Theoretical and experimental study of breaking wave on sloping bottoms

China Ocean Eng., Vol. 27, No. 6, pp. 737 – 750 © 2013 Chinese Ocean Engineering Society and Springer-Verlag Berlin Heidelberg DOI 10.1007/s13344-013-0061-5, ISSN 0890-5487

Theoretical and Experimental Study of Breaking Wave on Sloping Bottoms*

YANG Kuei-Sena, CHEN Yang-Yiha, b, 1, LI Meng-Syuec and HSU Hung-Chuc

a Department of Marine Environment and Engineering, National Sun Yat-Sen University,

Kaohsiung 804, China b International Wave Dynamics Research Center, National Cheng Kung University,

Tainan 70955, China c Tainan Hydraulics Laboratory, National Cheng Kung University, Tainan 70101, China

(Received 19 June 2012; received revised form 23 October 2012; accepted 28 March 2013)

ABSTRACT

This paper studies the continuous evolution of breaking wave for the surface water waves propagating on a sloping

beach. A Lagrangian asymptotic solution is derived. According to the solution coupled with the wave breaking criteria

and the equations of water particles motion, the wave deformation and the continuous wave breaking processes for the

progressive water waves propagating on a sloping bottom can be derived. A series of experiments are also conducted to

compare with the theoretical solution. The results show that the present solution can reasonably describe the plunging or

spilling wave breaking phenomenon.

Key words: Eulerian; Lagrangian; waves breaking; plunging; spilling

1. Introduction

In the shoaling process of wave propagation from deep water to shallow water, it will lead to an

increase in wave height and decrease in wave length. As a result of the steepening of wave crests in

shallow water, the wave profile gradually becomes asymmetric, unstable, and finally breaks.

Accordingly, many researchers paid more attention to solving the wave transformation on sloping

bottoms. However, since the sloping bottom was approximated by a large number of steps, the effect

of the bottom could not be fully exhibited and the wave breaking profile still cannot be evidently

described by theoretical analysis.

In previous studies, such as Sverdrup and Munk (1944) and Le Méhauté and Webb (1964), the

sloping bottom was based on a finite number of steps approximated by conservation of energy flux.

The effect of the bottom slope could not be fully exhibited using such an approach. Lewy (1946),

Stoker (1947) and Lowell (1949) described the exact solutions of the shallow water equations on

beaches of uniform slope for certain beach angles, but the linearized solutions cannot be satisfactorily

* The work was supported by the Research Grant Council of the Science Center, Taiwan, through Project Nos. NSC99-2923-E-110-

001-MY3, NSC99-2221-E-110 -087-MY3, and NSC102-2911-I-006-302.

1 Corresponding Author: [email protected]

YANG Kuei-Sen et al. / China Ocean Eng., 27(6), 2013, 737 − 750

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applied to arbitrary bottom slopes. Peters (1952) proposed an integral analytic form to express the

surface wave propagating along the sloping bottom and the solution has recently been calculated

numerically by Ehrenmark (1998). Keller (1958) derived an alternative linear form for progressive

waves on an uneven bottom, however, the solution was expressed as an implicit form and was

restricted to short deep-water wave on a gentle sloping beach. Most of the above-mentioned theories

cannot present the complete wave transformation on a sloping bottom.

To date, only few scholars have presented analytic solutions for the wave transformation on a

planar beach. Biesel (1952) suggested approximation method to account for the normal incident waves

propagating on a sloping plane where the bottom slope was first considered in the velocity

potential as a perturbation parameter. Chen and Tang (1992) modified Biesel’s (1952) theoretical

model and obtained the solution to the first order of . The two-parameter perturbation expression

based on the wave steepness to first order and the bottom slope to third order is used to derive

the expression of the velocity potential (Chen et al., 2005). These solutions can simulate the

subsequent wave profile before wave-breaking point. However, the behaviors of wave steepening and

overturning during and after breaking remain unanswered (Longuet-Higgins, 1981; Pomean et al.,

2008; Bridges, 2009). The objective of this study is to present an analytical method to clearly describe

the evolution of breaking wave process.

It is known that an Eulerian description at the free surface y can always be expressed as a

Taylor series of functions at y=0, which implicitly assumes that the surface profile is differentiable.

This assumption limits the applicability of Eulerian description in near-break condition. On the other

hand, in the Lagrangian description, the free surface is represented by a fixed level of the parameter of

the particle in the space of the Lagrangian domain. So it is more appropriate for the Lagrangian

description of limiting free surface motion. Thus, the Lagrangian approximation is more accurate and

reasonable than the Eulerian counterparts of the same order (Naciri and Mei, 1992; Chen et al., 2005;

Chen and Hsu, 2009; Hsu et al., 2010, 2009). Thus, we can expect that the solutions in the Lagrangian

form y can describe effects that cannot be captured by classical Eulerian solutions.

In this study, we use the Eulerian solution of Chen et al. (2005) and then obtain the Lagrangian

solution by using the Euler-Lagrange transformation. This Lagrangian solution can be used to describe

the wave profile prior to wave breaking point. We further assume that the particle speed must be at

least equal to the phase speed (i.e. kinematics stability parameter) and the wave profile becomes

vertical if the initial breaking stage is to occur. The equation of particle motion is also derived by using

the free falling concept to capture the post-breaking profile. Henceforth, the detailed breaking wave

profiles including the overhang characteristics of plunging breakers and spilling breaker will be

discussed. A laboratory experiment is conducted in order to validate the theoretical wave breaking

profile on sloping bottoms with observations.

2. Experimental Setup and Procedures

The purpose of this experiment is to investigate the wave breaking process for progressive gravity

waves on a sloping bottom. These experimental results can also be used to validate the present


739

theoretical solution. Wave breaking profiles can be verified and obtained by the theoretical solution

and experimental measurements. The experimental measurements were carried out in a glass-walled

wave tank, 35.0 m×1.0 m×1.2 m, in the Department of Marine Environment and Engineering, National

Sun Yat-sen University. A high-speed camera was set up in front of the glass-wall at the different

breaking point locations depending on the initial wave conditions. This method allows to successively

capturing the wave breaking profiles. The whole experimental frame is schematically shown in Fig. 1.

A piston-type wave generator was employed for generating monochromatic waves. Images were

captured by a High-Speed camera (MS55k2, Canadian Photonic Labs Inc.), which has a 1280*1080

pixel resolution and 1020 frames per second (fps) maximum framing rate. A transparent acrylic-plastic

sheet (0.9 m×0.625 m) was calibrated at 1-mm intervals in 5 mm×5 mm grids. Its function is a virtual

grid in the picture. The experiments were conducted at a constant bottom slope ( 1/10 ) and two

different water depth (d=79.6 cm, 80.7 cm) and two wave periods (T=1.2 s, 1.35 s). The incident wave

heights were 10.57 cm and 11.36 cm.

Fig. 1. Experimental frame and instruments setup.

3. Formulation of the Problem

In the present study, a two-dimensional monochromatic wave propagating normally over a

uniform gentle slope is considered, as shown in Fig. 2. The negative x-axis is directed outward to the

sea, while the positive y-axis takes vertically upward from the still water level, and the sea bottom is at

y h x , in which denotes the bottom slope.

Fig. 2. Definition sketch for surface-wave propagation on a gentle sloping bottom.


740

We assume that the flow motion is irrotational and that the fluid is incompressible and invicid.

The velocity potential ( , , )x y t satisfying the continuity equation leads to the Laplace equation as

follows: 2 2

2

2 20

x y

. (1)

The wave motion has to satisfy a number of boundary conditions at the bottom and the free water

surface:

(1) On an immovable and impermeable sloping plane with an inclination (0 1) to the horizon,

represented by ( , ) 0, 0f x y y h y x x , the no-flux bottom boundary condition gives

2( ) 1 0,y xf f at y h . (2)

(2) The dynamic and kinetic free surface boundary conditions are respectively 2( ) 2 0,t g at y ; (3)

d d ,y t at y . (4)

By eliminating the free surface displacement , the kinematic and dynamic free surface

conditions can be combined into one equation: 2 2[( ) ] ( ) / 2 0,tt y tg at y . (5)

3.1 Approximation Solution in the Eulerian System

For a small bottom slope , Eqs. (2) to (5) can be solved order by order. The detailed

computation has been discussed by Chen et al. (2005) and will not be repeated here. Here we quote the

theoretical solution up to 3 order derived by Chen et al. (2005) in the Eulerian system listed as

follows: 3

2 3

1, 1,0,1 1,2,1 1,1,1 1,3,1 10

( ) sin ( ) cosn

n pn

A A S A A S E

; (6)

32 3

1, 1,2,1 1,1,1 1,3,1 10

( ) cos ( )sinn

n pn

a a S a a S E

, (7)

where the full expressions of the coefficients in Eqs. (6) and (7) are given in Appendix A. S is the

phase function di

x

xS k x t , 3 3

1 3 3exp d exp di i

x x

p x xE e x e x , 3e being the wave amplitude

influenced factor; ( 2π / )T is the wave angular frequency, T being the wave period; and

,0

n

i nn

k k

is wave number . The subscript i express the physical quantity at incident wave at

x .

3.2 Transformation to the Lagrangian System

As a wave reaches its limit, the crest is fully developed as a summit which can be calculated as

the spatial surface profile by the system of Lagrangian coordinates. On the contrary, the wave profile


741

in the Eulerian coordinates can give a sinusoidal shape which is always approximately symmetric. The

Eulerian and Lagrangian wave profile prior to breaking as a wave propagating over a gentle slope

1/10 are demonstrated in Fig. 3.

Fig. 3. Comparison of wave profiles between Lagrangian

and Eulerian coordinates.

Based on the velocity potential given in Eq. (6), the horizontal and vertical Eulerian velocity

components of the fluid particle eu x and ev y can be derived. These Eulerian

motions can be linearly transferred into the Lagrangian system according to the trajectory of the fluid

particle and the corresponding velocity components lu and lv are

2 3

0 2 1 3

2 3

2 1 3 1

d cosh[ ( )](1 )cos ( )sin

d sinh( )

cosh[ ( )] cos ( )sin ,

sinh( )

l e

p

X h yu u a C S C C S

t h

h yS S S S S E

h

(7)

2 3

0 2 1 3

2 3

2 1 3 1

d cosh[ ( )]sin ( ) cos

d sinh( )

sinh[ ( )] (1 )sin ( ) cos .

sinh( )

l e

p

Y h yv v a S S S S S

t h

h yC S C C S E

h

(8)

where 1C , 2C , 3C , 1S , 2S and 3S are variables whose full expressions are listed in Appendix A. Then,

the displacement components of water particle, X and Y to the 3 order approximation can be

written as:

2 3

2 1 3

2 3

2 1 3 1

cosh[ ( )]d d ( 1 )sin ( ) cos

sinh( )

sinh[ ( )]sin ( ) cos ,

sinh( )

t t

l x

p

h yX u t t a C S C C S

h

h yS S S S S E

h

(10)

2 3

2 1 3

2 3

2 1 3 1

cosh[ ( )]d d cos ( )sin

sinh( )

sinh[ ( )](1 )cos ( )sin .

sinh( )

t t

l y

p

h yY v t t a S S S S S

h

h yC S C C S E

h

(11)

The free surface position , can also be expressed as 0y and the results are


742

0000 )()0(,)( yy YyXx , (12)

where ),( 00 yx is the original mean position of a water particle.

4. The Breaking Condition and the Equations After Wave Breaking

4.1 Kinematic Stability Parameters During Wave Breaking

Because of the change of water depth, the wave shoals and is refracted in the propagation process

from deep to shallow water. The celerity is reduced, hence, when the particle horizontal velocity at the

wave crest is larger than the wave celerity, the wave breaks. In order to describe the breaking wave

mechanism, the critical condition of wave breaking is applied and the breaking criterion is that the

particle horizontal speed u at the wave crest must be at least equal to the phase speed (or wave celerity)

c , i.e.,

1u c , (13)

where c is the wave celerity and u is the horizontal velocity of the particle at the wave crest. The

horizontal velocity of the particle on the surface can be obtained as 0( , 0, ) /u x t t and ck .

Hence, the wave breaking condition up to ( )O is

1 1cos coth( ) sin coth( ) sin 1ka S h C S h S S . (14)

And the surface elevation has an extreme value at the breaking point, thus we have

01 1

0 0

d ( ,0, )dcoth( ) cos cos sin 0

d d

x tka S h S C S S

x x

. (15)

From Eq. (15), we can obtain the wave phase at the breaking point

1

b 1 1tan coth( ) ,S C S h (16)

where the subscript b denotes the breaking condition. By substituting Eq. (16) into Eq. (14), the

breaking point can thus be determined.

4.2 The Geometric Condition at Wave Breaking

By using Eqs. (12), (14) and (16), the surface wave profiles near the wave breaking point can be

evaluated and sketched in Fig. 4. The figure shows the surface wave profiles prior to breaking on

different wave steepness and wave phases for bottom slopes of 1/ 5 and 1/10 , respectively.

From the theoretical analysis, it can be found that the surface wave profile at breaking point is similar

to the plunging breaker in Figs. 4a–4c, however, the surface profile is similar to the spilling breaker in

Fig. 4d. The above-mentioned conclusion is coincident with the classification of wave breakers

proposed by Galvin (1968). The great rapidity of surface waves change near the breaking point can be

seen in the figure, and the results support the following findings. (1) When a wave shoals on a sloping bottom, the wave profile becomes asymmetric gradually from

deep water to shallow water. The front surface wave profile is narrowing and steepening; however, the

back surface wave profile is lengthening and flattening. Eventually, the wave crest curls forward

leading to breaking.


743

Fig. 4. Successive wave profiles prior to breaking plotted by linear solution (up to the order of ).

(2) As wave steepness becomes larger, wave breaking takes place in a deeper water region. However,

the breaker type depends on the bottom slopes. In particular, for a given wave steepness, the spilling

breaker is usually found in a gentle bottom slope. On the contrary, the plunging breaker is usually

observed in a steep bottom slope.

From Fig. 4, it is obvious that there exists a vertical tangent plane like shock surface before the

overturning face of the wave. This condition is also quoted as a criterion for wave breaking. Assuming

there are two particles, 0 0( (1), )x y and 0 0( (2), )x y , shown in Fig. 5, the position of 0 (1)x is closer to

the shoreline than the position of 0 (2)x . After time t, the particle at 0 (2)x will run to catch up the

particle at 0 (1)x and now the position of 0 (2)x will be closer to the shoreline than the particle at

0 (1)x . During this case, the wave profile will not be supported by the water and the particle at

0 0( (2), )x y will move by free-falling motion.

Fig. 5. Illustration of water particle overturning.


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4.3 The Post-Breaking Wave

Influenced by shoaling effect, the wave celerity is slowed down near the wave breaking. After this

time the water particles meet the critical condition of breaking wave, which means the fluid particle

velocity on the wave crest is larger than the wave celerity, then the fluid particle will begin to leave the

wave. The water particles will move in the motion of free-falling jet. Then, the horizontal and vertical

displacements of water particles after the wave breaking can be described as:

0 0 b 0 0 b 0 0( , , ) ( , ) ( , )x y t x y u x y t ; (17) 2

0 0 b 0 0 b 0 0( , , ) ( , ) ( , ) 2x y t x y v x y t gt , (18)

where b 0 0( , )x y and b 0 0( , )x y are the positions of particle ),( 00 yx at wave breaking, b 0 0( , )u x y

and b 0 0( , )v x y are the fluid particle horizontal velocity and particle vertical velocity which arrives at

the critical condition of breaking wave, g is the acceleration of gravity, t is the passing time after the

particle leaving the wave. According to Eqs. (12), (14), (17) and (18) of the present theory, we can

obtain the subsequent evolutions of spilling and plunging breaking waves.

The duration of the overturning wave is less known by analytic solution. Longuet-Higgins (1980,

1981) gave a solution having the shape of hyperbola with asymptotes that enclose a steadily reducing

angle. Pomeau et al. (2008) proposed a generic singular solution for spreading of the crest during wave

breaking to predict the occurrence of an overhanging region. Bridges (2009) derived a new set of PDEs

governing the surface dynamics for inviscid gravity waves with vorticity. However, those solutions

cannot predict the complete wave transformation on a sloping bottom. The present theory can well

describe the wave shoaling, as well as the occurrence of breaking wave and wave overturning, similar

to the plunging jet in the breaking water waves over slopes.

Figs. 6–8 show the comparison between the sequence of computed wave breaking profiles and

snapshots of breaking wave behavior over a sloping bottom ( 1/10 ) captured by the high-speed

camera for three different wave heights (Hi=11.24, 10.79 and 10.54 cm) and wave periods (T=1.35,

1.32 and 1.2 s) at water depth d=80.7 cm. These profiles in Figs. 79, plotted the instant the wave face

becomes vertical, show the crest toppling over towards the water below. It shows a good agreement of

the theoretical wave breaking profile with laboratory measurements. The breaker jet creating a forward

moving jet is well observed in the present theoretical results. This indicates that the theoretical solution

is accurate for modeling such post-breaking waves. From these figures, it is obvious that the

overturning wave first has a vertical tangent plane, the emergence of the overhanging jet.

The overturning wave does not necessarily occur in spilling breakers which is shown in Figs. 9

and 10. Figs. 9 and 10 show the comparison between the sequence of computed spilling wave breaking

profiles and snapshots of spilling breaking wave behavior captured by the high-speed camera over a

sloping bottom ( 1/10 ) for two different wave heights (Hi=9.95 and 11.36 cm) and wave periods

(T=1 and 1.35 s) at water depth d=79.6 cm. It should be pointed out that the turbulent effect is not

considered in this solution which should be assessed in the future. It shows a good agreement of the

theoretical spilling wave breaking profile with laboratory measurements.


745

Fig. 6. Eight consecutive snapshots of plunging breaking wave behavior over a sloping bottom (α=1/10, d=80.7 cm,

T=1.35 s, Hi=11.24 cm, theoretical breaking wave height Hb=12.8 cm and breaking water depth db=10 cm

experimental breaking wave height Hb=13 cm and breaking water depth db=12 cm). The red line is the present

theoretical solution. “The.” is the theoretical result, “Exp.” is the experimental result.

Fig. 7. Nine consecutive snapshots of plunging breaking wave behavior over a sloping bottom (α=1/10, d=80.7 cm, T=1.32 s,

Hi=10.79 cm, theoretical breaking wave height Hb=12.3 cm and breaking water depth db=9.6 cm, experimental wave

height Hb=12.3 cm and breaking water depth db=11.7 cm). The red line is the present theoretical solution. “The.” is the

theoretical result, “Exp.” is the experimental result.


746

Fig. 8. Ten consecutive snapshots of plunging breaking wave behavior over a sloping bottom (α=1/10, d=80.7 cm, T=1.2 s,




Fig. 9. Seven consecutive snapshots of spilling breaking wave behavior over a sloping bottom (α=1/10, d=79.6 cm, T=1 s,





747

Fig. 10. Six consecutive snapshots of spilling breaking wave behavior over a sloping bottom (α=1/10, d=79.6 cm, T=1.35 s,

Hi=11.36 cm, theoretical breaking wave height Hb=12.9 cm and breaking water depth db=10.1 cm, experimental

wave height Hb=12.3 cm and breaking water depth db=9.4 cm). The red line is the present theoretical solution. “The.” is

the theoretical result, “Exp.” is the experimental result.

5. Conclusions

This paper provides a new Lagrangian solution for modeling the whole wave breaking processes

over a sloping beach. The effects of bottom slope and wave steepness on breaking wave profile are

discussed. In particular, this asymptotic solution can be used to simulate the overturning and a

prominent jet falling process of the plunging breaker or spilling breaker. A series of experiments

measuring the properties of nonlinear breaking water waves propagating over a sloping bottom were

conducted in a wave tank. Good agreements have been obtained on comparing the measured wave

breaking profiles with the theoretical trajectories predicted by the proposed Lagrangian solution. By

varying the slope and wave steepness, we can obtain plunging and spilling breaker types by using the

present theoretical solution. The post-breaking stages can also be simulated. The present solution thus

allows to compute the characteristics of breaking and post-breaking waves on slopes. However, it still

needs pointing out that a higher order solution and turbulent effect should be included in the future.

References

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Bureau of Standards, Circular 521, 243–253.

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Appendix A:

1,0,1 ( ; , ) cosh[ ( )]i iA A h a h y , 2i ia H ,

2, 1 ,

cosh( ) sinh(2 )tanh( ) cosh( )ia g ag h

A Dh hD h h

2 2

1,1,1 2

( ) cosh[ ( )] ( ) sinh[ ( )]( ) ,

sinh(2 ) cosh( ) tanh( ) cosh( )

ag h y h y h y h yA h y

D h h D h h

2 2 2

1,1,1 2

coth ( )

sinh(2 )

h ha a h h

D h D

,

4 4 3 3 4 22 2

1,2,1 2 4

23 3

2 3 2

2

2

( ) ( ) 4 5coth ( )( )

sinh(2 ) 22 sinh(2 )

coth( ) ( ) coth[ ( )] 4 10cosh ( ) ( )

cosh( ) 3 sinh (2 )

2 cosh ( )

sin

ag h y h y D D hA h y

D h DD h

h h y h y D hh y

D h D h

D h

D

2 2

0,2

sinh[ ( )]( ) ( ) ,

h(2 ) cosh( )

h yh y k h y

h h


749

4 3 2 3

1,2,1 2 3 2

4 2,22

4 2

( ) ( ) [4 10cosh ( )]( )

sinh(2 ) 3 coth( )sinh (2 )2 sinh(2 )

4 5coth ( ) coth( ) ( ) 1 coth( ) ,

2i

h h D h ha a

D h D h hD h

kD D h hh h h

D D

5 4 2 2 2

,2 5

2 2

118 32 15 cosh ( ) 2

3

3 15 15 coth ( ) ,

ik D D D D h DD

D D h

1,3,1,1

2 1

1,3,1, 1,3,1,21,3,1,1

3

tanh sec h

tanh

j

j jj

M jM j h h h N hN

eA DA h

,

6 6 5 5

1,3,1 3 3 2 2

2 2 4

4 4

5 3

2 2 4

4

( ) ( ) 1

6 sinh (2 ) 2 sinh (2 ) 2 sinh(2 )

12 40cosh ( ) 16 sinh (2 ) 100cosh ( ) ( )

6 sinh (2 )

4 10cosh ( ) 2 sinh (2 ) 10cosh ( )1

6 s

ag h y h yA

D h D h D h

D D h D h hh y

D h

D D h D h h

D

3 3

2

2

,2 ,2 2 2

,22 4 2

2 25 5

4 3 3

( )inh (2 )

3 (4 2 ) cosh ( ) cosh[ ( )] ( ) ( )

sinh(2 ) 2 cosh( )

20 35cosh ( ) 10 10cosh ( ) ( )

15 sinh (2 ) 3 sinh

i i x

i

h yh

D k h k h yh y k h y

D h h

ag D h D hh y

D h D

4 4

2

2 4 2,2 ,23 3 2 2

2 3 4

2

32

( )(2 )

3 cosh ( ) ( ) 4 5cosh ( ) ( ) ( )

sinh(2 ) 6

2cosh ( ) sinh[ ( )] ( ) ,

sinh(2 ) cosh( )

i ixxx

h yh

D h k D kA D D hh y h y

D h D

h h ye h y

D h h

26 5,2 ,22

1,3,1 3 3 2 2 2 4 2

2 2 4

3

,24 2

3 (4 2 )cosh ( )( ) ( ) 1( )

6 sinh (2 ) 2 sinh (2 ) sinh(2 ) 2

4 10cosh ( ) 2 sinh (2 ) 10cosh ( )1 ( ) ( )

6 sinh (2 )

i i x

i

D k h kh A h Aa a h A

D h D h D h

D D h D h hh A A k h

D h

2 4

4

5 3

2 2 25 4

34 3 3 3 2

12 40cosh ( ) 16 sinh(2 ) 100cosh ( )1 ( ) cosh( )

2 sinh(2 ) 6 sinh (2 )

20 35cosh ( ) 10 10cosh ( ) 2cosh ( ) ( ) ( )

15 sinh (2 ) 3 sinh (2 )

D D h D h hh A h

D h D h

D h D h ha h A h A A e

D h D h D

2 4 2

,2 ,23 2

2 3 4

3 cosh ( ) 4 5coth ( ) ( ) ( ) sinh( ),

sinh(2 ) 6i ixxx

D h k D kA D D hh A h A h

D h D


750

2 tanh( ) tanh( )i i ig h g h ,

22

1 2

1=

sh(2 ) th( )

h yC h y

D h D h

,

4 34 3

2 2 2

22 2 2 4 2

4 2

20,2

2

2 sh (2 ) sh(2 )

1 1 4 10 ch ( ) 2 sh (2 ) 10ch ( )

2 sh (2 )

4 6ch ( )1 ,

sh(2 )

h y h yC

D h D h

D D h D h h h yD h

kD hh y

D h

6 56 5

3 3 3 2 2

2 2 2 444

5 3

2 2 2 433

4 2

2

0,2

1

6 sh (2 ) 2 sh (2 ) 2 sh(2 )

52 10 ch ( ) 16 sh (2 ) 100ch ( )

6 sh (2 )

1 52 70 ch ( ) 6 sh (2 ) 30ch ( )

6 3 sh (2 )

12 3 7ch (

h y h yC

D h D h D h

D D h D h hh y

D h

D D h D h hh y

D h

D k D

220,20,2 2

2 3 4 2

4 2 20,2 3

4 2

) 2 ch ( )

sh(2 ) 6 2

32 8 10ch ( ) 2ch ( )1 ,

sh(2 )

A kh k hh y

D h

k eD D h hh y

D D h

2

1 2

2 23 23 2

2 3 2 2

20,2

4 2

2 25 45 4

2 4 3 3 2

2 ch ( )1,

sh(2 )

10 ch ( ) 5 2ch ( )

3 sh (2 ) sh(2 )

5 42 ,

35 35ch ( ) 35 20ch ( )

15 sh (2 ) 6 sh (2 )

5 ch

D hS h y

D h

D h D hS h y h y

D h D h

kI D Ih y

D D

D h D hS h y h y

D h D h

D

2 2 2 40,2

2 5 3

( ) ( ) 24 80 ch ( ) 32 sh(2 ) 200ch ( ).

sh(2 ) 3 sh (2 )

h k D D D h D h h

D h D h

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Theoretical and experimental study of breaking wave on sloping bottoms