Wavemaker Theory

7/30/2019 Wavemaker Theory

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HANDBOOK OF COASTAL AND OCEAN ENGINEERING

World Scientific Publishing Co. Pte. Ltd.

http://www.worldscibooks.com/engineering/6914.html

Chapter 2

Wavemaker Theories

Robert T. Hudspeth

School of Civil and Construction Engineering

Oregon State University, Corvallis, OR 97331, USA

[email protected]

Ronald B. Guenther

Department of Mathematics

Oregon State University, Corvallis, OR 97331, USA

[email protected]

The fundamental solutions to the wavemaker boundary value problem (WMBVP)are given for 2D channels, 3D basins, and circular basins. The solutions are givenin algebraic equations that replace integral and differential calculus. The solutionsare generic and apply to both full- and partial-draft piston and hinged wave-makers; to double-articulated wavemakers, and to directional wave basins. TheWMBVP is solved by conformal mapping and by domain mapping. The loads ona wavemaker are connected to the radiation boundary value problem for semi-immersed bodies and demonstrate the connection of these loads to the added massand radiation damping coefficients required to compute the dynamic response oflarge Lagrangian solid bodies.

2.1. Introduction

Wavemaker theories play several important roles in coastal and ocean engineering.The most important role is the application to laboratory wavemakers for bothwavemaker designs and wave experiments. A second role for wavemaker theoriesis to compute a scalar radiated wave potential to compute the wave-induced loadson large solid bodies applying potential wave theory. The displacements and rota-

tions of a semi-immersed six degrees-of-freedom large Lagrangian solid body arerelated to the displacements and rotations of wavemakers. The boundary between aplanar wavemaker and an ideal fluid requires special care because the fluid motion

25


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26 R. T. Hudspeth and R. B. Guenther

is an Eulerian field with time and space as the independent variables, and theplanar wavemaker is a Lagrangian solid body with time and the wavemaker as theindependent variables. Consequently, the kinematic boundary condition will be dif-ferent from the free surface boundary that separates two Eulerian fluid fields of airand water. The boundary between the fluid and wavemaker separates a Eulerian

field (the fluid) from a Lagrangian body (the wavemaker), and the wavemakerkinematic boundary condition (WMKBC) must convert the Lagrangian wavemakermotion to a Eulerian field motion in order that the independent variables for bothdependent motion variables are equivalent. This may be accomplished by multi-plying the Lagrangian motion of the wavemaker by the unit normal to the boundary.Because the unit normal is a function of space and the Lagrangian wavemakermotion is a function of time, the product will produce a motion that is a functionof both space and time that are the independent variables of the Eulerian fluidfield. Although this fact is not central to the WMBVP, it is an important con-

nection between the WMBVP and the radiation potential boundary value problemfor semi-immersed large Lagrangian solid bodies.1

The formulae for computing the two fundamental fluid unknowns for an incom-pressible fluid of the velocity q(x , z, t) and the pressure p(x , z, t) from a scalarvelocity potential (x , z, t) are given first. The classical linear WMBVP for dimen-sionless 2D planar wavemaker is reviewed for two types of double-articulated planarwavemakers. The sway X1(t) displacement of a full-draft piston wavemaker and theroll 5(t) rotation of a hinged wavemaker are connected directly to the sway dis-placement and the roll rotation of a semi-immersed large Lagrangian solid body. Inthis review, integral calculus formulae for computing the integrals that are requiredto compute the coefficients of the eigenseries for the fluid motion, to compute theloads on the wavemaker and the average power required to generate the propagatingwaves are replaced by generic algebraic formulae. For example, an integral equationthat is required to compute the nth eigenseries coefficient Cn for the nth eigen-function n(Kn,z/h) from a wavemaker shape function (z/h) may be computedsymbolically and expressed by a dimensionless algebraic formula In(,,b,d,Kn),that is given by

Cn = h01(z/h)d(z/h)

n(z/h)d(z/h) = In(,,b,d,Kn). (2.1)

The coefficient in (2.1) may then be computed very efficiently by substitution intoalgebraic formulae in all subsequent applications. Next a dimensionless theory forboth amplitude-modulated (AM) and phase-modulated (PM) circular wavemakers isreviewed. Then, a dimensionless theory for double-actuated wavemakers is reviewed.Following that, a dimensionless directional wavemaker theory for large wave basinsbased on a WKBJ approximation1 is reviewed. Next, a theory for sloshing waves

due to transverse motions of a segmented wavemaker in a narrow wave channel isreviewed. Then, 2D planar wavemakers are mapped to a unit circle by conformalmapping and to a fixed rectangular domain by domain mapping; and both the linearand nonlinear wavemaker solutions are computed numerically.


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Wavemaker Theories 27

2.2. Planar Wavemaker in a 2D Channel

Two generic planar wavemaker configurations are shown in Figs. 2.1(a) and 2.1(b).The fluid motion may be obtained from the negative gradient of a dimensional scalarvelocity potential (x , z, t) according to

q(x , z, t) = u(x , z, t)ex + w(x , z, t)ez = 2(x , z, t), (2.2a)

where the 2D gradient operator in (2.2a) is given by

2() = ()x

ex +()z

ez.

Fig. 2.1(a). Definition sketch for a Type I planar wavemaker.

Fig. 2.1(b). Definition sketch for a Type II planar wavemaker.


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The total pressure field P(x , z, t) may be computed from the unsteady Bernoulliequation according to

P(x , z, t) = p(x , z, t) +pS(z) = (x , z, t)

t 1

2|(x , z, t)|2 + Q(t) gz,

(2.2b)where Q(t) = the Bernoulli constant; and the free surface elevation (x, t) for zeroatmospheric pressure according to

(x, t) =1

g

(x,,t)

t 1

2|(x,,t)|2 + Q(t)

; x (, t); z = (x, t).

(2.2c)The scalar velocity potential must be a solution to the Laplace equation

22 = 0; x (z, t), h z (x, t), (2.3a)with the following boundary conditions:

Kinematic Bottom Boundary Condition (KBBC):

z= 0; x (h, t), z = h. (2.3b)

Combined Kinematic and Dynamic Free Surface Boundary Condition (CKDFSBC):

2

t2+ g

z

t 1

2

||2 + dQ

dt= 0; x (, t); z = (x, t).

(2.3c)Kinematic WaveMaker Boundary Condition (KWMBC):

A Stokes material surface for planar wavemaker is W(x , z, t) = x (z, t), and theStokes material derivative gives the KWMBC from

DW

Dt

=

x

+

t

z

z

= 0; x = (z, t),

h

z

(t). (2.3d)

Kinematic Radiation Boundary Condition (KRBC):

A KRBC is required as x + for uniqueness to insure that propagating waves areonly right progressing or that evanescent eigenmodes are bounded. For a temporaldependence proportional to expit, the KRBC may be expressed by

limx+

x iKn

(x , z, t) = 0. (2.3e)

A velocity potential (x, z) may be defined by the real part of

(x , z, t) = Re{(x, z)expi(t + )}, (2.4)


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where Re{} means the real part of {}; and = arbitrary phase angle. Thelinearized WMBVP for kh = O(1) is1

22(x, z) = 0; 0 x < +; h z 0, (2.5a)

(x, z)z

= 0; 0 x + ; z = h, (2.5b)

(x, z)

z k0(x, z) = 0; 0 x < +; z = 0, (2.5c)

limx+

x iKn

(x, z) = 0, (2.5d)

(x, z)

xexpi(t + ) = (z, t)

t; x = 0; h z 0, (2.5e)

(x, t) = Re

ig

(x, 0) expi(t + )

; x 0; z = 0, (2.5f)

p(x , z, t) = (x , z, t)

t; 0 x < +; z = 0, (2.5g)

where ko = 2/g.Because the boundary conditions defined by (2.5b)(2.5e) are prescribed on

boundaries with constant values of the independent variables x and z, a solution bythe method of separation of (independent) variables may be computed.1 The instan-taneous wavemaker displacement (z, t) from its mean position x = 0 is assumedto be strictly periodic in time with period T = 2/, and may be expressed by

(z/h,t) = Re

i

S

(/h)

(z/h)expi(t + )

=

S(/h)

(z/h) sin(t + ). (2.6)

The specified shape function (z/h) for the Type I wavemaker shown in Fig. 2.1(a)is valid for either a double-articulated piston or hinged wavemaker of variable draftand is given by the following dimensionless equation for a straight line2:

(z/h) = [(z/h) + ][U(z/h + 1 d/h) U(z/h + b/h)], (2.7a)where , = dimensionless constants; U() = the Heaviside step function with twoboundary conditions given by

[S/(/h)](z/h = 1 + d/h + b/h + /h) = S, (2.7b)[S/(/h)](z/h = 1 + d/h + b/h) = Sb, (2.7c)


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that may be solved simultaneously for the dimensionless coefficients , to obtain

= (1 Sb/S); = /h + (1 d/h b/h /h). (2.7d,e)The coefficients and for the specified shape function (z/h) in (2.7) may be

obtained for the Type II wavemaker shown in Fig. 2.1(b) by substituting

S = S+ S; Sb = S; = h b dinto the following boundary conditions2 in (2.7b) and (2.7c):

(S+ S)

1 b/h d/h

(z/h = b/h) = S+ S, (2.8a)

(S+ S)

1 b/h d/h

(z/h = 1 + d/h) = S, (2.8b)

that may be solved simultaneously for the constant coefficients , to obtain

=S

S+ S; = 1 d

h

S

S+ S

b

h. (2.8c,d)

2.2.1. Eigenfunction solution to the WMBVP

Because all of the boundary conditions defined by (2.5b)(2.5e) are now prescribedfor constant values of the independent variables (x, z) and the dimensionlessparameter kh = O(1), a solution by separation of independent variables1 issuggested according to

(x, z) = X(x) Z(z). (2.9)The eigenseries solution may be written compactly as1,3,4

(x , z, t; Kn) = n=1

Cn cosh Kn(z + h)exp+i(Knx

t + ), (2.10a)

where Kn = k for n = 1 and Kn = +in for n 2 provided thatkoh kh tanh kh = koh + nh tan nh = 0; n > 2. (2.10b)

The eigenseries (2.10a) may be separated into a propagating p(x , z, t; k) andevanescent eigenmodes e(x , z, t; n) or local wave components3 according to

(x , z, t; Kn) = p(x , z, t; k) + e(x , z, t; n)

=

C1 cosh k(z + h) +n=2

Cn cos n(z + h)

exp[nx + i(t + )].

(2.10c)


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The wave number k = 2/ where = wavelength. Because the numerical value ofkhmust be computed from an eigenvalue problem in the vertical z coordinate, equiv-alence of the eigenvalue k to the wave number 2/ requires a pseudo-horizontalboundary condition of periodicity given by k = 2/ and (x + , z) = (x, z). Itis computationally efficient to normalize the eigenseries in (2.10a) according to

n(Kn,z/h) =cosh Knh(1 + z/h)

Nn; n = 1, 2, . . . , (2.11a)

where the nondimensional normalizing constant Nn is

N2n =

01

cosh2 Knh(1 + z/h)d(z/h) =

2kh + sinh 2kh

4kh; n = 1, (2.11b)

2nh + sin 2nh

4nh; n 2. (2.11c)

The eigenseries in (2.10a) may be written as an orthonormal eigenseries by

(x , z, t; Kn) =

n=1

Cnn(Kn,z/h)exp i(Knx t ), (2.12)

where the orthonormal eigenfunction n(,) is dimensionless.

2.2.2. Evaluation of Cn by WM vertical displacement (z/h)

The following dimensionless coefficient computed from (2.5e) will replace integral

calculus with algebraic substitution for the coefficients Cn in the eigenseries (2.12):

In(,,b,d,Kn) =

b/h1+d/h

[(z/h) + ]n(Kn,z/h)d(z/h)

=

(Knh)2Nn

Knh

1 d

h

sinh Knd Knb sinh Knh

1 b

h

cosh Knh

1 bh

+ cosh Knd

+ (Knh)Nn

sinh Knh)

1 b

h

sinh Knd (2.13)that is dimensionless when and are given by (2.7d) and (2.7e) or (2.8c) and(2.8d). The coefficients Cn may be computed algebraically by (2.13) from theKWMBC (2.5e) to obtain

Cn = iSh

KnIn(,,b,d,Kn), (2.14)

and the orthonormal eigenseries (2.12) is given by

(x , z, t; Kn)

=

n=1

iSh

KnIn(,,b,d,Kn)n(Kn,z/h)exp i(Knx t ). (2.15)


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2.2.3. Decay distance of evanescent eigenmodes n 2

Numerical solutions and experimental measurements of ocean and coastal designsrequire that the KRBC (2.5d) be applied far enough away so that only the prop-agating eigenmode for n = 1 in (2.12) is measurable. The evanescent eigenseries

in (2.12) for n 2 will decay spatially at least as fast as the smallest evanescenteigenvalue 2. This eigenvalue must be 2h > (n 3/2) = /2. If the smallestvalue for 2h > /2, then 2 > /2h and (x, z) exp(x/2h). For the valuesof the evanescent eigenseries to be less than 1% of their values at the wavemaker,(x, z) exp(xd/2h) = 0.01 and xd/(2h) = 4.6 3/2, and the minimumdecay distance is xd 3h.

2.2.4. Transfer function for wave amplitude

from wavemaker stroke

The average rate of work or power done by a wavemaker of width B is1

W = P = B+1

h

01

p(x , z, )u(x , z, )d(z/h)d, (2.16a)

where the temporal averaging operator is defined by

=

+1

(

)d, (2.16b)

and

W = P =

3S2Bh4

22kh

I21 (,,b,d,k), (2.16c)

so that all of the average power from a wavemaker is transferred to only thepropagating eigenmode. The average energy flux in a linear wave is given by1

E =

gBA2

2

CG, (2.16d)

where the group velocity CG is given by1

CG =C

2

1 +

2kh

sinh2kh

. (2.16e)

Equating (2.16c) to (2.16d) gives the following transfer function for a planar

wavemaker:

A

S=

koh

kh

1(k, 0)I1(,,b,d,k). (2.16f)


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2.2.5. Hydrodynamic pressure loads (added mass

and radiation damping)

The wave loads on a planar wavemaker may be estimated by integrating the totalpressure over the wetted surface of the wavemaker, i.e.,

F

M

=

0S

P

n

r n

dS, (2.17a,b)

where the outward pointing unit normal n points from the wavemaker into the fluid,and the pseudo-unit normal n for the rotational modes is given by

n = r n = (z + h d)nxey = nyey. (2.17c)

Force. For the Type I pistonwavemaker of total width B, the horizontal componentof the pressure force on the fluid side only may be computed from the real part of

F1(t) = Re

iBh

n=1

Cn

b/h1+d/h

n(Kn,z/h)d(z/h)expi(t + )

= F1 cos(t + 1), (2.18a)

where the static component of the pressure force on the fluid side only is

Fs = gBh22

[1 2(d/h) + (d/h)2 (b/h)2]. (2.18b)

The hydrodynamic component ofF1(t) may be separated linearly into a propagatingand an evanescent component that are related to the piston wavemaker translationalvelocity and acceleration, respectively, from the real part of

F1(t) = Re{[11(S) + 11(iS2)]expi(t + )}= 11(S2 sin(t + )) 11(S cos(t + )) (2.18c)= Re{11X1(t) 11X1(t)}, (2.18d)

where the added mass coefficient 11 may be computed from the evanescenteigenmodes only, and the radiation damping coefficient 11 may be computed fromthe propagating eigenmode only. The average power may be computed from

F1X1t = 11(S)2

2. (2.19a)

Equating (2.19a) to (2.16d) yields

11 =

A1S1

2Bh

kohCG, (2.19b)


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that relates the radiation damping coefficient to the square of the ratio of theradiated wave amplitude to the amplitude of the wavemaker displacement.

Moment. For the Type I wavemaker of width B, the dynamic pressure momenton one side only of the wavemaker may be computed from the real part of

M5(t) = Re

iBh2

n=1

Cn

b/h1+d/h

1 +

z

h d

h

n(Kn,z/h)d z

h

exp(t + )

= M5 cos(t + 5), (2.20a)and the static component of the pressure moment on the fluid side only is

Ms =gBh3

6

1

d

h

3+ 2

b

h

3+ 3

d

h

2

b

h

23

d

h

1

b

h

2.

(2.20b)

The pressure moment M5(t) in (2.20a) may be separated linearly into a propagatingand an evanescent component that are related to the rotational velocity and accel-eration from the real part of1

M5(t) = Re 55i S2(1 + b/)+ 55

i S

(1 + b/)

expi(t + )

,

M5(t) = 555(t) 555(t), (2.21a)where

55 = Bh4

n=2

I2n(,,b,d,n)

nh; (2.21b)

55 = Bh4 I

21 (,,b,d,k)

kh. (2.21c)

2.3. Circular Wavemaker

Havelock5 applied Fourier integrals to develop a theory for surface gravity wavesforced by circular wavemakers in water of both infinite and finite depth. The fluid

motion may be obtained from the negative gradient of a scalar velocity potential(r,,z,t) according to

q(r,,z,t) = (r,,z,t), (2.22a)


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where the 3D gradient operator () in polar coordinates is

() =

rer +

1

r

e +

ze3

(). (2.22b)

The total pressure field P(r,,z,t) may be computed from the unsteady Bernoulliequation in polar coordinates according to

P(r,,z,t) = p(r,,z,t) +pS(z)

=

(r,,z,t)

t 1

2|(r,,z,t)|2 + Q(t)

gz, (2.22c)

where Q(t) = the Bernoulli constant, and the free surface elevation (r,,t) for zeroatmospheric pressure according to

(r,,t) =1

g

(r,,,t)

t 1

2|(r,,,t)|2 + Q(t)

;

r b + ( , , t); z = (r,,t). (2.22d)

The scalar velocity potential (r,,z,t) must be a solution to the continuityequation

2

=

1

r

r

r

r

+

1

r22

2 +

2

z2 = 0,

r b + ( , z, t); 0 2; h z (r,,t), (2.23a)

with the following boundary conditions:Kinematic Bottom Boundary Condition (KBBC):

z= 0; r b + (,h, t); 0 2; z = h. (2.23b)

Combined Kinematic and Dynamic Free Surface Boundary Condition (CKDFSBC):

2

t2+ g

z

t 1

2

||2 + dQ

dt= 0;

r b + ( , , t); 0 2; z = (r,,t). (2.23c)

Kinematic WaveMaker Boundary Condition (KWMBC):

r+

t 1

r2

z

z= 0; r = ( , z, t);

h

z

(b,,t). (2.23d)

Two types of circular cylindrical wavemaker displacements ( , z, t) may be ana-lyzed, viz., amplitude-modulated (AM) and phase-modulated (PM) wavemakers.


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The distinction between these two types is in the azimuthal dependency of thewavemaker displacement ( , z, t) from its mean position r = b, given by

( , z, t) = mS

(/h)(z/h)

cos m sin(t + )

sin(t + + m)

. (2.23e)

(2.23f)

Kinematic Radiation Boundary Condition (KRBC):

limKnr

r iKn

(r,,z,t) = 0; r . (2.23g)

Finally, physically realizable solutions to (2.23a) must be periodic in , i.e.,

(r,,z,t) = (r, + 2 , z , t). (2.23h)

The dimensional WMBVP may be scaled and linearized1 by expanding the variables

in perturbation series with a dimensionless perturbation parameter = kA. A scalarradiated velocity potential (r,,z) may be defined by the real part of

(r,,z,t) = Re{(r,,z)expi(t + )}. (2.24)A linearized WMBVP may be obtained by setting the dimensionless parameterkA = = 0 and by requiring that kh = O(1). This linearized WMBVP is

2(r,,z) = 0; b r < +, 0 2, h z 0; (2.25a)

(r,,z)

z = 0; b r < +, 0 2, z = h; (2.25b)

(r,,z)

z ko(r,,z) = 0; b r < +, 0 2, z = 0; (2.25c)

lim|Kr|r+

r iKn

(r,,z) = 0; (2.25d)

(r,,z)r expi(t + ) = ( , z, t)t ; r = b, 0 2, h z 0;(2.25e)

(r,,t) = Re

i(r,,z)g

expi(t + )

; b r < ; 0 2; z = 0;(2.25f)

P(r,,z,t) = {p(r,,z,t)} +ps(z)= Re{i(r,,z)expi(t + )} gz ; (2.25g)

(r,,z) = (r + ,,z); (r,,z) = (r, + 2, z). (2.25h,i)

The specified wavemaker shape function (z/h) is valid for either a double-articulated piston or hinged circular AM or PM wavemaker of variable draft that


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Fig. 2.2. Definition sketch for circular wavemaker.

is shown in Fig. 2.2 is identical to (2.7) for a 2D planar wavemaker with thedimension b replaced with a and the stroke S replaced with the azimuthal stroke

mS. The solution to the WMBVP (2.25) may be compactly expressed by the fol-lowing orthonormal eigenseries:

m(r,,z) =

n=1

Cmnn(Kn,z/h)H(1)m (Knr)MA(P)(m), (2.26a,b)

where the azimuthal mode function is

MA(P)(m) =

cos m

expim

; m 0 and integer, (2.26c,d)

and where (2.26a) represents an AM wavemaker; (2.26b) represents a PMwavemaker; n(Kn,z/h) = the orthonormal eigenseries defined in (2.12);

H(1)m (Knr) = the Hankle function of the first kind. When K1 = k and Kn = in for

n 2 and integer,

H(1)m (inr) = Jm(inr) + iYm(inr) =2

i(m+1)Km(nr), (2.26e)


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where Km() = the Modified Bessel (or Kelvin) function of the second kind oforder m. The coefficients Cmn may be computed by expanding the KWMBC in aneigenseries following the procedure in (2.14) and obtaining

Cmn

=

mSjh

Kn

In(,,a,d,Kn)

Ln(H(1)m (Knb)); n

1 and integer; (2.26f)

Ln(Zm(n)) = dZm(n)dn

=1

2{Zm1(n) Zm+1(n)};

Zm(n) = Jm(n), Ym(n), Km(n), H(1)m (n). (2.26g)

The solution to (2.25) is given by the real part of the following eigenseries expansion:

mj(r,,z,t)

[mSjh] =mpj(r,,z,t) + mej(r,,z,t)

[mSjh]

= Re

I1(,,a,d,k)

k

1(k,z/h)H(1)m (kr)

L1(H(1)m (kb))

+

n=2

In(,,a,d,n)

n

n(n,z/h)Km(nr)

Ln(Km(nb))

MA(P)(m)ei(t+)

. (2.27a,b)

Because the asymptotic behavior of the evanescent eigenseries Km(nr) dependson the mode m(1), it is not possible to specify a minimum distance from the wave-maker equilibrium boundary at r = b where the evanescent eigenvalues are lessthan 1% of their value at the circular wavemaker boundary. The wave field must becomputed far away from the wavemaker, and it is understood that far away mustbe computed uniquely for each radial mode m for either an AM or PM circularwavemaker. The evaluation of the power, forces, and moments, and added mass andradiation damping coefficients for both AM and PM circular wavemakers are givenby Hudspeth.1

2.4. Directional Wavemakers

Directional wavemakers are vertically segmented wavemakers that undulate sinu-

ously and, consequently, are also called snake wavemakers. Segmented directionalwavemakers may be driven either in the middle of each vertical segment or at thejoint between vertical segments. Because of these two methods of wave generation,parasitic waves are formed along the wavemaker due to either the discontinuity of


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the wavemaker surface (middle segment driven) or of the derivative of the wave-maker surface (joint driven). A dimensional scalar spatialvelocity potential (x,y,z)may be defined by the real part of

(x , y , z, t) = Re

{(x,y,z)exp

i(t + )

}. (2.28a)

The dimensional linear fluid dynamic pressure field p(x , y , z, t) and 3D fluid velocityvector field may be computed from

p(x , y , z, t) = (x , y , z, t)

t, (2.28b)

q(x , y , z, t) = 3(x , y , z, t), (2.28c)

3() = ()x

ex +()y

ey +()z

ez. (2.28d)

The dimensional WMBVP for directional waves is given by

23(x,y,z) = 0; x 0; B y +B; h(x, y) z 0, (2.29a)

(x,y, 0)

z ko = 0; x 0; B y +B; z = 0, (2.29b)

y= 0; x 0; y = B; h(x, y) z 0, (2.29c)

(x , y , z, t)

x= (y , z , t)

t;

x = 0,B y +B,h(0, y) z 0,

(2.29d)

limx+

x iKn

(x,y,z) = 0, (2.29e)

(x,y,z)

z= 2(x,y,z) 2h(x, y); z = h(x, y), (2.29f)

22() =

2

x2,

2

y2

(), (2.29g)

(y , z , t) = Re

i

U(y, z)

[U(y, a)][U(z, b , d)] expi(t + )

, (2.29h)

U(y, z) =

s

/ho(y)(z/ho), (2.29i)

U(y, a) = U(y + a) U(y a+),U(z, b , d) = U(z + h d) U(z + b),

(2.29j)


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Fig. 2.3. Definition sketch for rectangular directional wave basin.

where ko = 2/g and where a denotes the (possibly nonsymmetric) transverse ends

of the directional wavemaker in the transverse y-direction in Fig. 2.3. The solutionto the WMBVP in (2.29) is given by the following set of orthonormal eigenfunctions:

(x,y,z) = ig

n=1

n(x, y)n(Kn,z/h), (2.30a)

n(Kn,z/h) =

1(k,z/h)

1(k, 0); n = 1, (2.30b)

1(n,z/h); n 2, (2.30c)

koh = Knh tanh Knh = 0; n = 1, 2, 3, . . . , (2.30d)

where K1 = k and Kn = +in for n 2 and n() is defined in (2.11). Theorthonormal eigenfunctions (2.30b) and (2.30c) are applicable strictly only for con-stant depth wave basins; however, they may be applied to slowly varying depth wavebasins if (2.30b) and (2.30c) are considered to be evaluated only locally over rela-tively small horizontal length scales (e.g., several wavelengths ), where the depthmay be considered to be locally equal to a constant by a Taylor series expansion ofthe depth.1

Substituting (2.30a) into (2.29a) yields the following 2D Helmholtz equation:

22(x, y) + K2n(x, y) = 0; x 0, B y +B. (2.31)

Alternatively, for wave basins with mildly sloping bottoms, the mild slope equationmay be applied according to1

2 (CCG2(x, y)) + K2nCCG(x, y) = 0, (2.32)


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where the wave group velocity CG is given by (2.16e). If the product CCG is aconstant, (2.32) reduces to the 2D Helmholtz equation (2.31). Applying the WKBJapproximation1 for the x-dependent solution in the method of separation of variablesto (2.32) yields the following solution1:

(x , y , z, t) = Re

i g

(x, y)n(Kn,z/h)expi(t + )

= Re

ig

Mm=0

Amm

[C(x)CG(x)]x=0

C(x)CG(x)m(m,y/B)

n(Kn,z/h)exp ix

Qmnd

exp i(t + )

(2.33a)

m(m,y/B) =cos m(y/B 1)

Mm; Mm = 1 + m0; m =

m

2B(2.33bd)

Qmn =

K2n 2m; m > 0. (2.33e)

The coefficients Am may be computed from (2.29d) by expanding the wavemakershape function in orthonormal eigenfunctions1 and are given by

Amnk

QmmS

(/hxo)

= n(Kn, 0)In(,,b,d,k)

+a+/Ba/B

j(qi,y/B)m(m,y/B)d(y/B). (2.34)

2.4.1. Full-draft piston wavemaker

The prescribed transverse y-component of the snake displacement of a full-draft

(b = d = 0) piston ( = 0 and = 1) wavemaker may be expressed as

j(qj ,y/B) =+

j=cj exp i[kyB(y/B + y)], (2.35a)

where the coefficients cj may be computed from the integral in (2.34) by

cmj

= +a+/B

a/B

j

(qj

,y/B)m

(m

,y/B)d(y/B)

=Ra+,a + iIa+,a

mB(q2j 2m)(1 + m0). (2.35b)


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If the full-draft piston snake wavemaker spans the entire width of the wave basinso that a = B, then (2.35b) reduces to the integral in (2.34) and

cmj =

4qiBsin[qjB(j 1)] + (1)m sin[qjB(j + 1)]

i{cos[qjB(j 1)] + (1)m

cos[qjB(j + 1)]}((qjB m2)2)(1 + m0) . (2.35c)

Values for cmj for (possibly nonsymmetric) values for a are given by Hudspeth.1

2.5. Sloshing Waves in a 2D Wave Channel

A long rectangular wave channel with a horizontal flat bottom, two rigid vertical

sidewalls, and a wavemaker may generate either 2D, long-crested progressive wavesor two types of transverse waves, viz.,

(1) sloshing waves that are excited directly by transverse motion of the wavemakeror

(2) cross waves that are excited parametrically by the progressive waves at asub-harmonic of the wavemaker frequency.

The WMBVP for 3D sloshing waves is identical to (2.5) for planar 2D wave-makers except for the KWMBC at x = 0 and an additional kinematic boundary

condition on the sidewalls of the 2D wave channel at y = B in Fig. 2.4. Thekinematic and dynamic wave fields may be computed from a dimensional 3D scalar

Fig. 2.4. Definition sketch for a sloshing wave channel.


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velocity potential (x , y , z, t). The fluid velocity q(x , y , z, t) may be computed fromthe negative directional derivative of a scalar velocity potential by

q(x , y , z, t) = (x , y , z, t). (2.36a)The dimensional fluid dynamic pressure field p(x , y , z, t) may be computed from

p(x , y , z, t) = (x , y , z, t)

t. (2.36b)

A spatial velocity potential (x,y,z) may be defined by the real part of

(x , y , z, t) = Re{(x,y,z)expi(t + )}. (2.36c)The WMBVP for sloshing waves is given by the following:

2(x,y,z) = 0; x 0; B y +B; h z 0; (2.37a)

(x,y,h)z

= 0; x 0; B y +B; z = h; (2.37b)

(x,y, 0)

z ko = 0; x 0; B y +B; z = 0; (2.37c)

(x , y , z, t)

x= (y , z , t)

t; x = 0; B y +B; h z 0; (2.37d)

y

= 0; x 0 y = B; h z 0; (2.37e)

limx+

x iKn

(x,y,z) = 0; (2.37f)

(x,y,t) = Re

ig

(x,y, 0, t)

;

x 0,B y +B,z = 0.

(2.37g)

A solution to (2.37) is given by the following eigenfunction expansions1

:

(x , y , z, t) = Re

n=1

n(x, y)n(Kn,z/h)expi(t + )

, (2.38a)

(x,y,z) = n(x, y)n(Kn,z/h), (2.38b)

(x,y,t) = Re

n=1

n(x, y)expi(t + )

, (2.38c)

n(x, y) = i

g n(x, y)n(Kn, 0), (2.38d)

n(x, y) = ig

n(x, y)

n(Kn, 0), (2.38e)


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where n(x, y) is sometimes referred to as a displacement potential. The scalarpotential (2.38a) may be expressed from (2.38d) and (2.38e) as

(x , y , z, t) = Ren=1

ig

n(x, y)

n(Kn,z/h)

n(Kn, 0)expi(t + ) . (2.39)

The WMBVP may be expressed in terms of a displacement potential n(x, y) by

2n(x, y)

x2+

2n(x, y)

y2+ K2nn(x, y) = 0;

x 0;B y +B;h z 0.

(2.40a)

n=1n(x, y)

x

n(Kn,z/h)

n(Kn, 0)= i

gU(y, z)

x = 0;B y +B;

h

z

0.(2.40b)

limx+

x iKn

n(x, y) = 0. (2.40c)

n(x, y)

y= 0; x 0; y = B; h z 0, (2.40d)

where (2.40a) is the 2D Helmholtz equation.9,10

Because the boundary conditions are prescribed on boundaries that are constant

values of (y,z), a solution to the WMBVP (2.40) may be computed by the methodof separation of variables and is given by1

(x , y , z, t)

= Re

ig

Mm=0

Cm1m(y/B)1(k,z/h)

1(k, 0)exp iPm1x

+

m=M+1

Cm1m(y/B)1(k,z/h)

1(k, 0)exp

m1x

+m=0

n=2

Cmnm(y/B)n(n,z/h)

n(n, 0)exp

Qmnx

exp

i(t + )

,

(2.41a)

where

m(m,y/B) =cos mB(y/B

1)

Mm ; M2m = 1 + m0; (2.41b)

m =m

2B;


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Pm1 =

K21 2m =

k2 2m; k > m; m M; (2.41c)

m1 =

2m k2; k < m; m > M; (2.41d)

Q2mn = 2m K2n; m 0; n 1; (2.41e)

n 1: K1 = k > m : Qm1 = i

k2 2m = iPm; m M; (2.41f)

n = 1: k = m : Qm1 = 0

k < m : Qm1 =

2m k2 = m1 > 0; m > M; (2.41g)

n

2 : Kn = in : Qmn =2m + 2n > 0, (2.41h)

where M is the maximum integer value for m in order for m < k. Substituting(2.41a) into (2.40b) yields the following solutions for Cmn1:

Cm1 =

g

1(Kn, 0)

Pm1

+11

d y

B

01

d z

h

U

y,z

h

1

Kn,

z

h

m

m

y

B

;

m M (2.42a)

Cm1 =

g

1(Kn, 0)

m1 +1

1

d

y

B 0

1

d

z

hU

y,

z

h1

Kn,z

hm

m,y

B;

m > M + 1 (2.42b)

Cmn =

g

n(Kn, 0)

Qmn

+11

d y

B

01

d z

h

U

y,z

h

n

Kn,

z

h

m

m,

y

B

;

m 0, n 2. (2.42c)The first three transverse eigenmodes are illustrated in Fig. 2.5.

2.6. Conformal Mappings

Conformal and domain mappings are applications of complex variables to solve 2Dboundary value problems. Conformal mappingis an angle preserving transformationthat will compute exact nonlinear solutions for surface gravity waves of constant

Fig. 2.5. First three transverse eigenmodes in a 2D wave channel.


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form that may be treated as a steady flow following a Galileantransformation from afixed inertial coordinate system to a noninertial moving coordinate system. Domainmapping is a transformation of the wavemaker geometry into a fixed computationaldomain where a solution may be computed efficiently.

2.6.1. Conformal mapping1

Conformal mapping of the WMBVP provides a global solution that accuratelyaccounts for the singular behavior at all irregular points. The irregular points inthe physical wavemaker domain are transformed into both weak and strong singularkernels in a Fredholm integral equation. The two irregular points on the WMBVPboundary are located at (1) the intersection between the free-surface and the wave-maker boundary and (2) the intersection between the horizontal bottom and thewavemaker boundary. These two irregular points exhibit integrable weakly singularkernels. The far-field radiation boundary exhibits a strongly singular kernel and sig-nificantly affects the solution. The irregular frequencies3,4 are included in the globalsolution by the Fredholm alternative. A theory for the planar WMBVP computesa global solution by a conformal mapping of the physical wavemaker boundaryto a unit disk that includes the motion of an inviscid fluid near irregular pointsthat are illustrated in Fig. 2.6. A numerical solution to Laplaces equation in atransformed unit disk may be computed from a Fredholm integral equation. TheWMBVP defined by (2.5) is transformed into complex-valued analytical functionswhere the complex-valued coordinates are defined as z = x + iy. The coordinates

for the semi-infinite wave channel in Fig. 2.1(a) must be transformed to complex-valued coordinates z. The fluid velocity q(x,y,t) and dynamic pressure p(x,y,t)may be computed from a scalar velocity potential (x,y,t) according to

q(x,y,t) = (x,y,t); p(x,y,t) = (x,y,t)t

. (2.43a,b)

The WMBVP and Type I wavemaker shape function are given by (2.5)(2.7).There are both Irregular (I) and Regular (R) points at the intersections between

the Smooth (S) and Critical (C) boundaries B1 and B2 in the WMBVP as illustratedin Fig. 2.6 where these two boundary intersection points are identified as P1 and P2.The classification of the boundary points P1 and P2 in Fig. 2.6 depends on (1) theboundary conditions i(Pj) and (2) the continuity of the boundaries Bm and theirderivatives where i, j, and m = 1 or 2. A conformal mapping of the semi-infinitewave channel strip in the physical plane will yield a Fredholm integral equation,6,7

where these critical points may be transformed to singular points that are integrableover a smooth continuous mapped boundary.

2.6.1.1. Conformal mapping to the unit disk 1

Two conformal mappings are: (i) the physical z-plane to a semi-circle in the Z-planeshown in Fig. 2.7; and (ii) a semi-circle in the upper Z-plane to a unit disk in the


23/32





Fig. 2.6. Combinations of Irregular (I) and Regular (R) boundary points P1 and P2 betweenSmooth (S) and Critical (C) boundaries B1 and B2 intersections in the WMBVP.8

Fig. 2.7. Mapping of the semi-infinite strip in the lower halfxy-plane in the physical z-plane to

the upper halfXY-plane in the Z-plane.8


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Fig. 2.8. Mapping of the upper half-plane in the Z-plane to the unit disk in the Q-plane.

Q-plane shown in Fig. 2.8. The SchwarzChristoffel transformation

dz

dZ=

C1Z+ 1

Z 1 (2.44a)

may be integrated to obtain

z = x + iy

= h

Log[Z

Z2 1], (2.44b)

where the Log[] denotes the principal value of Log[] and the argument of theLog[] is arg < . Inverting (2.44b) yields

Z = X+ iY = cosh(z/h), (2.44c)

where

X = coshxh

cos

yh

; (2.44d)

Y = sinhx

h

siny

h

. (2.44e)


25/32





The radiation boundary in the z-plane transforms to a semi-circle in the Z-plane by

R2 = X2 + Y2 =1

2

cos

2x

h

+ cosh

2x

h

, (2.44f)

tan = YX = tanh

xh

tan

yh

. (2.44g)

Details of the transformation of the WMBVP are given by Hudspeth.1

2.6.1.2. Mapping Z-plane to a unit disk

The upper-half-plane of the Z-plane may be mapped into a unit disk shown inFig. 2.8 by the following bilinear transformation:

Q = + i = exp(i0) i Zi + Z , (2.45a)

that may be inverted to obtain

Z = X+ iY = i

1 Q1 + Q

, (2.45b)

and the mapping function coordinates are

=1

X2

Y2

X2 + (Y + 1)2 ,

=2X

X2 + (Y = 1)2,

(2.45c,d)

that may be transformed into the cylindrical coordinates for the unit disk inFig. 2.8 by

r2 = 2+2 =(X2 + Y2 1)2 4X2

(X2 + (Y + 1)2)2; = arctan

= arctan2X

1

X2

Y2.

(2.45e,f)

Details of the transformation are given by Hudspeth.1 A numerical solution tothe transformed WMBVP may be computed from the following Fredholm integralequation1:

2

(r, ) =

+

(r, )

G(r, r,, )

r+ G(r, r,, )

(r, )

r

rd,

(2.46a)

where

G(r, r,, ) = ln[(r, r,, )]; 2(r, r,, ) = r2 2rr cos( ) + r2.(2.46b,c)


26/32





Fig. 2.9. Nodal points on the unit disk in the Q-plane and the corresponding nodal points on the

wavemaker in the physical z-plane.8

Numerical solutions to (2.46a) may be computed by discretizing the unit diskboundary as shown in Fig. 2.9. The numerical details regarding the evaluation(2.46a) at the two weakly singular irregular points at B and D in the physicalz-plane in Fig. 2.7 and the strongly singular point at that is the verticalradiation boundary AE at + in the physical z-plane in Fig. 2.7 are tedious.8Global numerical solutions may be computed for both the linear and the nonlinear

WMBVPs.

2.6.1.3. Conformal mapping to the unit disk 2

The wavemaker geometry shown in Fig. 2.10 is mapped to the unit disk by twotransformations. The WMBVP is given by (2.5) and the WM shape function is

(y/h) = [(y/h) + ][U(y/h + 1 b0/h) U(y/h + a0/h)]. (2.47)In order to transform the wavemaker geometry to a Jacobian elliptic function,

it must be rotated and translated as shown in Fig. 2.11. The 90 rotation to thez-plane is given by

z = x + iy = iz = y + ix. (2.48a)


27/32





Fig. 2.10. WMBVP11 with the six critical boundary points at aa0bb0cd.

Fig. 2.11. Rotation and translation of the physical wavemaker rectangular strip in the z-plane to

the w-plane.11

The horizontal shift to the left in the z-plane is given by

z = x + iy = w h/2 = y h/2 + ix. (2.48b)In order to map the WM geometry in the z-plane to the semi-circle in the Z-planeshown in Fig. 2.12 as a Jacobian elliptic function, the rotated and translated stripin the z-plane must have the dimensions ofK +K and 0 K, whereK = h/2 and K = 3h = 6K. This requires a coordinate amplification given by

w =2K

h (x + iy)

=2K

h

y h

2+ ix

. (2.48c)


28/32





Fig. 2.12. Mapping of the wavemaker semi-circle in the Z-plane to the unit disk in the Q-plane.11

The SchwarzChristoffel transformation from the w-plane to the Z-plane is

dw =

CkdZ(a z)(b Z)(c Z)(d Z) . (2.48d)

The following change of variables:

Z = aZ; dZ = adZ : = a/c; C = c,

modifies (2.48d) to the following Jacobian elliptic function:

w =Z0 dZ

[(1 Z2)(1 2Z2)] 12 = sn1[Z, ], (2.48e)where sn[Z, ] = the Jacobian elliptic function of modulus or sine amplitudefunction.9 Define

k = sn1[1, ], (2.48f)

and the mapping of the rectangle {x1, x2; y1, y2} = {0, 3h : 0, h} is given by

Z = X + iY

=

sn 2Kh y + h2 , dn 2Kxh , k

1 dn2 2Kh y + h2 , sn2 2Kxh , k

+ i

cn 2Kh y + h2 , dn 2Kh y + h2 , , sn 2Kxh , k cn 2Kxh k

1 dn2 2Kh y + h2 , sn2 2Kxh , k ,

(2.48g)

where sn[, ] in the copolar half-period trio in (2.48g) is defined in (2.48e), andcn[, ] and dn[, ] are defined by cn2[, ] = 1sn2[, ], dn2[, ] = 12sn2[, ].The mapping from the semi-circle in the Z-plane to the unit disk in the Q-plane


29/32





Fig. 2.13. Transformed boundary conditions mapped to arcs on the perimeter of the unit disk.11

is shown in Fig. 2.12, and the mapping to the unit disk in the Q-plane is shown inFig. 2.13. The mapping of the Z-plane to the Q-plane is given by

Q =i Z (i + Z)i + Z (i Z) =

(1 ) Z2(1 + ) + 2iZ(1 )2(1 2) + Z2(1 + )2 , (2.49a)

where 1 < < +1. Changing variables to circular cylindrical coordinates by

R2(X, Y) =(1 2) 2Y(1 2) + (X2 + Y2)(1 + )2(1 )2 2Y(1 2) + (X2 + Y2)(1 + 2) , (2.49b)

(X, Y) = arctan

2X(1 2)

(1 )2 (X2 + Y2)(1 + )2

, (2.49c)

the unit disk may be transformed into functions of the copolar trio of Jacobianelliptic functions. The transformed WMBVP in circular cylindrical coordinatesis given by Hudspeth.1 A general solution to the transformed WMBVP may be

written as

10

(R, ) =N

n=0

Rn

an1 + n0

cos n + bn sin n

, (2.50)

where ij is the Kronecker delta function. Substituting (2.50) into the genericboundary conditions on each of the six arcs on the perimeter of the unit disk illus-trated in Fig. 2.13, multiplying each of these six boundary conditions by a memberof the set of the orthogonal series in (2.50), integrating over the interval of orthogo-

nality + yields the following matrix equation for each of the coefficientsan and bn

AB = H. (2.51)


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Fig. 2.14. Physical fluid domain.12

2.7. Domain Mapping

Domain mapping of the WMBVP12 follows the theory by Joseph.13 The physical

fluid domain shown in Fig. 2.14 for the fully nonlinear WMBVP is mapped to afixed computational fluid domain, and the discretized coupled free-surface boundaryconditions are computed by an implicit CrankNicholson (CN) method.14,15 Ateach iteration of the CN method, the potential field is computed by the conjugategradient method.15 The wavemaker motion (y/h,t) is assumed to be periodic withperiod T = 2/, and the WMBVP with the surface tension T is given by

2(x,y,t) = (x,y,t) = 0;

0 y (x, t),(y/h,t)

x

L.

(2.52a)

(x,y,t)

t+

1

2|(x,y,t)|2

T

2(x, t)

x21 +

(x, t)

x

23/2 + g(x, t) = 0. (2.52b)

(x,y,t)

y (x, t)

x

(x,y,t)

x+

(x, t)

t= 0;

((x, t), t) x L, y = (x, t). (2.52c)

(x,y,t)

y= 0; ((x, t), t) x L, y = 0. (2.52d)

(x,y,t)

x= 0; x = L, 0 y (L, t). (2.52e)

(x,y,t)

x= (y/ho, t)

t+

(y/ho, t)

y

(x,y,t)

y;

x = (y/ho, t),

0 y (0, t). (2.52f)


31/32





The initial conditions for t = 0 are

(x, 0) = H(x); (2.52g)

(x, t) = 0; (x, 0) x

L. (2.52h)

(x, (x, 0), 0) = 0; (2.52i)

The physical fluid domain shown in Fig. 2.14 may be mapped into a dimensionlessfixed rectangle of dimensions 0 1 by 0 1 by the transforms

=x

L; =

y

(x, t); = t; (, ) =

(x, t)

h, (2.53ad)

and dimensionless variables by

q( , , ) = (x,y,t)A

; p( , , ) = P(x,y,t)A22

, (2.53e,f)

( , , ) = (x,y,t)Ah

; w(, ) =(y/ho, t)

S; T =

T

ALh2. (2.53gi)

Because is a function of both x and y in (2.53b), transforming partial derivativeswith respect to x must be done with some care.12 Details of these lengthy transfor-mations and the transformed WMBVP in the fixed mapped domain are given byHudspeth.1

References

1. R. T. Hudspeth, Waves and Wave Forces on Coastal and Ocean Structures (WorldScientific, Singapore, 2006).

2. R. T. Hudspeth, J. M. Grassa, J. R. Medina and J. Lozano, J. Hydraulic Res. 387(1994).

3. F. John, Commun. Pure Appl. Math. 13 (1949).

4. F. John, Commun. Pure Appl. Math. 45 (1950).5. T. H. Havelock, Phil. Mag. 569 (1929).6. P. R. Garabedian, Partial Differential Equations (Wiley, Inc., New York, 1964).7. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book

Company, New York, 1953).8. Y. Tanaka, Irregular points in wavemaker boundary value problem, PhD thesis,

Oregon State University (1988).9. G. F. Carrier, M. Krook and C. E. Pearson, Functions of a Complex Variable, Theory

and Technique (McGraw-Hill Book Co. Inc., New York, 1966).10. R. B. Guenther and J. Lee, Partial Differential Equations of Mathematical Physics

and Integral Equations (Dover Publications, Inc., New York, 1996).11. P. J. Averbeck, The boundary value problem for the rectangular wavemaker,

MS thesis, Oregon State University (1993).12. S. J. DeSilva, R. B. Guenther and R. T. Hudspeth, Appl. Ocean Res. 18, 293 (1996).


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13. D. D. Joseph, Arch. Rational Mech. Anal. 51, 295 (1973).14. B. Carnahan, H. A. Luther and J. O. Wilkes, Applied Numerical Methods(John Wiley

and Sons, New York, 1965).15. R. Glowinski, Numerical Methods for Nonlinear Variational Problems (Springer-

Verlag, 1984).

Documents

Wavemaker Theory