Littoral drift computations on mutual wave and current influence

Communications on Hydraulics Delft University of Technology Department of Civil Engineering

COMMUNICATIONS ON HYDRAULICS

report no. 71-2

LITTORAL DRIFT COMPUTATIONS ON MUTUAL WAVE AND CURRENT INFLUENCE

by

Eco W. Bijker

Delft University of Technology

Department of Civil Engineering

1

Littoral Drift Computations on mutual wave and current influence

by

Eeo W. Bijker, professor on Coastal Engineering Delft University of Technology.

1) Synopsis

In "littoral drift as function of waves and current", presented at the 11th Conference on Coastal Engineering in London 1968

[2] ,

the author presented a method for computing the littoral drift starting from the longshore current velocity as this is generated by the waves and with the assumption that the material is stirred up by the waves.

In this paper measurements in a model basin are described by which the procedure is further tested.

A comparison with littoral drift rates computed by this procedure and derived from accretion and erosien rates along the coast of Queens-land (Australia) are presented.

For a hypothetical coast the littoral transports with varying bed roughness are computed. The influence of this roughness on the assump-tions forming the basis of the computation procedure is discussed.

2) Short outline of procedure

(1) The general accepted bed load formulae can be written in the

following form

1

Sb

=

5 D (Il't"

I

P

)"2

exp (-0.2711 DPg/Il't" )

Where Sb

=

bed load, D

=

graindiameter,11

=

relative apparent density,

p

=

flU1d density, g

=

accelerations of earth gravity, Il

=

ripple coefficient, indicating that part of the total bed shear that is available for transporting material and 't"

=

bed shear.

For the conditions of waves and currents, the bed shear resulting from the combined motion of waves and currents can be written as:

1'"

=

[1

+

(~u

Iv)2 I

2] •

't" ,

r 0 c

Where 't"

=

resultant bed shear, 1'"

=

bed shear due to the mere current, uc

=

am~litude of orbital velocit~ at the bed, v

=

mean current velocity and ~

=

coefficient.

The coefficient ~ has been determined theoretically and empirically [1] and is expressed in the resistance coefficient C of the bed, by:

1

~

=

0.45 rt

c

I

g2

=

0.0575

c.

In formula (1) the first term after the

=

sign can be regarded as the transporting factor. For the value 't" will be intro~uc2d, therefore,

at this place 1'" ,which can be written as 't" = Pg v

Ic ,

this represen-ting the transpo~tation of the material by th~ current.

Also in the case where the velocity vector changes in value and direction due to the orbital motion, the resultant velocity will remain equal to the original current velocity. However, due to this fluctuating velocity vector, the material is stirred up more intensively, which is expressed in the exponential term of equation (1) by writing for 't", 1'" •

The formula of the bed load transport by the combined effects r of waves and currents can now be written as:

[ (1 - y

Ih) I

(y

Ih) ]

Z ln y

Ih

d (y

Ih )•

2

1 { 2 2 2

.L

Sb =

5

D

(v/c)

g2 exp - 0.27 L DC

Il,lv

[1 + (;;uo/v)

12]J.

If it is assumed that the bed load takes place within a certain layer, the concentration in this layer can be computed from the value of the computed bed load and the current velocity in this layer. In [

2]

for this layer a thickness equal to the bed roughness (half the ripple height) has been assumed. From this concentration at a height "a" above the bed the concentration c(y) can be expressed by the well-known relationship:

c(y)/c

=

[(h - y).a/(h - a). y

JZ

,where

Z

=

w/rt

v , with w

=

fall

a . 1

*

velocity of the sand grain and v = ( 1: Ip)2 = bed shear velocity.

*

c

According to the procedure developed by Einstejn

[3

J,

the suspended load can be written as Ss = 1,83 Sb [1₁ In 33

h/r

+ 1

2] , where h =

\"later-depth and r

=

bed roughness.

1

1 and 12 are two integrals, viz.:

I

z-1

I

I

z

f

1 [ Z

I

1 =

o.

216 (a h) (1 - a h)

_a/h

(1 - y

Ih) I

(y

Ih) ]

d (y

Ih ).

1

0.216 a/hz-1 1(1 - a/h)z

f

a/h

Although Einstein has given with his original paper a graph for the integrals 11 and 12' the computation is still rather laborious. The Delft Hydraulics Laboratory and the Delft University of Technology have developed computer programmes for this computation, however.

In the following chapters all computations are executed with the assumption that the bed load takes place in a layer with thickness "r"

(= bed roughness) above the fictive bed. This fictive bed lays half way between crest and trough of the ripples and the roughness is half the ripple height. HOvlever, in chapter 6 the deviations which can occur with various magnitudes of the bed roughness are demonstrated.

3) Description of the model basin and the set up of the measurements The tests have been executed in the basin as shown in figure 1. The sand trap was covered by a screen of small bars in order to diminish as much as possible the disturbance of this sand trap on the current and wave pattern. Although it has not been possible to avoid disturbance completely, it has been possible to reach an equilibrium stage in which the model bed and beach line remained reasonably constant with a sand supply at the upstream end of the model which was equal to the average catch of material in the sand trap.

The angle between wave crests and the beach line in the deep part of the model was

7t

o • Due to these waves, which had an average height of 0.14 m and a period of 1.55 sec., a longshore current was generated. In order to avoid that a return flow would develop in the deeper part of the model, the total value of the estimated longshore current was supplied at the upstream end of the model, and discharged at the down stream end.

The average beach profile as it developed after

43t

hours is shown together with the initial profile on figure 2. This profile is rather stable, due to the fact that the rather high wave of 0.14 m breaks in the form of a plunging breaker at the point of sudden decrease of the depth at about 7 to 8 m from the reference line, that is about 4 m from the reference line, that is about 4 m from the coast line.

3

for instance 0.10 m, a profile as indicated by T8 will develop. In this case the recession of the beach line will continue over a much longer distance. The beach sand had a mean diameter of 0.22 10-3 m and a I90% of 0.3 10-3 m. Immediately at both sides of the longshore bar the

ripples have crest~ ~arallel to the shore. According to tests, described by the author in

8

J,

the value of uo/v* in this region should be greater than 20. The values for these tests are at these places about 100.

Further shorewards, the ripple pattern is less clearly defined which would indicate values of uo/v* between 10 and 20. The values in these tests are in this region about 14. After comparison with other tests in which the resistance could be determined via measurements of the energy slope and which had comparable ripple patterns, the bed roughness has been determined as 0.01 m.

4) Computations of the littoral drift in the model basin

According to the procedure described in chapter 1, the longshore current should be known for the computation of the littoral drift. In the model the current has been measured by means of floats.

The results of the computations are given in tables I and 11 for cross-section 3 and 10, 10 hours after the beginning of the tests.

The measuring points (which are also indicated in figure 2) are spaced 0.5 m apart. From the measurements follows that the longshore transport of water was 120 to 130 10-3 m3/s, whereas the discharge which was adjusted in the model was only 80 10-3 m3/s. Apparently the waves generated a higher current than adjusted in the model, so that a slight return flow must have occured.

In order to compute the longshore current according to Eagleson [4 ] , the place of breaking and the breaker height has to be determined. In a model this is somewhat difficult due to the surface tension.

For cross-section 3 the breaking point is assumed to lay at measuring point 9 (waveheight 0.~55 m and depth 0.18 m) and for the cross-section 10 at measuring point 8 (waveheight 0.165 m and depth 0.21 m). For these circumstances the longshore current according to Eagleson is for the cross-sections 3 and 10 respectively 0.38 m/s and 0.37 m/se This is even higher than has been measured in the model which supports the conclusion that the longshore drift which is adjusted in the model is somewhat too small, so that part of the longshore current has to return in the deeper part of the model. This could decrease the magnitude of the longshore current. The reason for the fact that the current adjusted in the model is lower than which has actually been generated by the waves, is that the development of the beach profile could not be predicted beforehand. For the elaboration of the measure-ments and the computation of the littoral drift this fact is of no importance since the computation is based on the actually occuring velocities.

In figure 3 the distribution of the littoral drift is shown as computed for cross-sections 3 and 10, 10 hours after the start of the tests, together with the quantities caught per hour in the sand trap between 3t and

9f

hours. The values indicating the transport in de model are the quantities passing through that part of the profile which is

represented by the relevant measuring point. The values in figure 3 give the transport per hour per meter of width.

The results show a distribution of the computed transport which is somewhat wider than the distribution of trapped material. It is

questionable, however, if this latter distribution is reliable, due to the disturbance that the sand trap offers to current and wave motion.

4

The distribution of the quantities caught in the sand trap also changes with time as indicated in figures

4

and

5.

In these figures also, the average location of the longshore bar is indicated. It looks like that the littoral drift becomes more concentrated in the area between bar and coast line as the duration of the test increases. This could be caused by the fact that with a more pronounced longshore bar a greater concentration of the longshore current between this bar and the coast occurs. Since no detailed measurements vf distribution of the longshore current are available at these duration times, no comparison with computed littoral drift distributions can be made.

5) Computation of littoral drift along the coast of Queensland, Australia Fig. 6 shows the coast of Queensland with the lines at which various coastal measurements have been executed. From these measurements wave roses along the coast have been determined taking refraction into account. From these waves roses the longshore current within the breaker region has been computed withbthe formula of Eagleson

[4

J.

Outside this area the current has been determined from measurements.

For the computations a bed roughness of 0.17 m has been assumed. Although this value seems rather large, it is in accordance with reported ripple heights at Palm Beach and Coolangatta. Computations of the littoral drift for the profile south of Tweed River (alpha lines) with an assumed bed roughness of 0.1 m and 0.25 m show a variation of the total transport of + 21% and -17%.

In fig.

7

the final results are presented. ~rom the accretion and erosion computations only relative values can be obtained. The actual values of the transports based on erosion and accretion are determined starting from the computed littoral drift at Tugun. This graph proves therefore, only that the ratios between the computed littoral drift at the various sites are in agreement with those resulting from the

accretion and erosion computations.

6) Influence of bed roughness on the computation of littoral drift

!or a sandy beach with a slope of 1 : 30 and a grain diameter of 4 10- m the littoral drift has been computed for roughness values of 0.05, 0.03 and 0.01 m. The velocity within the breaker region follows from the formula of Eagleson. The velocity profile perpendicular to the coast is shown on fig.

8.

In this figure the wave heights at the various points in the cross section also have been indicated. The transports as they have been computed (table Ill) must be multiplied by 40 m, the width of the coastal stretch for the relevant depth contours.

The results show a rather small variation of the bed load with changing bed roughness, but a strongly increasing amount of suspended load. The explanation for this phenomena is straightforward, because the suspended load is determined by the concentration near the bed. This concentration is determined by the thickness of the layer in which the bed load is assumed to take place. Since, according to the assumption made by the author in his original publication [2J , this layer is equal to the bed roughness, decreasing values of the bed roughness lead to strongly increased rates of transport in suspension.

In order to determine how the bed load is transported, motion pictures of wave motion over a sandbed with ripples have been made. Although in this case the current was zero or in the direction of the wave propagation, it was thought that useful information about thickness of this layer could be obtained.

5

Two characteristic pictures are shown as figures

9

and

10.

The pictures have been evaluated by measuring the transparancy of the nega-tives. The result of several measurements above the crest and the trough of the ripples are shown on figure

11.

Until this moment no calibration of the transparancy with respect to the concentration of suspended

material has been made. However from these measurements it becomes clear that there is a sharp drop in concentration at a height of

1.3

to 2 times the roughness above the fictive bed. From this follows a mean thickness of the layer in which the bedload is concentrated, equal to

1.6

times the bed roughness.

With this value the computations of table III have been repeated. The results are shown in table IV, and give a much smaller variation of the total transport with the assumed bed roughness.

7) Conclusions

The results of the tests in the wave basin and of the computations along the coast of Queensland show that the suggested procedure gives reasonable results. However, the results are very dependent upon the assumed bed roughness. The original criterium that the bed load takes place in a layer above the fictive bed with a thickness equal to the bed roughness, therefore, will be reviewed. First results of tests show that this thickness must be in the order of magnitude of

1.6

times the bed roughness and that with this assumption a smaller variation of total transport with the assumed bed roughness is obtained.

Acknowledgements.

Mr. J. Stuip executed flume measurements of the sand movement above the ripples, as a research assistant of the Delft University of Technology. The tests have been executed in the Delft Hydraulics Laboratory with the assistance of the technical staff of this laboratory.

The tests in the large wave basin and the evaluation of the motion pictures of the suspended load have been executed by ire K. Hulsbergen in the Delft Hydraulics Laboratory. (Laboratory de Voorst).

The results of the study on the coast of Queensland executed by the Delft Hydraulics Laboratory are published with the kind permission of the Coordinator General of Queensland.

References.

1968

1968

London

1. BIJKER, E.W.

Some considerations about scales for coastal models with movable bed. Delft Hydr. Lab. Publ. No.

50, 1967

2. BIJICER, E.W.

Littoral drift as function of waves and current Froc.

11

th Conf. on Coastal Eng. Vol I p.p.

415-435,

Also Delft Hydr. Lab. Publ. No.

58

3.

EINSTEIN, H.A.

The bed load function for sediment transportation in open channel flow. U.S. Dept. of Agr., Tech. Bull. No.

1026, 1950

4. EAGLESON P.

Theoretical study of longshore currents on a plane beach. M.I.T., Dep. of Civ. Eng. Hydr. Lab., Rep. No.

82, 1965.

Fig. 1 watersupply

I

I

sand slJpply

•

scate 1: 20 J 14 lS 16 Cross sections 13 12 11 10 9 8 breaker line 7 ~

long shore ClJrrent due to wa.ve moflon

6

LAYOUT OF TEST BASIN

,10 E c: .c: ,20-0. lU "0 ,30

j

0,38 m

o

Fig.2 distance

..

a::: ~ ~ " 0. ci. Q.

6.

E E E E Q) 0') 0 ci. ci.

d.

E E E (D ~ ci. cL

E E

MEAN BEACH PROFILE

N C""'I -.:t Lt\

0. ci. ci. ci. ci.

E E E E E

---~ I

~

initial profile T3 and T8

V

I I

-'-~!

~

'~

"-"

~

-

-

Mean waterlevel

~

....

-

..

"'Il.--

0

-

...

_'

_... ~-,-

_~

~~

- __I

Ta mean. profile after 37% h.; H~10

I

cm

>\

_~

...

-

0

T3 mean profile afterL.3%h. HzlL.cm

...

,

I

:",

_..

I

~

~

0

~

-0 I I .-'"

...

I

__1..",-

r' 0 1 2 3

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_{1 5}

I

I

₆

I I

₇

I·

_{1 8}

I

1

₉

I .

J.

_10\

_l"

12 ....~-1~ ... 15

o

0.2 0.1 0.3 O.L. 0.5

DELFT UNIVERSITY OF TECHNOLOGY

m 0.9 0.7

~

~ 0.6

8

c: lU l-Q)

-

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Cl) > o .0 t1l III ... .c: en Q) .c:

50

DELFT UNIVERSITY OF TECHNOLOGY T3 Fig. 3A

0 3 4

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5

I

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s

I

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10 11

m

N ("l"'l -..:r LO CD c-.. co 0") S2 ::: ~ .. distance

~

g-

E

g-

~

g-

g-

~

t

g-

~

g.

DISTRIBUTION LlTTORAL DRIFT discharge l/m/h 11 0.---r----,----~--.'~r__--___.----,._---_r__--____,---__.

1

.'\ '

100

f \ " \

~trd.pped:

31/2-93/" hours _ '" total discharge 21Sl/h 90I - - - + - - - - t - - - - + - - + - - - f - If,_\_' - I - - - ' . t - - - l - - - - + - - - t - - - i / \

~

I

~ mea~ured

801---:--:-::-+----++---\---jf--r-t---F.----t----+---;---I cross sect. 10 / \

I

\

I 10 hours I

I

.---w--7

computed ro~~t~d.ldiS~M~nOvh~/~~-~--~V~~_I~'"~~-"+IY~~-~----~'---~--~ 601---+1 +'

---I-'-I---l---lL

i

~.\

/

...-4

'.I."," \

t\

~

cross sect.3

it

1/ \~ 10 hours

I

/

""V \

f-\-

total discharge19711h I

~

, \ 40 f - - - i - - - - t - J ' - - - - + - # - - t - - - t - - - + - + - - + - - r - - - - ' t - - - t - - - t - - - i

I

!_"_o"('/

I

\1 \\

I

301---j--- /; I \ ~

:.,'-+---11---/ J , I

~

\,"\.

20r----+---.-/-+--j~---~.--+f-I---t---t-'\\--"

-"0.--'''''1-'

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:~

., "

.----+I~---'"

....

~-_t_----+---____l

~

... _ J.,"r : .... -

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~"':::::"

...

1

L.o./

r---:" - - -

r-Fig.38 5

..

T3 o __ N or-: ....-: or-: 0- 0- 0-E E E N (Yl -..:r 1.£\ <.0 r--. 00 C7l

ci ci. ci. ci. 0- ci. ci. ci. ci.

E E E E E E E E E

DISTRIBUTION OF WAVEHEIGHTS AND LONGSHORE VELOCITIES IN CROSS SECT. 3

U..:S I----,---~-0.2

TT-/T/"-T/r="\\T~~-r--'--1

-Ve-'-lO~i-ties~l-- - -

waveheights-\

0.1

X---...-1/1 _._-- --- -_/ "

~

__ V /

"'~

~

0.1

--T~'t-,-t-+--+--J-_e-_·_-··_--I-+---j~-I----.J

' " 0.2

"=-~"'"

water depths 0.3 " " m '

-

.

_ .

. .

.

"\

m,m/s

DELFT UNIVERSITY OF TECHNOLOGY _{Fig. 3C} T3 ts 2 N

....

a. E 0. Ci.

E

E

0. 0.

E

E

0. Ci. Ci.

E

E

E

0. 0. 0.

E

E

E

DISTRIBUTION OF WAVEHEIGHTS AND LONGSHORE VELOCITfES IN CROSS SECT. 10 I u.,j

/

_\

I

I veloci ties

/

- _ _ _ wave heigh 0.2 ~

V

_\

--

-

... ~

...

...

...

1 ____

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....

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i'.. I 0.1

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_~

---~

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water depths 0.2 '

-""~

0.3 m 1 2

I

₃

I

4

I

₅

I

6

I

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11 .... t'.J rn ...T .,.", ,n "-

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CIO

cl

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.- C"'4 t""i ~ In ID

ci. Cl. r.i. ci. cl 0..

E E E E E E 401 I

--+-10 I I 1.,- I Y I I ~ trapped sand 0/0

• so,

--l--l--T--,--i---'---I

I i i I

,

i "

VARIATION OF DISTRIBUTION OF TRAPPED SAND WITH TIME. T 3

Fig.4 0/0

ro

I

30 I I I I I I I ! I 11 m ~~

o

t::::::

.--•

3

·--r

10

~ C"! t""i. ~

u:

~

r-:

CIO O"l ~ ;::: ~

E E

~ ~

E E E

~

~ ~

~

~

VARIATION OF DISTRIBUTION OF TRAPPED SAND WITH TIME.

23'12-33'/2 h

33'12 -43'12

h

201 I I I _{• •}'0 I

)A:/~

_{\ . .};;!".43

'12 -

SS_I h _{I I I}

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T3

\1

'

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"

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\1

o c [T1 [T1 z (fl r l> z o

\1

•

•

0 _x

x-.f

I

/

/r-/XJ-X/

~~

t-x x - I XX..••lO\

Xl

-

- t-- ~

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Xt---~1 - - -1 - - -

- - -

r - - -1 - - - -I

-

f'X"- I-'- .¥ ~ >. .¥ a tl/ U

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..0 U tD O'l J: ..0 ..0 tI ci c:: ..!: 0 0 Z 0 - - - - 0 ..0 _~--zf-2,

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c::_;:, ;:, .; .t:.

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600000 Cl/ >. ...

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>'50000 :i u c

o

2 4 6 Scalce 8 10 of miles 12 14 16 18 20 x-x Based on Erosion/Accretion • Comput~d

LITTORAL TRANSPORT ALONG COAST OF QUEENSLAND.

heights S.l.

--~ _.-_..

""'-.

~ _O,S5m/s

""'-.

/ ,

"

""'-~

O,28m/s

O,'8m/S~~

I

H=O,78m H=1.86 m H=t68m

I

wave

I

_m. _____ , m

---

3m

---

Srn _6m ~30

--- ---

~ 4 6 2

o

o

20

VL

I

60 40

Figure

9.

Current pattern above ripples.

Fig. 11

CD

DELFT UNIVERSITY OF TECHNOLOGY

+'

~

4

r---t---r--~-l~-k-~--+--+--+----+---+---l

...

+~

i

""--=::-'--3

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CV ...

~

. , t...

2

r - - - - t - - - - t - - - + - - - + - - - + - - - - t - - - f - - -...

-+~~.--=--

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I

' , , ' ' ' '

. , -t;

"T

4 11.6r / , roughness ..

I

/ /

r ripple

o

;;---!---1.----.J..----L.----:";f:----.l-__

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~~tJ_-__r_.-:.:.~L__L-~th~e:i~gh~t

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50 100%1

!

.'J

////[777//

SED I MENT CONCENTRATION Transparency of pi cture . • fictive

I

Do bottom

!

r.r:

.

/ /

-1

cm above fictive bottom

8

r.-\--!-_--\~··~~~~~~:~~~~~~~~~I~~~~l~~~~--.ll---,'~--,---,

7

\\~

\ Cond itions- - - I - - - + _ - - - - l T

=

1.49 sec H== 10 cm 6 " h

=

20 cm . \ . \ Dm= 22.0fJ,

p

=

2.65 '- ripple height

=

1.S cm

_",1\.

ripple length

=

8 cm 5 I--t---,....-~"'~~,--i----t---+---+----r----..,---I---J

~

t

TABLE I

T

3

Cross section 3 at 10 hours

t:. :::

1.65

D ::: 2.20.10

-4

m

T ::: 1.55 s

-4

_{w :::}

_{0.030 m/s}

_{)(. ::: 0.40}

_{r ::: 0.010 m}

D

₉₀

%:::

3.00.10

m

v

h

vxh

H L u

C

C _Sb S

_Stot

0 0 .1.

D90

%

s

[10-

2

m

2

/s ]

l.

[10-3m3/h] [10-3m3/h]

[10,",3m3/h ]

[m/s]

[m]

[m]

[m]

[m/s ] [m

2

/s

J

[m

2

/s]

1

.05

.045

0.23

.020

1.00

.141

31.18

58.59

.008

.0027

.0110

2

.10

.070

0.7

.035

1.25

.198

34.64

62.05

.710

.4146

1.1250

3

.20

.100

2.-

.060

1.47

.276

37.43

64.84

7.802

8.5542

16.3564

4

.25

.130

3.25

.060

1.68

, .241

39.48

66.89

7.990

8.6924

16.6823

5

.25

.150

3.75

.070

1.80

.250

40.59

68.01

9.156

11.1189

20.2745

6

.30

.175

5·25

·075

1.93

.253

41.80

69.21

11.796

15.5936

27.3895

7

.30

.170

5.1

·075

1.90

.257

41.57

68.99

12.179

16.2759

28.4545

8

.15

.145

2.2

.110

1.76

.412

40.33

67.74

11.177

22.7754

33.9527

9

.08

.180

1.45

.155

1.95

.512

42.02

69.43

7.338

22.4334

29.7718

10

.05

.240

1.2

.160

2.23

.445

44.27

71.68

3.788

10.2292

14.0168

11

.03

.285

0.85

.160

2.40

.397

45.61

73.02

1.905

4.4168

6.3214

12

.02

.350

0.7

.155

2.60

.331

47.22

74.63

.895

1.5568

2.4515

26.68

74.744

122.0640

196.8075

-T

3

Cross section 10 at 10 hours

=

1.65

D

=

2.20.10

-4

T

=

1.55 s

-4 _w

₌

_{0.030 m/s}

_'H.

₌

_0.40

_r

₌

_{0.010 m}

D

₉₀

%

=

3.00.10

m

v

h

vxh

H L U

o

C

₀

C

Sb

S

_Stot

1

D90

%

s

[10-

2

m

2

/s]

1

[10-3m3/h] [10-3m3/h]

[10-3m3/h]

[m/s

J

[m]

[m]

[ m]

[m/s] [m2js]

[m~/s]

1

.08

.065

0.5

.030

1.20

.175

34.06

61.47

.214

.1034

.3172

"2

.20

.110

2.2

.055

1.55

.242

38.17

65.58

5.494

5.3281

10.8224

.25

.145

3.6

.070

1.75

.261

40.33

67.74

9.330

11.3323

20.6624

.30

.155

4.65

.075

1.83

.273

40.85

68.26

13.149

19.2661

33.0150

.33

.160

5.3

.085

1.85

.302

41.10

68.1)1

18.684

30.8782

49.5623

.20

.155

3.1

.085

1.83

.309

40.85

68.26

9.472

13.5641

23.0357

.15

.130

1.95

.120

1.68

.481

39.48

66.89

13.671

33.2131

46.8839

.05

.140

0.7

.150

1.75

.580

40.06

67.47

5.272

17.0600

22.3325

.02

.210

0.4

.165

2.10

.499

43.23

70.64

1.745

5.4213

7.1719

.02

.260

0.5

.145

2.30

.381

44.89

12.31

1.199

2.5170

3.7163

.02

.320

0.6

.135

2.50

.306

46.52

73.93

.752

1.1200

1.8718

.01

.350

0.35

.140

2.60

.299

47.22

74.63

.351

.5067

.8572

23.85

79.932

140.3163

220.2486

TABEL III

t::.

=

1.65

D

=

4.00.10

-4

m

T

=

12.00 s

- Beach slope 1

30

--4

_w

₌

_{0.053 m/s}

_{)(. =}

_0.40

_r

₌

_{0.050 m}

D90

%

=

7.00.10

m

v

h

H

L

u C C

_Sb

S

_Stot

0 0 1

D90

%

s

1

[m

2

/h]

[ m

2

/h]

[m

2

/h ]

[ m/s]

[m ]

[ m

J

[m

J

[m/sJ

[ m

2

/s]

[m

2

/s]

.28

1.00

.78

37·50

1.213

42.84

76.21

.124

.6388

.7631

.55

3.00

·1.86

64·50

1.643

51.43

84.80

.222

3.6666

3.8887

.18

5·00

1.68

82.00

1.120

_55·43

88.79

.062

.5796

.6412

.408

4.8850

5.2929

t::.

=

1.65

D

=

4.00.10

-4

m

T

=

12.00 s

-4 _w

₌

_{0.053 m/s}

_{)(. =}

_0.40

_r

₌

_{0.030 m}

D90

%

=

7.00.10

v

h

H

L

u C

C

_Sb

S

_Stot

0 0 1

D90%

s

1

[m

2

/h]

[ m

2

/h]

[m

2

/h

J

[m/s]

[m

J

[ m ]

[m ]

[m/s]

[ m

2

/s ]

[m

2

/s]

.28

1.00

.78

37.50

1.213

46.84

76.21

.116

.7888

.9048

.55

3.00

1.86

64.50

1.643

55·43

84.80

.208

4.6613

4.8691

.18

5.00

1.68

82.00

1.120

59.42

88.79

.058

.6683

.7268

.382

6.1184

6.5007

t::.

=

1.65

D

=

4.00.10

-4

m

T

=

12.00 s

-4

_w

₌

_{0.053 m/s}

_{)(. =}

_0.40

_r

₌

_{0.010 m}

D

₉₀

%

=

7.00.10

v

h

H

L

u C 1

CD

Sb

S

Stot

0 1 0 S

[m/s]

[ m]

[ m ]

[m

j

[m/s]

[m

2

/s]

[ m

2

/sjO%

[ m

2

/h]

[ m

2

/h ]

[m

2

/h ]

.28

1.00

.78

37·50

1.213

55·43

76.21

.101

1.1566

1.2580

·55

3.00

1.86

64.50

1.643

64.01

84.80

.182

7.6232

7.8057

~

18

5.00

1.68

82.00

1.120

68.01

88.79

.053

.8647

.9173

.336

9.6446

9.9809

TABEL IV t::.

=

1.65 D

=

4.00.10

-4

T

=

12.00 s - Beach slope 1 30

--4

_w

₌

_{0.053 m/s 'K}

₌

_{0.40 r :: 0.050 m} D 9

0%

=

7.00.10 v h H L U_o C₀ C Sb S _Stot :J. D9

o%

a 1 [m2/h ] [m2/h] [m2/h] [m ] [m] [ m] [m/a] [m2/s] [ m2/a] .28 1.00 .78 37.50 1.213 42.84 76.21 .12 .33 .46 .55 :' .00 1.86 64.50 1.643 51.43 84.80 .22 1.92 2.14 .18 ~.• OO 1.68 82.00 1.120 55.43 88.79 .06 .30 .36 .41 2.56 2·97 t::.:: 1.65 D

=

4.00.10

-4

T :: 12.00 s

-4

_{w :: 0.053 m/a 'K :: 0.40 r :: 0.030 m} D 9

0% ::

7.00.10 v h H L U o C0 C S S Stot 1 D90

%

b a 1 [m2/h ] [m2/h] [m2/h]

[m/a] [ m] [m] [ m] [m/a] [m2/a]

_[m

2/a ]

.28 1.00 .78 37.50 1.213 46.84 76.21 .12 .41 .53 .55 3.00 1.86 64.50 1.643 55.43 84.80 .21 2.44 2.65 .18 5.00 1.68 82.00 1.120 59.42 88.79 .06 .35 .41 .38 3.20 3.59 t::.

=

1.65 D

=

4.00.10

-4

T :: 12.00 a

-4

_{w :: 0.053 m/s 'K}

₌

_{0.40 r :: 0.010 m} D 90

%::

7.00.10 v h H L U C C _Sb S _Stot 0 0 1 D90

%

a 1 [m2/h] [m2/h] [m2/h]

[m/a] [m ] [m] [ mJ [ m/a] [m2/a ] [m2/a]

.28 1.00 .78 37.50 1.213 55.43 76.21 .10 .61 .71

·55 3.00 1.86 64.50 1.643 64.01 84.80 .18 3.99 4.17

.18 5.00 1.68 82.00 1.120 68.01 88.79 .05 .45 .51