Coastal changes, caused by a shallow water sanddam in front of the Delfland coast

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COASTAL CHANGES, CAUSED BY A SHALLOW WATER SANDDAM

IN FRONT OF THE DELFLAND COAST.

by

ir. W.T. Bakk€r

G. Delver.

september

1986.

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COASTAL CHANGES, CA

U

SED BY A SHALLOW WATER SANDDA

l'-f

IN FRO~T O

F

THE DELF

LAN

D

C

OAS

T.

cont

ent

s

I

n

tr

od

uc

tion.

2 Coastal changes calculated with two-llne theory

3 G

l

o

ba

l b

ehav

io

ur

o

f

the coas

t

lin

e

4 Cross-shore transport

5 Rat

e

of ch

a

n

ge

of sedimen

t

transport

6 Nu

m

erical solution

7 Critica

l

ev

a

luation of the theory and cons

e

quences

8 Con

c

l

u

s

i

ons

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9

R

e

c

o

mm

end

ati

ons for fut

u

re resear

c

h

Appendix A One-

l

lne theory

Ap

p

end

i

x

B

T

wo-lin

e

the or-y :

analytical solution

Appendix C Two-line theory; numerical solutions

Apj:'e

n

dix

_{D Computation of the distribution of a sandfi 11}

over the coastal profile

A

p;:Je

n

d

ix

E

Input in the KC-comput

e

rprogram

A

ppen

_{dix F Consideratlons on the magnitude of s}

_,

Lit

erat

ure

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Annex

Ann

~

x 2

Annex

3

Ann

o:x

4

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2

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3 4 6 10 13 16 17 18 19 20 27 49 50 59 61 65 67 68 69 71

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1•

INTROD

U

CTION

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Along tht coastline at Ter Heijde, near Delft, a dam parallel to the

coast in shallow water will be constructed havlng a horizontal crest

at the level NAP

-3

m

on the landward side, merging into the existing

coastal profile and having on the seaward side a slope 1:20. The dam

will have a length of 1.5 km (fig. 1). This work will be

carried out

as a part of a large experiment with sanddams including a larger dam

in deep water, r

ea

ching from NAP -10 m to NAP

-3

m.

The following

questions are to be answered in order to predict and judge the effects

that this work will have on the erosion and accretion process of the

shoreline:

How mu

c

h 1s trieamount of sand wi th which the beach above

the

NAP

-3

m depth line will accrete.

What will be the location in time of the amount of accretion of sand

a long the

coastline.

r---x---~

_x

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fig

u

r

~

1. Location and dimensions of the sandsuppletion into shallow

water along the coastl1ne at Ter Heijde. The numbers ··109

"

and "110" refer to the Rijkswaterstaat range system.

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3

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2. COASTAL CHANGES CALCULATED WITH TWO LINE THEORY

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In order to predict th~ behaviour of a co~st, this coast might be schematized by its -coastline-: the behaviour of this coastline can be predicted, assuming a uniform coastal profile.

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In a case as outlined in the introduction however, the change of the coës:al prof!:~ is esser.tial fcr ar. ad~quate prediction of the coast~l developments and therefore a two-line theory is used. Mathematics about onc- and two-line theory are given in appendix A and B; here

only some principles and global results wil1be treated.

On2- and two-line theory a~ply for wavedominated coasts; effects of

tides are essentially neglected.

The essentials of the on~-line theory are the following. As the

transport in posltive x-direction is larger in A than in B (fig. 2) a

co~cave coastline will accrpte. One readily sees, that this is alse the case when the waves propagate from rlght to 1eft instead of from

l",ft to rfght,

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-(J)o..vt. -

crests

----

-

..

...

-

-

---

---

-~r-.

_I""-r--

-~~~

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figure 2.

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~ith the assumptions cf th~ one -1in€ theory, tte accr~tion of the

coast is proportional to the curvature (appendix A):

'Uy s

z _

(1)

at

h

whc~t S is the r&te of chGnge of sediment transport in m3/(yearradian)

and h is the tota1 profile thickness In m.

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In equation (1) the factor s/h is a coastal parameter which is

considered to be constant in time along the coastline. The factor s is one of the parameters that needs to be calculated for this special Cáse at Ter Heijde.

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In the two-line theory the line of the beach Yl and the l1n€ of the inshore Y2 replace the coastline y (fig.3).

Eq. (1) is replaced by t vo similar equations, in wh Icn "sl/h," and

"s2/h2" replace s/h. However, an extra term "Sy(Yl-Y2)" enters the equ~tions, by which the cross-shore transport 1S introduced. Here Y2+W is the y-coordinate of the inshore, where W is the equilibrium

distancc b€tween beach ar.d inshorej Sy is another propo~tionality constant, treated in ch. 4. This term indicates, that when the profile

is stc0per th~n the equilibrium profile, the sand -drops down- and on the other hand, when the profile is flatter, sand moves upwards by the impuls~ of the waves.

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figure 3. Two-line schematization.

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3

G

LOBAL

BE

H

AVIOU

R

OF THE COAST

L

I

N

E

In the

d

ef

or

m

atio

n

p

r

o

ce

ss of th~ s

a

n

a d~m

th~ initial shap

t

y.f

(

x

)

of

the suppletion is of no importance for t

h

e

··

fina

l

··

state of it.

In

d

epend

e

nt of th

e

f

u

n

c

tion y-f(x) at t-

O

, t

h~

supp

l

etion will defor

m

to a G

AUS

S curve as ti

m

e passes bYe This curve is described as follows

(fir:. 4):

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(2 )

T

h~

facto

r

a ,

call

e

d -standarddeviation- , changes in time as yp

does.

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n progress of time,

a

will increase and yp will decrease. This

change is described as follows:

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w

h€

r~ S* is the amount of sand supply;

S*/h

is the area covered by th

~

supply. In the case of a rectangular supply over an initial length L

the initial

00

equals

L/(2/3).

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figure ~. GAUSS curve.

o(t) -

1(002+2ts/h)

yp(t) -

ypoooll(002+2ts/h)

(3 ) (~ )

wh

e

re

00

and ypo arc the initial values of a an yp

:

+00

(5 )

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In this cteformation process, the area coversè by this GAUSS curve stays the same. Generally, in the case that the orl~inal shape had been a GAUSS curve, the starting valu~ ypo can b~ calculated with:

S* /h

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(6 )

!~ th~ fi~al st~g~, th~ v21u!.of 00appea~s to be unimportant, because: S*/h Y - for t... p 1(Il1Tts/h) (7 ) o - 1(2ts/h) for t... (8)

Of course, the time af ter which the shape of the supply starts to resembIe to its final shape, given by (7) and (8), does depend on initial shape. Substituting the Ter Heijde va lues, to be explained later on, in eq. (3) one finds (L-1500 m, S*-450000 m3/yr, h-l0 m,

s-650000

m

3

/yr,

yp and 0 in m, t in years.):

the o - 1(15002112+2065000t)

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o - 563/(1+0.693t) (9 )

Af ter substitution of 0 according to (3) into (6) one finds for yp:

yp • 45000/1{2w(15002/l2+2065000t)}

yp • 4l.5/1(1+0.693t) (10)

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The values are in reasonable accordance with the ones, found in more accurate computations, to be treated later on.

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In the two-line theory differences in shape between the y-f(x) curves

will Cduse tránsportation of sand fro~ beach to inshor~ and visa versa. In case of a supply on the beach. sand will start to move tOw2rds the inshore.

Wher.only sanè is suppliec on the inshore, apart from the general

coastline change, an internal restoring of the equilibrium profile tax es pLacs, The "tLme sceLe" of this pr oc ess -ë. conception to be

elucicated later on- equals:

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('

,

)

For Ter Heijde To equals 2.3 year. when the upper level of the inshore

is cbo sen at hAP

-

3

m.

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~hs~ the mentioned Gaussian shape gives a mean shape of beach and inshore (call

y),

the shape y, and Y2 of beach and inshore itself have deviations according to:

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h, _ y,.,_.:. - ('+--e-tlTo)y (12) h2 y, • (l-e-t/To)y ₍₁₃₎ y '" (h1Y1+h2Y2)/h (14)

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Eq. (1~) and ('3) apply for the case that sl/h, equals_s2/h2 (1). From (13) it 5hows. that at a time t-O, y, equals zero and yequals

(h?/h)Y2: ~he time To gives a measure for the velocity with whlch the profile returns to the equilibrium profile.

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(I) Ir s,/h, -s2/h2' also when beach and inshore should have exactly the same shape, some redistribution of material between beach and inshore will take place; however, this is a complication which is not very essential and can be calculated numerically.

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As the offshore transport Sy equals Sy(Yl-Y2)' the initial total transport from inshore to beach can b~ calculated. Assuminb that

before the construct ion of the sanddam Yl and Y2 are zero, integration

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in x-direction givE's:

+ ... + ...

r

JSyCX -SyJY2dX

-00 _00 where: +00

r

j

Y2dx

=

S /h2* at t"'O -<Xl (15)

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(16)

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Thus the total initial onshore transport becomes:

( 17 )

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As s will be found to be 0.9 m/year at NAP -3 m, and as ha equals 3m, it shows that it may be expected, that in the first year (') 0.3

times the total amount of sand will be transported in onshore dir€ction. One might us~ the

sa

me

way of thinking in offshore

direction. Therefore, define -for a minute- the "beach" as the area abov€ NAP -6 m (reaching up to NAP +4 m, bcing a level, from where onshore transport wil1be neglegible). Then the total initial offshore transport will be (Sy/h1)S*, wh ere for Sy on the level NAP -6 will be found: 0.13 m/year; therefore (as hl -

ro

m in this case) about 1.3% of the sana will be initially transported in seaward direction. This amount will be neglected.

In appendix D the distribution of the material over the profile Is considered in more detail. For further information about one- and two-line theory thcre is refcrred to appendix A,B and C.

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(1) The transport in the beginning of the first year is 0.3 Sp

*

er year, but i t may be less at the end of the year.

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U. CROSS-SHORE TRANSPORT

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In this chapt~r the magnitudE of the const~nt Sy is related to the

wave climate. According to Rijkswaterstaat data, the waves, attacking

the Dtlfland COêst can be divided roughly in thc classes given in table 1 (Pr denotes probabili ty of occurrence).

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dcE.pwater wave waVE he I ght pcrlod H [mJ T [s

l

0.85 5.48 1.25 6.24 1.75 6.99 2.óO 7.9Ó 3.70 8.92 Pr

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20% 0.218 13% 0.336

9%

0.512 2% 0.666

1%

1.408

table 1. Wave classes considered for the cross-shore transport

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An estimate of the constant Sy as function of the depth,is deriv€d from two sources:

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1. Frû~ analysis of prototype data (Investigations on the coastal development of Sylt [lJ, Scheveningen [2J and Zeebrugge) a value for sy.at a level NAP -2 m of 1.5 m/year seems reasonable.

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2. Experiments of Swart

[3J

in the Delft Hydraulic Laboratory give a distribution of Sy over the depth. These data show, that thc depth below Still Water Level (SWL), where the maximum of Sy occurs, is found at a rather high level. This depth is called ""hm-62m"", in Swart's notation (tabIe 1). The value of Sy for waves of 0~5 m and 1~5 m, according to Swart, is a factor 10 to 100 smaller than for the higher waves in table 1. Therefore those waves have been neglected.

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1 0

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The relative magnitude of Sy for the higher wave classes at various levFls -with r~sp€ct to S~~ is given in table 2. Thes~ valu~s have been derived fro~ the Swart theory [3]. The average grain size has been assumed 200 ~m.

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H=1.7,) m H=2.6 ~ Hz3.7 m Total Tc6.99 s T-7.96 s TaB.92 s

depth rel. depth rel. depth rel. depth rel. relative _Sy

to to to to SHL Sy SWL Sy SWL Sy NAP [m]

[m]

[m] [m] setup; setup+ no tide tide -1 4B.4 -1 57.6 1 -1 2'::>.7 -2 26.6 -2 55.0 0 338.7 -2 6.6 -3 11 .9 -3 38.0 -1 86.82 143.4 -3 2.7 -4 6.3 -4 24.0 -2 61 .21 61.5 -4 1.6 -5 3.9 -5 15.0 -3 36.99 37.7 -6 2.7 -6 10.1 -4 15.44 18.8 -7 7.2 -5 7.23

8.8

-8 5.4 -6 5.44

table 2. Relative magnitudes of Sy as function of depth; af ter Swart [3].

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In table 2 on each line values of Sy are shown at depths with respect to SWL; the first two columns are shifted with respect to the other columns. In this way the assumption has been taken into account, that for waves of 1.75 m a setup of 1 m may be expected, where for waves of 2.6 and 3.7 m a se tup of 2 m is taken into account. The third column from the right shows the levels where Sy is calculated with respect to

NAP. In the second column from the rigHt the values of Sy on the same line have been added, where each of the three values per line has been given a weight, proportional to the occurrence of the wave (i.e. a factor 9, 2 and 1 respectively). In the last column the effect of the tide is taken into account; it has been assumed that 1/4 of the time the tide reaches a level of 1 m above, resp. below the mean level. In this way the va lues of the second column to the right have been

averaged in a vertical sense in proportion 1:2:1. Figure 5 shows the relative values of Sy without the effect of tide and with the effect

of tide respectively. Th~ absolute values are determined by taking the valu€ at NAP -2 m equal to 1.5 m/year. Thus, for a level NAP -3 m a value of 0.9 m/year results.

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1 1

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>

_t

_I)

"

'+-I

0 Q) -0

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~ -+-I t-C 0")

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0

E

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.-

Q)

>

-+-I 0

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Q.)

cr:::

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figure 5. Calculation of Sy 12 o;t 'l)

,...,~

E:;:;

...

+

~!

"'?<li ~Çfl

.s

(1)+

> :;:; 0 ~ ~ Cl.. Q.) Cl C\I Q.) ~ :;:; 0 c Cl. ~

...

_<li Çfl 0

0

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5.

RA

T

E OF C

H

ANGE OF SEDIMENT TRANSPORT

In this chapte

r

th

e

factor s, occurring

i

n eq.

Cl)

wil

l

be c

a

lc

u

l

a

ted

starting from the wave cli

m

a

t

e. T

h

e comp

ut

er progr

au.

KC

(I),

develop

e

d

by Rijkswaterstaat an the Coastal Engineering Group of the Department

of

Ci

vil En

gi

neering, Delft

U

nive

r

sity of T

e

c

h

no

l

ogy is used to

deter

mi

ne t

h

e sediment trans

p

ort and t

h

e rate of c

h

ange of sed

iment

t

ransp

ort a

l

o

n

g t

h

e be

ach

.

The

wav

e

c

l

i

m

ate

h

as be

e

n determined by

splitting the wave characteristics into classes, each characterized by

c

e

rt

ai

n v

al

u

e

s of (

H

,T,~) (where ~ is t

he

a

ngle

of wave inc

i

denc

e)

and

a certain pro

b

ability of occurrenc

p.

; each value of H,T,~ includes a

c

e

rt

a

in "bandwith

"

H-t.Hto H+t.H •••

etc. The wave heights are div

i

à

e

d

In

t

o cl

a

ss

e

s of 0.0-

0

.5

m

, 0.5-1.0 m ••.

etc; the angle

Q

in sectors o

f

-9

0

° ••

_6

0

°,_6

0

° ••_3

0

°, .•.••

,6

0

°••

90°; for Taperiod

of 6 s has b

ee

n

assu

m

ed. The probability of occurrence of each of t

h

e wave classes

(from

Ri

jk

swa

t

erstaat da

t

a) is given i

n

appendix D.

... ,',:, .. ... ",

figure 6.

(I)

KC stands for -kustconstanten- ( - Coastal Constants).

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One of t

h

e facilities of the KC-program -the one used here- is to

c

a

lculate the transport along a

-

bea

ch

-, having parallel d~pth

co

n

t

o

u

rs , after the waves are refracted over an

.

.

Inshor-e':,

hav ing

para

LleI

depth contours as weIl; no

vev

e

r-

t

h

e contours of t

h

e b

ea

c

h

ca

n

m

ake an angle with the ones of the inshore (fig.6).Shoaling waves are

ass

um

ec, w

h

ich _dependen

t

on their heig

h

t- break either on the bea

ch

,

eit

h

er on beach and inshore. In t

h

e last case only the sand transport,

c

au

s

0

d by th~ w

av

e

br

~

ak

i

n

g on the beach is taken into account. With

res

p

ec

t

to t

h

e sand transport, t

h

e assu

m

ption of SVASEK [4J has bee

n

a

ssumۏ:

s

3

nd

tr

an

s

p

o

r

t is p

r

oportional to t

he

longs

h

ore compo

nen

t o

f

the energy flux (CE

R

C) and the sandtransport between two depth

con

t

o

u

rs is p

r

o

p

or

t

ion

al

to t

he

loss o

f the l

c

ng

s

h

ore component of t

he

energy flux between t

h

ose two depth contours

(r.e..

a wave-induced

tr

a

ns

p

or

t

-for

m

ula

)

.

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T

h

e K

C

-

p

rogra

m

also

p

rov

id

es th

e

possi

b

ili

t

y

t

o calculat

e

th

e

transport along the e

dg

e of a shoal (Le. a .

.

submerged

beach'"),

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T

he

fo

l

lo

w

ing calculations h

a

ve be

e

n carried out:

1.

Transport

"

aIong the total profile

"

between NAP

-6.5

mand

NAP +3.

0

m.

(Contour lines in t

h

is area p

a

r

a

ll

e

l to the ones in the r

e

gion

(

l)

de eper t

h

an

N

AP -6.5 m.)

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2.

Transport -along t

h

e beach- between NAP -3.0 mand

NAP +3.0 m.

3.

Transport "along the inshore" between NAP -6.0 mand

NAP -3.0 m.

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For all t

h

os

e

three cas

e

s the follc

w

ing botto

~

topographies where

assumed: respectively:

a.

Contour lines in the

"

t.r-

a

nspor-t

area

"

(Le,

NAP

-6.5

to

+3

m

etc.) parallel to the contour l1nes in the deeper region.

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b.

Contour lines in the

-

transport area

-

make an angle of +5° with

the contour lines in the deeper reg ion.

c.

Contour lines in the

"

transport area

-

make an angle of -5° wi th

the contour lines in the deeper region.

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(l)

T

he

"angLe

of wave incidence" ~ is defined

at a depth of NAP -15 m.

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The calculations b. and c. yield the constants sl and s2' showing how

the transport changes with the direction of the beach, or inshore

changes. The calculation a. yields the resultant transport, when all

contour lints are parallel. Curiously enough, according to the

computation, the coastal development does not depend on this resultant

transport. Only transport gradients are of importance for erosion end

or sedimentation. Table 3 shows the results.

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c

o~putà

tion

description

resultant transport variabIe magnitude

(.-southward) m

3

/yr

[m

3

/yr/rad

]

total profile

-~1~1

s

64848

2

2

beach

-213682

_sl

3591753

3

inshore

-186886

_s2

314002

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tablc 3. Values of

sl and

s2 and the resultant transport.

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It is remarkable, that sl is much larger than s. The reason is

explained in appendix F. Following the considerations of appendix F,

the results of computation 2 will be ignored, and the following values

of sl'

52

and s will be assumed:

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s - 650000 m

3

/year

s2 - 300000 m

3

/year

sl - s-s2 - 350000 m

3

/year

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15

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6.

NUMERICAL SOLUTION

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Appendix C gives the explicit numerical procedure according to which

the coastal development is computed. The Jower- layer C' Inshor e") is defined between NAP

-3

m anà NAP -6 m (offshore losses are negl~ctea). The upper layer

C

'

oeach") is assumed between NAP

-3

mand NAP

6

m , as thF efft'ctof duns accretion is neglected, given thE relative1y short

p€riod of time. lt is assumed, that in the initial situation the

inshore protrudes 100 m in seaward directien over a stretch of 1500 m, thus making the content of the sanddam 450000 mI. As boundary

condition, it is assumed, that the beach and inshore keep their position at a distance of 4 km fro~ the centre of the sanddam. The width of the numerical grid is taken 100 m; the time step 1/40 year (a time step of 1/20 year gives a numerical instábility). The numeri cal

"sancJos s" is less than 0.016% for the î irst 5 years and less than

0.5~ for the lOth year of the computation.

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The calculated lines of the beach and the inshore are shown in a sche~atical way in annex 1, in order to give clear view of the development. Numerical values are given In annex 2. In annex

3

the results are plot in the shore plan at Ter Heijde.

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1

7.

CRITICAL EVALUATION OF THE USE OF THE THEORY AND CO

N

SEQUE

N

CES

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Ir.the use of the theory and the use of

a

ctual site infor

m

ation

several matters have been simplified.

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he

o

cc

urr

e

nc

€

of groynes is not included

in

th

e

calculations. A

gro

y

n

e

causes

turbulent current around the head of the groyne and is

a

r.o

bs

tä

c

le for

t

he transpo

r

t of sedi

m

ent along t

h

e beach. The

turbulent current around the head of the groyne will cause a process

of e

r

os

i

on at this point. T

h

is erosion will affect the process as

described in previous chapters.

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In t

h

e calculation of transport along th

e

beach profile the t

i

dal

influence is not included. Investigations of Bakker [5J and Opdam

[5J

show, that in the brea

ke

rzone the effect of radiation stress overr

ule

s

the tidal force already for a small angle of wave incidence.

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h

e attention is drawn to over-sche

m

atizations, fcllowing to the

theory, as reported in appendix F

:

effects of directional changes of

the inshore on th

e

transport along the beach are not taken into

account.

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The theory does not explain the behaviour of offshore bars in the

surfzone. It was found by Redeker

[7J and de Vroeg [8J, that those

bars moved near Katwi

j

k as periodical waves in offshore direct

i

on with

a propégation velocity of 60 m/yr. In fact, one might argue that a

s

3

n

ddam

as constructed will act as an offshore bar and that the theory

a3

given in the present report applies more to Iarger scale processes.

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In appendix 0 the two-line theory is elaborated somewhat furt

h

er, in

order to investigate its consequences with respect to the distribution

of the sup~lied sediment over the profile. Annex 4 sho

w

s a result. A

uniform distribution (equal seaward shift of all depth contours) of

the supplied sand over the depth is indicated by the zero-line in this

figure. It shows the transition of the sandfill-material, being

concentrat

e

d initially between NAP

-3

rnand NAP -6 m, to its Ii

m

itin

g

state of uniform distribution. This figure shows. that initial onshore

sedi

m

entation is not in contradiction with a visually seaw

a

rd

propagation of a decaying bar.

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8.

SUH

M

ARY AND CONCLUSIONS

T

hc

computations ptrformed with the actual site information leads to

th

e

following conclusions:

1.

The

co

n

str

uc

tion of t

he

s

ha

llow-~

a

tèr sandda

m

as shown in fig.

1

will

,

according to the computations, lead to

an initial

transport of sand

i

n two directions:

1.3

%

of the content of the sanddam in the offshore direct ion.

3

0

%

of t

h

e con

te

nt of t

h

e sand

dam

in onshore direction.

These figures indicate amounts of transport per year and are

valid in the beg

inning

of t

h

e f

i

rst ye

a

r. Because of this,

initially the contour lines along the beach (including the zone

above NAP -

3 m)

will move offshore with a estim

a

ted speed of 9

meter per year, at the centre

of the sanddam (at ··x-O·

·

).

2.

In th

i

s report the coastal behaviour is simulated by schematizing

the coast by a -line of the beach- (representing the coastal

profUe

above

N

AP

-3 m) and a

-

line

of the Inshore

'

schematizing

the lower part, including the sanddam (NAP

-3

m to NAP

-6

m).

Figur

e

3

gives a three dimensional pr~sentation of this

schematizatlon. Annex 1 glves the expected development of those

lines in the course of 10 years in a scherr.aticalway. In annex

3

those lines are plotted in the shore plan of Ter Heijde.

3.

According to the co

m

put.a

t

Icns, the depth con tours of the beach

will move in seaward direction up to a maximum

amount of 16 meter

totally in 1989 anè afterwards the beach

will

retreat again

ö

in

19

9

6 a resultant shift of 14 meter is left.

4.

Annex 4 shows a calculated relative

(1)

distribution of the sand

over the profile in the course of time. A description of the way

of presentation of this figur€ is given chapter 7: the way of

calculation and the assumptions in appendix D. Onshore

sedimentation goes together with a visually seaward motion of the

top of the decaying shallow-water sanddam.

(1)

N.B. no significance may be giv

e

n to the absolute valu~s indi

c

ated

along the y-axis.

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9.

RECO~

~1

ENDATIONSFOR FUTURE RESEARCH.

1.

The present report deals -in correspondance with the

commission-about the shallow water sanddam in front of the Delfland coast.

The

approach according to the one- and two-line theory can be

vEry weIl applied in order to investigate the implicatlons of the

deep water dam on the coast wlth respect to the effects of

r

e

fr

act

ion around the rounded edges of the dam. Furthermore, saná

might accumulate behind the deep waterdam because the wave height

in thE leeside of the dam can be somewhat lower than outside the

area of effect of the dam. This accumulation even can be caused

by the changing of the

"

direction of the waves only. This sand is

derived from other places along the coast. Before the dams are

constructed, the ord

e

r of magnitude of this accretion and erosion

should be known. The Delft University of Technology could be

charged with this commission.

The next points are confined to research with respect to the shallow

water dam.

2.

In the present study, the effect of the changing of direction of

the

inshore on the transport along the beach has only been taken

into account in an indirect way Cby the choice of the coastal

constants). This is rather unsatisfying and caused by lack of

time. A more accurate investigation, based on the work of Bakker

[:2] and Jas [13] is recommended. This study has the same

th

eor

etical scope as the investigation, mentioned ad 1.

Furthermore, more accurate dynamic equations throw light on the

magnitude of the inaccuracy of the mathematical results. For

instanee, the symmetry of the coastal shape and the fact that the

to~ of the Gauss curve keeps its position in the course of time,

independant of the wave direction are correct in the present

linear approach, in which the accretion only depends on the

curvature of the coast. When higher order terms in the dynamical

equations are taken into account, symmetry and fixed position of

the top are not any more conditio

sine qua non.

3.

The effect of tidal current should be taken into account. One

might think on some calculations, using the Bijker formula.

4.

It would be useful to make cross- connections with the

investigations of the Delft Hydraulics Laboratory concerning the

same subject. The DHL uses Crosstran for calculation of the

cross-shore transport. It is worthwile to investigate the

relation between the Baillard concept, occurring in Crosstran and

the constant

Sy

for offshore transport occurring in the present

study. Also other physical approaches migt be considered. It

might be pointed out, that also the DHL predicts a fast decay of

the shallow water dam.

5.

It is wortwhile to lnvestigate the sensitivity

ot

the

calculations. One might think of treating the sensibl1ity for Sy

and simulation of the groynes by taking 51 equal to zero.

6.

Further research, concernlng the wave climate, has yielded more

accurate wave data. Recalculations with this new data is recommended.

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APPENDIX A ONE-LINE THEORY

From:

Manual on Artificial Beach Nourishrnent[l] (draft report)

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21

-A.!::'EX VI COASTAL HORPHOLOGY THEORIES

1. One-line and two-line theory

1.1

One-line theory

1.1.1

Introduction

Based on a longshore transport and a coastline configuration. the theory of

Pelnard-Consid~re 1956 gives the basic equations describing the morphological

processes of coastline evolution due to longshore sand transport. These

equations lead to the well-known diffusion equation.

The fundamental equation may be solved point by point (both in space and time)

by means of numerical methods if an accurate description of the wave climate

and of its influence on the longshore transport is known.

A computation program has to include two parts:

a. computation of the longshore transport along a straight coastline and of

the variations in this transport as a function of the coastline direction;

b. computation of the morphological evolution of the coastline by means of the

basic equations.

1.1.2

Basic equations

For the one-line theory the coastal profile is schematized according to Figure l.I.

The equat10n of cQntinuity is derived from Figure

1.2.

The x-axis is chosen

along the original coast11ne and the y-ax1s in a direct ion perpendicular to

the original coastline in offshore direction.

y

cl.

x

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22

-When considering an infinitesimal element with length dx, the equation of continuity y1elds:

a

SI (Sl+~. dx).dt + p.h.dx.dt - Sl·dt - -

tf .

h.dx.dt or: 1

as,

a

__ +.Ê.l+p_O h •

ax

at

(l.1) where:

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SI - longshore sand transport

h - height of the schematized beach profile

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p - transversal sand transport t - time

The equation of mot ion may he derived from Figure

1.3:

y

..

.,

.ot

~/_c •••, _,./'/

\

/

1

I

I

---+

s.,

~\---~---+

•.

eIWP ...

Figure

1.3

Longshore sediment transport

The following condltions are posed:

1. The longshore transport is a function of the coastline direct ion only and of no other parameter~.

2. This function can he differentiated for any of lts values.

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- 23

-The first condition leads to:

0.2)

The theorem of Taylor-Mac Laurin, together with the second condition, ylelds:

dS

l

d

2S1

Sl(a) - Sol

+ a(~)

a-O

+ ;a

2

(daZ)

_Cl-O

+ •••••

The thlrd condltion then gives:

(I.3)

When deflnlng:

Cl Ir

tg a

_.ll

and

ax

Equation

(1.3)

ylelds:

Sl(a) • S

_ol

- sr!l.

_ax

where:

0.4)

Sl(a)

- longshore transport as a function of the coastline direction

Sol

_ longshore transport along a straight coastline parallel to the ~axis

SI •

variation of the longshore transport as a function of the coastline

direction

Cl -

coastline direction with respect to the ~axis

in radians

Equation (1.4) is the equation of Pelnard- Con8idlre in which the coastal

constants Sol and SI may

be

functions of

x.

When the equation of motion (1.4)

is substituted in the equation of continuity

(1.1)

the following equation is

obtained (with constant Sol and 81'

a

nd p - 0):

!I _~

a

2

y

at

h

W

(1.5)

This is the equation of Pelnard-Con8idlre, well-known

a

8 the equation of

diffusion and for which analytical 80lutions can

be

found.

The advantage of the numerical approach i8 that 80lution8 of the complete

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24

-equation yieIds:

.!l _

1

1'::1

.!l

+ 6 a

2y _

aSol

_ p

at

h

1

a

x • Cl

x

1

ä"i'T

ax

}

(1.6)

1.1.3

Numerical solutions

A numerical solution of Equation (1.6) can be found by .tarting again with the

Equations (1.1) and (1.4) and writing them a6 finite differenees:

àt

{

Y(n,t+l) - Y(n,t) - àt.p(n) - 2.h.àx

S(n+1,t) - S(n-1,t)}

(1.7)

S(n,t) _ S (n) _ sen) fY(n+I,t) - Y(n-l,t)}

o

2.àx

(1.8)

Af

ter substitution of (1.8) in (1.7), the complete equation may be written as

follows:

Y(n,t+I) - Al(n) + A2(n).Y(n-2,t) + A3(n).Y(n,t) + A4(n).Y(n+2.t)

(1.9)

with the eoeffieients:

A1(n) - -

àt

{S

(n+1) -

S

(n-l)} - p(n).àt

2.h.àx

0

0

àt

A2(n) - 4.h.(àx)2 • s(n-1)

àt

A3(n) - 1 - 4.h.(àx)2 {s(n-1) + 8(n+1)}

At

A4(n) - 4.h.(àx)2 • 8(n+l)

It may be observed from Equatlon (1.9) that the computatloD of the coastllne

position at point n and at timestep (t+l) i8 performed by atarting from the

coastline position at the points (n-2), n and (n+2) end at the former timestep t.

1.1.4

Boundary conditions .

The starting condition Y(n,O), is given by the coastline position at time t-O.

Any p08ition may be introdueed at eaeh point n of the x-axi•• The boundary

condition at n-O, Y(O,t) must be aiven as • funetion of time. A time-p08ition

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25

-function may he given at n-O, or a longshore transport may he given (constant

in time). Both possibilltles reduce to a glven coastllne direction at point

n-O.

The boundary condition at the other end of the computed coastline may he given

in the same way. The boundary conditions may thus he summarized as in Figure

1.4. R - , R ---~... x • cUli. YCl,Ol Y (R,OI 50 Ut) 50 (Ol Y - GIlIt.

Figure 1.4 Schematizatlon of boundary conditions

As the longshore transports Sol (0) and Sol (R) are time constants, the

angles S and y of the coastline at the boundaries are time-constants as weIl.

It has to he noted that the boundary longshore transports could easily he

introduced as time functions, if this function is known.

1.1.5 Computation of the coastal constants

!he essential role played by the coastal constants Sol and sI could he

noticed in the mechanisms described above. Hence, an accurate computation of

those values is important and it can he noted that they have to he constant in

time but not basically constant in space (along the x-axis).

The longshore transport Sol-along a straight coastline can be computed as a

function of the wave climate by adding up the transports due to the waves from

various directions, taking into account the frequency distribution. The

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-equation yields:

(1.6)

1.1.3

Numerical solutions

A numer1cal solution of Equat10n (1.6) can

be

found by .tarting again with the

Equat10ns (1.1) and (1.4) and wrltlng them as flnlte differences:

fit

{

Y(n,t+1) - Y(n,t) - 6t.p(n) - 2.h.6x

S(n+1,t) - S(n-1,t)}

(1.7)

S(n,t) _ 50(n) _ sen) {Y(n+1,t) - Y(n-1,t)}

2.flx

(1.8)

Af

ter substitutlon of (1.8) in (1.7), the complete equation may

he

written as

follows:

Y(n,t+l) - Al(n) + A2(n).Y(n-2,t) + A3(n).Y(n,t) + A4(n).Y(n+2.t)

(1.9)

with the coefflclents:

A1(n) - -

fit

{S

(n+1) -

S

(n-1)} - p(n).6t

2.h.6x

0

0

6t

A2(n) - 4.h.(6X)Z • s(n-1)

6t

A3(n) - 1 - 4.h.(6x)2 {s(n-1) + .(n+1)}

fit

A4(n) - 4.h.(6x)2 •• (n+l)

It may

he

observed from Equation (1.9) that the computation of the coastline

position at point n and at timestep (t+l) is performed by .tarting from the

coastline posltion at the points (n-2), n and (n+2) end at the former timestep t.

11

.

1.1.4

Boundary conditions

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The starting condltion Y(n,O), is glven by the coastline po.ition at time t-O.

Any position may

be

introduced at .ach point n of the x-axi•• The boundary

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26 -Ih()~O I.no /4% '//')j//'/ "//~/,(%/é/,~/(P/(//((/(/((/(&/(///ff4 50,2 ~ I • Ö,

Fig

1.5

Sediment transport due to different wave directions

The longshore transport may

be

computed for each wave direction by means of

the ~el1-known CERC formula. lt is, however, advisable to check the resulting

transport Sol oy means of other methods, .uch as volumes of erosion or

accietion buseà on measurements.

It may be nQted finally that the longshore transport is a function of the wave

he1ght and d1rection in the breaker zone. These values can be deduced from

deep water data by means of refraction and diffraction computations.

The second coastal constant • can be computed by applying a rotation 6a of the

coastline and calculating the difference in resulting longshore transport 6S

caused by this rotation.The coastal constant s then is equal to:

(1.10) 60

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APPENDIX B

27

-TWO-LINE THEORY; ANALYTICAL SOLUTION

From:

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- 28

-1.2

Two-line theory

The limitations of the one-line theory are numerous.

Of

ten it results in too

strong a schematization of reality. This can apply to initial and boundary

conditions, as weIl as the physical wave and beach characteristics. In

particular Bakker (1968) was concerned about a coast upon wh1ch the longshore

sand transport is only partially blocked by groynes which were shorter than

the width of the breaker zone. Bakker proposed a so-called two-11ne theory for

the solution of this problem. lnstead of schematizing a coastline with a

single curve, two curves are used as expla1ned in Figure 1.6.

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29 -S.,

-Fig

1.6

Profile schematization

The choice of reference axis for the y distances is arbitrary. The

areas of each of the pairs of shaded areas are equal. The longshore transport 5i ; is directed parallel to the coast out of the plane of the paper, and describes the littoral sand movement in its zone. The horizontal planes are usually selected at elevations, which correspond more or less to flat portions of the total profile. If special structures, such as groynes, are involved in the schematization, the horizontal planes are of ten chosen 10 that the limits of the boundary condltions correspond with the llmlts of a transport zone.

Just as with the one-llne theory 1t will be necessary to develop a contlnuity relationshlp and equations of mot1on for each of the Ichematization zones. The equation of continulty 1s more complex than that for the one-llne theory,

since there is supply and removal of land from two directions. Figure

1.7

shows an element in plan; the net transport of material into the element is equal to the volume retained for each of the 2 elements - one for each line of the schematizatlon. Thus, for i-I and 2 the following relation is obtained:

(1.11)

Sand is only transported transversely between the first and aecond zone so that:

(1.12) S - 0 and S - 0

yo y2

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-,S

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I

-~4

I

d

'

I-Si

-es,

S

i

-

;

l

~

~

_{I Y,.,}

I

_I K K+d. K

Fig. 1.7

Plan view

The above Equations (1.11) and (1.12) can be WTitten as:

a51

5

ay 1

0

-+

_{y1+ hl}

at-ax

a52

5y1

aY2

0

äX-

+ h

2

at-(l.13)

0.14)

The cross-shore transport Sy1 is defined by Bakker (1968) as:

0.15)

where

s

- cross-shore transport constant

y

W

- equilibrium distance between beach and inshore.

Similar to the one-line theory the equation of motion for each of the

schematized zones is:

SI

- S

-3Y1

Ol

sI

ax

S2

- S

-aY2

02

s2

ax

(1.16)

0.17)

The above set of differential equations (1.13) to (1.17) can

be

80lved

numerically or analytically; the latter only for simplified boundary

conditions as described in Chapter 2 of this Annex.

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31

-2. Analytical solution of the diffusion equation

2.1 General

Two main classes of solutions of the Equation (1.5) are related to the following type of problems:

initial value problem boundary value problem.

In case of an initial value problem the shape of the shoreline f(x) at time t-O is given. The question is how the shoreline changes with time. The solution of this problem is given by Smirnow (1964).

y(x,t) - f(t) • _~1 __ e /W/4at _ q;-x)2 4at (2.1)

dt

-x - longshore coordinate·

y _ actual position of the shoreline at point x and at time t f _ initial position of the shoreline at point x and at time

0

a - diffusion coefficient - s/h t - time

t

-

integration parameter running parallel to the x-axis s - coastal constant

h - profile thickness

This solution can be uaed for the study of the evolution of a sand supply.

An

example of a boundary value problem is the shoreline evolution due to sediment yield of a river. The .olutlon of thi. problem i8 also glven by Smirnow (1964). t y(x,t) -

f

t(t) o (2.2) 4a(t-t) _dt

2+(t) _ sediment yield of the river as function of time

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32

-lt is interesting to note the close resemblance of the two solutions. In both solutions it can be seen that the effect of the disturbance, +(t) or f(t) t is

instantaneously transferred along the complete x-axis. The effect of the disturbance is damped with the negative exponential function. This effect is essentially different from solutions of the wave equation, where the effect of a disturbance propagates with a prescribed celerity c. Points located at a distance larger than c.t from the disturbance are in case of the wave equation not influenced by the disturbance.

For the solution of (2.1) the shape of f(t) is not prescribed. Suppose that f(t) is constant between (x-+6x) and (x+16x), (6«1), and zero outside this range, then Equation (2.1) represents the effect of a spike. Bowever, in principle any f(t) can be regarded as the sum of an infinite number of these spikes. The total effect of all these spikes is according to (2.l) simply the superposition of the effects of the individual spikes. This auperposition follows from the linearity of the diffusion equation. The aame .uperposition principle ean also be applied in case of a eombined problem of an initial value problem and a boundary problem, e.g. the supply on a river delta.

A fioal remark concerns the factor

l1+ät •

It can be aeen from both aolutions, that the value of this factor determines the actual ahape of the ahorelines as a function of x. It will be shown, that future ahoreline developments due to human interference, can also be related to thi8 factor. In the next paragraph a few methods will be pre8ented, by which the value of

l4ät

can be determined.

In the following ehapters thi8 knowledge will

be

used to analyse the future shoreline evolution as a function of

l48t •

2.2 One-line theory

2.2.1 General

For a designer of a replenishment 1t i8 important to have an answer to

questions, 8uch as how long"a given replen18hment will last, and wbich part of the beach in longshore dlrection will be protected by the repleD1lhment. In this Section these questions will be dealt with for the one-11ne theory.

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33

-2.2.2 Deformation of a replenishment as a function of time

The deformation of a replenishment by waves is also governed by the diffusion equation. The solution for a general shape f(t) of the initial replenishment is given by (2.1). In the literature solutions for at least three given shapes are known. These shapes are the Dirac-function, the rectangle and the

triangle.

Pelnard-Considêre (1956) has applied the solution for the Dirac-function to coasts:

y - _--:l.Q_ e

l4ffät

(2.3)

where

Q -

2 b Y the area of the replenishment. o

~e value of the diffusion coefficient a can be obtained by considering the nourishment on two times.Suppose a Gauss-shaped nouri8hment i8 placed

at t - T, where t is the time-axis of the Dirac-function.A new time exis is

*

started at the moment of dumping. The total fill volume i8 equal to S .Further the diffusion coefficient a in

(2.1)

i8 equal to a/h.The 80lution of the

Dirac-function can be written now as:

-x2 ]

exp [ 4 a/h (t+TI)

(2.4)

The unknown constanta e (coastal conetant) end Tl (initial period between t-O of the Dirac-function and time of placing) can he 80lved. if the deformation of the nourishment is known on two times, for example at t •

0

and t- tl, preferably in the centre where x • 0:

*

y(O.O) - S 14'1rhsTI

*

y(O,tl) - S " 4 'Ir h s (tl+ Tl)

With the resulting value of a, of course 4 at.

(2.5)

(2.6)

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34

-lf the rep1enishment is p1aeed in a rectangu1ar shape, sueh that it extends

from x - -b to x - +b and up to a distance of y

from the original shore1ine

o '

the solution can

be

written as

y - Yo/2 [erf(R)(l-P) + erf(R)(1+P»)/2

(2.7)

where

2

JU

-u2

erf (u) -

-=

e

du

{TI 0 (2.8)

u - - x/{4at

R - b!l4at

(2.9)

P - x/b

(2.10)

For a triangu1ar shaped replenishment extending from x - -b to x -

+b

and with

its maximum y - 2y

at x - 0, the shape of replenishment as a function of time

o

is given by:

y _

y

{(I-P) erf (R(I-P»

+ (I+P) erf (R(I+P»

- 2 P erf (PR) +

o

-R2(I+P)2

_R2(I_P)2

-(PR)2

~

(e

+ e

- 2 e

) / ywR

1

(2.ll)

The solutions for the rectangular and triangular Bhaped repleniBhment in the

present study are in principle obtained from the paper by Walton et al.

(1979). However the solutions for the reetangular and triangular

rep1enishment, and a1so for the Dirac-function, differ from the Bolutions

given by Walton in that respect, that the area of all three types of

rep1enishments is equal to 2by •

o

The effect of the initial shape of the repleni8hment on the longterm behaviour

of the shoreline ean

be

derived from Figure 2.1. In tbis figure the Bhape of

the shore11ne is given for (2.3), (2.7), and (2.11) for the following values

of b/;-48t - 5, 1 and 0.5. From this figure it follows that the initial 8hape

of the replen1shment is hardly noticeable in the longterm Bhoreline evolution.

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35 -24 pqlnard C onSldèlrcz

crec :

2

e

Yo 20

.-.

, , '6

I

6

~ _b_:5 2Yo V4at' ~ 0-0 -b .b '.2 YIYo ,

I

~ Yo _0-0 -b .b

1

08

,

0

•

0

04

,

0

•

-,

'\.~

1 2 3

°0~---~1---~~==~~~~~---~3

0

.

8

b :0.5 V4at'

I

3:o:

.!

::;

Q-~

1 -.c

...

.

.

,..

.,." 1 2 3

Fig. 2.1

-- ...~ xlb

The effect of the shape of the initial eand eupply on the evolution

of the supply in time

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36

-This conclusion justifies the use of approximations such as the Dirac-function for the longterm shoreline evolution.

2.2.3 Evolution of specific points of the replenishment

In Section 2.2.2 the general evolution of the shoreline has been presented at three specific stages. In this paragraph the evolution in time and space of the following points of the replenishment will be studied:

the maximum width of the replenishment at x • 0

the longshore position of the point. that bas acccreted over a distance of 0.1 of the original width of the replenishment

the longshore position of the point, that bas accreted over a distance of 0.1 of the present maximum width of the replenishment.

The evolution of the maximum width of the replenishment as a function of

l4it

is shown in Figure 2.2. Also in that case it can be noticed, that the effect of the initial shape of the shoreline on the evolution reduces with ime. It is interesting to note the half decay time. If it takes ti years to have a reduction of the maximum of the replenishment to y/y • 0.8, then it

o

takes 4tl to reach y/y • 0.4 and 16tl to reach y/y - 0.2. This means that

o 0

as far as time is concerned y is related to t as

(2.12)

The calculation of the longshore position of those points of the shoreline where the accretion is equal to 0.1 of the width of the beach fill at x • 0, that is y(x,t) - 0.1 y(O,t) and y(x,t) • 0.1 y(O,O), requires some algebra and elaborations. This will Dot

he

presented here. !he results are given in Figure 2.2. In line with the previous results it follows from this figure that the position in time and Ipace of the point that accretes over a width 0.1 y(O,t) is af ter the initial phase almost independent of the original

shape of the replenishment.·!he position of the point y(x.t) - O.ly(O,t) can simply be derived from the solution for the Dirac-function.

x

,~

-.

b b

t

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37 - t2r---.---~---~---.----..---0.6 y

Y

o

1

0.6 0--0-<:1 04 6 6 6 02 0--0--0 0.8

I

I ~

•

_J

._

_D

Yo

I

-I I ~I

.~

-. Zu:

..

~

,P"-..

•

ti ~

"

-.,

~ y y ' ....

Tt

:0.' I _~:0.1 :a... J'O I

!...

Ill.

,-tr----

~

1

li '4_~...

_--.

-

--

._.----I :

---~

I

,

l

!

.

1.6 1.0 )(

-

b

o

.

~

0.2

o

o

0.2 0.4 0.6 0.8 ,.0 U:~

V4at'

'

.

2

1.~

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38

-Equation 2.

"

13 shows the dependence of x on tt. In Figure 2.2 also the position

in time and space is shown of the point with an accretion of y(x,t) - 0.1

y(O,O). For the solution of the Dirac-funetion y(O,O) is infinlte, and hence y

(x,t) - 0.1 y(O,O) for that funetion is located at the origine It is

interesting to note that for the reetangular and the triangular replenlshment

the position of points, where y(x,t) - 0.1 y(O,O) do not converge for small

values of

bIl

4at.

This is caused by the fact that the constant y(O,O) is for

a triangle twice as large as for a rectangie. Searching along the shoreline

for points with an aceretion of 0.1 y(O,O) will therefore yield different

values for a reetangle and for a triangle. It can also be noted in Figure 2.2,

that with the inerease in time the replenishment amoothes to auch an extent

that no points can be found with an accretlon of 0.1 y(O,O).

The results in the Figures 2.1 and 2.2 can

be

generally used for the design of

beaeh replenishments, because they are presented in dimen8ionless terms and

they are not related to a specifie coast, wave climate or type of .. terial.

2.3

Two-!lne theory

In this section formulae are developed for the case that the height hl over

which the sand is originally dumped (beach nourishment depth) 18 .maller than

the transporting depth hl

+

h2 ' and therefore the aand will apread in the

course of time more or less uniformly over the profile (aee Fig. 1.6).

Starting point is the two-llne theory (see Chapter

1

of this Annex).

The equation for the transport SI

along the beach 18 according to Equation

(1.16):

(2.14)

where 501 ls the "stationary transport", i.e. the transport vhen the coastal

direct ion ls parallel to the x- exis and 8 ls a proportionality constant. The

y- coordinate of the line of the beach is Yl.

Using similar definitions for 502 and 82 ' the tran8port S2 along the inshore

is according to Equation (1.17):

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39

-In this section, eontrary to seetion 1.2. it is assumed that (Y2

+

W) is the

y-eoordinate of the inshore.

W

is the equilibrium distanee between beaeh and

inshore.

For the offshore transport S it is assuoed:

y

(2.16)

The present case will be simplified as:

(2.17)

I

where:

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(2.18)

Continuity equations for beaeh and inshore are:

aYl aSI

hl

F+ax-+

Sy (Yl -

Y2)-

0

aY2 aS2

h2

F

+ ax- +

sy (Y2- Yl) - 0

(2.19a)

(2.19b)

Combined with the equations of motion (2.14) and (2.15) one finds:

(2.20)

Let

Y

be the weighted value of y aecording to:

y -

(2.21)

11

Then (2.20) can be written as:

I

11

11

ay

s

---

at

h 2-~

ax

(2.22)

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40

-Subtraetlon of (2.19b) divided by Y2

uslng (2.14), (2.15) and (2.17):

from (2.19a) divided by h, results into,

(2.23)

in whieh:

(2.24)

The two-line problem now bas been redueed to a one-line problem aeeording to

Equation (2.22), where (2.23) gives an offshore transport equation. Equation

(2.23) is of the same kind as (2.22), whieh follows from the .ubstitution:

t/T

Y

-

v • e 0

or

e

.*

(2.25)

wlth:

(2.26)

11

!hen Equation (2.23) obtains the same .hape as Equation (2.22).

Af

ter .olving

y

and y. one finds Yl and Y2 from:

11

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_ h2 Y1 - Y

+

h

Y.

hl

Y2 - Y -

h

Y.

(2.27)

(2.28)

Suppose the original shape of the beaeh nourishment is given by:

(2.29)

For Y

o

one may think as well about a Gauss-curve as about a rectangular beaeh

nourishment. Bath examples will be elaborated later on.