Topics in coastal engineering

TOPICS IN COASTAL ENGINEERING

Compiled and Edited by

The Staff of ~oastal Engineering

for

lectures given by

Prof. Dr. Ir. E.W. Bijker

Delft University of Technology Delft, "I'heNetherlands

Table of Contents Introduction iii ChapterI: Harbors 1 Design Criteria 1 Depth of approachChannels 1 Width of approachChannels 13 Definitionsof Terms 19

Chapter11: Developmentsof LongshoreCurrentFormulas 20

EmpericalCorrelations 20

Energy Considerations 22

Consèrvation of Mass 22

Conservationof Momentum 23

Chapter111: RadiationStress 25

MomentumFlux in Still Water 25

MomentumFlux in Waves 26 Definition 28 Evaluation 29 TransverseComponent 34 Shear Stress 36 Transformations 37

ChapterIV: Determinationof Currentsalong a Coast 38

Wave Forces 38 Tidal Forces 42 FrictionForces 43 TurbulentForces 45 Comparisonof Forces 45 Results 48

ChapterV: LongshoreSand Transportation 49

EmpericalFormulas 49

Formulasfor SteadyCurrents 51

Influenceof Waves 60

Formulasfor Waves and Current 63

Verificationof the Formulas 65

Simplificationof the Formula 75

Results 75

Chapter VI: Local Coastal Accretion Introduction

Equation of Motion Continuity Equation

Solution of Pelnard-Considère

Sandtransport Around End of Breakwater Pertinent Philosop.hy Non-Parallel Accretion 79 79 80 81 83 86 93 94 96 97 100 108 109 109 110 112 114 117 119 127 135 References Numerical Example

Chapter VII: Beaches with Groins References

Chapter VIII: Wave Forces on Piles Drag Forces

Dynamic Forces Lift Forces

Chapter IX: Offshore Constructions Chapter X: Offshore Mooring Structures Chapter XI: Submarine Pipelines

List of Symbols Bibliography

Introduction

This set of lecture notes is intended to supplement the lectures of Prof. Bijker covering Topics in Coastal Engineering. In some cases, information in these notes will be amplified in the lectures.

These notes are written in American rather than English. The reader will see some words spelled differently, for example.

A complete list of references literature has been compiled and included at the back of this volume. Some references are listed, additionally, in the chapter where they are important. An attempt has also been made to compile a complete list of symbols.

Chapter I Har-bor-s ,

In general, a harbor is an area where ships can load and unload their cargo, and where they can enter under almost all conditions of weather and sea.

~~~~g~-Ç~~!~~~~

For the design of a harbor project various design criteria must adopted.

a. For the lay-out of the harbor with respect to geomorphological developments (coastline, dépths in the entrance etc.) conditions which occur more-or-less regularly are decisive. Generally the average conditions govern in this case. The choice of values describing these conditions is rather difficult. Before one can do this, a thorough knowledge of the physical phenomena involved

(such as sand transport) is r-equi.r-ed,

b. For the lay-out of the harbor with respect to wave penetrations and navigability of the entrance, conditions that occur one or only a few times per year are decisive. In this case possible economic

losses due to the fact that ships cannot be loaded or unloaded or are unable to enter the harbor play an important role.

c. For the design of breakwaters,etc. the criteria are governed by conditions that occur only rarely; severe breakwater damage is not acceptable each year. Severe damage is only acceptable once in every ten to fifty years.The severityof the damage is determined by the ratio of damage to building costs. To do this the method of optimum breakwater design has to be used; a subject covered in a separate volume.

~!9!~_~~~_9~E!~_~f_~EE~~~~~_~~~~~~!~_~~~_~~~e~E_~~!E~~~~~.

The depth of the entrance is governed by the following factors:

a. water level

b. draft of the ship c. movement of the ship d. safety margin

a. The waterlevel to be taken into account is determined by the

frequency of entering of ships of a certain size. When the biggest ships; say 250,000 OWT, enter the harbor only once every few days, it is acceptable that ships of this size enter only at high tide. A ferryboat, on the other hand, must be able to enter the harbor at all time~ even during low water spring tide. A good estimate of the required depth can be made only after all factors are considered, including economic losses due to time ships have to wait before they can enter.

b. The draft of a moored shi.pis not only determined by its own characteristics and its cargo, but also by the physical aspects of the water such as its density,:salinitYJ and temperature. A moving ship, moreover, is subjected to squat. Squat is the increase of the draft with respect to S.W.L. due to the speed of the ship. This phenomena can be explained easily with thè help of Bernouilli's principle. / / ut! /1/(/(//(/(////(//(//// (//////

-c

'>

>

-77/77777/777777777777777777777777777

The velocity of the water beside the ship increases due to the ship's motion. According to Bernouillhthe water level goes down as its

velocity increases. Squat is most pronounced, of course, in a relatively narrow channel, but occurs also in a infinitely wide channel. Some-times the draft increase is more at the bow that at the stern (this is the case mostly with carriers having a large block coefficient) and sometimes the reverse is true. Two other factors determining the amount of squat are the depth of the fairway and the speed of the ship. Velocities ranging from four to fifteen knots cause a squat in the order of magnitude of 0.1 to 1.5 m.

c. Waves cause the following ship movements: The Dutch translations are given in parentheses.

___

~ h__

t

_

heaving (dompen) pitching (stampen) rolling (rollen) swaying (verzetten) surging (schrikken) yawing (gieren) wave crest

These ship motions will increase the required depth and width of the channel depending upon the ship size and wave motion.

The increase in draft caused by motion can amount to several meters. For large carriers having great bearns, the effect of rolling can be very important; especially when the natural rolling frequency of the ship is the sarne as the wave frequency. Resonance in this case can

o

increase the rolling of the ship to values of about 10 . For a large tanker with a beam of 60 m the draft increase is

~o

x sin 100

=

+ 5 m! However, when the depth is limited, the mot ion of the ship is damped because the water between_the bottorn of the ship and the sea bed cannot escape fast enough. This is called the cushion-effect.

d. Fluctuations in depth of a channel are the result of the continuous mot ion of sediment on the bed due to waves and current and also of the allowed dredging tolerances. Als~the way in which the dredged profile is measured has an influence on what minimum depths are finally marked on the navigation charts. The depth that is marked on the charts is usually relative to LLWS. It is of the utmost importance that only the most recent issue of the chart be used. A special remark should be made here in case sea charts are used for a first reconnaissance for a harbor and dredging project. A sea,è~is an aid for the mariners, and therefor, the shallowest depths are indicated. It indicates, we might say, a surface connecting the summi ts of the undersea Landacapa ,

In order to determine what depth a channel should have, or what draft a ship in a channel of a certain depth under certain conditions is allowed, the stochast ic character of the ship's mot ion and the actual depth of the channel should be taken into account.

It is evident that the ship's motion, resulting, for the greater part, from the wave mot ion has a stochast ic character. The actual position of the bed, as it is introduced in this computation is, however, a Lso a stochast ic va riable. This is caused by the following circurnstances:

The soundings as weIl as the dredging are not accurate. The depth which is obtained has a certain standard deviation. Apart from this, ripples or dunes of an important dimension caused by waves and current may pass through the channel; this increases the standard deviation.

A continuously decreasing depth due to shoaling should be taken into account seperately.

The composition of the actual movement is indicated in the following figure: design ship at rest desi datum water level water level beside ship Actual

1- -

-

--G

I

vertical mot ion of deepest point Squat

=

Z ship. h I I + A + C C bed surface

The total depth is

h

₌

D + Z + G + I + A + C.

r max

h

₌

total depth of channel D

=

draft of the design vessel

r

Z

=

max.squat of the design vessel at the speed allowed in the max

channel

G

=

deviation of the water level from the predicted value

A

=

allowance for the bed fluctuations around the mean bed level of the channel

C

=

under-keel clearance that should be available to ensure a convenient steering and propulsion of the ship

I

=

allowance for vertical motions of the ship due to wave action. The following remarks,can be made about the various components of this equation.

D : Although normally the summer draft in sea water can be

r

'taken, sometimes,when there is such a very important outflow of fresh water that even the sea area is influenced, the draft in fresh water should be taken.

G: This may be either gust oscillations or wind set up (wind from sea) and wind set down (wind from land).

A: These are the fluctuations as they occur in the bed of the channel. This is illustrated below.

Present design

bottom elevation -'-- charted bottom elevation

---

- ---x configuration Normal Gauss Distribution

C: In order to guarantee good steering and propulsions capabilities of the ship this under-keel clearance should have a certain minimum value. In the case of a stochastic process such as this one, this value may be less during a certain percentage of the time. The determination of this percentage is partly guess work and can also be determined from tests with self-propelled and free sailing and controled roodels.Either remote control or a helmsman in the model ship may be used.

I: This movement, which is normally only of interest in the case of wind from sea or with a swell irrespective of the wind direction, should be considered together with A. The actual motion is determinèd from the response curve of the ship to the wave motion.

This response curve depends on the ratio between the period of oscillation of the ship itself Tand the encounter period of the

s waves T with respect to the ship.

e

This T is expressed in the period and direction of approach of the e

waves and the velocity of the ship in the following manner.

·

v

cos a)

=

l/(l - .2.. cos a) T À T

=

T/(l -e c

v

s where :

T, À, c are wave period, length, and celerity of propagation, respectively.

a is the angle between wave direction.and direction of the ship's motion.

V is velocity of ship.

s

directio

The response curve between ship and waves can be defined as

R

=

amplitude of the ship mot ion amplitude of the wave

For an irregular wave pattern, the resulting ship mot ion is also irregular. The vertical mot ion of a certain point of the ship can be expressed

-

analogus to wave mot ion - in terms of a "spectrum of ship motion".

The spectral density of this spectrum is given by

where:

f i(T)

=

spectral density of ship motion

[!_TI2I~ at period T

f

n(T)

=

spectral density of wave mot ion

[m2Is] at period T

R(T)

=

response factor of the given point at period T

HI~

n(t)

--:t>

Wave mot ion

Vertical ship motion

f n R T 1 T

n (t) is the excursion of the water surface from the still water level as a function of t.

Since it is assumed that the local (or instantaneous) wave ordinate n (t) is distributed normally, the variance of this excursion is

00

ff(n) . dt which is also equal to the total energy of the o

wave spectrum.

The probability of nt between Hand H + öH can be written according to the normal distribution as

K + ÖK q (K < n (t) < K + ÖK)

=~

I-K 2 -T /2E e dT

In this expression the total energy equals p g E.

The total energy of the spectrum of the vertical motion, it' of the ship is, when this total energy is again pgm, given by:

00

m

₌

_f

f, dT 1. 0

and i is also normally distributed according to: t

q K < it < K + t::.K)

=

K

Here the mean value is zero, and the standard deviation

=

cr

=

~.

s The relationship between wave mot ion and ship motion can be indicated by the following scheme,

Wave Motion Ship Motion

~

rJVVVV'

f1(T) ~~I~R_e_sp_~_n~(;_:~F_a_c_t_o_r~I~---~

3r-E

1

dl.strl.o /butl.on l.strl.° ~d' Ibutiutl.on

i(t) s

/~

distribution distribution

H

For the actual computation of the depth a distinction should be made between the situation with and without waves,

Case with no waves.

In this case the keel of the design vessel moves in a horizon-tal plane and only the irregularities of the channel bed have to

be taken into account.

If it is assumed that only during a certain percentage of the

time (that is also during an equal percentage of the length of the

channel) the açtual under-keel clearance may be less than the

accep-ted under-keel clearance, then an extra allowance, A for the bed

irregularities has to be taken into account. If this percentage is 0.1 then this extra allowance is

A

=

3.09 (J

a

where the value of 3.09 can be found from the tables of a standard -dized normal distribution.

Case with wave motion.

An equivalent procedure is followed. However, in place of

stan-dard deviation (J of the bed irreqularities, now the combined

stan-a

dard deviation of ship motion and bed irre&ularity has to be used. Since the two movements are independent stochast ic variables, the variance of the combined movement can be written as:

2 2

(Jk

=

(J_a + E_s

For an equal chance that the under-keel clearance is smaller than the accepted value G, the total extra allo~ance A + Y will be in this case 3.09 (Jk'

In both cases there exists still the chance, however, that the ships keel hits the bed under very extreme circumstançes. This should be calculated via the encounter probability of such an event during the passing time of the vessel. If there is no wave motion, it is necessary to know the average length of the bed irregularities in order to determine how many irregularities may be encountered during the passage of the channel; Let this number be N.The encounter pro-bability of a bed undulation with probability of exceedance PA is

E

=

1 -

G -

PA]N

When (J

=

0.15 and the accepted keei clearance

=

0.4, then in the a

case above, the tot al acceptable value of A is 3.09 x 0.15 + 0.40

=

0.86

=

5.6 cr. The probability of cxceedence of this value is a

1.2 x 10-8. With an assumed irregularity length of 200 m, and a channel length of 4000 m, then N

=

20. Here, we have assumed no wave motion.

The probability of hitting the bed is in this case: E

=

1 - [1 - 1.2xl0 -8

J

20

=

2.3 10-7, which is, indeed, extremely low.

In the case of combination of wav.es and bed undulations, the number of motions of the ship will normally be the only decisive factor, since the bed irregularities are assumed to be rat her long. N is, in th is case, the number of oscillations performed by the ship during the journey of the vessel in the approach channel. Analogous to wave motion, the average period of the ship Qscillations is:

['"f i dT ~ -T

₌

0

I®

f.

i

dT 0

For a value of T of 10 sec, a length of the channel of 4000 mand a speed of the ship of 12 kn

=

6 mis,

N

=

4000/6

10

=

67

From the above discussion about the case for waves the total ac-cepted value for A is:

A

=

3.09 ok + 0.40

For 200,000 DWT ships and H

=

4.5 m, cr ~ 0.4

s m

Ok

=

I

0,0225 + 0.16'

=

I

0.182'

=

0.426

The total value of A (ship movement, bed undulation, and under-keel clearance) is 3.09 x 0.426 + 0.40

=

1.718

=

4.03 Ok'

-5 The probability of exceedance in th is case is 3 x 10 .

The encounter probability E

=

1 -

G -

3 x 10

-sj

of hitting the bed is in this case

67 -3

=

2 x 10 .

depth of the channel should be greater if the channel bed is so hard

that damage has to be feared.

e. In case the bed is rocky so that serious damage has to be feared,

th is last derived encounter probability must be made extremely low,

10-8 to 10-10 for instance;if the bed is muddy, on the other hand,

the encounter probability can be rat her high, and only the steering

capabilities count.

For very soft beds consisting of soft mud or sling mud even

negative under-keel clearances can be accepted.

In general, it is reasonable to assume that in channels in

shallow water with an average wave height, the desired water depth for

large ships is 10 to 20 percent in excess of the draft of the ship.

Most recent information:

Wicker, C.F. Economie Channels and Manoevring Areas for Ships

Proc. A.S.C.E., Vol. 97, No. W.W.3 Aug. 1971

p.p. 443 - 453

Waugh,Jr. R.G. Water DepthsRequired for Ship Navigation

Proc. A.S.C.E. vol. 97 No. W.W.3 Aug. 1971 p.p. 455 - 473.

Eden, E.W. Vessel Controllability in Restricted Waters.

Proc. A.S.C.E. Vol. 97 No. W.W.3 Aug. 1971 p.p. 475 - 490.

The width of the channel can be determined in comparable way, by considering the mot ion of the ship in the horizontal plane also as a stochastic variable. By this method the chance that a certain excursion from the idea_lcourse line will occur , and therefore, the chance that a ship hits the bank of the channel or another ship can be determined. In this way, we determine the width of the channel in which the risk of collision is brought down to an acceptable value.

There are a few principle differences with the case of the vertical motion of the ship.

i. The probability distribution of the deviations of the theo-retical (ideal) course line cannot be determined as easily since the period of the movement of the ship around the theoretical course line is normally long compared with the sailing time of the ship. The deviations from this course line are, therefore, not stochastic, in-dependent variables.

ii. As soon as two ships are so near to each other that a chance on collision exists, the two movements are no longer independent. Tests are necessary for investigation of this problem. Three possi-bilities are available:

a. From observations in the prototype the probability distribution of the deviation from the theoretical course line d:.an be determined. With this distribution the probability of a certain deviation from the theoretical course line can. be determined.

With this distribution the probability of a certain deviation can be computed. In this solution the mutual influence of the ships is not yet taken into consideration. It is, unfortunately, difficult to obtain sufficient data under extreme circumstances.

b. With the aid of a model study it is more easily possible to obtain sufficient data under extreme circumstances.

This model can be an hydraulic one with real ships on a scale of about 1 : 25 to 1 : 50. In this case the mate and pilot will be nor-mally in the model ship or the model ship will be steered with an

automatic pilot. The great advantage of this method is the very true way in which bank influences and the mutual influence of the ships on each other can be determined.

~. Another possibility is the use of a steering simulator. In such a

simulator the movement of the ship is made visible by solving the

dynamic equations of this motion using a computer. All required cir-cumstances of wind, waves and current can be introduced into the equ-ations. The pilot has all normal navigational aids at his disposal so that a very realistic reproduction of the movement is obtained. The great advantage of this method is the much greater number of trials that can be made at acceptable costs. The reproduction of the mutual influence of the ships and the influence of the banks of the channel may still cause difficulties.

s:. Q) 2.0 .--f .--f '.-1 +-' en ~ '.-1

ti::

_m s:. '"Ó _1.5 +-' Q) '"Ó s:. Q) +-' m ;3: 1.0 unacceptable coïrib~nat~ons Directional stability critical

:~~. ___.._Lateral forces and

";'~ moments critical Acceptabie

..

:

I

.

combinations ~,:":;,,,: Bottom clearance _ critical 6 •• , '•• 0", •• 1· !.:..: :'o!I. . 2 3 4 5 6

channel bottom width ship beam

The widthof the approach channel is also determined largely by the hydraulic conditions. Of courses a channel with a cross current of importance must be wider than a channel in still water. Alsos the number of ships that are expected to sail at the same moment in the channel will determine the width. It is difficult to give fixed data. As a general rule one can state that the path width required by a ship is about 1.8 times the beam of the vessel. Between two vessels meeting each other a distance of about one beam should be kept between the two paths. In a channel with banks almost up to the waterlines a distance of 1.5 beams should be kept between the side of the bottom of the channel and the path of the ship (see: C.F. Wicker: Evaluation of Present State of Knowledge of Factors Effecting Tidal Hydraulics and Related Phenomena; Chapter X, Design of Channels for Navigation; Department of the Army: Corps of Engineers.

The width of the harbor entrance itself should, in principles have the same width as the approach channel just in front of the entrance. However, in a long approach channel a meeting of two ships may be inevitable, whereas this may be avoided in the entrance itself. On the other hand, touching of the breakwater ends will involve greater damage to ships and possibly give greater hinderance to navigation than the grounding of a vessel in the approach channel. In order to prevent the ship from completely blocking the channel of the entrance a width slightly greater than the length of the ship may be used. When a ship sails into a harbor and there is a current crossing the approach channel, the ship will follow a course as indicated on the sketch below.

1

I

I

~

I

I

,

I

~

I

I

~ ~ ~

(

I

\

_,

\

Due to the fact that during the passage through the entrance the bow will be in still water while the stern will be in the current, a moment will be acting on the ship, forcing it to turn. Sufficient width must be available inside the entrance.

Another general rule is that the approach line of the bigger ships should be as straight as possible.

When a ship has passed the harbor entrance, it needs a certain distance in which to stop. If there is some wave motion and current in front of the harbor, the minimum velocity with which the ship can enter will be in the order of magnitude of 3 to 6 KT. In a harbor, it is not possible to give full astern, since the ship will then swing to starboard (when the normal revolution direct ion of the propeller is clockwise when looking forward).

It is, therefore, necessary that tugboats assist the ship in keeping the proper course.

..../ ~ ","

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=='+' _I mlH "Ó .-I m lil Gl .-I lil ~ m _I "Ó _==' ~ m _I 0 ·M X m ILo ·M I GlN

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~ .c: _::::. (j)~:;3: +'+' Gl 000 H 0 _.c: 0- m 000

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0::il::3

tl()Gl ~ 000 OM "Ó Gl ft ft ft C!)E-<<( lil _> 000 _I z ::x:: ~ ~ Gl OU') 0 H (j) _+' Gl ~ U')N.-I I

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~ p., m +' I

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p., Z lil 0 ₁

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0 H «:;3: z I ° E-< ~ (j) _I I I

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1/ " /

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L-... 1

1./

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l--::

~ I ~ ""...

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The usual procedure is that the ship slows down, while the propeller turns forward dead slow or is stopped. Reverse power is not allow-able since the ship becomes uncontrollable in this case. When the speed is brought down to 3 knots it is assumed that assistance of

tug boats is obtained. This will say that tug boats have connected to the ship and have manoeuvered themselves into the position to give

as-sistance. At this moment reverse power can be applied to the ship's screws.

The valves above are normal stopping distances keeping a straight course. For example, when a 130,000 ton oil carrier'sailing at

four-teen knots has to make an emergency stop, giving continuous full

astern, the stopping length is only 3 or 4 km (see the figure below.)

In this case, however, the direction (heading) is uncontrolled.

24 22 E-< 20 :J: Q lH ₁₈ 0 (/) ₁₆ -e ~ (/) 14 ;j 0 ..c: 12_+' c 'H ₁₀ Q,) N ₈ 'H (/) ~ ₆ Q) ..!<: ~ 4 E-< 2 u

/

n n ~~ f-,D

tr>!

;;,

\

/

0 ~

"I

/

0 0

ei

C(

rIJ

~

IJ

VSi 0

;-

I ...~

_'"

b' .-i-: 1.:1 'U

-V

'--1 ""erG.

.

.,

_.t:/

0

,_

;y

~

V

0 " J( I!J

_1/

'j(~ I Q 0

/

/'

0 riJ

/'

/

./

t.>

0 1[,

a

,

0

o

.

2 .4 .6 •8 1.0.1.2 1.41.6

La

.202 .2 2.4 2.6 2.8

Stopping distance in Nautical Miles under extreme emergency conditions

Some general ratios for ships are:

for normaI s aps BRT ~ 1.5h· DWT

for very Iarge cru e canr-i.er-sd . DWTBRT::=2

for all Total displacement to 1.4

ships DWT ~ 1.3

The ratio BRT varies from 1.7 for freighters to 1.3 for VLCC s.

NRT

Deadweight tonnage (DWT): the vesseï.!sUfting capacity or the number

of tons of 2240 lb. (= 1016 kg) that a vessel will lift when loaded

l.nsalt water to her summer freeboard marks.

Deadweight includes: crew, passengers, luggage, provisions, fresh

water, furniture, coal in bunkers,fûel'::oilin tanks and so on.

~oss register tonnage (BRT) of a vessel consists of its total

mea-sured cubic contents expressed in units of 100 cu.ft. or 2.83 cU.m.

Net register tonnage (NRT) of a vessel is the carrying capacity arrived at by measuring the cubic contents of the space intended for revenue earning. One hundred cubic feet is the standard spaqe taken as the accomodation for one ton of goods.

Disp1acement: the number of tons of water displaced by a vessel afloat

(1 ton

=

2204 lb). The sum of light weight and dead weight is equal to

the displacement.

In accordance with Archimedes' principle, displacement and weight are equivalent quantities for floating bodies. The displacement of a ves-sel i'S" the weight of water displaced at a given draft, and also the weight of the vessel and its contents.

Chapter 11 Historical Development of Longshore Current Formulas

Amphibious landing during World War 11 were handicapped by currents parallel to the coastline.

These longshore currents were generated by obliquely approaching breaking waves. The war effort stimulated the start of investigations into this phenomona. Later~ it was realized that this current is also of importance to the transport of sand along the coast.

In this section, we attempt to examine some of the development history of longshore current formulas. One of the more recent ones will be developed in detail in later sections.

Both Galvin and Thornton have written good reviews of longshore current formulas. Galvin uses a practical approach, while Thornton takes a more theoretical point of view. The table on the following

page gives a summary of the most common formulas. No claim on completeness is made. Most investigators used at least one of the following concepts in their formula:

1. Emperical correlations~ 2. Energy considerations, 3. Conservation of mass, 4. Conservation of momentum. There are now discussed separately. 1. Emperical Correlations

The emperical formulas may be derived in either of two ways: a. Develop a formula based, to some degree~ on methods 2, 3, or 4,

with undetermined coefficients. These coefficients are then evaluated using available data. The Inman/Quinn~ Brebner/Kamphuis, and Galvin/ Eagleson formulas were derived in this way. These emperical formulas except for that of Gal~in/Eagleson are not dimensionally correct, i.e. the coefficient are not dimensionless; this is not necessarily'

a practical disadvantage.

b. Linear statistical corre1ation of all available emperical data can yield an equation; Harrison used this approach.

There are varying opinions about the elegance of these correlation methods. By accepting these methods, one is accepting addition of

·

"f

·

• ,: :

..

•

.

..

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Putting a wave period T

=

10 sec, a beach slope m

=

50, and

a wave height H

=

2 m into the Harrison equation yields a longshore current velocity of 0.78 mis when the wave crests approach parallel to the beach. This result is, of course, queer even though the input data is realistic. Harrison himself noticed this problem and warned that his formula may only be used when the angle of incidence of the waves is between 20 and 150. Thus, in the above example, we are in error by using his formula outside its desired range.

2. Energy Considerations

The authors of these formulas assumed that a percentage of the incident wave energy is used to generate and maintain the longshore current. Galvin comments that there is no theoretical justification for the assumption that the ratio of current energy flux to incident energy flux should be constant. He calculated this ratio from experimen-tal data. He found that the ratio s small, often less than 0.01

and usually less than

p.l,

and certainly not constant.

3. Conservation of Mass

The conservation of mass approach is based upon the idea that the mass of a solitary wave swept over a bar is distributed in some way to the longshore current. This title "conservation of mass" is a bit misleading. It suggests that the other approach methods ignore

continuity. This is not true; all of the equations shown in the table satisfy the continuity equation.

Bruun presents two methods: a rip-current theory and a continuity theory based upon the Chézy Equation. His rip-current theory is based upon the assumption that the entire mass flux of the solitary wave is concentrated in the rip-current. These rips are spaced 300 m to 500 mapart along the shore. Bruun's continuity theory, on the other hand,is based upon an assumption of uniformly distributed return flows. Here again, he assumes that the entire mass flux of a solitary wave comes over a bar. This raises the water level uniformly between the bar and shore over a longshore distance equal to the distance between

wave crests measured along the beach. Making a further very arbitrary assumption that the other water surfaces remain undisturbed, yields a value for longshore current when the Chézy Equation is applied.

The Inman/Bagnold approach differs in that they consider that only the longshore component of the wave motion contributes to the current,

with the result that their predictions are much smaller than those of

Bruun.

Galvin/Eagleson emperically correlate the mass flux of the longshore

current with a fictious mass flux related to the wave. Since it turns

out that both fluxes are proportional to the square of the breaker height,

an equation results in which the longshore velocity is independent of

the breaker height.

The main obsection to all conservation of mass approaches is that

the equation of motion is completely ignored. One can say that only

a portion of the available information has been used. It is relatively

easy to postulate a flow system satisfying continuity which

really exist in practice.

4. Conservation of Momentum

Methods using conservation of momentum seem to be the most reliable.

cannot

Several are compared here. One of the more important is derived in

detail in the following two chapters. Both Bakker and Battjes have

published comparisons of various formulas using momentum conservation.

One will find in practice that the formulas of Thornton, Longuet

-Higgins, and of Bakker vary very little in result. The differences

among them are caused by a subjective preference for some second-order

terms, simplifications, or different bottom friction formulas.

Battjes shows that the numerical value of the friction factor, f, is

of the same order in all three formulas. Other formulas belonging

to this group, with the exception of Bowen's, are more or less out of

date. Bowen, however, does not specify his friction factor.

Putman, Munk and Traylorused solitary wave theory in their formula~.

They used a bottom friction term proportional to v2, but did not include

the effects of waves on the bottom friction. It soon became obvious that

their friction coefficient was much larger than the normal Darcy-\oleisbach

coefficient. Inman/Quinn proposed to take the friction factor as:

f

=

0.384

v

lo5 •

Unfortunately, this did not solve the problem and violated the dimensions

Eagleson did better by taking the friction proportional to V2/sin ,. This is better, but still a bit questionable •• is the angle of wave incidence.

After Bowen applied radiation stress and Jonsson, Bijker, and

Kajiura explained bottom friction in separate papers, the formulas by Thornton, Longuet-Higgins, and Bakker ware developed independent of one another.

Radiation Stress is explained in the following chaper. After

Chapter 111 Radiation Stress

Radiation stress is a term applied to a pressure force in excess of the hydrostatic pressure force caused by the presence of waves. This pressure is revealed only when second-order terms (proportional to the square of the wave amplitude) are considered. The wave profile considered, however, remains a simple sinusoidal wave.

The effect of radiation stress is most commonly observed as a force acting on a wave reflecting boundary or as a force difference causing a water-level set-up~ This force difference results from wave height differences. We shall use the results of this radiation stress derivation to determine the longshore current caused by waves.

The discussion which follows is adapted trom that of Longuet-Higgins and Stewart (1964) published in "Deep Sea Research", Vol. 11 p. 529-562.

In all of the following discussion, these conventions will be observed:

a. The origin of coordinates wi11 be placed at the still water surface with the X axis directed positive to the right, in the direction of

.,cl wave propagation. The positive Z axis extends upward trom the still

water surface.

b. The problem will be considered to be two-dimensional; a width of 1 meter perpendicular to the plane of the drawings will be assumed (Y direction, initially)

c. The water density, p, is constant.

One should also remember that according to Newton's Law, a force is equivalent to a rate of change of momentum. A pressure, or stress, is equivalent to a flux, or flow of momentum. Tqis is also rate of change of momentum per unit area.

Momentum Flux in Still Water

If we first consider a water body of uniform depth (h) having no wave action, then the pressure at any point in the fluid is given by

p

= -

p g Z

o (1)

where g

=

gravitational acceleration and p is the density. This is independent of x and is the flux of horizontal momentum

across a vertical plane (x

=

constant) per unit of vertical distance, since a width of one unit is assumed.

The total flux across the section is obtained by integrating (1) o Po

=

f

Po dZ

=

-h o

f

p g Z dZ -h (2)

This is, too, independent of x and, therefore, there may be no net

change in momentum as we progress trom one plane x

=

x to another

o

x

=

x + dx.

o

This constant total flux (or force) P is rea11y the hydrostatic force

o

which would be present on a rigid vertical wall extending to the depth

Z

=

-ho This hydrostatic component, P , will latèr be subtracted from

o

the total computed flux with waves to reveal that component which

results trom the waves. This resu1tant will then yield the Radiation

Stress.

One will diseover, later, that this radiation stress does not

have the units of force per unit area. This is because a true stress

(pressure) will be integrated over a constant depth h, giving a force

per unit.length. Since the depth remains constant at each point wh ere

we shall examine the radiation stress, then transformations lègitimate

for true stresses can still be used.

Momentum flux in Waves

Consider a sinusoidal wave in water of finite constant depth h.

From short wave theory we find:

n

=

a cos (kx.- wt) (3 )

aw

u

= --,

--:-

~-:--~

sinh (k h) eosh k(Z + h) eos (kx - wt) (4)

w

=

sinh (k h)

aw

sinh k(Z + h) sin (kx - wt) (5)

where:

n

=

wave profile

a

=

amplitude of the surfaee wave

k

=

wave number

=

2n/wave length

w

=

eireular frequeney

=

2n/wave period

x

=

position along X axis

t

=

time

Z

=

distance (+up) from water surfaee

h

=

water depth

u

=

instantaneous particle horizontal velocity w

=

instantaneous particle vertical velocity (see figure 1) Z

=

n u w

t~

2 pu -h

FIGURE 1 Definition Stretch

A rather general expression for the flux of horizontal momentum across a unit area of the vertical plane in the fluid is:

(7)

The second term, p momentum as follows:

in a time dt the volume of water passing an element of the verti-2

u , represents a bodily transfer at horizontal

cal surface is u dZ 1dt

its mass is: pul dZ dt

the horizontal momentum then (mass • velècity) is p • u . 1 . dZ • dt • u

Dividing by the area (dZ • 1) and time (dt) to get flux per unit area yields the desired term.

We might note here that u may be positive or negative and may have even a zero time average; while u2 is always positive and will also have a positive time average. This fact will be of importance in later discussions.

Fluid crossing this plane also, in general, will have a vertical velocity component w.

stress and is commonly called a "Reynold Shear Stress" in theory of turbulence. These Reynold ShearStre~will not be considered further here~however.

Returning to our more immediate problem~ we integrate (7) over the depth to get the total flux of horizontal momentum across the plane x

=

constant. Formally:

PI

=

J

(p

+

p u2) dZ -h

(8)

Note here that the upper limit of integration is the actual water surface, and not zero is often used in first-order theory.

Definition of Radiation Stress

The principal radiation stress component, denoted Sxx~ may now be defined as the mean value of P1 with respect to time minus the mean value of P with respect to time.

o

n 0

Sxx

=

P1

P

0

=

I

(p + p u2) dZ -

I

Po dZ (9)

-h -h

The bar is used to denote a time average. In the first integral, we must be sure to take the time average afte!'-cómplèting-the

integration, since the limit n of the integration is, itself, a function of time.

In the second integral, the bar may be omitted, since that integral is constant, see equation (2).

(9) now becomes:

=

J

-h 2 (p + p u ) dZ o

I

Po dZ -h (10)

There remains the problem of evaluating SXX.

We can now state a definition of radiation stress in more

sophisticated terminology as: The radiation stress is the contribution of the waves to the time average of the vertically integrated

Evaluation of Radiation Stress

As an aid to evaluation of equation (10), it is separated into three parts as follows:

(1) (2) (3) SXX

=

Sxx + Sxx + Sxx (11) 0,)

--J

p 2 S

=

u dZ (12) xx -h (2) 0 S

=

_f

(p - p ) dZ (13) xx 0 -h (3)

J

S

₌

_P dZ (14) xx 0

It may be verified that (11) through (14) are equivalent to (9) by substitution. These terms will now be considered individually.

Equation (12) will be split again as:

(1) 0 n

f

2

I

2 S

₌

p u dZ + p u dZ (15) xx -h 0 order

the integrand (p u2) in both terms of (15) is already of second

proportional to a2; see equation (4). Since n is also a function

of a, then the second integral in (15) will yield only a term of third order.

Since only first and second order terms are considered here, then (15) may be approximated (to second order) by:

(1) 0

S_xx

=

f

P u2 dZ -h

(16 )

Now that both limits of integration are constant, the time average may be moved inside the integral

(1) S xx o

=

f

-h p u2 dZ (17)

This is a Reynolds Normal Stress integrated trom the bottom to the still water level. It is, obviously, generally positive.

Since both limits of integration in equation (13) are constant, we may again apply our previous technique. Equation (13) becomes:

(2) S xx o

=

f

-h

(p -

p ) dZ o (18)

Po is excluded from the time average since it constant. (2)

S is caused by changes in the mean pressure p when compared xx

with the hydrostatic pressure p found in the absence of waves. p can o

be evaluated by completing a second order analysis, but

p

may be more easily evaluated indirectly from vertical momentum considerations as follows:

Using arguments similar to those already used, the mean flux of vertical momentum across a horizontal plane must be aqual to the weight of water above that plane. The average water lev.el elevation is Z

=

o.

Therefore:

p + p w2

= -

p g Z

=

Po (19 )

or

2

P - Po

= -

p w (20)

(20)substituted into (18)yields: (2) 0

=

f -

p 2 S w dZ xx -h

this is, obviously, less than zero in general.

(21)

The third radiation stress term, equation (14), is the pressure integrated from the still water level to the wave profile with this integration averaged over time.

This integration leads, strictly, to difficulties when n is below Z

=

asince, then, p is undefined in the range n < Z ~ a. This may be most easily overcome, according to Longuet-Higgins, by "extending the velocity field upward to the mean level" Z

=

a. Near the free surface, p is very nearly equal to the hydrostatic pressure measured from the instantaneous surface

n.

The pressure fluctuates in phase with the surface elevation.

p

=

pg (n - Z) (22) Substitution in (14) yields: (3 ) S xx

=

I

pg (n - Z) dZ (23) o

Evaluating only the integral (neglecting the time average) and noting that n is independent of Z (eqn (3) ):

I

pg (n - Z) dZ o

=

pg

[I

n dZ -

I

Z dZ }

J: ]

o

Taking the time average yields: (3)

S

xx (24)

This is generally greater than zero. Since

n

=

a cos (kx - wt) then

2 n

= 2

1 a ;2 n 1

f

(-lr 2 cos x dx

=

1:.)

2 o

and (3)

s

_xx (25) Interpretation (1) Adding S xx of S xx (2) and S

xx yields, using (17) and (21:):

(1) (2) 0 0

J

2

J

w2 dZ S_xx + S

=

p u dZ p xx -h -h 0

J

2 w2)

=

p (u - dZ -h (26)

From short small amplitude wave theory (equations 4 and 5)=

2"

2

u > w • Therefore, (26) is generally > O.

Before attempting further solution of (26) directly, we note that for irrotational, incompressible flow:

a

2 2 -

_a

_Z (u - w )

=

2 (u au _ w aw)

az

az

=

2 (u aw + w au)

ax

ax

a

=

2

ax

(u w)

=

0

Therefore, (u2 - w2) is independent of Z, even though u and w are both, themselves, functions of Z.

(26) becomes:

o

=

p (u2 - w2)

f

dZ

-h

Formal substitution of (4) and (5) in (27) gives (1) (2) S + S xx xx 2 2 a w

=

ph _ __;,__-sinh2 kh

Since on1y the trignometric functions depend upon time:

[ cosh2 k(Z+h)cos

2

Ckx-wt)-sinh2k(Z+h)sin2(kx-wt)]

=

p h a2 w2 [COSh2k(Z+h) cos2(kx-wt) _ sinh2k(Z+h) sin2(kx-wt)1

sinh2kh 'TT Since 1

f

_cos2 x dx ₌ 1 2: 11 0 then this becomes: 2 2

= ~

p h a w [cosh2 k(Z + h) - sinh2 k(Z + h)] 2 sinh2 kh 2

with cosh x - Sl.nh2 x

=

1, this becomes:

2 2

=~pha w

2 sinh2 kh

for small amplitude waves, w2

₌

g k tanh k h,

yielding: (1) (2) 2 k h S + S

₌

P g a xx xx sinh 2 k h (28) (29)

If we remember that the total energy density of the waves, E, may be

defined as:

E

=

~

p g a2 then we can determine

2 (30)

Sxx from (11), (25) and (29) using (30):

2kh 1

SXX

=

E(sinh 2kh +

2)~

0 (31)

In deep water, 2 k h/sinh 2 k h + 0 yie1ding:

In shallow water, 2 k h/sirih2 k h ~ 1 yielding:

(h< L/25) (32b)

Sxx has units of force per unit length of wave crest.

Transverse Radiation Stress Component

It now becomes necessary to examine the flow of momentum in the Y Z plane. The Y axis is in the plane of the still water surface directèd parallel to the wave crests. In contrast to the previous section, a unit.

thickness in the X direction will be assumed. This new radiation stress component will be denoted by SYY' lts derivation closely follows that for Sxx'

We consider the total flux of

Y

momentum parallel to the wave crests through a plane Y

=

constant. lts mathematical definition corresponds to (9) and is:

=

Ï

-h o (p + p v2) dZ -

f

-h

P

dZ o (33)

Just as with SXX ' equations corresponding to (11) through (14) are:

SYY

=

S(l)_yy + S(2)_yy + S(3)_yy s(l)

1

2 dZ

=

p v yy -h 0 S(2)

=

_J

(p - p ) dZ yy 0 -h n S(3)

=

_J

p dZ yy 0 (34) (35) (36) (37)

Here, the analysis becomes somewhat simpler, since for long-crested waves, v

=

0 by identity.

S(l)

₌

0

yy

Also, comparing (36) and (37) with (13) and (14):

0 S(2) S(2)

f

2 dZ

=

=

- p w yy xx -h S(3) S(3) 1 2

=

=

p g a yy xx 4

using (21) and (25) respectively.

Substitution of w from (5) in (38) yields:

S(2)

=

yy 2 2 a w 2 sinh kh o

f

_-h cash k(Z2 + h) dZ - p 2

using the time average of cos , we get :

2 2 0 S(2) a w

f

2

=

- p . 2k cash k (Z + h) dZ yy 2 s1.nh h -h the integral yields: kh

r

. 2 dZ 1

f

2 cash k (Z + h)

₌

cash q dq k -h 0 with q

=

k(Z + h) . 1 [ sin~ 2 q

I

k\

%

c

]

=

k 0 1

[

sinh 2 kh ~h ]

=

_{k 4} + (41) now becomes: S(2) a2 w2 sinh 2 kh kh { }

=

- p ₊ yy 2 k . 2 4 2 s1.nhkh 2 2 p a2 w2 sinh 2 kh p a w h

=

8 k sinh2 kh 4 sinh2kh (38) (39) (40) (41)

or , using (28) : 2

=

-p g a 4 2 p gak h 2 sinh 2 kh (42)

This does not look like much, but when we add S(3) to get Syy

(remembering that S(l)

=

0): yy yy

=

S(2) + S(3) yy yy 2 2 kh 2 p g a p g a _{+ p} g a

=

_{4 2 sinh 2 kh 4} 2 kh P g a

=

-2 sinh 2 kh

= -

E kh (43) sinh 2 kh using (30).

In deep water, kh/sinh 2 kh ~ 0 yielding:

(h > L/2) (44)

In shallow water, kh~sinh 2 kh ~ 1/2 and: 1

Syy

= 2

E (h < L/25) (45)

Syy has units of force per unit of length of wave orthogonal.

Shear Stresses

Finally, for completeness, we must investigate the possibility of the transfer of x - momentum across the plane y

=

constant. Since this momentum manifests itself as a shear stress, the pressure at the point does not contribute. By definition, pressures act only in a normal direction. This results in an equation sornewhat simpleI' than (9).

=

[" p u v dZ -h

(46)

Since our waves are still long-crested, v - O.

Therefore, quite simply, (46) becomes:

(47)

Since the shear stress SXY is zero, then, from strength of materials, we can conclude that SXX and Syy must be principal stresses.

Transformations of Radiation Stresses

In the preceeding sections we have found the horizontal stresses acting on vertical planes through our point oriented parallel and perpendicular to the wave crests and extending from the water surface to the bottom. These compoeents have been found to be principal

stresses.and may be expressed and transformed using the methods of strength of materiais. Two common methods use tensors or the Mohr Circle.

In tensor form, the total stress S may be expressed as:

[:xx

sJ

[ 2 k h + 1 0

J

sinh 2 kh 2" S

₌

₌

E (48) k h 0 sinh 2

The transformation via the Mohr's Circle will be illustrated and used in the following chapter on determination of currents along a coast.

Chapter IV Determination of Currents along a Coast:

Computatdon of current velocities parallel to a coastline is required in order to properly estimate the sand transportation along a coast. Four force components, together, determine the magnitude of the resultant velocity. These forces are:

1. Wave Forces resulting from the radiation stress 2. Tidal Forces

3. Friction Forces on the bottom, always acting to reduce the current velocity

4. Turbulent Forces resulting from differences in velocity between adjacent streamlines.

In the case of a fully developed long-shore current, these four forces will determine a state of dynamic equilibrium with the current velocity being constant.

The determination of each of the force components referred to above will be explained separately in the following sections. The axis and sign conventions used in the following wil1 be the same as those used in the earlier derivation of radiation stress.

1. Wave Forces

The wave forces result from a shear component of the radiation stress. The work of Longuet-Higgins presented earlier has been adapted by Bowen (The Generation of Longshore Currents on a Plane Beach;

Journalof Marine Research, V.27, 1969, p.206-216) for our application (a sloping bottom, shallow water, and breaking waves).

x coastline ~Q)

-'-~§'----~N .

- -~-

~-- .

.- y_---- __

...J

~::E

~."I---.--.---

-_."2--._--Figure 1

Coastal plan view showing axis notation and angles. The positive vertical axes Zand z are directed positive up from an origin at the still water surface elevation. The depth contours are assumed to be parallel to the coastline. The X Y Z axis system is the same as was used in the Radiation Stress derivation. The y axis is parallel to the coast with the x y plane in the still water surface.

Using Mohr's Circle, figure 2, the radiation stress components acting in the x and y directions (perpendicular and parallel to the coast) are: S

=

xx + cos 2 4> ( 1) S

=

yy cos 2 4> (2) S_xy

=

sin 2 4>

=

(Sxx - Syy) sin ~ cos ~ Note S

=

S in magnitude

xy yx

~ is the angle between the wave crests and the shore.

S

Shear Stress

Figure 2 MOHR'S CIRCLE

(3)

Normal stress

Except in zones where the longshore current is not fully developed the shear stress component S yields the only force of importance here.

xy

Remembering that we are now in shallow water, making the proper substitutions from the radiation stress derivation in equation (3) yields: S

=

E

(1

xy 2 _ 1) _Sl.n4>• _cos 4> 2

=

E sin 4>cos 4> (4) 1 2

where E

=

8

p g H , with H equal to the height of the waves at the point in question (in the breaker zone).

As noted in the radiation stress derivation, S has units of force per xy

unit length (length of coastline, here). In order to be consistent with the units of relationships developed later, this must be transformed into a force per unit horizontal area. This can most easily be done by considering the derivative of equation (4) with respect to x. In order to carry this out, the following modifications are made to expose the functions of x:

Since 4>is a function of depth, h, and hence, x, we substitute:

cos 4>_-

=::

cos 4>b_r

o ~

4> < 4>br« 1rad. (Sa)

sin 4>

=

sin

with c

=

wave speed in shallow water

c

=

rg-ï1

sin sin '"_{'f'br (Sb)}

Further, we assume a linear relationship between water depth and wave height within the breaker zone.

8ubstituting (5) and the definition of E in (4) yields:

1 2 2 ~

8

= -

p gA h

v~-xy 8 ~r sin 4>brcos 4>br (6)

having one variable, h •

Differentiating using the chain rule

F

₌

d (8 )

r- dx xx

F

₌

5 p g A2 h3/2 sin _{4>br cos 4>br tan ah} r

16 _hbr1/2

(7)

dh

where tan ah is the bottom slope dx at depth h. This has the desired units.

2. Tidal Forces

From the theory of long waves associated with tides, we find that the tide force per unit of volume may be given by:

a

h'

=-pgay-(8) where

V is a unit water volume

is the tide force acting on the unit volume

a

h'

Cl Y is the slope of the surface of the tide wave. (measured on a profile parallel to the coast).

h' comes from:

h'

=

Z

cos (~ t - K y) (9)

where ~ and K are associated with the tide period and tide wave length respectively.

Clh' "

=

K Z sin (n t - K y) substituted in (8) yields: Cly

,

FT

K Z sin (0 t - K y) (10)

V

=

- p g

In order to compare this to the radiation stress force component, the units of the equation must be modified to give force per unit of horizontal area. This can most directly be accomplished by taking the volume Y as a unit of area, Ab' times the local water depth h.

Equation (10) then becomes:

=

(11)

Multiplying through by h yields: F '

T

-

=

FT

= -

P g h K Z sin (n t - K y)

Ab (12)

where FT now has the desired units (force p.erunit area).

3. Friction Forces

Bijker has derived an expression for the bottom friction force in a zone with waves. This derivation is valid provided that the breaker

angle ~br is less than about 200•

T'

=

T C [ 0.75 ub

+

0.45 (~

V )

1.13

]

(13) where

T' is the bottom shear stress (total)

T is the shear stress caused by a current alone.

c

y2

=

p g

C2

y _{is the stream velocity}

~ =

0.45 K C

rg

K is the von Karmán Constant

=

0.4, yielding

~ =

0.0575 C (in metric units)

ub is the water velocity component along the bottom caused by the waves.

The computation of ~ in the breaker zone is nearly impossible theoretically. However, in order to determine a solution to our problem it will be assumed that a sinusoidal wave still exists.

From short wave theory:

w H cosh (kh) 2 sinh (kh)

(14)

where H is the wave height.

\

with k

=

21T

L

w

=

21T

T and the

approximations for shallow water, (14) becomes

21T

T

=

__;;----,.--2 • __;;----,.--2~ h H c H

=

_{2 h} (15)

where c

=

L

= l""g1i'

for shallow water. T

finally û

_b-2'

-' H

rt:

h (16)

Substituting all of this into equation (13) yields i

[0.75

+

0.45 (~ V H 2 r;;" 1.13 {~ )

]

(17)

[

~

81°1

s/

d S

( (Ll) pUE (L)mOJJ) ~u1AEaI auoz Ja~EaJq a4+ u14+1M anJ+ ÁIIEJaUa~ s1 s141 °saoJoJ ~u1u1EmaJ Ja4+0

OM+ a4+ 0+ aA1+EIaJ IIEms s1 aOJoJ ap1+ a4+ 'saoEId ÁUEm uI °Á+poIaA a4+ 0+ u01+1soddo u1 S+OE sÁEMIE 'ÁIsn01Aqo 'aoJoJ u01+o1JJ a4+ ~aA1+E~au JO aA1+1sod Ja4+1a aq ÁEm aOJoJ ap1+ aq+ ~u01+oaJ1P Á aA1+1sod a4+ u1 +OE sÁEMIE saoJoJ aAEM a41 °saoJoJ u01+o1JJ pUE 'saoJoJ IEp1+ 'saoJoJ aAEM :Jap1suoo 0+ s+uauodmoo aOJoJ aaJ4+ ÁIuO sU1EmaJ aJa4+ 'aIq1~1I~au

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saoJoJ +uaJJno a4+ JO UOS1JEdm00

°pa+oaI~au ÁIa+aIdmoo ÁIIEnsn s1 +1 'aJoJaJa41 °aJa4

s

paJap1suoo saoJoJ aaJ4+ Ja4+0 a4+ 0+ paJEdmoo uaqM IIEms ÁJaA s1 j aOJoJ +uaInqJn+ s14+ JO s+oaJJa aq+ 'UMOU~ MOU s1 SE JEJ sV

(81) Á_V

₌

s_j

: SE uaA1~ aq UEO aOJoJ JEaqs +uaInqJn+ s141 °ÁJOa4+ JaÁEI ÁJEpunoq u1 pasn +E4+ 0+ IaI1EJEd ~u1UOSEaJ E s1 s141 0(S1XE X a4+ ~UOIE) +SEOO a4+ 0+ Jasol0 saAom auo SE sa1JEA +SEOO aq+ 0+

IaI1EJEd +uaJJno a4+ +E4+ +OEJ a4+ mOJJ +lnsaJ saoJoJ +uaInqJn+ a41 .

saoJoj +uaInqJn1 oh

°Á+100IaA JO u01+ounJ pa+Eo1Idmoo Ja4+EJ E s1 +1 'OSIV °au1IaJ04s a4+ mOJJ aouE+s1P 'aoua4 pUE '4+dap Ja+EM JO u01+ounJ E s1 I~ +E4+ a+oN

-Since we want to find the velocity distribution as a function

of distance from the coast, and hence h, it would be convenient to solve (19) for V in terms of h. Examination of the right hand side indicates that this will be extremely dirficult, at best.

Bakker has made simplifications to the right hand side of (19), (egn. 17), in order to more easily obtain a solution. In place of (17) Bakker begins with, (from Bijker):

2 TI

=

P g V C2 (20) (21) as an equivalent to equation (13).

This is simplified by assuming that in the breaker zone ~ ~ » V; this yields: sin w t (22) where ~

=

a

b sin w t Substituting :

~ =

0.45 K C

;g

and taking an average with 11"

1

f

sin

e de

₌

2 11" 11"

yields: p g V 0.45 K C _2~ TI

₌

₌

C2

rg

1T (2)( .45) P K /g'V ~ (23) C Substituting (16) in (23) : TI

=

( 2 )( .45) P K

r;

V H

rg

=

C 2

;-ti

1T .45 P K g V H 1T

rb'

C (24)

Using (5c) this becomes :

TI

=

.45 P K g V A

Iïl

1T C

(25)

or, in another form

TI

=

P g3/2 Ç, A

/h'v

ff C2

(26)

This is now much simpier to work with than equation (17).

We can now return to the problem at hand --_comparison of the

forces -- byequating (25) with (7),

5 P g

sin ~br cos ~br tan ah

=

.45 P K g V A

l1i' (

27 )

1T C

16 h 1/2

br

Solving this for V as a function of h yields:

V

=

5 1T A h C

(16)(.45)K hbr1/2

sin ~br cos ~br tan ah (28)

Svasek and Koele have found that for the Dutch coast A

=

0.4 to 0.5

for the significant wave and that A ~ 0.3 for the root-mean-square

•

Substituting for the constants

A

=

0.3

K

=

0.4

we get

v =

1.63 he'_{~~2 S1n ~br cos ~br tan ah} (29)

which gives the velocity distribution as a function of distance from

the coast, provided that hand tan ah' the beach profile parameters,

are known.

It should be pointed out that in the example just considered, only the wave forces and the friction forces were considered. It is entirely possible that under certain circumstances the tide force, for example, might also be important. This could obviously be added to the analysis, but shall not be done here.

Result of this development

We have determined the velocity profile along a horizontal line extendipg out f~om the beach. This velocity profile will be used later

in conjunction with a sand transport formula in order to develop a

sand transport profile.

It has been found that when this is done for the Dutch coast, for example, the sand transport at a distance of 200 m from the coast is about three times as much as at a distance of 600 m. This is a result

of the combination of increased longshore current velocity and increased

wave forces on the bottom material which cause more severe stirring. A sand transport formula will be developed in the following chapter.

Chapter V Longshore Sand Transportation caused by Waves and Current

1. Emperical Formu1as derived from an energy balance

For some time now, sand transport along a coast has been related in some way to the component of the wave energy along an axis parallel to the coast. These methods have found wide application and are based upon·sound physical reasoning.

In its most general form, such a relationship is:

S

=

A E

a (1)

where

S is the total sand transported along the coast.

E is the energy flux component parallel to the coast, measured a

in the breaker zone.

A is a proportionality constant.

Unfortunately, in this purely emperical formula, A is not dimension-less but has dimensions [ L T2 M-1

1.

The energy flux component, E , is given by: a

E

=

E K2 sin ~b cos ~b

a 0 r (2)

where:

~b is the angle between the breaking wave crests and the beach line.

E is the energy flux in deep water in the direction of wave o

propagation.

K is the refraction coefficient. r

From short wave theory,

E

=

o

1

with

p

=

the water density

g

₌

acceleration of gravity H

₌

deep water wave height

0

c

₌

wave speed in deep water

0

Back substitution of (3) in (2) and then in (1) yields

(4)

where all of the constants have been combined, and the constant, A, evaluated using data such as that available from CERC. Conveniently, a bit of dimensional analysis reveals that the coefficient 0.014 is now dimensionless.

Formula (4) although reasonably trustworthy, does have a few limitations. These are:

a. Only the total sand transport is computed. No information on the sand transport profile along a line perpendicular to the coast can be obtained. Especially on coasts subjected to spilling breakers, or where more than one offshore bank is present, this limitation can be serious.

b. This formula is independent of the type or size of bottom material.

This is, of course, not true on real beaches. This formula is still valid, however, provided that it is used only for beach materials similar to that for which it is derived, namely, uniform sand ranging in diameter from 0.2 to 0.5 mmo

c. The slope of the beach does not enter the equation.

d. This formula computes transport caused by waves alone.

Influences of superimposed currents are not considered. This limitation can be very important in river deltas, for example.

e. This formula may not be applied to shoals, dumping grounds, or near dredged channels.