‘ 1 +4 ele
3. The inverse transforms of Equations D7 and D8 are (Appendix A):
so4 co e,t -x' [et ave Mal = “Oey 2 ao e + x erfe (D9) 2ve,t
Tn = Bae ee t -6 Vie, t ee
D2
APPENDIX E: SHORELINE EVOLUTION DOWNDRIFT OF A JETTY IF AN ARBITRARY NUMBER OF SOLUTION AREAS IS USED TO MODEL DIFFRACTION
1. The area downdrift of a jetty is divided into N distinct solution areas of assumed different sand transport properties. In an arbitrary solu- tion area j , the amplitude of sand transport rate is denoted as 265 and the incident breaking wave angle as ood - The shoreline evolution is denoted as y, in the solution area bounded by the shoreline coordinates Eg and ae Equations 95 to 99 (main text) mathematically describe the shoreline evolution in one solution area. Using the Laplace transform technique, the
governing equations take the following form:
a’y, ie Fa a OO (El) dx” q J ¥4 7 Yet ene Ez) v5 = V5-1 x. x5 (E3) dy, dy, ls & ues ( Sx62 L dx j-1 dx zs “of Holwoqel s 8) mS oS, J dy. dy. dae) Sar aad ~ & 4 dx j dx x “oft! joj s (6) Sea ol
El
where
(0). Roe WE (E6) J oj+l
The solution to the ordinary linear differential Equation El is
Sans j j
= A.e + B.e (E7) y jj J where
qo = (E8)
in which Hs and 8 are constants to be determined through the boundary conditions. Since the shoreline evolution in each solution area is connected via the boundary conditions with the neighboring areas, an equation system with 2N unknowns (two constants for every solution area) is obtained. The
boundary conditions E2 and E3 give the following relationships:
q.X. -q.x. @a 98s Ga ohn Rete awed en atten a dos (E9) j j goul sj q.X. -q.X. Gane —q.,,X. j jt+l Pedal jt+l jt+l j+l jt+l A.e + ec y+ + Bel (E10) 2. Furthermore, Equations E4 and E5 give q.x -q.x @la 38 Gla 92S. B. awd Jee alae Sah | ee aj Welle teoet dics He Bie egal Oto ie + ae (E11)
E2
ate Cel - Ore oi (E13) 3. Equations similar to E9 to E13 may be written from solution area 2 to solution area N-1l . In the first and last solution areas, two other con- ditions prevail at the outer boundaries, namely, no sand transport in the first solution area (area 1) and no shoreline change as x->» in the last solution area (area N). The Laplace transforms of these boundary conditions
are
dy,
as x = 0 (E14)
= tan a
4. Equation E15 implies that the constant Ay is zero. The resulting system of equations to be solved in order to determine the value of the con- stants is conveniently written in matrix form. A general system of N_ solu-
tion areas gives rise to 2N - 1 equations as follows:
-1 0 0 0 0 0 mule2 92% 2 749% -e 0 0 -q,x 45x -q5x Aeae 2*2 Ane 2 A 6 q5x 7G 5x q4x —-q,x 4 see aaa Ban3 eke 4 x 0 Qy-1*N “4y-1*n ~Iy*N e e e q yes xX, Set = x, -q x, ~5 e NeIN § @ NHITN _@ WN
(E16)
E3
It is seen that the solution corresponding to even a small number of solution areas involves intensive algebraic calculations. Furthermore, the inverse transformation is difficult to perform, necessitating use of the Fourier
inversion theorem.
E4
APPENDIX F: SHORELINE EVOLUTION BEHIND A JETTY FOR LINEARLY VARYING BREAKING WAVE ANGLE
1. In the case of shoreline evolution behind a jetty for lineraly varying breaking wave angle, the amplitude of the sand transport rate is regarded as constant downdrift of the jetty, and the incident breaking wave angle varies linearly from the jetty (with value a.) to the value hey in the region undisturbed by the jetty. Two solution areas are needed for describing shoreline change, one in the shadow region and the other outside the shadow region (illuminated area). Equation 101 (main text) describes the variation
in breaking wave angle in the shadow region which is of length B . The
Laplace transformed equations and boundary conditions are
ae Ch 1 Pes Hever OS SR (Fl) e A B Ss dx ay, Si — 0 x > B (F2) 2 Eo 2 dx dy tan a 1 4 a ee x = 0 (E3) Yq 2 Vp xa (F4) dy, dy 1 2 dx x "Nine oP 2. The solution to this system of ordinary linear differential equations is a =a (pe i =ax = (aos alle q(B-x) “ye q (Btx) eho, Ceo ee an V9 2B 2 D 2 qs s s s
Fl
A a, - a -—q(x+B) -q(x-B) -qx , = ( H P ) 5 e e e (F7)
- - tan a 52 $2 v qs
where eS clipes (F8)
3. Equations F6 and F7 are easily transformed term by term (see
Appendix A) to yield Equations 103 and 104 (main text).
F2
APPENDIX G: SHORELINE EVOLUTION BEHIND A JETTY FOR EXPONENTIALLY VARYING BREAKING WAVE ANGLE
1. The breaking wave angle varies exponentially with the distance be- hind the jetty from zero at the jetty to the undisturbed value ce far from the jetty. The mathematical formulation of the boundary condition at the jetty is expressed by Equation 105 (main text). A varying breaking wave angle along the x-axis is described in terms of the diffusion equation by a distrib- uted sink with a decaying strength with distance. The transformed equation
and boundary conditions are
dey Omny, ay eB ems ane x > 0 (G1) 2 € Ss dx dy _ . a 0 x = 0 (G2) y = 0 x > © (G3) The solution to Equation Gl is fs athe t ale I igereGge G22) sen qs\Y - 4q S\Y - 4 Equation G4 may be written as partial fractions: —qx -yx 722i ee Sa ee ne! (cs) 2 - + qs Gey me Gact ny, Y ht ws s In Equation G5, the last term may be inverse transformed to yield ae 2 wo ( Bast et) (C6)
Gl
2. The first part of the first term is inverse transformed according to
Appendix A and gives
Vas on 2 fer er /Aet 1 - yx x Y cit Mea ey Nines i i\aenazear a) aa Y Y 2Vvet
2 + 1. Sr PC (Oe rarersr| (eee ve) (G7) Y 2vet
In the same way, the inverse transform of the second part of the first term in
Equation G5 gives
Ts) mY |2 fet eee 1 + yx x oY, iO ae Y a = = ime ita! sa ) Y 2vet 1 xtet 2 x + 0) ey Y erfe ( + ie) (G8) y 2vet
3. The complete solution consists of Equations G6, G7, and G8 as given
by Equation 106 (main text).
G2
APPENDIX H: NOTATION
Length (m)
Amplitude of periodic beach cusps (m)
Cross-sectional periodic beach area (m2)
Constants in general functions of the Laplace transform variable Length of shadow region downdrift of a groin (m)
Wave group velocity to breaking point (m/sec)
Depth of closure (m)
Error function
Arbitrary initial shoreline shape (m)
Acceleration of gravity (aafaee)
Constant
Significant breaking wave height (m)
Integer number
Integral of the error function
Slope of a line segment
Nondimensional constant
Geometric length (m)
Laplace transform of a function y
Nondimensional groin length
Integer number
Integer number
Number of solution areas or reaches
Pitch height of a circle segment (m)
Loss percentage from a beach fill
Sand transport rate per unit length of beach from a source or sink Ge faveae)
Constant sand discharge from a river acting as a point source eee)
Time variable sand discharge from a river acting as a point source Glee) s constant sand discharge from a river with a finite mouth (ae Jalloce)
Amplitude of sand discharge from a river acting as a point source Ge face)
s/e
H1
(Gti Gt Gta Co ae]
a SS] flo &
Longshore sand transport rate Gases
Maximum value of bypassing sand transport rate GOs) Amplitude of longshore sand transport rate (meee)
Laplace transform variable
Time (sec)
Dimensionless time
Time when bypassing of a groin starts (sec)
Time (sec)
Time (s)
Time in the matching solution when groin bypassing starts (sec) Modified time in matching solution (sec)
Time period of an oscillation (sec)
Volume of sand released from an instantaneous source Ge) Distance between two groins (compartment length) (m)
Space coordinate along axis parallel to trend of shoreline (m) Dimensionless alongshore distance
Distance alongshore (m)
Laplace transform of a function y
Shoreline position (m)
Dimensionless shoreline position
Geometric length
Integration variable
Angle
Angle between breaking wave crests and shoreline Angle between breaking wave crests and coordinate axis Constant
Rate coefficient (a or aa)
Ratio between the amplitudes of longshore sand transport rate in two neighboring solution areas
Change in quantity Coastal constant (diffusion coefficient) GE linee)
Integration limit in the complex plane having all singularities of the integrated function to the left
Porosity of sand Integration variable
Density of water Gale)
H2
fe) Density of sand (coy)
i Wave number of periodic beach cusps (rad/m) T Integration variable b Phase angle W Angular frequency (rad/sec) Subscripts: Denoting various specific values of a variable or various solution areas Ibn By Boa akg Sp im H, v
Superscripts: Denoting various specific values of a variable or various solution areas
ee 2s Skcushe Ai, lp tle iW
Ro Ib
Mathematical symbols d Differentiation e) Partial differentiation
| | Absolute value
H3
} Saige) frime Sq VILAGE ot
i rve ening ‘ eo ibeheae ptr, roa : <-
Aj plage, ppp shope sal * “elaie aed
‘jamvhes) Womedpet?. Thkugml
Tite area ee pare hts — arf “th sotasion i
® of " ee ie: slsiad , mn ES ilinieay @ tb jealbey’ aut hare. westri antsaien” } . 7 | abe ey, "S ay via anes ows ame pa | Mis ie veo ni Vnseig the} Tab wo eT ‘LH eye edratiel to: trend in shania
n ha
okaatanyany nat ie i