CROSS-SWELL - A comparison of mathematical derivations with the results of laboratory tests

ROS

Ame.aus-nows. Feu 015 - 781838

A comparison of mathematical derivations with the

results of laboratory tests

Dik Ludikhuize

Henk Jan Verhagen

a

CROSS

SWELL

A comparison of mathematical derivations with the

results of laboratory tests

.Dik Ludikhuize Herd( Jan Verhagen =

January, 1381

-TABLE OF CONTENTS

LIST OF SYMBOLS

.1. SUMMARY .

! I!! '.° '!'

2. INTRODUCTION 11. ,e4 lee 2

3_ THEORETICAL DERIVATIONS ₃

3

3.,a4 General theory ,

3,bt. Special case theory . AA _{'4 1.0}

4. DIFFRACTOM MODELS . . . . . . . 11 0 .

,

Ek tit i.

,

,ii. , 13 16 5. EXPERIMENTAL RESULT'S i . I.

5,a. DescriptiOn Of the experinents 5.b. Measurement methods , . . !a .0

5.c, Wave height measurements r,. 4

., % 1

,

.-/ .: ,"' ni

,

.

4.

.

*

.

.a; ,.!! ., «3 eie ..: e. 1. 4 16 17' 18

,5.d

St-up c

neasiuTements 4 .

.

f* ,i6 4 A 23

5,e.

Various abservati,ofit, . 0 I 1.E qa 41 ht 4. 1 It5 30

6. CONCLUSIONS .

'1,! 32

6.a. Wave heights 1, 55.5 iii 51, II5 We

6-b, Breaker1ndex 4 ,k, 1 15 :1 1 "W ... oa il ' :!! ,N '1. *.

.

32, 32 6.c. Set-up , , , . , . A 0! 4; ea 4.

.

.. a 39! 32 1

CONTENTS (continued)

Page

REFERENCES 34

PHOTOS

ANNEXES I Calculation of the energy of cross-swell

II Radiation stress caused by cross-swell

Vector of unity

E Wave energy

g Acceleration of gravity

Water depth

Wave height

17I Mean wave height

k Wave number (27/X)

KD Diffraction coefficient

Wave length within the cross-swell system

S Radiation stress

T Wave period

Velocity perpendicular to the coast Velocity parallel to the coast

Y Breakerindex

6 Phase angle

Surface elevation

_

Mean surface elevation, set-up

X Original wave length

P Density of water

)

a Standard deviation

T Friction force

(I) Direction of the wave (angle between wave crest and beach)

1. SUMMARY

The results of some initial laboratory tests regarding cross-swell are compared with theoretical considerations.

A first order linear wave model describes the surface of cross-swell quite accurately.

The breakerindex proved to be nearly equal to the breakerindex of standing waves.

Set-up is calculated with the theory of radiation stress.

A set-up formula for cross-swell is presented. Test results fit rather well with the data from this formula.

The tests were executed in the wave basin of the Laboratory of Fluid Mechanics, Delft University of Technology.

2. INTRODUCTION

The phenomenon of cross-swell occurs when two wave fields with different wave height, wave period and direction come together

in the same area. When such a cross-swell reaches the coast

it is not possible to calculate lcngshore current (and thus littoral sediment transport) with the usual formula. In

previous reports (Ludikhuize & Verhagen 1973) some theoretical derivations regarding cross-swell were presented in order to calculate littoral sediment transport.

These derivations are summarized in this report.

Theoretical derivations and considerations about cross-swell were not found in literature. There was no proof at all that the derivation presented in Ludikhuize & Verhagen 1278 give-an acceptable description of the phenomenon.

One of the basic assumptions is that both wave fields do not influence each other. Especially this assumption had to be proved

experimentally. To prove this very extensive laboratory tests have to be made.

However, the authors got the opportunity to do some short tests in the Laboratory of Fluid Mechanics of the Deift University of

Technology. _{Because of the limited time available (one and a half} week) certainly not all the aspects of this phenomenon were examined.

But these tests give some indications about cross-swell.

This report can be regarded as a starting point for further investigations, theoretical considerations and derivations on the subject of cross-swell.

The authors wish to thank ir. _{P. Visser of the Laboratory of Fluid} Mechanics for his comments on this report.

3. THEORETICAL DERIVATIONS

3.a. General Theory

In this chapter the theory of cross-swell will be discussed. In

the derivation of this theory several assumptions have been made, viz.: the two wave fields do not influence each other

each wave field consists of stationary, long crested, regular waves each wave field can be described by the linear Airy-wave theory

The consequence of these assumptions is that at a certain point the surface elevation is the sum of the two original surface elevations (of the original wave fields).

cocyjci,4e

Fig. 3.1.

The surface can be described by:

3

-In which:

6

n'

'

n1 + n2 = al sin (w1t

-I:PI)

+ a2 sin(w2t

- -1<-2.; + 6)

total surface elevation

amplitude of the original waves

angular frequency of the original waves

wave number and wave direction of the original waves

in which A is the local wave length of the original

4-waves and e the vector of unity in the original direction of propagation.

the phase angle, to fit the choosen coordinate system

The internal product

k.p

can be written for a carthesian coordinate system as:

= kx x + k

y

The surface elevation is visualised in fig. 3.2. The maximum elevation (in C) is the sum of the original amplitudes:

nmax

=a

1 +

a2

This maximum will only occur at points, and not along lines, like crest lines in long crested waves. Because of the special shape of the surface the term 'wave height has to be defined somewhat more detailed than in case of long crested waves. For cross-swell the wave height is defined as the difference between the maximum surface elevation in the wave field and the minimum surface elevation (i.e. a negative elevation) of the same field but not necessarily in the same point.

27

=. .

e 2 X2 2 (3 . ) (3 . 2 ) (3.3)

Fig. 3.2.

The distance between two wave crests can be measured in the direction of propagation (L) or transverse to this direction (Ls). This

second distance is called

Ls because in this direction the wave system

resembles somewhat a standing Wave,

HT = (al + a2) - (-al -a2) = 2a1 + 2a2 = H1 + H2 (3.4)

Thus, the 'wave height' in cross-swell is the sum of the two original wave heights.

Fig,. :3

The distance L

is:

P 2'

L2 22L11_2

i L . A + )1 s r ncp nci 1 2 (L L - 21L L

cos)

s in24) 1 2 2 2 n co SO

(3.5)

(3.6)

The distance L

L2 , 2 2L1 L2

Ls2

=simp )

"4- (si.in¢)2 2 cos¢

sin 4)

.12

(L12 + L24 +

2L12

cos)

tin

ci) =

-is:

L1 +

When the incoming waves are perpendicular (c'i, = 900) then

2

L = L12 + L22 and also Ls2 = L12

2 + L2'

L and _{Ls are} called virtual wave lengths, because the direction of wave propagation does not necessarily match to the direction of L and

Ls. They match only when L1 = L2.

This will be discussed in the next chapter.

The energy of cross-swell is:

1

E = _{pg (H12 + H22)}

1

and not E =pg

HT2. The derivation of this formula is presented

in Annex 1 to this report.

Another important aspect is the radiation stress in cross-swell. In Annex II is shown that:

S

_{= (S)1 + (Sxx)2}

xx xx

Syy =

(S) +(S

_yy1

_yy)2

S _{= (Sxy)1 + (Sxy)2} xy

These formulae are derived under the following restrictions: The periods of both wave fields are not identical

The radiation stresses are an average over an interval T (see below).

The longshore current and the set-up can be calculated from the averaged radiation stresses. At a fixed point the radidtion stress itself

will

fluctuate in time. This fluctuation has a certain return period T, which depends on the periods of the

two wave fields.

(3.7) (3.8) (3.9) (3.10) -7.

This period T is the smallest whole multiple of T/ and

T2

T = a

T1 = b T2

in which a and b are integers which are as small as possible.

When the waves reach the coast breaking will occur. Parallel to the coast the envelope of the surface resembles somewhat a standing wave,

so it can be expected that the breakerindex (y, relation between water depth and wave height; H = yh) will be higher than in case of only one wave field.

. H 27

The breaking criterion for standing waves Is = 0.22 tanh h), which leads to H =

1.38

h in shallow water.

When the wave heights of both waves differ very much, it is possible that the waves of each field break independently of each other. This is certainly not true when the wave heights of both fields are almost identical.

The position of the mean water level (set-up or set-down) and the currehts in x- and y-direction can be calculated by solving the

following equations: Xy aSyy 3v 3n + 71) + u + v + a + Dx _BY

-y

x ay turbulence terms + Tby = 0 aS aS Du aU an\ XX xy p(h

+ h)k-- + U

+ v+ g) +

+ + .

at

ax ay ax ax ay . + turbulence terms + Tbx = 0 (3.11) (3.12) (3.13) p(h YY

in which:

Tby' Tbx

= position of the mean water level = velocity perpendicular to the coast = velocity parallel to the coast = mean friction forces

These equations can be simplified assuming that:

the velocity u, perpendicular to the coast, is zero:

3u n u at Du Du

=0

--=0

3x

the velocity v, parallel to the coast is constant in time and is constant along the y-oxis:

the turbulence terms are negligible.

The equations 3.12 and 3.13 become:

DS 3n pg (h n) + + IY

+-by

= 0

3y 3x ay

3S3S

pg (h dr1 XX xy

-+bx

= 0 ' 3x 3x 3y

Note: The second simplification is not correct. From the laboratory tests it appeared that the velocity in y-direction is certainly

not constant. This simplification has to be a subject for further studies. 9. (3.14) (3.15)

at

xy +

h,

Near the coast ( _/L < 0.5) the functions of S , S and S

xy xx _YY

become very complex because of:

shoaling (both wave height and wave length will change) refraction (the angle of incidence of both wave fields will change)

It is beyond the scope of this study to describe the effects of shoaling and refraction for cross-swell mathematically.

In this study is tried to show that for a cross-swell situation the generally used formula:

nmax = 165 yHb (3.16)

is not correct.

3.b. Special case theory

In this chapter is discussed the theory of a special case of cross-swell, viz, when the wave periods of the two wave fields are identical.

This special case is investigated in the laboratory tests.

In this case the surface extention (n) becomes a function of the location. This can be seen by means of formula 3.1 which becomes:

n = ni + n2 = al sin (wt - + a2 sin (wt - 1(-2.; + d)

(wave number 1ZI and angular frequency (w) of both wave fields are identical).

The phase-lag between n1 and n2 depends only on kl.p and

which are functions of x and y. The surface extent ion n at one place becomes a sinusoidal function of the time; it will repeat itself after T (T = 21T-) seconds, thus after a period T the elevation (:). the surface will be the same as it was a period T before.

(3.17)

-In this pattern lines with maximum surface elevations (n

max = a1 + a2) can be distinguished. At these lines the phase-lag between n1 and n2

is constantly zero. Also lines with minimum surface elevations can be distinguished (nmin = lal -

a21),

for these lines the phase-lag

is constantly 7 (see fig. 3.4).

(if al = a2 then the maximum surface elevation at 'phase = 0'-lines is 2a1' at 'phase

= 7'

the surface elevation is zero).

The crest and trough points propagate along lines which are the same as the 'phase = zero-lines, because the celerities of both wave

systems ara identical. Also the other points of the surface propagate along the phase-lines.

The energy and radiation-stresses can also be calculated and it appears that they are place-dependent.

The radiation-stresses cannot be described with simple relations as in the general case. In Annex gill these radiation-stress formulae are derivated (form 29-32 in that Annex).

In general the results can be summarized as:

S = f (H1'

H2' cf,

cl)2'

(p - q))

in which (p - q) varies from

-ft

to +7 and represents the location in respect to the phase-lines in fig. 3.4.

The radiation-stress formula is sinusoidal in (p - q) and therefore a sinusoidal set-up distribution along the coast and a sinusoidal

longshore velocity distribution can be expected.

4. DIFFRACTION MODELS

The tests were executed in a basin in which the two-wave-fields were generated by a snake-type wave generator.

Each half of this wave generator produced a wave field. Because the width of the generated wave field was smaller than the width of

the basin, diffraction occurred.

Two diffraction models are made for the analysis of the measurements. By means of these two diffraction models diffraction coefficients and phases can be computed.

The two diffraction models are based on the following assumptions:

constant depth (which is true until the toe of the slope; fig. 5.1). both wave fields do not influence each other

no reflection occurs against the sides or against the slope in the diffraction model of each wave field it is assumed that part of the wave board which produced the other wave field was an impermeable wall with a fixed position.

Diffraction model I

13.

In this diffraction model is assumed that the width of the basin was much larger (theoretical infinitively times larger) than the width of the wave generator.

Because of the above-mentioned assumptions the diffraction of each wave field can be calculated separately. _{In order to get the total} wave heights, the results of diffraction models of the two wave

Fig. 4.1.

Diffraction model II

/

r--L_ 1

In this model also two diffracted wave fields are added (see fig. 4.2) In wave field A is assumed an impermeable wall at the left side of the waves, and perpendicular to the wave crest, while in wave field B an impermeable wall is assumed at the right side.

In both fields the other part of the wave board is assumed to be inactive and the impermeable wall which guides the other wave field is neglected.

Remark: In the model tests these two impermeable walls were placed in the wave basin, but they were not exactly perpendicular to the wave crests.

VC7Vefea /2

VOW, 2,Orn/,

a

Ivor, r" 15. Fig. 4.2.

For both diffraction models (I and II) the diffraction coefficients and phases were calculated with a computer program based _{on the}

theory of Sommerfeld, using an approximation of the Fresnel-integrals. The maximum amplitude of the surface elevation at certain points

is also calculated in this computer program in the following way:

a = Kpi . a1 . sin (wt - Phase 1) + KD2 a2 . sin (wt - Phase 2) in which: KD1' KD2 = diffraction coefficients (-)

Phase 1, Phase 2 = phase differences caused by diffraction (rad)

al' a2 = amplitudes of incoming waves (m)

= angular frequency (rad/sec)

= time (sec)

I

5. EXPERIMENTAL RESULTS

5.a. Description of the experiments

The experiments were made in the large wave basin of the Laboratory of Fluid Mechanics, Delft University of Technology. For the layout of

this basin see fig. 5.1. The main dimensions of the experiment area were approx. 15 x 34 m. The water depth was 0.40 m. Along one

long side a snake-type wave generator was available. This generator was adjusted in such a way that two wave fields were generated,

both directed to the centre of the basin. Two series of testst were run:

we-we

,

Fig. 5.2.

a' = (P2 = 43°

b. pi = 43°, p2 = 28°

On the other side of the basin was a 1:10 slope. The waves were guided by two guide boards on both sides of the basin.

The basin had a fixed cemented bottom.

The period of the waves was in all the tests constantly 1.62 sec. During the test the wave height was measured in various places, and the set-up was measured in two rays perpendicular to the slope. Additional qualitative observations were made of the longshore velocities with dye.

cle75 T - WAVE BOARD-34 1

12.8

16.40 : I 6 11 16 21 26 , 31 36 I. 1 I I I I... ....-...!:!....i. . :: ... :::: . ..,... . i ...:_,:.,:._:.:._. . : . : I : 43 . MATE IA 1:::.1::: . : . j-: T... , ' ... . .... 7-1; : T7-7 . 77 '; : :7: : I . ... . ; . ' ; DST OF THE WP, E BOARD ' 1 ' 1 '.. .! 1 RIGHT RAY . I :. I ; RAY-0 1..;:l.:1" . ' .:-. . . : . !:, .

..

1:-::,..::: ::::; . ::: : ' ::: . .1.... ; 1... I 1 ; !

:

8.1.' 53 ' ;;;'58-1: 62 67 1.; 7 .:-, 77 1 A2. 82 92 r.: 97

102 107 1./12/17 120

RAY I , , . 1 . RAY .II H 1 : 1-7--" RAY TrTITT ; i . , 1 : !I : ; t I I '- -'' RAY .2=E; i i -i: .. . : ... , !

'IL'

1 '. :I' RP.,Y..1G2 ....-1::-.1 ... HI .(1 . 1. 1 1 .1 i ; .J''' 4--; RAY . ... ....:::,_: II: : i : ::::' :::;:.

_

. ; 1 :1 : , ,::!: ! : : : . . . -RAY UT ,

TOE !OF THE SLOP

. .... . ... ... . ;.! ": . .. 7: T. T. 17-7f 7:, . . .. .. . ! . I. . . I .

'01

; 13 09 :

SCHEMATIC PLAN OF THE WAVE -:; BASN (sciaie 1:125

0-.Vi1 LEFT RAY

; NIIDST RAY' ::I ; ! : : ! ; ... : : , . . . IN DA , ... . .., ; --I. . .77".

..,r.,.. -, I .."'. :: . 0 :I _--. 15 . . r ' ::.: . L : t::: . -::-1.-7-'. :

5.b. Measurement methods

The wave heights were measured with a wave height surveyor during a period of 30 seconds (for each point). This period is chosen,

because during initial tests appeared that this measurement period gave reliable information with an acceptable standard deviation

of + 5%. In the basin certain measurement-rays were chosen,

viz. (see fig. 5.1):

Ray 0 : 4.04 m from the wave board

Ray I : 6.04 m from the wave board

The Rays II, IV, VIII, XII, XVI, XVIII and XX are situated on the connection lines between the points number 2, 4, 8, 12, 16, 18 and 20 of the set-up measurement rays.

Between the left set-up measurement ray (L.R.) and the right set-up measurement ray (R.R.) the distance between the wave height

measurement points was 20 cm. In the remaining part of the basin this distance was mostly 60 cm.

The set-up was measured through holes (no. 1 up to and including 25)

in the set-up measurement rays. These holes are connected with small tubes, placed against one .side of the basin. The water level in these tubes, represents the water level in the basin at that point. The water level is measured in a pot, because of the water level in

the (small) tubes varied due to wave pressures. The surface of this pot is many times larger than the cross-section of a tube so

damping will occur and the water level will stay almost constant. In the right set-up measurement ray (R.R.) set-up could not be measured, because several holes were clogged.

18.

With the mean depth and the wave height (at that point) the

breakerindex y can be calculated. Because it was not possible to measure the set-up in the Right Ray, it was also not very useful

to measure the wave height in that Ray. The wave height measurements of Ray IV until XX are used for the determination of the breakerdepth

the measurements of the other rays (0 up to and including II) for checking the envelope of the surface extensions with the computed one (which could not be done with Ray IV until XX because of refraction).

The set-up measurements are also important for the indication that, in this case, set-up does not depend on breakerheight and index

5

b '

) but probably on the radiation stress caused by the two

lb

wave systems.

5.c. Wave height measurements

The measured wave heights are compared with the computed wave heights. When the formula, which can be used to compute the wave height at a certain point (form 3.1. and 3.4.), is examined, it appears that

the two incoming wave heights Hi and H2 are not known. Because of technical reasons, a part of the wave board had to be removed, these wave heights could not be measured. The computation is, therefore, reversed: the wave heights H1 and H2 are calculated by means of the measured wave height. The problem rises that two

variables (H1 and H2) had to be calculated from the following equation:

The maximum value of the left side of the equation depends on the ratio between H1 and H2. In the computations it is assumed that this ratio is known, viz.:

ot When the angles of incidence of both systems are absolute the same (F11 = - F12) than H1 = H2

When Fl1 - Fl2 than:

H2 = H1 ./cos (FI1)/cos (FI2)

(FI1 and Fl2 are angles of incidence of both systems).

This can be explained as follows:

Assuming that the energy is equally distributed over the wave board the energy-flux through bl and b2 is the

same.

2 1 , 2 ,

/8 Pg ml .U1 = 18 Pg m2 .°2

-3-b1= b.cos (FI1 ) ; _{b2 = b.cos (FI2)}

H2 = H1 . /cos (F11)/cos (FI2)

Two types of tests were made:

the wave systems had opposite angles of incidence, viz. +430,

43

the wave systems had different angles of incidence, viz. +280 ,

-43°

(Both tests

with

the same wave period,

T =

1.62 sec).

At first the incoming wave heights H1 (respectively H2) are computed; he results are presented in table 1.

Table 1:

(+43°, -43°)

Table 2:

20.

When the column of the maximum wave heights is examined it appears that very high wave heights

Oil

_{respectively H2'} in this case H1 = H2) are

computed. This occurs mostly on 'phase = 7'-lines. These are locations where the surface extension is theoretically almost zero, which does not

happen in these tests (this is explained later). The very high wave heights are therefore excluded; than the following results are obtained:

(+43°, -4-°;

3 pxcludiny Ray I: no. 10, 111; Ray II:

45, 55, 67

and 77)

Ray Points no. Amount of points Mean wave height

(ri)

Standard deviation (0) CY/74*100% H max Hmin 0

6- 33

10 7.31

2.84

38.8

12.03

3.46

0

112- 85

10

_7.19

1.66

23.09.22 4.84

I

1- 37

13 8.66 5.71

65.9

21.95

1.97

1 81-120 14

8.22

4.91

59.8

22.941 3.32

II

38- 80

43 14.81 32.33

218.4

208.51

3.07

Tot. All above 90 11.22

-

-

208.51

1.97

Ray Points no. Amount of points Mean wave height iF-1) Standard deviation (a) a /D1:100% max Hmin 0

6- 33

10 7.31

2.34

38.8

12.03

3.46

0

112- 85

10

7.19

1.66

23.09.22 4.84

I

1- 37

12 7.55

4.27

56.5

17.12 1.97 I 81-120 13 7.09

2.59

36.5

12.44

3.32

II

38- 80

39

7.44

2.10

28.3

13.60

3.07

' Tot. All points 814 7.36 =2.57 34.9 17.12

1.97

When the wave heights of Ray 0: no. 53; Ray I: no

39, 51, 62, 63

and Ray II: 61 are excluded the following result is obtained:

Table 4:

(+28/-43);

using H2 = H1 * /cos (RI/cos

(FL));

excluded points

H

mn

3.69

3.03

.37

6.03

2.92

3.56

4.69

3.51

2.92

Ray Points no. Amount of points Mean wave height (H) Standard deviation (c) c)-/1.4*100%., H Hmin 0 33- 6 10 7.65

4.17

54.4

18.36

3.69

0

36- 83

48

9.04

7.50

82.9

54.00

3.03

0

85-112

10 7.11

0.89

12.5

8.46

6.37

0 85-118 12 7.30

0.98

13.4

9.36

6.03

I

1- 37

13 7.66

2.67

34.8

12.50

2.92

I

38- 80

44 9.08

6.54

72.0

32.05, 3.56

I 84-120 13 7.96 2.51

31.5

11.901 4.69

II

38- 81

44 8.29 4.20

50.6

30.351 3.51

Tot. All points 194

8.44

-

-

54.00

2.92

1

1

Ray Points no. Amount of points Mean wave height (17I) Standard deviation (a) c7/*100% Hmax 0

33-

6 10 _7.65

4.17

54.4

18.36

0

36- 83

47

8.08

3.48

143.0 17.17 0 85-112 10 7.11

0.89

12.5

8.46

0 85-118 12 7.30

0.98

13.4

9.36

I

1- 37

13 7.66

2.67

34.8

12.50

I

33- 30

40

7.29

2.17

29.7

15.22 i 84-120 13 7.96 2.51

31.5

11.90 II

38- 81

43 7.78

2.49

₇₁

,,.

0 16.24

Tot. All points 188 7.68 _=2.67

_34.8

18.36

Tabel 3:

(+281-43°);

using

_{H2 = H1 *}

/cos

22.

In the case with different angles of incidence the incoming wave heights are also calculated assuming HI is H2, the result becomes:

Table 5:

_(+281-43;

using H1 = H2)

When chc wave heights of Ray 0: no.

53, 63, 64;

Ray I: 51,

62, 63;

and Ray II: 61 are excluded the following result is obtained:

Table 6:

(+28/-43;

using H1 = ; excluded points)

Ray Points no. Amount of points Mean wave height (RI) Standard deviation (a) a

/100%

Hmax Hmin 0

33-

6 10 6.90 3.51

50.9

15.84

3.45

0

36- 83

48 8.30

4.67

56.3

25.84

2.86

0 85-112 10

7.09

0.93

13.28.56 5.37

0 85-113 12 7.29 1.03

14.19.44 5.95

1

1- 37

13

6.97

2.23

32.7 10.93

5.47

I

38- 81

44

8.44

5.31

62.9

31.62

3.43

1 84-120 13 8.00

2.65

33.1

12.381 4.64

11 38-81 44 7.95 3.55

44.6

25.41

3.33

Tot. All points 194

_7.95

=3.98 50.1

31.62

2.86

1

Ray Points no. Amount of points Mean wave height (P) Standard deviation (a) a/D*100% Hmax _ . Hmin 0 33- 6 10

6.90

3.51

50.9

15.84

3.45

0

36- 83

45 7.43 3.15

42.3

15.59

2.86

0 35-112 10 7.09

0.93

13.28.56 5.37

0 85-118 12 7.29 1.03

14.19.44 5.95

1 1- 37 13 6.97

2.28

32.7 10.93

5.47

1

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41 7.21

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33.4

14.52

3.43

1 84-120 13 8.00

2.65

33.1 12.38

4.64

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31.0 14.29

3.33

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2.86

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By means of the above tables the wave heights of both systems are determined. The wave height of the systems with an angle of incidence of plus or minus 430 is valued at 7.5 cm.

With this mean wave height (of 7.5 cm) the wave heights are computed (with formula 3.1) in two ways:

lit Assuming H1 = H2

* Assuming H2 = H1 . icos(F11)/cos(F12)

The results of the above calculation and the measured wave height are presented in figures 5.4. until 5.15; it appeared that the

differences between both calculation methods was very small, so only one line of computed wave heights is drawn.

The distances between two points with minimum wave heights (or maximum wave heights) is the distance Ls (see chapter 3.a.). This distance can be calculated (theoretically) for both cases:

Ls

(+43°/-43°) = 4.25 m

L (4.230/_430)

5.00 m

The measured values (in the middle area between L.R. and R.R.) are:

-5.d. Set-up measurements

The set-up was measured only in case of the wave systems with different angles of incidence (+28°/-43°). First the breakerindex (mean) will

be valued, using the mean depths and the wave heights of Ray XVIII and

XX. The wave heights of Ray IV until XX are presented in fig. 5.16

until 5.21. The mean depths are calculated by means of the measured (mean) set-up values.

Ls (+43°7-43°)

= 4.15 m

, i : i

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-.. -...-t -..-...1:- .:::. . ..i. ... .... , -.1-:::-:-:-- :t: . I. 1 . .... .. _{..7t..-. .... :-} ... , . .. ; ... . ....1 ... ... ... .. . ... . .. ...,. ...,.. ... ., , I ::- ... ; 1 . ' ' ' I . ' . ' '' ' .... . :. -.... " ' . ... . ... : ' -;L: .. : 1 ' .' ' ' '''''. ' '. 777 ... ...7- .. .. ... ... i ...:.:: 7:: : : -. :I . ::: ... :1.. : -: :: 7 . ii I. ...iLid ... . .,

...

::. I : .. . .: !. ...*.: L. .. .. '', .. :-_:_:.:: .

:::"-1::::

. .. ... 4.- ... .... .. .._..::71::::--..1...,.. ... ,,..

.7...=....

, :: .. .-...-1 .. :: . ; "I: :: ,.... .. ...1. -.: ... .:.-, 7111 ,... ,. - I .. .. .. ..._-_,..4 .1- --. .1 .: .... , 1- i ... .F: . :: 7.:: 1 ---..-1. ... . -- --- .---.777---.- 7-7: -7-1 : ' .. .... :::: :1::: : ... .._'- ,. ... , ....;--.Ar.::: _{... 1::... .. ,.} I. ... - -- ----

1:.::.:

.. .,: 1 ...1.,...! I . ... .._. ..: ... ,. ...1: ; I 11. I

.... ... ..

..-1.:::::::. ::.;:i:...1 ... ..- .. 1-:::.:/-...:. . ., - - . :Li::: . : . : . : ::-..:-.:1:::... :7". ;. .. ...

L

.4 _._..._ ... . . ..,.... , I.::: .: ... 1 ... .. -... 7-' ...1.... . ...

.

-... . i, .:. .-.1, - 1 .,. . I -: ... . ::::t-.4,:::. _{L....,1 3} . --:

-

. I...:.' 0 :1 ' 6 0

f

-: :..6.2...,-.4._._. ... __ . ... . t.., -_. -/ 1 V . '7.--

:.--. -

T . i ... - --7-7.--- - .:.. :::. it... -,-. .. ... . .... h 4 .. " 1,7 ' 01, : RA y yr.; ;I/ 3. ) LI 'i") 6.4 t .ot r d .,a --.; . :;°"1,;.\.) . . . ; :: ... : : . 7. !! . . :7,1. - ; ... 4 -iI

..

.1 4 , l' --4.!* e +" .k* 1 -.---- - ::: 77' " 11., .

_

_* . ,.. ../ ' 11/ a 11 f ; ""' .:-.- ... . 7 .!'' -t 2. 0

.1: ..._{. .} e . .. :7-77:77:7 77.7 ... . 7:17: .. :1: ;1 ... ,. -

...

i

--....

I .1 1 * ":! ` ₄ .. .... .

I--- '

T :.:.: .... I

" "

' :: -;:: :. ::::t : !. -,.::: ... : ... ... .. ,. ... ...; ...II ... 1- - ----' :.:1-... 7 ... ... 7.7.7.,....:::.7,..:1..1f v ... 1 ' ... : ! : : i,. : : . 7: :1: :.-'. -1- :::: . :.::::-:r.::',I.:...,_4: _.__...-- _-7.-7:--,---....! ... .,,.. ,... ..:.,.... .... ;

..L... ,....k ...

. ...17.:...t.., . ___-.. I - j _ ., ... ..., . ...;,..,.. ... ... ... ;.... ... . ... 1 . ., ... I.:. : =1....--. ..t. .:. I ..'t :H:11 ...

"

, - 7. . . I !T to 1 4 , iT, t: I 1. . . .. . . .. ..,i,.,-...: .1° :11 TTT' A .1 . ,. .. . 4 ! k 1. t b.0 ;

-

c- -

fc,

t ;-r --- .1 : -1 ....1 . I " : ... ' ' .. . ' ' ' ... : : : ' : : : : :r .:

t

,

--I -7.-( L. K) 5

Left set-up measurement

ray (L.R.):

Middle set-up measurement ray (lt..

214.

,

Measurement Amount of Set-up"fl.

a_, a

Actual I a_

point measurements (cm) waterdepth

j

ih

fl. cT111.4 1

T*100%

( s? 4 (cm) 1 nt 4.1na , _-1 ₃ ! 01 0.10 i

4

1 !

40.03

,

-

39.13

-,

1 1

8

1 ='0.03 ---- 51,.07

-12.

i

-0.09

,

-

2291

16 1 -70,25 -- -- 14,75 ,.., 18 1

-0.27

-

-

10.73

-L

-20 H ₅ ,H3,20

0.02

1

t

6.70

6.3% 1

2i

+0.85

0.08

9 % ,

5.85

n

6

41.66

0.07

, 4 %

4.66

1 1...5% 23 ₅

_+2.57

_0.09

' 3.6%

3.57

2.6% , 24 5

+3.86

O'.05 H 1.3% , ,

2-86

, 25

H21

i

+4.64

J 0.04

0.9% ' 1.64

Measurement Amount of Set-up Fi c. .1 a; . Actual

point. measuremeirits:. (cm), (cm)n waterdepth h

. h (cm) L' .-- *1100,% n/ *1001%

a-4 f

3H 0

_10.10 , ,,. 40.00, 0.5% 1' --= ' - 39.05 '

-a

, . 12 1

-0.03

,

1

22.87

.1 - -, 31.00 , il , 16, 1

-0.03

-

-1j5.,G7 -, 18 1 ,

-0.11

,

,

10.89

,, 20 5

-0.21

0.06

27 %

6.69

10.8% , 211 7

-0.11

0.08

69 %

'4.84

1.6% 22 5

+0.54

0.03 iD %

3.34

H 1.0% 23 6 H

+1.22

0.20

16 %

2.22

₉ % 24 8 +2.11 0.21 9 % 1

i I

19 25 20

+3.30

0.20

6, %

0.30

, 67 % - 40.00 -- -- -6 9 1.3% 0

-

-0 -- -- - -- -

-Table 7: ratios between wave height and water depth in Ray XVIII

Point

Ray XVIII (set-up point 18) Measured

wave height

Ray XX (set-up point 20)

Point Measured wave height H/h 33

6.42

(0.95)

34

6.35

(0.94)

35 6.60

(0.90)

36

8.04

'1.19'

37 10.74

'1.59'

38

8.22

'1.22'

39

7.59

'1.12'

40

7.23

'1.07'

41

8.89

11.32'

42

8.98

_'1.33'

43

8.57

'1.27'

44

7.80

'1.16' 45

4.34

(0.64)

46

5.35

(0.79)

47

7.89

'1.17'

48

_9.98

'1.48'

49

9.37

'1.39'

50

8.83

'1.31'

51

8.23

'1.22'

52

8.69

'1.29'

53

8.53

'1.26' 54

8.82

'1.31'

55

9.42

'1.4o'

56 10.74

'1.59'

57

9.90

'1.47'

58

6.29

(0.93)

59

3.77

(0.56)

60

5.84

(0.87)

61

7.10

(1.05)

62

7.60

'1.13'

33

6.19

(0.58)

34 5.95

(0.55)

35

6.52

(0.61)

36

7.65

(0.7i)

37

9.22

(0.80)

38 14.74

'1.37'

39 15.00

1.40'

40 14.38

'1.34'

41 14.67

'1.36'

42 14.49

'1.35'

43 12.73

'1.18'

44

9.66

(0.90)

45

_8.39

(0.78)

46 6.61

(0.61)

47

8.64

(0.8o)

48

9.13

(o.85)

49 10.59

(0.99)

50 11.27

(1.05)

51 12.42

'1.16'

52 13.99

'1.30'

53 14.69

'1.37'

54 14.54

'1.35'

55 14.20

'1.32'

56 10.11

(o.94)

57 5.73

(0.53)

58

5.85

(0.54)

59

3.36

(0.31)

60

5.07

(0.47)

61

6.65

(0.62)

62

7.84

(0.73)

26.

By means of figures in above tables the water depth at Ray XVIII is valued at 10.8 cm (1- 1.5%), at Ray XX 6.7 cm (+ 2%). The breaker index,

breaker 'height

Ob)

devided by breaker depth (hb) can be 'calculated'. Because the wave heights differ parallel to the coast not each value of H/h (wave height devided by water depth) represents a value of Hb/hb (= ,r). The values of 1-1/h are presented in Table 7; all values which (probably) represent a value of Hb/hb are in quotes.

The breaker index is valued at 1.30 (4- 10%) by means of the figures

in quotes from above table. Because the shape of wave envelope

parallel to the coast resembles somewhat a standing wave, the breaker index is also valued by means of the breaker criterium for standing waves:

27h

(

/L)breaking 0.22 tanh

Using this equation it appears that the breaker index varies from 1.25 until 1.35 (depends on the chosen wave height).

The set-up measurements are also used for the determination of the

maximum set-up (I-)max). This value is a measure for the amount of (wave) momentum perpendicular to the coast. The maximum set-up can be

determined in several ways, viz.:

1. By extrapolation of the connection line between point 24 and 25

(last point at which the set-up is measured). The maximum set-up is determined in three ways, viz.:

mean slope of the connection line: (n25 - n24)120

-'max' slope of the connection line: f(n_{--25 0251)(-'n24 G24)}/2°}

'min' slope of the connection line:{(n2_

5

-

.

st::

... ... . . :7-771 :I 7: : i ..

L1].

...::.. .... .. ... ... L . ... L ... .... ... ... ... .. ... .... . .. ... _.. .--1... .... .:. ...

H

r:

1 .... ...1-...'...,t..._,L.. ...:7.1:::::. ---:.4...,.:=7. .

:-.-t

- .. , ...

, .. .. /- :. 4 ' ,Zo

I.

, . t '! '. t ... :7. ; ' 4.'7' -I' .: ... ',', '77f".- 1 P. - :I' ' 11 . : . I. : ! .. . . , .. ...

....

.;: : A , ' ::* .':' ' :: : I'. '

...

, ;

-t ::;: " ; . 1 ; " . . ... . 7/:..) ... :: .. .: ...::-.t.

_.i-...

--::t ::: .t.: :: , .

.. ...1-

..7... 1 71 -7"1-7-77-77-1:-:7:7.7,::.:1. ::::.: :".;: 7 .. f '''7L:7 'Li.' : ; : - ' *. ; ; :1.. :1. - , - . .. , ..1. . .: ' .. i-;.,,...::7.7/7-771'77.1-::7 ".:17.--77-1- '7-7 . -. , _{.. .} ...:". . t . 7 ' ' : ...-...:: `..: .- -L .,1-7--

.---.7---; - _,..._,----,. ..

. . .1 3-"' 7:::1:: .:.:-.__LI._-__ It__ . . .. . I .:!.. ..-:.-3----. . . .. , .:_...:._:__..1:___:1____L. _.:.!.:.

-'H:::.:.:1..

i _. I .

.:2.__I I.

' _.l... . . . .

.. ...

-.."..:1--- . .-.11 :

... I- ?

"

":''t"

. _{... ... 1} ' t.:....-'

---:;--7:-.."....1:i::::::-"':-I .. : .'. ....1. ;...! ... . .:.: :::: . :I: :.: g...--1 .. ::: : :: .::7::1:... :.:_,;.,

..$. r-NA-f,: ...-:mcaSte.re.:1-.,c to..y . .. .., .... ...

t... .1. ::: (..../...:' R.)7 :. .. ... :'._ ..1.... ... I . -' . ' .1:-

''''

' " ... -. ... .7.- : ... -1- -... ... ... . 1 I . .1 , I.:-...1... 1....,.... ...I... . V....

...

. ;: " _{1:._'...:::::::"...1...L:...1...:V.'}

7

_{i... C(.4...1...1:-St:Cel k :i..:2-* 'I.'i . .}

-. ... :::. n:, .t. , . .. . .. - . ... .... .. . ... . .. i. . ; _{.". ....} .1 AN: . : -:': . :.iil.... r .-.1-"1"' - _I. 60_7 t . - . . i _.1.,,,-,__e_c_...,. ,__.. ..,c_ _ v,___.__:_:.-:_. _._t ____ ... -

...

:..1:.:::

... - -t ... ::, .: .. ... .:`.. - " .... : ... :. .7_ t-=_.- i...:..::.:L.:: I.L_____ . ..., I- 1 -4 : ...:1:: .! ; t--- :-CI,* -I - - - .... ...I . :: . :: . .:: .. ..

...17:t- ...

...:

't , ..,. ...t.__ ...;:.:..., :::.:i:._:...:. -..-..:.:T.:.:1 . .:.::::::..:.::, -9. ..,.. -771 .-r-

..

....

. _,:

... --.-

... ..:i::::'. . _.1... '7777, 77-7-77:1777- -

..

7 / - .... _ , ....1::

...

... /.

-....

; ...

_... ... 7 ...---7.--7---- .7 7 .

..

... I'. - . ....7:1'7.. ... . : . ... ... . .... ..,.... -1.:... ... .1 t -_.= :

...

... t ... t.L ... ... . .... ... . . .... ..._... .... : =777177: t. .... :. I .... . ' - ... - 1-. .. : 7 '" : 7:77 ... .

Li::: .1

... ,_.1

i' I

...t. 1 I

--L---::17: :1:7 ::, ::

-

- :::7

: ... .... ... -: : ' " ... p. 4 , - - ... , . I .. . .... . ... .. . ,. .. ; ... _ . .. .. t .. _ 4 : .... ' "1 -...It..._ .. -...,..,....:.:::;.:.:r... :..:.'-i: .. ...,.. .. .... i. ; .. si ..7. ..i. . . ... I ... . ... ...- .. . ... ..1 r ... - I. .. .. , . ... 7 ... . ..., I . ..- ..-::: 1 ' .. 1 --'---1---..71: ...I...I. ... -.-.-r--

-

,. , _ .-..!....-.,..±..i.:_:_.:.1..._._....L.,__-_-..._...___._:...1.:_i_...__....---...7. .4,,.... .1: .... ... :. .... . .. : 7:7 : .... ... 17...:';:.: : : ':i. I - "- .. .-11 ... -- - 1 ... - - I ; . ... ... . ., ...,... ... _... , .. . . .. . -::i: 17 --I..

...

...t..., .... 11: . _ __4 . . .... . . ::::!....t. - . : . ... : :".; '7:77777.77 77-7.: 77 I . ...! . .1 .. . ....; . . .... : _Fre r.-Lup e cA s. e 1 .717 . ... I. ... . . . . IF V " : . ... t ... 3 7.7-71 I '7 77: 777 . ... 1... . , ... . ' : . . "

-S

$ . i . .11 r.

. ' ,74,. 7 7-71 . _. t . 1 . ...it.. .1( ,;1I . .7.7, 7,7 7 .. -'; . .. il. T,77 1.! "

, ,, . : ; 7 - i , . t .1 . '' : .,..,,, -i ;.: 11 ' 11, ' t 7. ft _{t ; t} 1 1 ,., - 7 P. P ,., 7,

-'I'

. .. 7. 11 . . , . ; : :' 11r.:: ::: . . . . .. :. .. .. ...4. . " : -7: - T77:1- . 777 -.77777 . : : : : - ; ... : :1:: . : . L... 4

,

. 4 7" , . a

I -_ ; . ; ; 1 . . ; . ;414.4 . . ... . . l'1 14 ... : . -4 1::4 , ... ... .. , . .... 1 . 7177.. 711.

...

' :;::: . . ' :

....

1:::: _1::: : t';71 : : : :" ...

...

...

..

.. . . ::::1:1,' 4":44144 . . ... . 7 ' 1. _Ii -.F.' ::,:i: -_LI .J...,L .._. . ..., . j...4... .::!:::: .: ,... 1....: -... .. . J. ,.. ,... i... 1.::: --_, 1.11.1.1' .

....

:

1::

LLE..'

...

: . . ... . ... .. ... ... _.:.:;:..

..

. , . .. . ; ... 1:;:. : '4,1 . :

rr.'

I

0

; 1. . : : ".."' : -: " -" ' I ; . : : ! ' ... '": ::.: . : .. :.7 7: . 7 . : .

: ,::

::::;::; ''. v.. i:: 'I. . 'I. is.... I ,... .._,..._. -,.7T--.--: - 77 ''--- --;

-

,:

--.

1 ..

..

-:

::

. :: . : i: .

.:,:

:::,.. :-.::.: .:-.: i ....:.:. :::::. "... :. ... ..,... .... ; . T:.... '.,

..

1:17 "!.:.''t-.7 '!:,:.: ".' ":-. .:-. '.:','.'-. '' .. -.. : .... ...:..1 .":: .0.C.),. ...c... -.c..;;JR

-I : -I-I r

' -r .. .. .. ; . , .. . . . .

...

r%2.o.r.V.4: s 4.64.1i: It...:...::: i ; 2 D : -... : , . . i : ' ' ; : 'I . " : . -' ' ; - ' . ; " ; ; " ' ; !,." 7::1!:: :: . ::

:'. ::...::

: :::.

I::

I.

...

::r

.... "

.. . ... , .": .

...

. ; .. ; ; .. ; :,..,..: :::::: ...:: ...-]:::::.: ....;'...i:::::: -... :.. 7,1 .... i. :., "....F... ::: 1 ...!: :::: ::: H.:L.. . ::-.1:,-:.: :,..-...: :::":

-

!:::: ,.: 1":.:.: _ :1:: I. ... .... " p. . . .1.

vt

. I :.:.1::::

I.."

:;:: ...Hi: ...-H.: :...-H.:H. . ...:,.... .. ...t.". ; .. ,:::.". :::: .:::::::: ::::i. :: i

...

: ":71T-S":; I

2. First the mean slope (of the water level) is calculated in several points and than a line with that slope is drawn through the point with the largest set-down.

The results of the above calculations are:

Left ray:

mean value:

5.7 cm (slope: 0.78/20

through

+1.64)

Max. set-up : max. slope: 6.0 cm (slope:

0.87/20

through +1.68)

'` min. slope:

5.4

cm (slope:

0.69/20

through +1.60)

Mean slope

0.99/20

max. set-up (through

-0.30 cm): ' 6.3 cm

The mean set-up is valued at

5.9

cm

(a= 0.35/6%)

Middle ray:

mean value: (slope

1.19/20;

through

0.30) ' 3.75

cm Max. set-up : max. value: (slope

1.6/20;

through

0.50)

' 5.5

cm

min. value: (slope

0.78/20;

through 0.10) ' 3.15 cm Mean slope

0.85/20

-+ max. set-up (through -0.11)

= 3.5

cm

(with max. value): 3.85 cm (a=

0.95/25%)

The mean set-up is valued at:

(without max. value): 3.45 cm

(c= 0.25/7%)

The mean set-up is valued at

3.6

cm (+ 15%).

The maximum set-up can be calculated assuming that the formula, derived by Battjes (n

= 116 yHb), is also valid in this case.

max Left ray :y

1.3 (+

10%); Hb = 14.8 cm (+ 15%) 5 nmax

= T

yHb = 6.0

cm (+ 12%) Middle ray :y = 1.3

(+ 10%); Mb = 3.8 cm (+ 5%)

=4

N(

= 1.5

cm (+ 12%) Imax ic Hb 1. ma

Another method for the calculation of the maximum set-up is the addition of the set-up of each wave system, assuming that the radiation-stress, perpendicular to the coast, can be added.

In this method is the radiation-stress assumed not to depend on the phase difference between the two wave systems. With other words,

Sxx' Syy and Sxy do not depend on (p-o) in form 3.18.

Assuming each wave system breaks indepenJently from the other (as if the other system does not exist), the following results are obtained:

Left ray : the diffraction factor KD has to be valued for each system. KD (-43°) = 1.25, KD (-28°) = 1.20; assuming y = 0.8 --, nmax . ,max + 71max = 2.5 + 2.5 = 5.00 cm " tot - -nmax = 2.9 + 2.9 = 5.80 cm Y = 0'9 ' Imax = + p,max toL

The mean value is ;

max = 5.4 cm (+ 100/). Middle ray : KD (-43°) 0.95, KD (-28°) = 1.05 assuming y = 0.8 ;max 2.0 + 2.2 = 4.2 cm y = 0.9 nmax = 2.3 + 2.6 = 4.9 cm

-The mean value is r = 4.6 cm (+ 10%)

.max

A third method is using formula 50 of annex III

- 1 y 2

Q-( ) h

- 1 +

in which h is the water depth of the breaking wave (11.4 cm).

a is the coefficient )43-,7-(----(1)1)/cos(,2), which is 0.89, and y is 1.3.

cp. is the angle of wave approach on the breakerline.

28.

For the waves in the

experiment1

= 160

cf?2br = 11

For a point on the phase = 0 line (p - q = 0) Q can be written as:

Q = 2,1)

+ 2a +

(1

-

a2)

= 8.72

0 = (cosct,/ + acos¢2)2 = 3.37

Fora point on the phase = Tr line (p - q = 4L) Q can be written as:

7 ,

Q = aiTcos¢1 coscp2 + 24> + + + a2)

= 4.37

cos2(I) + a2 cos2

=q)2

= 1.69

1

So, the maximum set-up is -1g. (11:389)2 .

8.72

.

11.4 = 5.87 cm

the minimum set-up is :Eir (11A9)2 .

4.37

.

11.4 = 2.94

cm

The results of the various calculation methods are summarized in the table below:

cm measured Battjes-formula Addition Annex Ill

maximum set-up minimum set-up

5.9

3.6

6.0

1.5

5.4

4.6

5.9

2.9

2br

5.e. Various observations

Because of the high turbulence it was not possible to measure longshore velocities with current meters. Also because of the short measurement section (only 8 m) and the high set-up (6 cm) large currents were caused by water level slopes. These set-up currents had velocities

of 1.5 to 2 m/sec. (fig.

5.25).

In the measurement section also a large influence of these currents has to be expected. Therefore only

qualitative results regarding longshore currents are presented in this chapter.

By the breaking waves a large quantity of water is brought into the breakerzone. But because only on the crest lines waves have

sufficient height to break. On

the 'H = 0' lines (lines with phase = Tr) no wave breaking will occur. On these lines a fast

C :.-? ."-,7). rn , no GEtv p se Z. C kosse

The waves break on the indicated breaker line. Breaking starts at point A and the breaking front extends then towards B.

30 .

I )

1 E.'

/1/

S

I\

-/

IN

I 1

area are indicated in fig.

5.26.

7

The velocities were in the order of 1 cre.s.t: 0.7 - 1.5 m/sec. like Cr

ii.

b

Idre. ), ie 10,5e 1pkzic

.0

r"Tr

5.56-\\E,, /A\

rip current was observed.

\

\,'

The currents observed in the breaking

S.;B

It)

of the Chezy-formula, suppose C = 30 and the average water depth on the slope

0.2 m,

then

v = C

₌

71;1- = 30 /0.2 * 6/500 =

₌ 1.5 m/s.

Sie,517

frir

One has to realize that the waves break alternating on crest line

X and Y (when there is a crest at line X there is a trough at line Y). Due to this alternating breaking the rip current on the 'H = 0' line is also moving

somewhat to-and-fro.

The breaking waves do not really plunge but they form some kind of

a vertical water front which propagates towards the coast and than suddenly collapses (fig. 5.27)

6. CONCLUSIONS

6.a. Wave heights

The shape of the surface envelope was predicted almost correctly in the middle section of the wave basin. The distances between two minima or two maxima could be calculated with an accuracy of 98%. The variations in these distances were partly caused by diffraction.

It was not possible to calculate the wave height in any arbitrary point very accurate because of:

diffraction; the applied diffraction is not totally correct. the waves became too high with regard to their (combined) length; therefore the maxima became somewhat lower', and the minima became higher.

Breakerindex

32.

The breakerindex is quite high (1.3) but resembles very much the breakerindex of standing waves.

During the tests it appeared that the waves did not start breaking at one line parallel to the coast, but a certain breakerzone developed. The ratio H/h was maximal at the points where the wave was just not yet broken.

Set-up

The assumption that the set-up of cross-swell can be predicted by addition of the set-ups of both individual systems appears to be not correct.

The Battjes-formula describes the set-up for cross-swell only in a realistic way for the maximum set-up.

An identical derivation as the derivation of the Battjes-formula leads to a very complicated set-up formula for cross-swell. It appeared that the set-up calculated with this formula agrees with the measurements for 80% and more.

REFERENCES

Longuet-Higgins & Stewart, 1964; Radiation Stresses in water waves; a physical discussion with applications; Deep sea research,

vol.

ti, pp

529-562.

Longuet-Higgins & Stewart, 1962; Radiation stress and mass transport in gravety waves, with application to surfbeats;

J. of Fl. mech., vol.

13, pp 481-504

Longuet-Higgins & Stewart, 1963; A nOte on wave set-up; J. of Marine Res., vol. 21, b.-10

Battjes, J.A., 1974; Computation of set-up, longshore currents, run-up and overtopping due to wind-generated waves, Delft University of Technology

Battjes, J.A., 1977; Golfspanning, T.H. Delft

Verhagen & Ludikhuize, 1978; Bengkulu harbour project, final report, Vol. B; the influence o cross-swell on longshore sediment transport

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