Application of side wall reflection type directional wave generator and laser beam type wave surface probe – especially on the linear characteristics of the one point concentrated transient water waves

Wave Generation'95

Yokohama, Japan

P1995-15

Symposium on Wave Generation, Analysis and

Related Problems in Experimental Tanks

-

especially on directional waves

-Yokohama National University

25 September 1995

TABLE OF CONTENTS

Preface

iv

Organizers

Photographs of Facilities

vii

List of Papers

xiii

Papers

1

Recently much progresses can be seen about the ocean wave

generation and related techniques in experimental

tanks or basins. In

those techniques, common problems exist for either naval architecture

or civil engineering field, so it will be fruitful to exchange knowledge

and opinions each other.

Considering such situation, members of both group

of ITTC SKC*

and Work Shop on EADWM' planned to hold a symposium about

wave generation on the occasion that ITTC SKC is held in Yokohama

National University.

We hope that this symposium will accelerate the progress of ocean

wave generation ,analysis and application techniques in experimental

tanks or basins in naval architecture , civil engineering and related

field.

PREFACE

Tsugukiyo HIRAYAMA

Yasumasa SUZUKI

Representative of the Organizing

Committee

" Sea Keeping Committee ofInternational Towing Tank Conference

** Work Shop for Engineering Application

of Directional Wave Maker

rrof. Tsugukiyo Hirayama

Dr. Yasumasa Suzuki

Prof. Shigeru Naito

Dr. Tetsuya Hiraishi

Prof. Ning Ma*

Organizing Committee

(Yokohama National University)

(Port and Harbour Research Institute)

(Osaka University)

(Port and Harbour Research Institute)

(Yokohama National University)

* Secretary

Organizers

21st ITTC Seakeeping Committee:

Dr. Raymond Cointe

(Bassin ft Essais des Carenes, France)

Prof. Alberto Francescutto** (University of Trieste, Italy)

Prof. Tsugukiyo Hirayama

(Yokohama National University, Japan)

Dr. Roumen Kishev

(Bulgarian Ship Hydrodynamics Centre, Bulgaria)

Mrs. Kathryn McCreight

(David Taylor Model Basin, U.S.A.)

Prof. Shigeru Naito

(Osaka University, Japan)

Prof. 011e G.A. Rutgersson (Royal Institute of Technology. Sweden) Dr. Deuk-Joon Yum

(Hyundai Heavy Industries Co., Ltd., Korea)

Ir. S.G. Tan*

_{(Maritime Research Institute Netherlands. The}

Netherlands)

*Chairman

_'Secretary

Workshop for Engineering Application of Directional Wave

Maker:

Dr. Yasumasa Suzuki

_{(Port and Harbour Research Institute, Japan)}

Dr. Tetsuya Hiraishi

_{(Port and Harbour Research Institute, Japan)}

Sponsor

Photo 1 Directional waves generated in the towing tank (L=100m, B=8m, d=3.5m) of Yokohama National University by individually controlled plunger type (24

segments) wave generator.

Photo 2 Short-crested waves are generated in the Marine Dynamics Basin (L=60m,

B=25m, d=3.2m) of the National Research

Institute of Fisheries

Engineering. The 80-segment plunger-type wave maker installed at the end

Photo 3 _{Deep Water Offshore Structure Experimental Basin of Port and Harbour}

Research Institute, Ministry of Transport. (Since 1984, several series of model tests in directional seas have been carried out. Basin 35m x 27m,

Generator 80cm x 35 paddles (piston type) ) _

iiii ii !!!!!!!!

iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

IN!

!IIIII

Photo 4 _{Oblique waves generated by C-shaped wave makers of Taisei Corporation.}

C-shaped wave makers are controlled by a new non-reflected wave maker

theory. The wave makers seen at the right front side in the photograph are

absorbing the oblique

_{waves, and the wave field is not disturbed by the}

reflected waves from the wave makers. The wave heights along the _wave

crests are spatially uniform and the crest lines of_{the waves are straight.}

Photo 5

Short-crested wave generation by directional

wave maker of Central Res. Inst. of Electric Power Ind.

Photo 6 A wave generated by a multi-unit wavemaker in the towing

tank of Ishikawajima-Harinia Heavy Industries Co.,

Ltd.

Photo 7

The photograph is a scene of the thy test of the multi- _{directional wave maker, viz., snake-type}

wave maker, in the Seakeeping Basin of Nagasaki Research and

Photo 9 _{Synchronized 40 pistons and generated regular waves which have straight}

wave crest line are found. This is the link type wave generator in

Penta-Ocean Construction Institute of Technology

_{( Number of pistons : 40,} Number of paddles : 39, Maximum wave height : 40 cm Wave period : 0.5-2.5 sec)

Photo 8 Department of Civil Engineering, Chuo University. The basin is 10.5 m

wide, 6.8 m long, and 0.6 m depth. The multi-directional wave maker consists of 28 piston-type wave paddles continuously linked or 29 sets of actuator rod

Photo 10 Upper: Directional spectrumwaves in a towing tank

Lower: Yacht model on an One Point Concentrated Transient Water Wave (Yokohama National University)

DIMENSIONS : 100 m x 24.5 m.

111111111.11i11111111111.1111111111111111111111111111111111111111111111 l'1111i111

lowsimmomiowliimmommimmoomuNmE NMI

Regular and irregular waves. Wave period 0.7 - 3 s.

Wave direction 180 -270 and 0-90 deg. and any angle in between.

Fig. 13. Cross section of the wave generator

Fig. S. Wave generator with a phase difference of400 Wave length

3 in waves -1111111 7 Workshop Working pond Beach Carriage Sub-carriage Auxiliary carriage Wave generator Wire mesh package Wave generator motor Support piles Rails

Pit (4.5 x 4.5 m. depth 3.5 m below basin bottom)

Fig. 8. Wave generator with a phase difference of 120°. Wave length

1.70 in

16. General arrangement ofthe platform alongside the adjusting-discs. Irregular waves

Photo 11 Generated regular and irregular oblique

waves (1956) in the seakeeping basin of NSMB (now Maritime Research Institute Netherlands

(MARIN). Quoted from International Shipbuilding

List of Papers

1. Generation and absorption of waves

S. Naito

2. Multi-directional wave basin and Mach reflection of

28

periodic waves

Y. Moriya & M. Mizuguchi

3

Nonlinear effect on the estimation of directional

₄₄

wave spectrum

T. Sek-imoto

Application of side wall reflection type directional

60

wave generator and Laser beam type wave surface probe

- especially on the linear characteristics of the one point

concentrated transient water waves

-T. Hirayama, N. Ma, -T. Harada & J. Lee

Effects of measuring positions on directional wave

83

spectrum in a basin with side wall reflection

Y. Feng, T. Kinoshita & W. Bao

Non-reflected multi-directional wave maker theory

and experiments of verification

K. Ito, H. Katsui, M. Mochizuki & M. Isobe

7 Characteristics of water waves generated by a multi

unit wave maker in a towing tank

M. Itabashi

Hydraulic model tests on floating type and offshore

man-made island type

power plants with directional

wave maker

M. Ikeno

Wave force and overtopping rate in directional sea

160

T. Hiraishi

Multi-directional wave maker and its real

time

175

wave control system applied to a seakeeping model basin

B. T. Nohara, I. Yamamoto & M. Matsuura

106

126

Wave Generation '95 Yokohama, Japan, 25 Sept. 1995

Generation and Absorption of Waves

Shigeru NAITO

Department of Naval Architecture

and Ocean Engineering Osaka University

2-1 Yamada-oka, Suita, Osaka, 565 Japan INTRODUCTION

It is a very important project for the researchers of naval

architecture and civil engineering to develop the theory to generate the ocean wave field in an experimental wave tank.

The real ocean wave field has the infinite area but the experimental tank does

not, therefore solving that problem is not so easy. With these points as

background, recently a wave tank has been equipped with the absorbing wavemaker

in many research institutes and universities. Many theoretical and experimental papers have been presented on the absorbing wavemaker with performance to absorb

and generate regular and irregular waves in wave tank.

They are concerned with the characteristics of absorbed wave energy and the

condition of optimum control in regular waves. Practically this is not

sufficient, and more interest must be focused on the problem in irregular waves.

The extension and integration of these theories to solve the problem mentioned

above is required.

The optimum conditions for complete absorption of wave energy in regular waves

are well known. However, to establish the system of wave energy absorption in

irregular waves, it _{is necessary to consider the problem on the basis of the}

time domain analysis, because the optimum conditions depend on wave frequency

and irregular waves contain many component waves having different frequencies.

This paper shows the theoretical review of wavemaker and absorbing wavemaker,

and generation and absorption of irregular waves. _{In addition to the theoretical}

research, experimental result are shown.

The new absorbing wavemaker has been built and experiments _{are carried out and}

also some simulations with a two dimensional model of floating type wave

absorber in irregular waves are carried out _{to confirm the validity of these}

formulations.

1. BASIC THEORY OF WAVEMAKER

1. 1 Theory

The basic theory of wavemaker has already established by _{Watanabe and Havelock,} which is two dimensional theory. The theory is extended _{to make the directionel}

wavemaker and absorbing wavemaker. That theory is summarized as follows.

The coordinate system is defined as Fig. 1.

At y-z plane the infinite length wavemaker is _{arranged. Water depth is h. We} assume that the fluid is irrotational and incompressible. Laplace equation _for

wave potential (1)(x,y,z;t) is satisfied.

0 < x m

V20(x,Y,z;t) = 0 in -co < y < m

(1)

The wave potential must satisfy four boundary conditions as follows. ao 0 : on z -h [Bottom condition] (2) az acD2 80

+ g 0 : on z - 0 [Free surface condition] (3)

at2 az

Radiation condition (4)

Condition due to the motion of wavemaker (5)

The elevation of free surface is

Ti(x,y,t) =

1 ao 1

at Jz.0

The potential is presented as 0(x,v,z;t)=4)(x,y,z).exp(-iwt)

Fourier transform and inverse are defined as follows

1

a,

$(x, z)- 0(x,Y,z)exp(-i4).dy : (h(x,y,z) = (1)(x.9.,z)exp(i4).(H. (7)

By using $(x,9.,z), the equations (1),(2),(3) are rewritten

(7) + - Q.2$ - 0

- 0 on z = -h

- (02$ + g. 0 on z 0

The form of the solution which satisfies the boundary conditions (2),(3) and (4) except (5) has been obtained as follows

0(x,k,z) = -iA,(9.).cosh{k(h+z)}.exp(iik2-9.2.x)

+ A0(9.).cos{kn(h+z)}exp(-ik2+9.2x) (9) nS I

The first term shows a progressive wave, second term shows a local wave.

Ap(9.), An(i) are unknown constant values obtained by the condition (5), which

are proportional to the generated wave amplitude. The values of k and k

determined by the dispersion relation of water waves.

(02 - g.k.tanh(kh)

w2 -g.ka.tan{k,h}

Let the displacement of wavemaker be

D(0,y,z).exp(-iwt) (11)

The velocity of it is

U(0,y,z) -iwD(0,v,z) (12)

At x=0, the x-directional velocity of wavemaker is the same one of the fluid velocity.

(6)

(8)

$.(0,k,z) = C(0,1.,z) (13)

where C(0,k,z) F[U(0,y,z)].

Calculating ck, we obtain

C(0,k,z) = AD(k)ik2-9.2 cosh{k(h+z)}-Z k(k)/kn2+k2cos{k(h+z)} (14)

Multiplying both sides by cosh{k(h+z)} and integrating -h to 0 with respect to z

and using the orthogonal relation between trigonometrical function and

hyperbolic trigonometrical function, the coefficients Ap(k),k(k) can be known

0 Ap(k)- I 0(k,z)cosli{k(h+z)}dz // Jk2-9.2 - h

i

0 A,,(k) = - C(k,z)cos{kr,(h+z)}dz//ikr,2+9.2 iw

no(x,v) =

- 0,1 cosh(kh) g

z0

27rg -3-R(kh,k) cosh2{k(h+z)}dz = (15) 1k2-k2 cos2{10h+z)}dz 6 - h k(k).exp(iJk2-9.2x).exp(iyk).dk (20)

The mutual relations among the _{wave number of directional wavemaker(ko), the}

wave number of progressive wave(k) and wave direction(8) are

- Q(k,, h, k),/

/lc

2+9.2 (16)

where the coefficients R(kh,U,Q(kji,k) are

R(kh,k) _{C(k,z)cosh(k(h+z)}dz / cosh2{k(h+z)}dz (17)}

6 - h 6

0

Q(k. h, 9.) - _{C(k,z)cos{ka(h+z)}dz / cos2{10h+z)}dz} (18)

The solution of wavemaker theory defined by equation (9) is obtained by the

inverse Fourier transform.

1 (b(x, y, z)- $(x,k,z).exp(iky)dk 27r 27r A0(k).exp(iJk2-k2x).exp(iyk)dk.cosh{k(h+z)} 27r 0.1 An(k).exp(-Jkfl2+0x).exp(iyk)dk.cos{k,(h+z)} qb, (x, y, z) + (x,y, z) (19)

The first term

C(x,y,z)

is a progressive wave potential, the second term

02(x,y,z) is a local wave potential.

2a is the length of wavemaker. Performing Fourier transform of the product of

eq.(24) and eq.(25).

F[U(0,v 7 j = [u(z)exp(ik0y).T(Y)].exp(-i9.0dy = u(z) T(y) = 1 0 a 2.sin(k0-k)a exp[i(ko-i)y].dy =

u(z).

C(.,z) (26) -a ko-11

Therefore, Ap(k),An(k) and

R1(kh) are obtained as follows

cb, (x, y,

z)-R,(kh)

Ap (Q.) s

2.sin(k0-k)a 12,(kh)

(Z) =

The underlined part is the end effect term of finite length of directional

wavemaker. As the result, far field wave potential is obtained

2sin(k0-i)a 2g , (k,-9.).Jkz-0 (ko Jk2-0 k,-9 Jk, +0

Iy5

a ) IY1

>a

j (25) (17)' .R1(kh).exp(iJk2-9.2x).exp(iWR.cosh{k(h+z)} 2/g coo = Jgko 2g/T0 (21) k w2/g w = Jgk = 2g/T (22)

k, k.sin(0) : _'1k2-1c,

2 -K 2-

k'slne kcos(0) : k>ko (23)

and the relations are shown in Fig.2

The theory mentioned above is the general directional wavemaker theory.

The characteristics of various wavemakers can be specified by defining the

velocity of the wavemaker as U(0,y,z) or the displacement D(0,y,z).

By separating variable y and z as shown

U(0,v,z) =

u(z)exp(ikoy)

: D(0,y,z) = - u(z)exp(ikoy)/iw (24)

and giving u(z), characteristics of various types of wavemaker can be discussed

as follows.

1.2 Finite length-directional wavemaker [Fig.3]

At first time, for discussing the characteristics of wavemaker, it is better to

define the finite length-directional wavemaker. We define the masking function as

u(z).cosh{k(h+z)}dz / cosh2{k(h+z)}dz

-5-q52(x,y,z)

1.3 Infinite length _{- directional wavemaker}

[Fig. 4]

The potential of infinite length

_{directional wavemaker is derived by}

_considering

the case a->- at eq.(31),(32) and the definition of Delta function

_{is used to}

perform an operation

as follows.

sin(ax)

a->o. 7rx

2.sin(k0-9.)a

exp(iik2-9.2x).exp(iy9iR

(29) .R1(kh).cosh{k(h+z)} 27r

(k0-) .4k2-9.2

Using the next relation

2.sin(k0-k)a

a

exp{i(ko-)}d

a (30)

(ko -9.)

the equation (29) can be simplified with Hankel function

Ho( 1) as

-iR,(kh).cosh{k(h+z)}

a exp{i(ko-WEexp(iJk -0x).exp(iy9.)dkdk

Zr 27r _a

A2-0

-iR,(kh).cosli{k(h+z)}

a

_{exp(iJkz-9.2x).exp{ik(Y-)}dk}

exp(iko [

]clE

27-c a

-iR,(kh).cosli{k(h+z)}

a

exp(ikoE).7r41,(1) [lax2+(v-02]..dt

₍₃₁₎ 27r _{_a}

In a

manner similar

_{to get the}

_{progressive wave potential,}

_the

_local

wave

potential is obtained with modified Bessel function

Ko

Q,(10).cos{k(h+z)}

E "-[

_.

a _aexp(ik0 a exp{i(ko _a k).2.K,

is

exp(-Jkfl2+9.2x).exp(iyQ)dk 27-c

Q1(10).cos{k,.(h+z)}

= 9.))E.dk].

[lin,/x2+(y-)2]dk

" a

exp(ik, 0.11,(

(32) (33) Jk02+9.' [kix2+(37-t)2]..dE 27r

The wave amplitude at far field

iw _{co.R1(kh).cosh(kh)} 770(x,y) = g 2g

,_a

1

'

-

2-sin(k0-Ua

Q,(knh) Z

_{exp(-An2+0x).exp(iy9.)Rcos{ka.(h+z)}}

27r _,.. Ica-Q. Jkfl2+0

Substituting eq.(34) into eq.(29), the progressive wave potential is derived Lim cb, (x, y, z)= -i.Ri(kh).cosh{k(h+z)} sin(ko-Q)a exp(iJk2-i2x)exp(iy9.)

27rf

lim dQ _. a-. Jk2-2 -i.lii(kh).cosh{k(h+z)} "- _{exp(iik2-0x)exp(iyQ)} -27r- o(k0-9). d9. 27r _ 03 ik2-0 exp(iJk2-k02x).exp(iyk0) = -i.lt,(kh).cosh{k(h+z)}. 10-k, 2 exp(i.k(x.cose+y.sin(9)} = -i.1?1(kh).cosh{k(h+z))* (35) k.cose

As the same manner the local wave potential can be obtained.

exp(-Jkfl2+ko2x).exp(iyko)

Lim 43,2(x,y,z)=Z Q1(10).cos(kr,(h+z)). (36)

a4w ./1( +k02

An orbital velocity is

qbly/C51. = iksin(0)/ikcos(0) = tan(0) =

Vror/V.,

(37)

The direction of an orbital velocity agrees with one of progressive wave. In

the case of finite length wavemaker, both values don't coincide. This is an end

effect and this difference must be considered.

1.4 Element wavemaker [Figs. 5, 71

The element wavemaker is defined by the limiting case a40 at eq.(31) and eq.(32).

Operating delta function (5(E-0) to the integrand of eq.(31) and eq.(32) yields

-i.lti(kh).cosh{k(h+z)} a

Lim (1),(x,y,z)= exp(i10).H,(1)

a->0 2

-i.R, (kh)cosh{k(h+z)} ( 1) [kJx2+y2] (38)

Lim (j), (x, y, z) Q1 (kn h)cos{kn(h+z) } K. [kn Jx2q2] /7r (39)

a>f)

This result shows

Cy/d),.

yix - tane, which is similar to the periodical

singularity making same amplitude wave at every direction.

The element wavemaker is useful to know the characteristics of wave field

surrounding the tank wall, which has an arbitrary configuration, by the

1.5 Infinite length Exponential type Directional wavemaker [Fig. 61

The displacement D(0,y,z) and velocity U(0,y,z) of wavemaker _{are explained as}

follows.

The infinite length exponential two dimensional wavemaker is defined in the limit case k040. When a->0, the infinite length piston type directional

wavemaker, and when k0--)0 and a40, the (infinite length) piston type two

dimensional wavemaker are defined respectively. At first time, perform the Fourier transform of U

F[1.1(0,y,z)] = _{uoexp(az)exp(ikoy).exp(-iky)dy 2gu0exp(az)6(k0-9.)}

o

Substitute these results into eq.(20), the free _{surface elevation is known.}

ri,(x) ido

sinh(2kh) + 2kh

-7-exp(ikx) 17701 - 2d0 ₍₄₃₎

as the results, R(kh,9.), Ap(k) are determined

0 R(kh,k) = 2g.u0.6(k0-9.). exp(az).cosh{k(h+z)}dz/ h cosh2{k(h+z)}dz - h 271..u0-6(k0-U.Rp(kh,a) Ap(k) _{27r.u0.6(k0-9.).Rp(kh,a)}

/

-co.cosh(kh) -27.u0 R, (kh, a) (5(k0-9.).exp(iJk2-ex)exp(iy9.) 770(x, y) -(19. 2gg w.cosh(kh) 1k2-9.2 exp(iik2-1c.2x).exp(iy10 .-iw.d..R.(kh,a). 1k2-k02 D(0,y,z) U(0,v,z) u. = = do.exp(az).exp(ikoT) u..exp(az).exp(ik.y) ₍₄₀₎ i.d0.k-sinh(kh)qp(kh,a)-exp[ikfx.cose+y.sin(911/(k-cose) (41)

Rp(kh,a) can be derived as follows

41c-Ea-exp(-ah) - a.cosh(kh) + k.sinh(kh)]

R,(kh,a)

-(42)

(k2 - a2)[sinh(2kh) 4 2kh]

1.6 Infinite length Piston type _{Two dimensional wavemaker [Fig.}

8] This wavemaker is defined as

ko,o(=e,o)

_{and a->0}

4.sinh(kh)

Lim Rp(kh,a)

-a->0 _{sinh(2kh) + 2kh}

The wave amplitude is

17101- -ido-sinh(kh)

kh.lexp(4kh)+4.kh.exp(2kh)-1) Calculated result of Incido I is shown in Fig. 16-2.

u(z) A,.k,.cos k,(z+h)

and integrate u(z), the displacement d(z) is

d(z) = i.A .k ,-cos 10z+h)/(0 (48)

The amplitude of d(z) is do at z=0, Ai is specified as follows

A, - -ico-do //k, .cos(k, h)

(49)

n, (x, - do .tan(k, h)exp(-k, x)exp(-iw t)

(50) 1.8 Infinite

progressive This wavemaker

Finding the function

R,(kh)=

length flexible type

wave

is special, but important. u(z) to make R,(kh)

0

u(z).cosh{k(h+z)}dz//

h ,_h

Two dimensional wavemaker without

zero, written by eq.(17)',

cosh2{k(h+z)}dz (17)'

the special wavemaker can be obtained. The function u(z) generally is

determined as follows

u(z)= Afl-kfl.cos kfl(z+h) : n is arbitrary number

(47)

This result shows that the motion is in existence innumerably which does not

generate progressive wave shown in Fig. 10. In case of n=j,

When kh the limiting value 2d0 is well known. Calculated result of A

(-17101/1d0 1) is presented in Fig. 16-1.

1.7 Infinite length Flap type Two dimensional Wavemaker [Fig. 911

This wavemaker is defined as follows

ko 0

Displacement of wavemaker : d(z) = (1 + z/h)do (44)

Velocity of wavemaker : u(z) = -iw.(1 + z/h).do

In this case, the analytical form corresponding eq.(17) is

0 R,(kh) = (1 + z/h)cosh{k(h+z)}dz// cosh2{k(h+z)}dz 4exp(kh).{(kh-1).exp(2kh)+2.exp(kh)-(kh+1)) (45) kh.{exp(4kh)+4.kh.exp(2kh)-11 4exp(kh){(kh-1)exp(2kh)+2.exp(kh)-(kh+1)}

The displacement of the wavemaker at an arbitrary depth z is

d(z) = do -cos ki(z+h)/cos(kih) (51)

The experimental result is shown in Fig. 16-5, which wavemaker's motion is the

lowest order mode.

1.9 Infinite length - flexible type - Two dimensional wavemaker without local

wave [Fig. 11]

The motion of wavemaker which does not generate a local wave, namely no added mass, is a very special case as An = 0.

An = -4 u(z)cos kn(z+h)dz//[sin 2k0h-t2k0h] = 0 (52)

It is easy to obtain u(z) satisfied eq.(52) as follows.

u(z) = i-k-An-cosh{k(z+h)) (53)

Integrating u(z), the displacement of wavemaker is determined.

d(z) = - kAncosh{k(z+h)}//w ₍₅₄₎

Let the amplitude at z=0 of d(z) be do.

Coefficient An : An= do-co//k-cosh(kh) (55)

Progressive wave : n = - i-do-tanh(kh)-exp[i(kx-cot)] (56)

Displacement of wavemaker : d(z) - - do -cosh{k(z+h)}//cosh(kh) (57)

kh-4-Wave amplitude ratio [Fig. 16-41: A = I 77 I

//

I do I = tanh(kh) 44-) 1 (58)

-A- shows the ratio of axes length of the trajectory of _{wave orbital motion which}

is elliptical in shape. This motion is unique shown in Fig. 11.

1.10 _{Vertical Element Wavemakers equipped at Wave Tank Having An Arbitrary}

Configuration

Two vertical element wavemakers P, and P, are considered in Fig. 14. As the

distance between P. Pj is Q, the phase difference between Pi and Pi is k cos

6 shows the direction of _{the regular progressive waves. Therefore, when the}

normal velocity of the Pi wavemaker is assumed Ui=u(z), the normal velocity of

the 13, wavemaker must be setted as Iyu(z)exp(kcos(69.)).

As the result, the envelop line of cylindrical waves generated with the two

wavemakers propagates to e direction.

Applying this idea, _{a regular wave can be generated by the many vertical element}

wavemakers fixed at the wave tank having an arbitrary configuration. _Calculated

wave contour examples in two wave tanks having different configuration _{are shown}

in Fig. 15. _{In case of these examples the fixed wavemaker has the performance to} make and absorb wave simultaneously.

1.11 Some numerical and experimental results

The new two dimensional wavemaker is built up.

It consists of ten horizontal element wavemakers. Each element _{wavemaker can be}

-9-driven independently with personal computer control.

Performance of five kinds of wave makers on A (the ratio of the progressive wave

amplitude and the displacement of wavemaker at free surface) are shown in

Figs. 16.

2. THEORY OF WAVE ABSORPTION

Wave absorption means the extraction of wave energy, namely the meaning of absorption and extraction are the same. The mechanism of it consists of usually the vertical plate(piston type and flap type) and floating body at the terminal

of wave tank. We discuss two type absorbing wavemaker.

2.1 Twodimensional absorbing wavemaker without local wave piston type

[Fig. 12]

In the region xM in Fig. 13, the velocity potential is assumed as follows.

-ikx ikx

= [A,.(co/k).cosh{k(z+h)}.e + Ar*(6)/k)cosh{k(z+h)}.e 1/sinh(kh)

Eknx-cos{k,,(z+h)} (59)

The first term is an incident wave, the second is a reflected wave, the third

is a local wave. A,, A, are an amplitude of incident waves and reflected

waves respectively. The velocity of the fluid particle on the wavemaker and the

surface of the wavemaker must be the same as follows.

i.co.d(z) = i.w.(A,-Ar).cosh(k(z+h)1/sinh(kh) E k.ko-cosika(z+h)I (60)

To determine the unknown coefficients, the orthogonal characteristics of cos and

cosh functions are used as

AJ2kh+sinh(2kh)}

d(z)cosh{k(z+h)}dz = (63)

4k.sinh(kh)

We consider two cases satisfied this equation.

[a] Piston type : d(z)=do

The displacement of wavemaker can be obtained easily from eq.(63).

no iw(k-Ar) 2kh+sinh(2kh) -h (61) d(z)cosh{k(z+h)}dz iwsinh(kh) 4k 0

Ak

sin(2kh)+2kflh (62) d(z)costkfl(z+h)/dz = i 4k -h

Substituting A,=0, namely the condition of no reflected wave, into eq.(67), the

0

d(z)cosh{k(z+h)}dz = -h

dt,

A1{2kh + sinh(210)} {2kh + sinh(2kh)}

cosh2{k(z+h)} A1{2kh+sinh(210)}

dz = (66)

sinh(kh) 4ksinh(kh)

This result agrees with the right hand side of eq.(63) precisely. Namely the motion presented by eq.(65) satisfies the condition which means not to generate the reflected wave. At the same time, eq.(65) satisfies the next relation by

substituting eq.(65) into eq.(62).

An - 0 ₍₆₇₎

This means that the local wave does not exist either.

Namely the unique absorbing wavemaker defined by eq.(65) does not generate a

reflected wave and a local wave. In the wave field surrounded with the

absorbing wavemaker of which motion is shown eq.(65), an incident wave only

exists. The movable wall like this is the rdreamlike wall].

It is important for numerical research on water _{wave to consider the radiation} condition at infinite far boundary precisely, because at the open boundary a coming wave must be transmitted perfectly and a local wave must be not to

generate simultaneously. _{The unique absorbing wavemaker mentioned above should}

be used on the research.

2.2 Infinite length directional absorbing wavemaker piston type

-We obtain the motion of wavemaker to absorb the incoming wave at an angle with

respect to the y-axis. The wave numbers are defined k 2

k2 + k2

k2 _

kn 2

k 2 ₍₆₈₎

Wave potential is presented in x0 except exp(-it) term.

y.y

((x,y,z) = [A .(w/k)-cosli{k(z+h)}.e-i(k. x+k )

+ An.(w/k).cosh{k(z+h)}.e-i(-k. _J'Ysinh(kh)

+1; An.e knx".cos{kn(z+h)}.e _{ikY "'}

It is interesting that the third term, local wave potential, has the

characteristics of progressive wave shown _{as exp(-iky). Assuming the motion of}

wavemaker to absorb an incident wave by considering _{eq.(69) as follows}

(64)

(69)

4sinh2(kh)

4k-sinh(kh) coshk(z+h)dz

[b] Exponential type

Assuming the displacement d(z) of wavemaker as follows

kh4

d(z) = A1-cosh{k(z+h)}/sinh(kh) A,..exp(kz) (65)

D(0,Y,z)=dcosh{k(z+h)}.e ikv.Y : U(0,Y,z)=-iw.d.cosh{k(zth)}.e ikY" (70)

Comparing eq.(7C with differentiate eq.(69) with respect to x, the relation is

derived as follows.

cosh{k(z+h)}

-iw-dcosh{k(z+h)}=ik.(A.-A,).(w/k). E Ankr,cos{k,(z+h)} (71)

sinh(kh)

Multiplying both side by cosh{k(z+h)} and integrating -h to 0, the result is

ik.(A.-k).(w/k)/sinh(kh) (72)

Substituting the condition A.=0, the relation

d - A (kjk)/sinh(kh)

is obtained. Using this relation, D(0,y,z) can be obtained.

D(0, y, + Ai

k. cosh{k(z+h)} -ik..y kh-- k. -ik..y

kz

44÷44 A.

e

.e

sinh(kh)

This equation shows the unique motion of wavemaker which does not generate a

reflected wave. If e is replaced by 0, that is ky-4,k.->k, the result agrees

with eq.(65) shown in case of the two dimensional result. This result teaches us

that the absorbing problem of directional regular s.aves is similar to two

dimensional one like strip theory shown previous chapter.

2.3 Optimum External Control Force in Regular Waves Floating body

-In general, the equation of heaving motion Z of a floating body is expressed by

A

[M+m(w)1z"+[N(w)+Ne(w)1z'+[Ke(w)+C]z = pg2 Vt) (74)

where M: mass of floating body, m(w): added mass, N(w): wave damping

coefficient, C: restoring force coefficient, Ne(w), Ke(w): damping and spring

coefficients of external mechanism, A = iX,eXP(iER):(AR, EN: amplitude ratio of

progressive wave amp. to heaving motion amp. and its phase). Let heave z(t) and

wave C(t) be expressed as follows,

z(t) zoexp(iwt), (t) = oexp(iwt) (75)

Frequency response function of heaving motion Hc.(w) is given by

zo pg2A/w2

Ke(w)+C-w2[14+m(w)lliwEN(w)+Ne(w)]

(76) (73)

Then, the conditions to obtain the maximum energy of regular waves are given by

These relations are written as word as follows.

The spring coefficient of external mechanism is selected so that a floating

body synchronizes with incident regular wave.

The damping coefficient of external mechanism is equal to the radiation wave

damping coefficient of a floating body itself.

These two coefficients depend on the incident wave frequency. Therefore, to

extract the energy of irregular waves composed of many wave components, it is

necessary to make a time variable external mechanism. A general view of this

system is shown in Fig. 17.

2.4 Representation of the optimum conditions in time domain

In irregular waves, it is necessary to satisfy the conditions shown by eq.(77)

in the time domain instantaneously. We define the Fourier transformation as

follows,

1

F(w) - f(t)exp(-iwt)dt , f(t) = f(w)exp(iwt)dw (78)

27 - 00

Hereafter, let Fl and F denote the Fourier transformation and the inverse one

respectively. The impulse function L(t) of fluid is given by

1

L(t) [N(w)+ico1m(w)-m.1]exp(iwt)dw ; m = m(-) (79)

27

Fourier transformation of this equation is

N(w)+iwfm(w)-m.1 = L(t)exp(-iwt)dt _(79)'

The real part and imaginary part of both side are compared respectively,

following relations are obtained.

1 CO CO

m(w) = ma, L(t)sin(wt)dt : N(w) L(t)cos(wt)dt

L(t) is real function, so eq.(79) is transformed as follows.

1 1 CO L(t) it 0 N(w)cos(wt).dw it 0 w[m(w)-m.,]sin(wt).dw (80)

The first term is even function, and _{the second term is odd function on respect}

to time t.

Usually, _{an arbitrary function can be expressed by using of the even and odd} functions, so let Le(t) and Lo(t) denote the even and odd parts of L(t)

respectively, then

-13-Namely,

As the result

2pg2

Fl[L(t)]= L(w) = .A2(w)

3

Eqs.(83) and (85) are the representation of the optimum conditions to extract

the irregular wave energy in the time domain.

In Fig. 18, the numerical simulation results on the control force of external

mechanism and extracted irregular wave power are shown in form of time history.

2.5 Equation of motion under the optimum conditions

When the external force is acting on the floating body, the equation of motion

becomes an integro-differential equation as follows.

1 L(t) = 71 -(M+m.)e(t)+ L(E)z'(t-)dCz+ - 00 pg2 pg2 . 2. A2(w)coswtdw : N(w)= A2(W) 0 w3 w' _fl(t)z'a-Od (82) K(t)z(t-E)dt = F,(t) (86)

The third and fourth terms of left hand side show the external control force

L(t) Le(t)+Lo(t) (81)

1 CO 1 00

Le(t) = N(w)cos(wt)dw : Lo(t)= w[m(w)-m.]sin(wt)dw (80)

71 0 71 0

L(t) must satisfy the causality, therefor the relation is obtained as follows,

L(t) = Le(t)+Lo(t) = 0, t < 0 1

L(t) = 2Le(t) 2Lo(t), t > 0 J

Let K(t) be F-I[IL(6)],then K(t) is given by

K(t) = F-1 [K, co)] = F-1[-C-E(02(M+m.) + (02{m(w)-m.}]

= -Co(t)-(M+m.),5"(t)+F- qw2(m(w)-m.}7- -CO(t)-(M+m.)(5"(t)-Lo'(t) (83)

1 CO

where Lo'(t) = w2fm(w)-m.)cos(wt)dw (84)

it 0

Next, let n(t) be F-1 [Ne(w)], then 11(t) is written as

1 1

E(t) = F-1 [Ne(w)] = N(w)coswtdw - Le(t) L(t) : t>0 (85)

Fec(t), and it is shown by changing two terms into other form as follows.

00 n.

Fec(t)= -(M+m.)z"(t)- Cz + Le(k)z'(t-)dk - Lo' ()z(t-)d

Substituting eqs.(84) and (85) into eq.(86), and by using next relation

kr.

CO CO

Lo'()z(t-)dE=[Lo(k)z(t-k)] - Lo(E)z'(t-k)(-1)dt= Lo()z'a-OdE

kr_. - CO

Eq.(86) is rewritten as

CO

L()z'(t-E)dk + _co[Le()-Lo(E)]z'a-Od F.(t) (86)'

The second term of left side is important physically, and it means the control force of external mechanism to extract the energy of irregular waves perfectly.

This term is transformed by eq.(82) as follows,

I=ITLe()-Lo(E)]z'(t-)dN L()z'Ct-Old + 0 0 -- r-+ ][Le()-Lo()Jz'(t-E)dk=1[Le()-Lo(k)]z'(t-k)d -- 0 j0

By variable transformation as and with the character of even function Le() and the relation shown in eq.(82), I becomes

2Le(-)z'(t+E)dE = L()Z'(t+k)dE (87)

0 0

Eq.(86)' is rewritten as a final form by

00 CO

L()z'(t+E)(1 F.(t)

0

This result means that the ideal system to absorb the irregular wave energy perfectly can not be realized because the second term of left hand side,

namely the external control force, does not satisfy the causality.

Eq.(86)" can be transformed into the equation in the frequency domain easily as

follows,

ipeexp[iEH()]

2L(w)ica(w) F (0, where F ()= A2(6)).(w) (88)

co2

As the results, _{the frequency response function with optimum control of external}

mechanism is derived. Z(w) exp[iEK(0] -15-(86)" 2A,() (88)'

where Fl[L(t)]=L(w) and Fl[z(t)]=Z(w).

Let's consider the time mean of absorbing power ET under the conditions

mentioned above. Because z' (t) and D(t) are real function, the following

expression can be applied.

1 Pg2 E = lim ET -T4 2 T/2 1 (t)D(t)dt -T/2 2gT 1 pg2 S?. (w) = 2 2 ,0 F,[D(t)]Z'*(w)dw 1 [k(w)]2 pg2 oo _S(w) [lim ]dw = 0 w 2gT 2 0

Whole power of irregular waves

2

do

(89)

where S(w) is a wave power spectrum. Usually, Pierson-Moskowitz type spectrum

S".(w) is used, therefore the relation between both spectra is given by

S(w) = S"(w) ; 0 < w < (92)

2

As the results, E becomes

The theory mentioned above is based on the assumption that a floating body has a

symmetric sectional form. In case of unsymmetrical body, k(w) and Z(w) can be

where D(t) is a force acting on the real energy absorbing part in the

mechanism, and * means a conjugate complex.

external

't 1

FI[D(t)] = F,[ ii(E)z'(t-k)dC = L(w)iwZ(w) (90)

0 2

Substituting eq.(90) into eq.(89),

1

2z(c1)z.(w)L(w)dw = w2Z(w)Z(w)L(w)(16) (91)

4gT 2gT

The wave power is

pg2a2(w)/(4w)

where a(w) is an amplitude of regular waves, frequency and the mutual relation

between a(w) and SP.(w) is

a2(w)=2SPK(w)dw

This relation is applied, and using the eqs.(88) and (88)', the absorbing wave

changed as follows.

pg2

A+ 2

k(W) = (XT24-2), [Z(W)]2 = .1 (W)12 (93)

3 ' HV2+T212

where A and A indicate the progressive wave amplitude ratio of weather side

and lee side of floating body respectively.

In this case, E becomes

pg2

SPI(w) A+ 2

dco

2 0 X72+A-2

If we could make the unsymmetrical body with the character of

A (w) = 0 ; through out co,

Eq.(94) is rewritten as

(94)

-17-2 0

This formula gives whole power of an irregular wave presented its spectrum as

SP.(w).

By the fact as described above, the theory on absorbing irregular waves power

can be clarified.

3. _{RANDOMNESS AND UNIFORMITY ON PHASE OF IRREGULAR WAVE}

When we generate an irregular wave numerically, to select the phase 61 mentioned

below of each component wave is important and its result affects on the

stochastic characteristics of the simulated waves. In this section the

randomnessand uniform distribution of the phase is discussed.

An irregular wave is usually presented with Fourier series as follows. 77(t)=E [a cos(w 0+1) sin(60 ] = fa 2+b 2 cos(

"

t6t-00 )

(95)

n " n

= arctan(bn/an) : phase

where the distribution function of _ar and bn is _{a normal distribution and the}

correlation of an and bn is

E[an bn I

= ₍₉₆₎

iE[an 2]E[b0 2]

Usually p=0 is assumed, namely an and bn are independent. By using this

assumption 0=0, it is derived the phase probability density function _{is the}

uniform distribution. But in the actual sea, is that assumption always satisfied?

If that assumption, p=0, is not satisfied, we want to know how the phase

distribution changes.

At first step, the probability density function of _{en(wn), must be obtained. The}

pg2

S,.(w)

joint probability distribution function of an and bn is shown as follows.

1 1 an 2-2pan bn fb, 2 1

f (an , bn ) = exp

27a2i1-p2 2a2(1-p2)

where a2 is a variance of stochastic process n(t).

The variable transformation is performed by using next relation.

bn

(98)

a,

As the result, a simple form on E can be obtained.

-R2(E2-2pE+1)

77exp dr? =

0 2a2(1-p2)

The right hand side of eq.(99) is Cauchy probability density function having the

peak point at E=p

Around p=0, the function is expanded as follows.

a 2a2

f(E) = (100)

g(E2+1) g(E2+1)

Where On=tan-I(E), this relation is transformed as follows.

fE(E) 1

f(e) = = + sin(28) + (101)

Id/de 7

When p=0, this result indicates that a, and b are independent each other, hence

the distribution of 6(w) becomes uniform like 1/g. In case of p#0, namely an and

bn are not independent, the second term of eq.(101) remains, therefore, that

term indicates the degree of dependency between an and bn.

The calculated time history of an irregular wave is presented in Fig. 19. At each

figure, the phase distributions are different as shown also in Fig. 19. When we

glance at these time histories, to distinguish the difference of phase

distribution among them is difficult. Of course these time histories have same

spectrum. Namely, it is not essential for the generated time history to make the

phase distribution uniform.

On the contrary, the order of the phase, that is randomness, is more important

than the uniformity for an irregular wave. An example of the most artificial phase order handling is an impulsive wave, namely transient waves shown in

Fig. 20. In this case the phase distribution is the uniform and the order of the

phase is arranged in necessary order to generate the impulsive waves. As the

results, these simulated impulsive waves do not have a stochastic steadiness but

have same spectrum.

To simulate the worst irregular wave for a floating structure or a ship in

ocean, it is acceptable to break consciously the assumption required the phase

(97) 1 RE) = 27(72./FP' (99) g(C-2pu+1)

of irregular wave.

4. GENERATION AND ABSORPTION

The basic theory on wave generation and absorption is shown in previous section.

When we consider an advanced wavemaker which has the performance to

generate

and absorb an irregular wave simultaneously,

the problem on the causality can

not be avoided.

On the realistic stand point of view,

the system does not need

to

satisfy

the

causality perfectly.

Therefore the

constant

coefficient

system of external mechanism is recommended.

We investigate the floating body with the external control mechanism.

The

equation

of

its

motion

z1 (t)which generates

a

regular wave(frequency

cc o )is shown as.

0

(Um.)z1"(01

1,()z,'(t-)dE+Cz,(t)=Fos(t)-[Ne(coo)z,'(t)+Ke(coo)z,(t)]

(102)

- 0,7

Fos(t)

is the oscillation force of the floating body with external mechanism.

The second and third term of right hand side are the external control forces.

The

equation

of motion

its

z2 (t)which absorbs

an

incident

regular

wave

(frequency co=coo)

is shown as

0 a2(6))+02(6) 0

where a(w)=Ke(c00)-(02[Vm(0], 13(c)= co[Ne(co0)+N(co)]

These systems satisfy the causality because the coefficients of

_{control system}

is constant.

These two motions are independent,

_{therefore the solution of the}

absorbing wave making

motion z(t)

of a

floating body can

be

_{superposed by}

solutions z,(t) and z2 (t)as follows.

z(t)=zi(t)+z,(t)

₍₁₀₅₎

Let incident irregular wave spectrum be S,n(co) and reflected

_{wave spectrum be}

S(w).

In case of narrow band spectrum,

_{the value of}

(co-coo)

becomes small,

hence Sre(co) becomes small.

Let Sge(co) be generated irregular wave spectrum and So.(co) be

_{composed spectrum}

of Sre(w) and Sge(w),

_{it can be presented as follows.}

S0

2(())+S, 2(6))+2S, . _(6.)S, . (co)cosEE(6))1 ₍₁₀₆₎

-19-(Mtm.)z,"(t)+

CO

L(E)z2' (t-E)dCz2(t)=Fw(t)-[Ne(co0 )z,

(t)+Ke(co, )z, (t)] (103)

In case of an irregular wave,

the coefficients Ne(coo) and Ke(co8) are selected as

the value at

co=co,

which is the representative frequency value of the incident

irregular wave,

which is usually the center frequency or mean frequency.

The

absorption efficiency of this system is given as

4N(co0)co2N(co) S(w) S(co)

When E(w)-7t thorough out co, the relation Sc0.(a)=1S,c(w)-Sge(6)1 is obtained.

And if Src(w)=0, Sc.(w)=Sge(w) is obtained. When the fact So.(6)=S,(co) is

confirmed by experiment, we can know the fact S_(co):4 is realized.

Namely the phase E(w) is the most important control variable.

In case of a regular wave, this phase E(w) can be determined to absorb an

incident wave and to generate a regular wave simultaneously having required wave

height.

5 CONFIRMATION OF THEORY BY EXPERIMENTS

The experimental concept on energy absorption by the floating body equipped at

the terminal of two dimensional tank are presented in Fig.21, and the results of

absorption efficiency of regular wave is shown in Fig. 22. This system is the

force control system which is very simple on the theoretical stand point of

view. Measuring the force acting on the floating body directly, the system must

be controlled so that its force always coincides with the theoretical forces

calculated with the measured velocity and acceleration of the body like

Ne(wo)z,'(t)+Ke(wo)z,(t).

In case of the irregular wave, the absorption efficiency of the experimental

results are shown in Fig. 23.

In spite of the constant coefficient system, almost irregular wave energy are

absorbed. We can expect to absorb the irregular wave energy like this because

the frequency band of irregular wave is usually narrow, the approximation (0-(00

is proper.

The experiment was carried out by using the transient package wave because it is

easy to separate into an incident wave and a reflective wave. Spectra of an incident wave and reflective wave are shown in Fig. 24. This figure indicates our system of wave absorb-maker can almost absorb incident wave energy. The time history of incident package wave and reflective wave are shown in Fig. 25.

The upper figure Fig.25-1 is obtained when the wave absorb-maker is not operated. The lower figure Fig. 25-2 is obtained when the wave absorb-maker

(constant coefficient system)is operated.

6 CONCLUSION

By giving the normal velocity on wavemaker surface, most kinds of wavemakers

are formalized.

Furthermore, we define the vertical and horizontal element wavemaker of which

the fixed wavemakers at an arbitrary configuration tank can be composed. That is, the wave field in the arbitrary configuration tank can be presented

by assembling the element wavemaker easily.

The optimum conditions in the time domain can be made clear to extract the energy of irregular waves perfectly. But the system consisted of a floating

body and external mechanism can't be realized theoretically, because it does not

satisfy the causality. On the realistic point of view, it is not so important.

In order to realize the actual ocean wave field in the restricted water area surrounded with wall like the towing tank, it must be solved to generate short

crested random waves having the expected wave spectrum and probabilistic

uniformity. This problem is not easier than the generating regular waves

written in this report. This is the subject for future study.

structure. Therefore it is acceptable to break consciously the assumption required the phase of irregular wave.

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Fig.l. Co-ordinate system

Wave-maker

Wave- maker

Fig. 3. Finite length - Directional - Wavemaker

Wave-maker

Fig. 4. Infinite length - Directional - Wavemaker

-23-W

,

T _Wave-maker

/A

_,,,

z x

Fig. 7. Horizontal element wavemaker Fig. 5. Vertical element wavemaker

Fig. 8. Infinite length

-Piston type - wavemaker

V

Fig. 9. Infinite length -Flap type - wavemaker Fig. 2. Relation among wave numbers of progressive wave K Fig. 6. Wave fields generated with the vertical element

wavemaker Ko and the propagation direction of wave 0 wavemaker (left figure) and finite length directional wavemaker (right figure)

n = 4

Fig. 10. Various kinds of motions mode of the wavemaker not to generate progressive wave

Fig. 11 Wavemaker not to generate local wave

A,

/111Ik

MI

Reflected Incident

wave wave

Fig. 12. Co-ordinate system for describing absorbing wavemaker without local wave

Fig. 14. Relation between arbitrary two vertical element wavemakers fixed at the wave tank wall

I

Fig. 13. Co-ordinate system for describing directional absorbing wavemaker

\.(

Fig. 15. Calculated wave fields in two wave tanks having different configuration with many vertical element wavemakers (upper : 90 element, Lower : 115 elements)

A 2.5 2.0 1 5 1.0 0.5 2.5 2.0 1.5 1.0 0.5 _oo ---0 2.0 4.0

Fig. 16-2. Comparison of A between experiment and calculation on Flap type wavemaker

2.0 4:0 3.3 Experiment --- 10 elements - infinite elements, =,^^ C ("1I 8.0 10.0 0 2.0 4.0 2.5 25-k h 0 Experiment --- 10 elements infinite elements! 3.0 10.0 2.0 4.0 6.0 8.0 10.0 h

Fig. 16-4. Comparison of of A between experiment and calculation on Wavemaker without local wave

-25-2.5 2.0 1.5 1.0 0.5 L 0 Expenment --- 10 eiements - infinite elements, _ 6.0 8.0 10.0 k h

Fig. 17. General view of waves energy absorption system

Fig. 16-3. Comparison of A between experiment and calculation on Wavemaker with snake motion to vertical direction (wave number of the snake motion of wavemaker)

2.5 (- 0 Experiment 2.0 --- 10 eiements - infinite elements ' 5 -- --- --- ---1.0 Co cf) 0.5

Fig. 16-1. Comparison of of A Abetween experiment and calculation _{Fig. 16-5. Comparison of A between experiment and calculation} on Piston type wavemaker _{on Wavemaker without progressive wave}

Wave Spring Damping

;(7) 10.0 3.0 lcti 9.5 2.0 1.5 1.0 0.5

Ii ill

I I

'.1

" '1,1111

11

Fig. 18, Control force of external mechanism (upper) and Extracted wave power of irregular waves

. A1111111116 at . aalih .10111 . ...41111111..

,..o zo. oc s. oc oo oo ao. oo So. oo S 0 .c ac

Phase A

A

'N

Fig. 19. Generated time histories of irregular waves having different phase:, (these time histories have the same spectrum)

Fig. 20. Impulsive waves (phase is controlled in necessary order

MOTOR) ACTIVATOR

FORCE L

0 ET E Cl ER

Fig. 21. System diagram of terminal wave absorb-maker

---

*******

00 2.0 a.0 5.0 S.0 10.0 12.0

L

1S

Fig. 24. Incident wave and reflective wave spectrum of the constant coefficient system

cal. exp. a FLOAT C) .= 1 A02 z 1.0 1.2 1.4 1.E5 1.3 ( 1/s ec)

Fig. 23. Absorption efficiency of irregular wave (Experiment)

= incident reflective ! .06'

i ,

u qv-vvvv- ,,v 7"\; ''v","-V \I

A A Phase B

'

..J-'

/,

A . ,,, v... ,..., .11, vA I J 1 PhaseC '' c

AA

i-Phase D D

' \

-7674--41---, .4

t

\I T=0 sec

.

A ) 11 V ' ij ' /' T = 5 sec :, ' \j jili T .10 sec I I I u 7 0.9 I I I 3 I T F,ec I

Fig. 22. Absorption efficiency in regular waves (Experiment)

X?

C3us al iystem cal

0 exp

const. coeft. system --cal ci exp Li

---incident wave group reflective wave group

20.0

Fig. 25-2. Measured wave form when the wave absorb-maker (constant coefficient system) is operated

-27-

27-35.0

35.0 (sec)

3.3 10.3 5.3 25.0 (sec)

Fig. 25-1. Measured wave form when the wave absorb-maker is not operated

71 A A A A A , . .A . A. . U )

)

j\ivvvy

n. A Ir\\ A. A, A, -\

VOI1JV

Wave Generation '95

Yokohama, Japan, 25Sept. 1995

Multi-directional Wave Basin and Mach Reflection of Periodic Waves

by

Yoichi MORIYA

Graduate school of Faculty of Sci. and Eng., Chuo Univ.

Kasuga 1-13-27, Bunkyo-ku Tokyo 112 Japan

and

Masaru MIZUGUCHI

Professor, Department of Civil Engineering, Chuo Univ.

Kasuga 1-13-27, Bunkyo-ku Tokyo 112 Japan

Y. MORIYA

ABSTRACT

First, the validity of both the end control method and the second order wave generation method, for generating uniform waves field of the target values of wave height and angle in

a multi-directional wave basin, are shown. Next, influence of the diffraction _{at the edge of}

a vertical wall on oblique reflection is studied both experimentally and numerically. _Finally,

stem waves generated by Mach reflection are discussed by numerical simulation of Boussinesq equation.

INTRODUCTION

In a multi-directional wave basin, generating a uniform wave field with the target values

of wave height and angle is important. However, it is well-known that both wave height and angle of generated regular waves show considerable spatial variation for the wave maker installed in a large wave basin (Takayama, 1984). The main cause is the finite length of the wave maker as a whole, or the discontinuity at both ends. The effect of the discontinuity may be suppressed by employing linear decrease of wave paddle amplitude at both ends of the wave maker.

Finite amplitude waves generated by a sinusoidal motion of wave maker in the _wave

basin of constant water depth is to break up to the main and the tail waves or to change the wave profile while propagating. Goring and Raichlen (1980) presented a wave generation method for one-dimensional finite amplitude waves, in which the finite displacement of the

wave paddle is considered in time series. However, multi-directional wave makers are usually controlled by the superposition of sinusoidal motion. Here, the second order wave generation method is developed by superposing two components of sinusoidal motion with Goring and

Raichlen's modification.

When oblique waves with small incident angle reflect from a straight breakwater, wave crests grow perpendicularly to the breakwater and these waves (stem waves) propagate along

the breakwater (Wiegel, 1964). This reflection is usually called Mach reflection. Mach

reflection of solitary wave has been studied experimentally (Melville, 1980), theoretically

(Miles, 1977) and numerically (Tanaka, 1993). Stem wave height of solitary wave may become four times of incident wave height (Miles, 1977). Mach reflection can be an important

problem on engineering. A numerical calculation of parabolic approximation of Boussinesq equation (Yoon and Liu, 1990) shows that Mach reflection of periodic waves may also occur.

However, stem waves of periodic waves is not well studied. Here we consider that Mach

reflection is a part of nonlinear diffraction. Linear diffraction follows Helmholtz equation

(Berger and Kohlhase, 1976). Weakly nonlinear diffraction in shallow water may be described Boussinesq equation.

The influence of diffraction at the edge of a vertical wall on oblique reflection is studied experimentally by measuring the wave height distribution, and the applicability of numerical

calculation of Boussinesq equation is examined. It turns out that stem waves can not propagate

long enough to fully develop in our medium-size wave basin. Stem waves after long

propagating distance are studied by numerical simulation with Boussinesq equation. Simulated

results are compared with analytical solution of K-P equation (Moriya and Mizuguchi, 1994) for symmetric hi-directional case.

MULTI-DIRECTIONAL WAVE BASIN IN CHUO UNIVERSITY

Figure 1 shows the multi-directional wave basin in Chuo University. The basin is 10.5 m wide, 6.8 m long, and 0.6 m depth. The multi-directional wave maker is located along one of the 10.5 m end-wall of the basin. The distance from wave paddles to the opposite end-wall is 5.2 m. The concrete floor of the basin is almost flat; bathymetry measurements yielded depth variations less than 0.5 cm. Wave absorbers are placed along the wall of the basin and around the side of the wave maker. The multi-directional wave maker consists of 28 piston-type wave paddles continuously linked or 29 sets of actuator rod. Each paddle is 30 cm wide and 50 cm high. Flexible sheets are inserted between each wave paddles as well as in a gap between wave paddles and the bottom floor.

We have recently expanded the wave basin to the length of 9.2 m. Experimental results shown here were obtained in the old wave basin.

THE END CONTROL METHOD

2.1 THEORETICAL BACK GROUND (Mizuguchi, 1993)

Wave maker of a piston-type in a infinitely wide and long basin is treated under the

assumption of small amplitude. After some mathematics, it is shown that water surface

elevation ri generated by a single sinusoidal stroke motion of continuous-type, asillustrated

in Fig. 2, is given by the following equation.

-29-IVA =a f'( -

i

)[N0(4))sin(ot)+,/,(4))cos(ot)]dq -kw kw 2sinh2kd a -2kd +sinh(-2kd) Akx._ ÷(icy)2

which is valid except very near the wave maker. Whcre A and a are the amplitude and

angular frequency of the stroke motion, respectively, k is the wave number for a and the water

depth d, and w is the width of a wave paddle. For multi-stroke motion, the water surface

elevation is given as the sum of the each paddle solution. The integral in the above equation is evaluated numerically.

The length of the wave paddles to be controlled for producing the uniform wave field

is studied for normally propagating waves. Figure 3 shows the relative wave height variation,

ER, along the line x = 0 against the length of the controlled paddles, New, where the

amplitude are reduced linearly to zero at both ends. It is clear that one wave length is the best choice. Numerical calculations were performed for a parabolic or cubic curve-type reduction of the amplitude with no improvement. For obliquely propagating waves, different length should be applied for each side, as the magnitude of the diffraction depends on the angle,

between the wave ray and the wave paddle. Figure 4 shows a correction factor which is the ratio of the control length for the oblique waves to that for the normal waves. This is obtained by using a simple diffraction diagram. When the angle between the wave ray and the wave paddle is acute, the controlled length is slightly shorter than the case of normally propagating waves. When this angle is more than 90 deg, the controlled length may become considerably longer than the case of normally propagating waves.

2.2 EXPERIMENTAL CONFIRMATION (Toita et al., 1993)

Experimental confirmation of the end control method is undertaken by using the multi-directional wave basin in Chuo University. Experimental conditions are of water depth d = 20

cm, wave period T = 1.2 s, S (= 2A ) = 2 cm. Surface profile was measured by_{some new}

capacitance type wave gages. The analog signals from these wave gages were digitized with the sampling frequency of 50 Hz. Measured data were analyzed by the zero-down crossing method. Figures 5 (a) and (b) show distributions of relative wave height in y and x direction for normal incidence. Number of controlled wave paddles are five each at both ends. Figures

6 (a) and (b) show distributions of relative wave height in y and x direction for oblique

incidence, for 0 = 20 deg. Number of controlled wave paddles arc, in this case, four at the right end and six at the left end as calculated from Fig. 4. It is clear that wave field generated

by using the end control method are very uniform. However, this method can not escape a

side-effect. The area of the uniform wave field is smaller than the originally-usable area, as

the controlled wave paddles at both ends do not give the _{same energy to the water as the other}

non-controlled paddle.

For a multi-directional wave field, each component wave of multi-directional waves should requires each relevant control lengths.

-31-y = (t). (4)

3. THE SECOND ORDER WAVE GENERATION METHOD 3.1 THEORETICAL BACKGROUND

A piston-type wave maker is assumed. Relationship between the position of the wave

paddle (t) and the corresponding velocity of water particle u(y,t) is

a(t)

-14(y,t)

at

For finite amplitude shallow water waves of permanent shape it can be shown from continuity

condition that the velocity averaged over the depth uw(y,t) is

cn(y, uw(y ,t

t))

)-

y = E (t), ₍₅₎

(y ,t

where c (= Vgd ) is phase velocity and g is the gravitational acceleration. Putting u =

lc

and

expanding n in Taylor series around y = 0, and retaining terms up to second order, we have

agar)

(0,02

a(t)

Er

at

\

=dill (0,0 + E

1 .

Next, n and E are supposed to be given by following expansions.

03 (0,t) = E a mcos(m In =0 (0,t) msin(nat)13 ax m=0

(t)=

E E

{amnsin(n at) -bnincos(n a t)}

kd m =On =0

Substituting for Eqs. (7), (8) and (9) into Eq. (6), and arranging the term in order, we have 2 1 1 a1l31

ai

(t)= {

a1sin(a t)+

2kd

- )sin(2 a t)} , (10) kd 2 2d

where a1, a, and p, can be calculated by expanding both a particular wave profile and its

differentiation with respect to x in Fourier series.

For oblique waves, the phase difference for the neighboring rods A is A -2TEwcos0

where L is the wave length and 0 is the wave angle. The phase shift of the second component waves should he calculated by using a half wave length of the first component waves as the second component waves arc bounded by the first component waves.

3.2 EXPERIMENTAL CONFIRMATION

Experimental conditions are, again, of d = 10 cm, T = 2 s, wave height H = 1.0 cm and

0 = 20 deg. Here, we employ the first order cnoidal waves which is calculated from the

monodirectional analytical solution of KP equation. Two component waves are calculated by the second order wave generation method, Eq.(10), for this condition. The first component is T, = 2 s, H, = 0.907 cm and 0, = 20 deg and the second component is T, = 1 s, H, = 0.295 cm and 02 = 18.9 deg. Measuring points of the surface elevation are y = 50, 100, 150 and 200 cm along the center of the basin (x = 0 cm). When the wave field is generated by a sinusoidal motion with H = 1.0 cm, the sinusoidal wave profile transforms to asymmetric wave profile as waves propagate (Fig. 7 (a)). When the wave field is generated by the second order wave generation method, the wave paddles move forward quickly and backward slowly, and the

wave profile shows little change while propagating (Fig. 7 (b)). The second order wave

generation method shows satisfactory performance in producing permanent finite amplitude wave field.

4. OBLIQUE REFLECTION OF PERIODIC WAVES AT A VERTICAL WALL 4.1 BOUSSINESQ EQUATION AND NUMERICAL METHOD

Boussinesq equation, which may describe nonlinear diffraction in shallow water, can be

written as (Peregrine, 1967),

+ a(2x+62Y -o

at ax ay

aQx a (Qx2)

_{a ("Y) Oar' =j-d2 a (a2Qx}

_a2

at

_{ax D}

_ay D ax 3 _at ax 2 axay

aQy a

+

(Q,Qy)+ a (Qy2

)1_0

d2 a (a2Q, .32(2y)

at ax D

ay D

ay 3

at axay

_0y2

where 0, and Qy are the line flux of x and y direction and D (= d + I") is the total water depth. Calculation of Eqs. (12), (13) and (14) is performed by employing the staggered

scheme of central difference in space, the leapfrog method in time _{stepping. Spatial grid}

width is 1/30 of wave length and time grid width is 1/35 of wave period. Along y = 0, that is along the wave maker, the flux calculated by the first order cnoidal waves is prescribed.

Along both sides and the other end, nonflux boundary condition is applied. _{In order to ignore}

the influence of the reflected waves from the opposite end wall and _{from the side walls, a}

calculation area is taken to be sufficiently long.

4.2 THE INFLUENCE OF DIFFRACTION AT THE EDGE OF A VERTICAL WALL ON OBLIQUE REFLECTION

The diffraction of finite amplitude waves is studied both experimentally and numerically.

A vertical wall is placed perpendicularly to the wave paddle. For normal diffraction case with

an eke, the wall was located 50 cm apart from the wave paddle. For the other_{case, the wall}