AD-A266 820____________N-1 857~

1II II111i1111l ll~111111t1111 iliiii l1111 __________________________________ ______________,__

April 1993

By Erick T. HuangNC EL Sponsored By

Technical Note Chief of Naval Research

NONLINEAR WAVE FORCESON LARGE

OCEAN STRUCTURES ELFc-rESJUL19 1993

E

ABSTRACT This study explores the significance of second-orderwave excitations on a large pontoon and tests the feasibility of reducing anonlinear free surface problem by perturbation expansions. A simulationmodel has been developed based on the perturbation expansion techniqueto estimate the wave forces. The model uses a versatile finite elementprocedure for the solution of the reduced linear boundary value problems.This procedure achieves a fair compromise between computation costsand physical details by using a combination of 2D and 3D elements. Asimple hydraulic model test was conducted to observe the wave forcesimposed on a rectangle box by Cnoidal waves in shallow water. The testmeasurements are consistent with the numerical predictions by the simula-tion model. This result shows favorable support to the perturbation ap-proach for estimating the nonlinear wave forces on shallow draft vessels.However, more sophisticated model tests are required for a full justifica-tion. Both theoretical and experimental results show profound second-order forces that could substantially impact the design of ocean facilities.

93-16180

NAVAL CIVIL ENGINEERING LABORATORY PORT HUENEME CALIFORNIA 93043-5003

Approved for public release; distribution unlimited.

LL. 0a

0 0 CE

> E

".0 0iCý 0CN 0.

E

E E. Eo E0 E0EýEE E -2--E

E( E

2 1 inches

00

0 w 00,EE ~ ýE~~~ E 24 2- - :

2~~ = zD0

80 r , E0 E E

-CC

.Lq 0 R0 ý ,k b0

SI CIT0

REPORT DOCUMENTATION PAGE J •Ood

Puohc reoorting burden Wer this collection of information is estimated to average 1 hour per res.oxnse, including the time for reviewing instrucions. searching existing data sources.gatherlnng and maintaining the data neeoed, and Cornleling and reviewing the collection of intormTation. Send comments regardinn this burden estimate or any other aspect o thiscollection nformationn,Including suggestions for rco.-cnq this burder. to Washington jeaequartlers Services. Directorate for Intormation and Reoorts. 1216 Jetterson Davis Hithway.Suite 1204. Arlington. VA 22202-4302. and to tMe Ztl~ce of Management and Budget, Paperwork Reduction Protect (0704-0188). Washington. DC 20503

1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

IApril 1993 Final: Oct 1991 through Sep 1992

4. TITLE AND SUBTITLE S. FUNDING NUMBERS

NONLINEAR WAVE FORCES ON LARGEOCEAN STRUCTURES___________________________________ PR - R000-N0-210

6, AUTHORIS) WU - DN661001

Erick T. Huang

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESSE(S) IL PERFORMING ORGANIZATIONREPORT NUMBER

Naval Civil Engineering LaboratoryPort Hueneme, CA 93043-4328 TN - 1857

9. SPONSORINGNdMONITORING AGENCY NAME(S) AND ADDRESSE(S) 10. SPONSORINGIMONITORINGAGENCY REPORT NUMBER

Chief of Naval ResearchOffice of NavalResearchArlington, VA 22217-5000

11. SUPPLEMENTARY NOTES

12, DISTRIBUTION/AVAILABIUTY STATEMENT 12b. DISTRIBUTION CODE

Approved for public release; distribution unlimited.

13. ABSTRACT (Maxnhmn 200 w•b)

This study explores the significance of second-order wave excitations on a large pontoon and tests the feasibility ofreducing a nonlinear free surface problem by perturbation expansions. A simulation model has been developed basedon the perturbation expansion technique to estimate the wave forces. The model uses a versatile finite elementprocedure for the solution of the reduced linear boundary value problems. This procedure achieves a fair compro-mise between computation costs and physical details by using a combination of 2D and 3D elements. A simplehydraulic model test was conducted to observe the wave forces imposed on a rectangle box by Cnoidal waves inshallow water. The test measurements are consistent with the numerical predictions by the simulation model. Thisresult shows favorable support to the perturbation approach for estimating the nonlinear wave forces on shallow draftvessels. However, more sophisticated model tests are required for a full justification. Both theoretical and experi-mental results show profound second-order forces that could substantially impact the design of ocean facilities.

141 SUBECT TERMS I& NUMBER OF PAGES

Nonlinear wave forces, perturbation expansion, finite element method, 56

large floating bodies AS PICE CODE

O17. =ECUTY CLASMSPICATION IS SECURITY CLASIFICATION I 12. SECURTY CI.ASSIICATION 2. UJMITATION OF A3TRACT

Unclassified Unclassified Unclassified UL

NSN 754041-2804-500 Standoa -o 2% her,, 2'69

CONTENTS

Page

INTRODUCTION ........................................ I

PROBLEM DEFINITION .................................. 2

PROBLEM FORMULATION ................................ 2

First-Order Boundary Value Problem ..... ...................... 3Second-Order Boundary Value Problem .......................... 3

FINITE ELEMENT FORMULATIONS .......................... 8

FINITE ELEMENT DISCRETIZATION .......................... 10

Stiffness Matrix and Nodal Force Vector for 3D Element .............. 11Stiffness Matrix and Nodal Force Vector for 2D Element .............. 11

SECOND-ORDER WAVE LOADS ON A FLOATING BODY ............... 13

EXPERIMENTAL STUDY .................................. 17

Test Setup and Procedures .............................. 17Test Results ....................................... 18Nondimensional Forces and Wave Heights

Versus the Ursell Number ............................. 20

NUMERICAL RESULTS ................................... 20

CONCLUSIONS ........................................ 21

REFERENCES ......................................... 21

APPENDIX - EXTENSION OF BERKHOFF'S WAVE THEORY TO THE SECOND-ORDER WAVE PROBLEMS ....................... A-I

V

NTIS CRA&IDTIC TABUnannouncedJustification

By........... ................

Distribution /

Availability Codes

Avail and°orDist Special

INTRODUCTION

Existing analytical models for the prediction of wave excitation of ocean structures assumethat the amplitudes of waves and structure motions are small in comparison with the wavelengthand the nominal dimension of structures. Thus, the diffraction waves from the structure may beapproximated by a simple harmonic function. These assumptions preclude their application insevere sea states. Large waves, in fact, present a much steeper profile than that of a simpleharmonic function. A precise description of the wave kinematics requires retention and propertreatment of higher order terms of the equation of motion and a proper specification of the waterbody boundary at the free surface. These two factors, which are ignored in linearizedapproximations, are the principle sources of wave nonlinearity. When these nonlinear effects areincluded in the diffraction of waves by a body there are, at second order, interactions at the sumsand differences of the component frequencies of the incident waves. Although the magnitudesof these nonlinear effects are, in general, only second order, they act at frequencies away fromthat of the ambient wave energy, and may therefore be of primary concern. This is especiallytrue when these excitations are near the natural periods of the body motions or where restoringor damping forces are small. With the presence of large waves, second-order effects areimportant corrections to the linearized results.

Theoretical developments of the second-order diffraction problem have until recently beenlimited. Studies on simple geometry of the uniform vertical circular cylinder were conductedby Issacson (Ref 1), Chakrabarti (Ref 2), Molin (Ref 3), Wehausen (Ref 4), Hunt and Baddour(Ref 5), Chen and Hudspeth (Ref 6), Rahman (Ref 7), and Ogilvie (Ref 8). The results are,however, controversial due to the difficulties in the correct treatment of the second-order freesurface boundary condition and a proper specification of the radiation condition for the second-order diffracted waves. Previous studies indicate that a weakly nonlinear diffraction problemmay be reduced to a set of linear boundary problems by the perturbation method. Solutions maybe obtained by Green's function method or the finite element method. Kim and Yue (Ref 9)presented a second-order diffraction solution for an axisymmetric body in constant water depthfollowing the former approach. The solution gives an explicit second-order flow potential interms of wave-source Green functions. The nonlinear forces, accurate to the second order,include slowly varying drift forces and biharmonic forces, which appear at a frequency twice thatof the incident wave. The present study pursues the finite element method approach, which givesthe possibility of addressing a sloping sea bottom of irregular geometry.

An immediate need for these analytical techniques exists at the Naval Civil EngineeringLaboratory (NCEL) in the design of connectors for a pontoon-supported waterfront facility. Therequirement of including a second-order diffraction force in the assessment of extremeenvironmental forces imposed by maximum design waves is an apparent trend in the near future.

• • i II

PROBLEM DEFINITION

A mathematical idealization of hydrodynamic couplings between waves and a group oflarge floating or fixed structures in water with a free surface is defined in Figure 1. Waveactivity around the structures, hydrodynamic loads on the structures, and the resulting motionof the structure are investigated. The water body may be open or partially sheltered, and thewater depth may vary. Relevant waves, including incident, scattering, and radiation waves, aredescribed in a way similar to Stokes' second-order theory. The water body is divided into athree-dimensional (3D) region 0, and a two-dimensional (2D) region 02' The 3D regionincludes areas near structures where the seafloor may change substantially and the 2D regionincludes the remaining areas where the seafloor is reasonably flat. In 3D regions, waves areaddressed in detail in 3D space to account for the effects due to irregular geometries of structuresand the seafloor. In 2D regions, waves are approximated by plane wave theories (Ref 10) interms of free surface parameters for better computation efficiency. Boundaries of these tworegions are illustrated in Figure 1. These boundaries are the radiation boundary, !r', which

truncates the water domain to a finite area, and the border separating the two regions, F, .Waves transmitted beyond the radiation boundary are accounted for in a collective form in termsof parameters on the radiation boundary. Incident waves are specified on the wave inputboundary rF, which coincides with the radiation boundary for convenience, when wind waveexcitations are considered. Other boundaries involved are the sea bottom, rb, the wettedsurfaces of floating structures, Fr, and fixed structures, Fr. Subdivisions of the water body andthe associated boundaries are illustrated in Figure 2.

PROBLEM FORMULATION

A Cartesian coordinate system (OXYZ) fixed to the earth, as shown in Figure 3, isemployed as an inertia reference, where the X-Y plane lies on the mean water surface and theZ coordinate is measured positive upward from the still water. The responses of a floating bodyare described in reference to a body coordinate system, Gxyz, attached to the body, with itsorigin G located at the center of gravity of the body. The xy plane is parallel to the staticwaterplane and the z coordinate is directed vertically upward. For the purpose of computingwave drift forces and moments exerted on the body, a third system of coordinate axes (Gi$i)is defined where its origin is at the center of gravity of the body and is always parallel to thefixed coordinate system OXYZ.

With the assumption that the fluid is inviscid and incompressible and that the flow isirrotational, the relevant wave problems can be addressed using potential flow theory.

Assuming weakly nonlinear waves, the total velocity potential 4 is expressed as aperturbation in the wave slope parameter e a kA < 1:

= 0(1) + C2 0(2) + 0(e 3) (1)

2

where:

f(1) = first-order velocity potential

0(2) = second-order velocity potential

Further, k is the incident wave number given by the dispersion relationship C02 = gk tanh (kh),and g is the gravitational acceleration. The associated free surface profile j7 is decomposedlikewise:

11 = eT•(1') + e21 a)+ 0(e 3)

This approximation leads to the decomposition of the otherwise nonlinear boundary valueproblem to a set of two linear boundary value problems. The first order problem has beendescribed in Reference 11. This discussion will be limited to the second order.

The first- and second-order potentials must comply with the Laplace equation andappropriate boundary conditions as follows.

First-Order Boundary Value Problem

Laplace equation (fluid domain): V2 0(1) . 0 (2)

Seabed and fixed structure boundary: = 0 (3)an

Floating structure boundary: n =) (4)

an

Free surface: g 1+ = 0 (5)

Radiation condition: lrn A - 0 - i k,0) = 0 (6)

Second-Order Boundary Value Problem

Laplace equation (fluid domain): V2 0(2) = 0 (7)

Seabed and fixed structure boundary: -(--2 0 (8)an

Floating structure boundary:

3

For radiation waves: an = vn (9)an

For scattered and incident waves:

-= -(x() V) VO(1) -n + (v) - V I)) -N 1) at body surface. (10)an

Free surface:

= + + 0()(00 + 10) onZ = 0. (11)

Radiation condition: outgoing waves from the floating bodies.

In the above, R = (x2 + y2)112 is the radial distance from the origin of the inertia frame,

and 1. is the normal derivative into the body. The second-order problem is complicated byan

the inhomogeneous forcing term in the free surface boundary condition (Equation 10), which isgiven in terms of quadratic products of the first-order potential. The first-order problem isclassical and a variety of numerical solutions are available. For example, a solution using afinite element scheme has been obtained in a previous study (Ref 11). Equations 10 and 11correlate 0(2) with 0(1). Hence, 0(2) is completely defined once 0(1) is known.

The first-order velocity potential may be expressed in component waves as follows.:

where:

0(1) = first order incident wave potential

0(') = first order diffction wave potential

B() = first order radiation wave potential

For monochromatic incident waves, the time dependency is separated and rewritten:

0) = [ ) ( ) ] ' (13)

4

Substituting Equation 13 into the second-order free surface boundary condition in Equation 11,a quadratic periodic forcing function is obtained which oscillates at twice the frequency of thelinear waves. As a result, Equation 11 may be written as:

(2) (2) Q(x,y,O)e-it (14)I~t+ g 0 =~~,~

where the quadratic forcing function Q(x,y,O) consists of nine components as follows:

Q(x,y,O) = QU + %I + QBB + (QIS + QS) + (QIB + QBI) + (QSB + QBs) (15)

where:

Q11 = plane wave forcing function which leads to the ordinary Stokes'second-order wave component at double frequency 2W

QSSQBB = self interactions of first-order scattered waves and first-orderradiation waves, respectively

Qis,...,Qas = cross interaction of the first-order terms

The details of these components are given in the following:

Qss= i 2 2(V4) -(1)2 (1) i(')s 21 (1) 4,()

r 2

I (1g) 2 (1) 2O ( (1)40(1)Q-sB WB 2 ( ZZ + -41 * SZ

t Wg

5

= (iO) (fV414) 4- (1) 4,(l) + (4) 2 ) ] (16)

S ,•(D '00) + (a .g ').{ ,

QIB = i 2(V og (4))) - Bzz ÷ 4,1 (Bzg

QDI = i 2(V4 •)(v) ) -()VZ + --- 4g)

Qs = i [2(VO$())(V4g)) - 44s),'z + a2 ( 1) 4)]

s hi2(V4og)(V44ý') -. g)4,(1 )z) +),0(1)1

where the subscriptions of spatial coordinates indicate partial differential. These quadraticforcing function terms behave like a nonuniform pressure field applied to the free surface in first-order problems as described by Wehausen and Laitone (Ref 12). They generate additionalcylindrical standing and outwardly going progressive waves at the double frequency.

Since Equation 14 is nonhomogeneous, the second-order potential consists of thehomogeneous solution f(2)H and the particular solution t(2)P as:

OCZ) =- M + 2H

-12 + 4(2Z)H)e-12 t (17)

These components satisfy, respectively, the homogeneous and inhomogeneous force surfacecondition (Equation 11), and jointly, the inhomogeneous body boundary condifion, Equation 9or 10. The particular solution may be further separated as:

(2)P 11)+ 0()+ 0 2. I

(18)

+0 +0 +(0 I0I(I

6

The homogeneous solution consists of:

4D()H = (2) + OD) (19)

By substituting Equations 18 and 19 into Equation 17, the total second-order potential becomes:

4t(2)+

B(20)

+ C) ()+ 0()+0 2

or

0(2) -=DZ .+ (")(2) + 4 () (21)

where:

4~ (2)s[s2) + 4D(2) + (12 +) 4 C2) + (2) P [ + 40 (2)

D*S Ss BB+(is+ SI)+ 'JB 4BI) + (tBS + )

The component 4•) represents the forced wave motion generated by the Q11 forcing function,

which is the usual second-order Stokes' wave. 0(2) is the second-order scattered wave potential

and L0C2) is the second order radiation wave potential generated by body motion. Both satisfythe homogeneous free surface boundary condition. Their far field behavior is given by:

1 3

*H r 2 ev +0(r 2),r1

where k2 is the double frequency wave number satisfying 4W2 = gk 2 tanh (kh). This means that,in principle, these components can be solved by the existing first-order method.

The second-order potential #(2) in Equation 21 consists of three parts: (1) the incidentwave~~~ cmoeto, ()tepud-diffraction OC)3 and (3) the radiation components OC2).

wave component 0 11 (2) the pseudo-ifcto D*

The second-order incident wave potential is given by:

7

(2) 3 3 H(irH' cosh[2 k(d +z e- -j2wt (212)ll 8 kT ýL ) sinh 4 (k d)

The radiation components satisfy the same equations as the radiation potentials of thefirst-order potential theory, at the double frequency 2W. Newton's law applies through the useof Bernoulli's equation in the computation of the second-order wave exciting loads, which resultsin the equations of motion at the second ord-r, to be discussed later.

The solution procedure for Equation 14 using finite element method is straightforward,regardless of the complexity of the forcing function on the right-hand side of the equation.

FINITE ELEMENT FORMULATIONS

According to the calculus of variations (Ref 13), the solution to a boundary value problemis the potential which minimizes a certain functional, in terms of the governing equation of theproblem. For 0(2), the functional 11(2) can be expressed as:

llQ fff- (Vi( 2))

2 dQ + f f(4ý)) dS (23)

where:

n1 = fluid domain

Sf = free surface

SB = body boundary

SR = boundary for the radiation condition

The integral (f'f.,) over the unknown free surface in Equation 23 may be approximated

by a Taylor series expansion, in terms of the variables at the mean sea surface.Solutions for the second-order diffraction and radiation boundary value problems may be

formulated using the standard finite element method. The radiatiop boundary condition isimposed on a finite boundary, r2 , at moderate distances from the structures of interest. Theentire fluid inside this boundary is divided into two regions. Each region has the functionalscorresponding to its respective governing equations and associated boundary conditions asfollows:

8

For the three-dimensional region:

i1(2) = ff f f (V )d, _ ff2-- ( )2dS01! Sf g

f f Q(xyO) 4)()dS -f f (vý 8m( dS,- Q- 1 f(vim81m dik (24)

+ ff [(F().V) V ,p),fi + (V,) - V',,O)• . 8 dSkS.

j = II,D+S,i,k,...i -1,2,...6k =1,2,...M

where M is the total number of floating bodies and i4k is that of the radiated waves due tomotion in the ith mode by the kth body, in which i=1,2,...,6 corresponds to the surge, sway,heave, roll, pitch, and sway modes, respectively. The quantity Q in the third term in the aboveequation is given in Equation 15.

For the two-dimensional region:

1,2) = f f= {F" (V ())2 - 4 k2 F( (4))2} dQ2

022

(25)-f A2as (Radiation Condition for 42), j=I,D+S,i,k...

-f , a n- - i-

The function F* in Equation 25 is the function defined by Equation A-22 of Appendix A.On r', the three-dimensional region is coupled with the two-dimensional region by the

following relationship:

4 2)(x,y,z) ,42)(x,y) cosh2k(d + z) j=lIS+D,i,k (26)sinh4 (k d)

where d is the water depth.

9

FINITE ELEMENT DISCRETIZATION

The integral Equations 24 and 25 define the velocity potentials of the water flow over theentire fluid domaii. The fluid and boundaries are thus discretized, as shown in Figure 2, toconform the finite element applications.

The potential 02)' within each element can be approximated in terms of the unknown

nodal values 14• of that element and interpolation functions IN]. In matrix notation:

02)' = [N] {(?0} (27)

The functional ]1(2) is now minimized with respect to the nodal values 4) for a singleelement to determine the flow potential which satisfies the boundary value problem. For tile 3-dimensional case, Equation 27 can be inserted into Equation 24 and written as:

aI) 3 = (2) = 0 (28)

This being true for any variation, 6 t) requires:

an__• = a0 a(4•))2 (29)

(2)

from which parameters ({o2I are to be determined. Because the functional is quadratic, elementEquation 29 reduces to a standard linear algebraic equation, that is:

= [k]{.,}2) -_fI = 0 (30)a (3O)

This transforms the integral equations into a set of linear algebraic equations in the unknownvalue of velocity potential at boundary nodes on all elements.

10

Stiffness Matrix and Nodal Force Vector for the 3D Element

The element stiffness matrix [k] and the element nodal force vector {f} in Equation 30are given as:

[k] = ff f (VN) T (VN)dxdydz - f f .12' N T NdS (31)(O)*sf) g

fff = -ff NT(v),AS .'bk(St)ý

+ffNT[(x(1)v)V)V40j"-n + (v(') -V4') "N(1)]dS ".6k

(32)+ f -NTQ dSSf g

vU = normal velocity

x' = the first order displacements

In the above equation, ne and Se denote that corresponding integrations are taken over a single

element only. The velocity potential, e ), at boundary nodes of a single element consists ofthree components. These are incident, scattered, and radiated potentials. Hence, the elementequations can be written as:

[k]{4?)} = {f}); j = I1,S+D,i,k (33)

Stiffness Matrix and Nodal Force Vector for the 2D Element

Similarly, the element equation for the 2D case can be derived by taking the

differentiation on the functional 11(2) given in Equation 25 with respect to 4).. Then:

11

aI• =f f {(VN)F'rF(VN)-4k 2F'N T N)dx dy -4' .2)

- f F N'T•dS (34)(Q.~ an

Tf F aNTN IP(2N a s4c = 0F-jaN N as)i as )JMO.

and a system of equations for the problem is:

[k(x,y)]{4t)}j = {f(xy))j; j =II,S+D,i,k (35)

with

[k(x,y)] ff{(VN)TF'(VN) -4k 2 F" NT N} dx dy

(36)

+f(xy)J = fF" N NTdS (37)

f an

Equations deducted from the 2D and 3D elements can be assembled to form a set of

global equations pertaining to the entire fluid region. These may be written in matrix form as:

[KJ{,M)} = IF) (38)

The global stiffness matrix [K] is symmetric and banded. Its half-bandwidth indicatesinteractions between nodes of an element and its immediate neighbors. Equation 38 may nowbe solved numerically.

12

SECOND-ORDER WAVE LOADS ON A FLOATING BODY

Once the second-order velocity potential is determined, the second-order wave loadingon the floating body can be calculated. It is assumed that the vessel undergoes small first-order

motions X(1) from its mean position R(o), and that the pressure can be expanded in a Taylorseries about the hydrostatic pressure at the mean position and the following expression is found:

p = p0) + ep () + c2pC2) + 0(e 3) (39)

where:

hydrostatic pressure: p = - p g Z(°) (40)

first-order pressure: p = _ p g Z(-) _ p dot (41)

(2 -_Ip1# 00 2 - (2 (I' . t 001)

second-order pressure: pC2) = - 2 pIt4() _ p - p(•) (42)

The total hydrodynamic force exerted on the body relative to the coordinate system GiS,is given by the following equation:

= -ffp,#dS (43)

where S is the instantaneous wetted surface and )7 is the instantaneous normal vector to the

surface element dS relative to system G0#.

• = ) + cNN')

Substitution of the pressure p given by Equation 39 and retaining the integration to thesecond order, one obtains:

P = P(o) + P(i) + e2P12) + 0(e 3) (44)

The instantaneous wetted surface S is composed of two parts, So and s (Figure 2). So is theconstant wetted surface up to the still water line and s is the surface between the still water lineand the wave profile along the vessel.

The second-order forces can be calculated by integrating all products of pressure p and

normal vector 9 which give second-order force contributions over the wetted surface So and s:

13

p2) -f f (p) + p2)f)dS - f f p'(ndS (45)so

Substituting

6(0) X r

into the above equation yields for the first term

_ffpR) = O -x -fp f (1)a"6 (46)so so

where P(1) is first-order wave exciting force and hydrodynamic reaction force. According to

Newton's second law, P(I) can be written as:

PG) [MI (47)

where [M' is mass of vessel in the air and X () is the first-order acceleration of the center ofgravity of the body. Thus, the first term of the second-order force may be obtained from thefirst-order motion dynamics as:

- ff p( 9"(" dS x x M R) (48)so

The second term of the second-order forces is integrated numerically. The following wasobtained from Equation 45:

_ffp(2)rxds = pf _ j$(1)I2 + (2 + jZO . ol)dS (49)So So

14

The integral of the oscillating surface s is calculated to the linearized free surface by substitutingp(M from Equation 41:

-ffp(')i•ids = -ff(-pgz(l) - P ID,))f )dS (50)S 3

where the surface element is written as:

dS = dZ(1) .dl (51)

At the still water line, the linearized free surface condition to first order is:

_ p 0!" = p g co) (52)

where (M is the first-order free surface elevation. Inserting Equations 51 and 52 into Equation50 results in the following:

_fpO) fidS - f fl& (53)

2 WL

in which 4(l) is the relative wave elevation defined by:

,()) = C -l)_ Z4L (54)

The final expression for the total second-order force thus becomes:

P(2)=- f _I Pg(C'l) 2fdl+ xiiX

WL2(55)

- - p - .

15

The second-order force given in Equation 49 can be divided into a steady component and a time-dependent biharmonic wave exciting force. The steady component is a second-order steady wavedrift force calculated from Equation 49 by taking the time average over one wave period. Sincethe flow is periodic, the time average value of Cr is zero. As a result, the second-order steadywave drift force becomes:

f(2) = (2) f (1 p o() i i+61WL

)

(56)

+ f f [! IV 00)1f2 + 0()+ (X0) V 01))]ildS (6SO2

For regular waves

p ff ),22dS = 0

Thus, the second-order force can be written as:

P(2) = ID + R,, {f) ei2ot) (57)

where f(2) is the second-order biharmonic wave force and is given by:

W 4g 0(58)

ff [_ p(V40())2 +2ipcp4P) +2iwo(')'V4(1))lfds

16

where:

4DO) = 0)(ei~ t

0(2) = 42) eiwt

=• * •ei~t

•(1)= 1)et

Z(O = i(l) riot

y =

The total hydrodynamic moment about the center of gravity of the vessel relative to thecoordinate system GiM is given by Equation 5:

f= -ffp(A )dS (59)S

The derivation is analogous to that followed for the forces. The final expression for the second-order wave moment is:

= (2)p g )(Ixfi)dl + W) x(1 []6)

WL2(60)

-ff[~P~(I -- 2 t I~I o(I)J x fi) dS

EXPERIMENTAL STUDY

Test Setup and Procedures

A hydraulic model was designed to observe the coupling process of a fixed barge withnonlinear waves in shallow water. The experiment was conducted in a two-dimensional wavetank at the Civil Engineering Laboratory of Texas A&M University. Figure 4 illustrates thegeneral setup of the experiment. The tank was 36 meters long, 0.91 meter wide, and 1.22meters deep. The water depth was maintained at a constant of 20.3 cm throughout the entireexperiment. A rectangular box (101.6 cm by 33.9 cm by 8.4 cm) was placed at roughly 11meters from the wavemaker and centered in the tank, to simulate a barge afloat at the water

17

surface. The box was mounted to a rigid aluminum frame with three legs as shown in Figure4. The box was further ballasted to a neutrally buoyant condition at the draft of 8.4 cm in thestill water

Two load cells were placed at the end of each of the three legs to measure the horizontaland vertical forces, respectively. Each cell provided a capacity of 50 pounds at an accuracy of+ 1 percent of the force applied. The total forces and moments were deduced from thesemeasurements.

A significant influence on the wave activity around the test model due to the closeproximity of side walls of a narrow wave tank is anticipated. The wave forces imposed on thetest model will be different from those experienced in an otherwise open water. However, thistest was intended for validating the numerical procedure. The side walls of the wave tank weresimulated in the numerical procedure to give a fair comparison between the physical andnumerical models.

The experimental waves were generated by a computer-controlled Commercial HydraulicsRSW 90-85 dryback, hinged-flap wavemaker. The water pumped by the flap traveled down thetank channel and evolved into shallow water Cnoidal waves before it reached the barge. Tworesistance-type wave gauges were used to monitor the wave profiles. One was placed at 75 cmforward of the bow along the center line of the tank, and the other at 47 cm aft of the bow,centered between the box and the side of the tank. The accuracy of the gauges was ±1 rmm.Four wave periods were selected intentionally to avoid the resonance frequency of the wave tank.Six wave heights in the range that promised results of well defined Cnoidal wave profiles werechosen for each wave period. The wave parameters used in the test are summarized in Table 1.The Ursell number of these waves varied from 0.5 to 70.

Data acquisition was accomplished using a Hewlett Packard HP-3852A dataacquisition/control unit with an HP-330 workstation. This system was equipped with a 10-channel relay multiplexer (HP-44718A) and an integrating voltmeter operating at a 5-1/2-digitresolution (HP-44701A). Details of the test setup and procedures are described in Reference 14.

Test Results

Twenty-four sets of time history of the wave profiles and the horizontal and verticalforces were recorded. A sample set of these measurements is ,hown in Figure 5. The incidentwave elevation shows good agreement with that predicted by the Cnoidal wave theory in bothwave height and period. However, it is noted that the experimental wave is asymmetric incontrast to the asymmetric mathematical form. This discrepancy grows as the wave heightincreases. This results from the wave generation mechanism used in the test. The force timehistories are similar in shape to that of the wave profile. The time histories of all measurementsare decomposed by Fast Fourier transform up to the third harmonic as follows:

3 X aX(t) = X0 + 1 •-cos (nlt + e)

U-1 2

18

where:

Xo = the mean value

X. = height of the nth harmonic component

en = the phase angle

w = base frequency

The symbols H, X, and Z represent the wave height and the double amplitudes of theforces in the horizontal and vertical directions, respectively.

It was found that all the mean values and the third harmonics were negligibly small incomparison to the first and the second harmonics, and are therefore not discussed. The resultsof decomposition of the waves are shown in Figure 6. The height ratios of each harmoniccomponent to the resultant wave height were plotted versus the resultant wave height. The sameratios deduced from the theoretical Cnoidal waves are presented in dashed lines in the same plotsfor comparison with the measurements. The measurements are almost identical with theirtheoretical counterparts. It is noted that, for the short period waves, the height ratios remainessentially constant over the range of the total wave height considered. For the long periodwaves, these ratios vary nonlinearly with the increase of total wave height with the first harmoniccomponent decreases and the second harmonic increases. This result clearly demonstrates thetypical nonlinear characteristics of the shallow water waves.

The force time histories are decomposed likewise. For convenience, the components arefurther nondimensionalized in the form of force coefficients as follows:

Cx = XipHilb

C ZIpHilb

where p is the specific weight of water and "I" and "b" are the length and the beam of the box,respectively. These force coefficients are equivalent to the ratios of the heights of the firstharmonic forces to the first harmonic wave height. Figure 7 indicates that these coefficientsremain constant at a specific wave period. This implies that the heights of the first harmonicforces are linearly correlated to the height of the first harmonic wave. The second harmonicterms H2, X2 , Z2 are plotted as functions of the first harmonic wave height H, squared as shownin Figure 8. The units are given in cm for H2 , Newtons for force harmonics, and cm2 for H1

2 .The results, which closely fit the first-order regression line, indicate that the second harmonicterms X2 , Z2, and H2 are linearly correlated to the square of the first harmonic wave height.

The proportionality constants for X2 monotonically increase as the wave period increases,whereas the constants for Z2 generally increase but slightly fluctuate. The proportionality

19

constants for H2 increase as the wave period increases, although the constants are generally very

small compared to those of the forces.

Nondimensional Forces and Wave Heights Versus the Ursell Number

The first and second harmonics of the forces and incident wave height are non-dimensionalized by dividing them by their respective resultants, then compared to the Ursellnumber,

Ur= HL

2

h 3

where h = water depth. The first harmonic ratios exhibit a linear trend and the second harmonicratios present some nonlinearity, as shown in Figure 9. HI/H and X1/X decrease linearly as theUrsell number increases while Z1 /Z remains constant. H2/H and X2/X increase initially thenapproach to a constant value while Z2/Z decreases slightly throughout the range of Ursellnumbers. This trend seems to agree with the previous finding shown in Figure 10 from Sarpkayaand Isaacsson (Ref 15) on nonlinear shallow water wave forces on a vertical circular cylinder.

NUMERICAL RESULTS

The finite element procedure derived in the previous section on theoretical considerationswas executed in a computer program, NAUTILUS, coded in standard Fortran 77 language. Thisprocedure discretized the water body under consideration with a combination of three- and two-dimensional finite element meshes. This feature permitted optimization of computer resourceswhile retaining the essential details of the physical problem under study. The discretizationresulted in a linear matrix, Equation 38, which was treated by a proven procedure based on thefrontal method developed by Irons (Ref 16). The solution gave a complete description of thefluid activity in terms of velocity potentials. NAUTILUS presents the results in a convenientform for practical application, including wave height distribution over the entire water surface,wave-induced forces on the fixed structures, and the dynamic motion of the floating structures.All computations are executed to the second harmonic.

NAUTILUS was used to reproduce the hydraulic model test described in the previoussection on experimental study. The results were used to verify the feasibility of the NAUTILUSprogram. Since NAUTILUS does not include a wave generation mechanism to simulate thewavemaker, the wave tank will be approximated with a channel of infinite length with a steadysecond-order Stokes' wave approaching from one end of the channel. The water body withinthree times of the barge length from the center of the barge is simulated with three-dimensionalelement meshes, while the rest is approximated with two-dimensional element meshes. Thewater body under consideration extends to a distance of ten barge lengths on each side.Figure II illustrates the discretization of the model test setup. Wave parameters used in themodel test are entered into the numerical model for comparison with the forces observed fromthe model test.

20

The results of the wave forces predicted by the numerical model (in black symbols) arepresented along with the hydraulic model measurements (in white symbols) in Figure 9. The fairagreement between the theoretical and the empirical results strongly supports the approach taken.

CONCLUSIONS

A second-order simulation model for three-dimensional wave-structural couplings inmoderate seas has been formulated. The results confirm the feasibility of reducing a weaklynonlinear free surface problem by the perturbation method. This treatment eliminates the needfor specifying the unknown free surface, and allows the nonlinear problem to be addressed at thewell-defined mean water surface in the perturbation components. Each component forms a well-defined linear boundary value problem that can be solved effectively. The present study solvesthe component boundary value problems with a finite element procedure. This procedure, whichuses a combination of two- and three-dimensional finite elements, gives a good balance betweencomputation efficiency and fair proximation of the physical problem.

The wave-induced forces on a rectangular box held steady at the water surface werestudied both numerically and experimentally to examine the significance of the second-orderforces. The second-order forces were found to increase with the Ursell number, LH2/h3, whichis a measure of the wave steepness and the relative water depth, in a manner closely resemblingthe second-order profile of a Cnoidal eave. This implies that the significance of the second-orderforces is similar to that of the second-order wave amplitude to the total amplitude of Cnoidalwaves. Hydraulic model measurement indicates that the second-order lateral forces induced bysteep waves in shallow water can be as large as 30 percent of the total lateral force. This...onfirms the importance of the second-order forces in high sea states. Actual wave forcesexperienced by ocean structures in rough seas can be much greater than those predicted by lineartheories. More importantly, the additional wave forces may occur at frequencies away from thedesign wave frequency, and coincide with the resonance frequency of the structure of itscomponents and cause adverse effects. Ocean structures for use in high sea states must bedesigned to withstand second-order forces.

REFERENCES

1. M.Q. Issacson. "Nonlinear wave forces on large offshore structures," Journal ofWaterways, Port, Coastal & Ocean Division, ASCE, vol 103, 1977, pp 166-170.

2. S.K. Chakrabarti. "Comments on second-order wave effects on a large diameter verticalcylinder," Journal of Ship Research, vol 22, 1978, pp 266-268.

3. B. Molin. "Second-order diffraction loads upon three-dimensional bodies," Applied OceanResearch, vol 1, 1979, pp 197-202.

4. J.V. Wehausen. "Perturbation methods in diffraction," Journal of Waterways, Port, Coastal& Ocean Division, ASCE, vol 106, 1980, pp 190-291.

21

5. J.N. Hunt and R.E. Baddour. "The diffraction of nonlinear progressive waves by a verticalcylinder," Quarterly Journal of Mechanical Applied Mathematics, vol 34, 1981, pp 69-87.

6. M.C. Chen and R.T. Hudspeth. "Nonlinear diffraction by eigenfunction expansions,"Journal of Waterways, Port, Coastal & Ocean Division, ASCE, vol 108, 1982, pp 306-325.

7. M. Rahman. "Wave diffraction by large offshore structures; an exact second-order theory,"Applied Ocean Research, vol 6, 1983, pp 90-100.

8. T.F. Ogilvie. "Second-order hydrodynamic effects on ocean platforms," InternationalWorkshop on Ship and Platform Motion, Berkeley, CA, 1983, pp 205-265.

9. M.H. Kim and D.K.P. Yue. "The complete second-order diffraction solution for anaxisymmetric body," Part I - Monochromatic Incident Waves, Journal of Fluid Mechanics, vol200, 1989, pp 235-264.

10. J.C.W. Berkhoff. "Linear wave propagation problems and the finite element method,"Finite Elements in Fluids, vol 1, edited by R.H. Gallegher, et al., John Wiley & Sons, London,England, 1974, pp 251-280.

11. Naval Civil Engineering Laboratory. Memorandum to files on the interaction of ships withfloating terwiinals, by T. Huang. Port Hueneme, CA, 1990.

12. J.V. Wehausen and E.V. Laitone. "Surface Wave," in Handbuch des Physik, Springer,Berlin, vol 9, 1960, pp 446-778.

13. F.B. Hildebrand. Method of applied mathematics, 2nd edition. Englewood Cliffs, NJ,Prentice-Hall, 1965.

14. G.S. Hook, C.H. Kim, and E.T. Huang. "Wave excitating forces on a platform fixed innonlinear shallow water waves," in Proceedings of the International Conference, CivilEngineering in the Oceans V, College Station, TX, Nov 2-5, 1992, pp 311-325.

15. T. Sarpkaya and M. Isaacson. Mechanics of wave forces on offshore structures.New York, NY, Van Nostrand Publishing Co., 1981, pp 455-7.

16. B.B. Irons. "A frontal solution program for finite element analysis," International Journalof Numerical Mathematical Engineering, vol 2, 1970, pp 5-32.

22

Table 1Wave Parameters Used for the Hydraulic Model Test

(sec) (cm)

0.75 0.54, 1.09, 1.66, 2.11, 2.58, 3.13

1.00 0.52, 1.07, 1.60, 2.08, 2.55, 3.05

1.20 0.54, 1.52, 3.19, 5.09, 7.23, 9.23

1.58 0.69, 3.90, 4.98, 6.30, 8.48, 10.82

23

~~~[-I /•""'"

INCIDENT %WAVE.,, ~INPUT".

BOUNDARYFIXED STRUCTURE

2 D REGIONS•X

Si COUPLING

MECHANISM

FLOATING STRUCTURE -

SOLID WALL RADIATION BOUNDARY

Figure 1Definition sketch of a compound structure and the

ambient fluid domains.

24

E

L . " ,- -i • I

/..' -•::

4"' /'I

* . E, , , ., .,, , . . \ • , , ,. ., ,

25a I I I I

LD 4

04

rxC

Q)

47

QCaNO M-

02

load cells

SEE ( c )FOR DETAIL

0wave gauge

wave gauge a

(a) top viewLOAD

CELLSWAVEABSORBER

M 3 cm 8.4ca

75cm 476. cm 1.9m

.3 cm - 3c

(c) locations of load cells.

Figure 4General layout of the hydraulic model.

27

5---4-f (a) Wave Profile

Expt.3--- Theo.

2--

E

0

-4

21 21.5 22 22.5 23 23.5 24 24.5Tune (See)

16-12" (b) Horizontal Force

4428.

2o.S12-s

CI

21 21.5 22 22.5 23 23.5 24 24.5

Tune (See)

44 "r- se

-12

-20.-28 III::,

ii 215 22 22.5 23 23.5 24 24.5T"une (Sec)

Figure 5Profiles of sample incident wave height, horizontal force,

and vertical force.

28

1.4

1.4 Tm = 0.75 SEC Tm= 1.0 SEC

1.2 0H-i/H1.2J.- Hl/ H

o H2/H1.0 0 . 0 0 .

0.8 - 0.8i> 0.8

> *0HI/H0.6 -0 - Theory0.6- z H2/H

-4- Theory

0.4- 0.4

0.2 0.2

0.0 0 0 0 0,

0 1 2 5 0 1 2 3 4Wave Height, H(CM) Wave Height, H(CM)

(a) (b)

1.4 1.4Tm = 1.20.Sec Tm = 1.58 Sec

1.2- 1.2.

1.0" * - 4-4 - -

US0.8- 0.8."0 H /H 0 Elb/H

0.6 -+- Theory 0.6 -+- Theoryo OH2/H O H2/H

._ -i- Theory -4- TheoryS0.4- 0.4. _.t

0.2 0 0.2 q 0

4 e--Q--", _.Ql ____-"__________"_____"_

0.0 -Q -M 0.0 , _0 2 4 6 8 10 1 5 6 7 8 11

Wave Height, H(CM) Wave Height, H(CM)

(c) "(d)

Figure 6Decomposition of Cnoidal waves.

29

0.30 0.20

Tm = 0.75 SEC Tm =1.0 SEC

0015 O.Cx

,5 0.05 0-Cz0 Cz Theory

U 0.20 - Theory

00 0 0 0 000.10r~b.

0.10. 0 00 0 00

0.05

0.05 '

0.001 0.00 , 0 , 1

0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Wave Height, HI (CM) Wave Height, H1 (CM)

(a) (b)

05 0.8Tm= 1.2 SEC Tm= 1.58 SEC

0 Cx 9 Cx

0.44 Theory

-- Theory 0.6 0 ryS0.3 • O.~C--.O- -C. --L _

~0.30 0 * .=-

00.4

0.20

0.1 * * •0.2

00. l , • ,@ 00 0 - 0

0.0 *

0 2 4 6 8 10 2 4 5 6 7 8 9 10

Wave Height, H1 (CM) Wave Height, HI (CM)

(c) (d)

Figure 7Hydraulic model measurements of the wave force coefficients.

30

--2.6

-~2.0Tm =0.75 SEC T=..SS /C •.Tm =1.0 SEC

•2.0 --0 X2 -XS5O- 2

-O-- Z2 1 -0-Z2

F21.5 -+H2 -E -2�I -- Tbeory (H2) /

o / 1.0

i1.0"/0

0.5/ 0.5 Q,0.5 / ,.i/ 0

0.0 ,/ o -- - -o~~~~~o00 ..-.-r• I_., - .oo

0 - 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

H2 C2 H2 CM22 cm2 H2 cm2

(a) (b)

" E 16 25-16 Tm = 1.2 SEC T- 1.60 SECUE

14 T=.OE-4-O-" X2 209 X

// ;20-Z 12 -0-ZZ / -- Z2

10 H7 -7 H2s'. • 15 _ .Teoy(-2

-I- Theory (H2) Thory (H2)

o / 0 72 0C / .10O§6 /z

0 c

4 / Z,5

2 '9- o

10 20 30 40 50 60i0 k 1020- 030 405060

H n2H 2 MH1 cm 2 H 1 cm 2

(C) (d)

Figure 8The second harmonics of the incident wave and wave forces.

31

1.0 ,,,

0.90.80.7

0.6 Theory

-0.5

S0.40.3 n =2 C]

0.2 =20.1

0.0

,

0 10 20 30 40 50 60 70 80In?2/h3

1.0 O b I I I '

0.9 - 80.81, 0 00.7 *, 0

S0.6

0.4 0n 00.3 n =2

0.20.1 [

0.00 10 20 30 40 50 60 70 80

HL2/h 3

1.0 1 1 1 1 1

0.9 " 0 8 8 0 On=1 00.8 0 0

0.7

0.6

0.5

N 0.4N

0.30.2 n =20.1. .0 0•0.01 T, Ij 0, I

0 10 20 30 40 50 60 70 80jL2/ h3

Figure 9Fourier decomposition of incident waves and wave-induced

force as a function of the Ursell number.

32

4

k*o-- = 0.2

3

FcnoidalFlinear

I

0 200 400 600 800 M000

U (=HL 2 /d3 )

Figure 10Nonlinear shallow wave forces on a vertical circular cylinder.

Ratio of Cnoidal to shallow sinusoidal wave theory forcepredictions as a function of Ursell number U for various

values of ka (from Ref 15).

33

E

LIM-

34

Appendix

EXTENSION OF BERKHOFF'S WAVE THEORY

TO THE SECOND-ORDER WAVE PROBLEMS

BASIC EQUATIONS

Field equation:

V2 4€ - o (A-1)

Free surface equation:

a4 40 Q (A-2)4€o €() + _ = -- , z=O

g az z

where:

Q = -i2w -(V4(Q)2+T -

Bottom Condition:

9+(2) ah a3,C) ah +a¢O)-3+_ + L _ = 0, z =-h(xy) (A3)

ax ax ay ay az

The derivation of the reduced equation starts with basic Equations A-I through A-3 by

introducing dimensionless coordinates with the aid of the characteristic length L = g/W2 at the

free surface and the mean water depth H.

A-I

L = wave lengthH = mean water depth

z h(x,y) = water depth

V z+h(x,y) = 0 = bottom surfaceL x = characteristic length

w = wave frequency

H h(x,y)

Apply

x ,z -xL H I

y _Y h/ ,-h Y=-YL H I

and drop the superscript (2). The nondimensional governing equations given in Equations A-I

through A-3 are given as:

±!t + +_2 -2o4 (A-4)ax' 2 ay12 (H)

a4_ 4W2 H * = f*(4(l)) H Q H (A-5)

az' g g

__ !! (a4 Lh+ 4,h 0, zO h' (A-6)

a8z' L ax' a! + ay' aý 0

A-2

Introducing:

H W2 H H•=•- a= , £-g

Equations A-4 through A-6 can be written as:

. + I 0 in fl (A-7)ax12 ay' 2 P 2 azt 2

a4 _ 5 4o = f', z =0 (A-8)az'

/4 + e2(a O h + i h = 0, z/ =-hH -h/ (A-9)az, ax' ax ay/ ay=

Assuming the potential function 40 can be written in the form:

40 (xy~z) = Z * (h, P z /) (P(x, y/, z'C)

and developing the function 4p into a Taylor-series with respect to e z/:

41(x,y,z) = Z*(( +ez/i + C12 ( +

Assuming a mild slope:

H EH <<[

Therefore:

4ý(x~y,z) - Z(h, z') o(X, y')(A-10)

A-3

Substitute Equation A-10 into A-7, and assuming a mild slope (E << 1), we obtain a simplified

equation as

z-,29 -0,29. - = 0

-3x' ay / 2 i2 aZ/ 2

That is,

a 2,9. 0 -a2P) _ 1 0-2Z41( ax /2 ay / L2za/

The left-hand side of the above is a function of (x',y') only, while the right-hand side is a

function of z' only. Therefore:

I o °-P. = -4k 2 (A-11)

(P. a3x/ 2 ay f2

and

= 4k 2 , or = - L2 Z.4k 2 (A-12)42z a z' 2 az12

Substituting Equation A-10 into A-9:

Z jL*° + 2 z/ ZI h/S + e(.-) 0 , z=-hazI

Therefore:

a--.- = 0, z'-.h, e 2 .-0 (A-13)az'

A-4

Substituting Equation A-10 into A-8:

az 0o - 48Zq0o = f or az - 48Z = f'/q0 , Z/=O

8z' az'

Therefore, the simplified equations (assuming , << 1) are given as:

Laplace Equations:

I [± 9.. = -4k 2 (A-14)

a2Z = j22 Z-4k 2 (A-15)az'2

Free Surface:

-E - 4 8 Z = f7 9P, z/=O (A-16)az'

Bottom Condition:

aZ 0, z= -h/ (A- 17)azt

Solution of Z

From Equations A-15 and A-17, the following result can be obtained:

Z(h',z') = cosh2kI(z'+h') (A-18)sinh4(k p h)

A-5

or

cosh 2(t( +h)

Z~~)= sinh 4 (! h)

Set k/L = K ... wave number

Z(h,z) cosh2i(z +h) (A-19)sinh4(.h)

Solution of p0

From Laplace Equations A-7 and A-10, we obtained:

z C z ~0' 4P. (P &z =0ax,2 ay/ A 2 az /2

a-x" ÷ ay/2 v 2 az/2

Therefore:

Z2dZ'J(PO + y (. +± f Z(pZ4k2)dZ' = 0fiz -X, -2-h/ ay' h

Z2dz/ V).+" P.- + (4k2 2(p.) dz 0(A20)-h'f a' ay /2 f

A-6

0

Value of f Z 2 dz/_h/

From Equation A-19, the integration of Z2 can be written as:

f odz2 _ f cosh2(2k)pz'+h') d,-h/ -h' sinh' (k p h')

f [cosh4kp(z/+h') + 1]dz'2sinh(k ph) -h'

I 1 ---1 sinh 4kl Wz+hJ) +lI

2sinhY(kLgh') [4ki k*pIhj

I [ sinh4ktLh' ]h

2 sinh" (k tL h 1) 2(2 k) p I

sinh 4Itp h' _1 + 2(2 k)ph'4kjpisinh(kjph') 2 2sinh2(2k)jih/I

= sinh2(2k)Ih' [1 + 2(2k) ph'4kjisinhW(k ih') 2 2sinh2(2 k) p h'

or

fZ2dz2 = sinh2(2ic)h [1 + 2(2K)h ]

-h' 2 (2 K) H sinh(r.h) 2 2sinh2(2ic)hI

The group velocity, C., is:

1[+ 2%h ]C

C2 = CI)2 /K 2

A-7

Let

C = 2)-=II+ 2(2x)h ]C(2)9 2 sinh (2 (2 ic) h)

and

[C(2f =(2 Wo)2

(2 i) 2

Therefore:

f Z2dz/ = sinh 2(2 ic) h C(2

fz~z g (A-21)

]1/ 22(2 K) H sinh'S(r h) C(2)

From free surface, bottom condition Equation A-16,

az -46Z = -- , z/=O

az' (p0

az (2k) sinh(2 k) p (z' + h')az' ih( Lh) I/

(2 k)psinh(2 k) p h']

Therefore:

az 4_ Z (2kp) sinh(2 k) IL h' cosh 2 k p hlaz/ sinh 4(k p h') sinh4 (k g h')

(2 k IL) sinh(2 k) IA h/ - 4 8 cosh(2 k) p h'

sinh4 (k I h')

f.

A-8

Therefore:

1 f" 1

sinh4 (k p h) 20 2k p sinh (2 k) p h' - 4 8 cosh(2 k) p h'

Therefore:

1 (f) 2 1

sinhs (k p h') o2 [2 k p sinh(2 k) p hh - 4 8 cosh(2 k) p h']2

or

1 (f') 2

sin (K h) (p 0 2KHsinh(2K)h- 4 H cosh(2ich)jgJ

Therefore:

0f Z /dz sinh 2(2 K) h [C.IC2)]fh 2(2 v) H 22 )J2

hi22)HKrKHsinh(2 Y) h 4 2 H cosh(2 .h)

I. g

24P0

where:

[ " " = inh2(2 ) h(f -)2 Ca)/C(2)2(2x)H 2 h]2

(A-22)2 ZxHsinh(2K)h- 4w2H cosh(2xh)

A-9

Therefore, Laplace Equation A-20 becomes:

I- !fL. - .L2 + 4k 2q0o V 0(p2 aX2 ay2 2

0 vp0

Since , * 0,

a-'V-x + -) y F"±y) + 4 2 F'* = 0 (A-23)

Equation A-23 is the reduced diffraction-refraction equation.

A-10

DI•TRIBUTION LIST

ARMY / HQDA (DAEN-ZCM), WASHINGTON, DCCNA / TECH LIB, ALEXANDRIA, VACNO / DCNO, LOGS, OP-0424C, WASHINGTON, DCCOMNAVBEACHGRU TWO / CO, NORFOLK, VANAVFACENGCOM / CO, ALEXANDRIA, VANAVFACENGCOM / CODE 03T, ALEXANDRIA, VANAVFACENGCOM / CODE 04A3C, ALEXANDRIA, VANAVFACENGCOM / CODE 06, ALEXANDRIA, VANSWC / CODE 1235, BETHESDA, MDONT / CODE 226, ARLINGTON, VA

DISTRIBUTION QUESTIONNAIRE

The Naval Civil Engineering Laboratory is revising Its primary distribution lists.

SUBJECT CATEGORIES

1 SHORE FACILITIES 3D Alternate energy source (geothermal power, photovoflac1A Construction methods and materials (including corrosion power systems, solar systems, wind systems, energy

control, coatings) storage systems)18 Waterfront structures (maintenance/deterioration control) 3E Site data and systems integration (energy resource data,1 C Utilities (including power conditioning) integratn energy systems)10 Explosives safety 3F EMCS design1 E Aviation Engineering Test Facilities 4 ENVIRONMENTAL PROTECTION1F Fire prevention and control 4A Solid waste managementIG Antenna technology 48 Hazawous/toxlo materials management1 H Structural analysis and design (including numerical and 4C Waterwae management and sanitary engineenng

computer techniques) 4D 011 pollution removal and recoveryi. Protective construction (including hardened shelters, shock 4E Air pollution

and vibration studies) 4F Noise abatement1K Soil/rock mechanics 5 OCEAN ENGINEERINGIL Airlields and pavements SA Seafioor soils and foundations1 M Physical security 5B Seafloor construction systems and operations (including2 ADVANCED BASE AND AMPHIBIOUS FACILITIES diver and manipulator tools)2A Base facilities (including shelters, power generation, water 5C Undersea structures and materals

supplies) 50 Anchaors and moorngs28 Expedient roadulairfields/bridges 5E Undersea power systems, electromechanical cables, anc2C Over-the-beach operations (including breakwaters, wave connectors

forces) 5F Pressure vessel taciftles20 POL storage, transfer, and distribution 5G Physical environment (including site surveying)2E Polar engineering 5H Ocean-based concrete structures3 ENERGY/POWER GENERATION 5J Hyperbaric chambers3A Thermal conservation (thermal engineering of buildings, 5K Undersea cable dynamics

HVAC systems, energy loss measurement, power ARMY FEAPgeneration) BOG Shore Facilities

38 Controls and electrical conservation (electrical systems, NRG Energyenergy monitoring and control systems) ENV Environmental/Natural Responses

3C Fuel flexibility (liquid fuels, coal utilization, energy from solid MGT Managementwaste) PRR Pavements/Raikoads

TYPES OF DOCUMENTS

D - Techdata Sheets: R - Technical Reports and Technical Notes: G - NCEL Guides and Abstracts; I - Index to TMS: U - User

Guides; 0 None - remove my name

Old Address: New Address:

Telephone No.: Telephone No.:

INSTRUCTIONS

The Naval Civil Engineering Laboratory has revised its primary distriution lists. To help us verityour records and update our data base, please do the following:

"* Add -circle number on list

"* Remove my name from all your lists -check box on list

"* Change my address - line out incorrect line and write in correction(DO NOT REMOVE LABEL).

Number of copies should be entered after the title of the subject categoriesyou select.

Are we sending you the correct type of document? If not, drce the type(s) ofdocument(s) you want to receive Wsted on the back of this card.

Fold on line, staple, and drop in mail.

DEPARTMENT OF THE NAVYNaval Civil Engineering Laboratory560 Laboratory OrivePort Hueneme CA 93043-4328

t NO POSTAGEOfficial Business NECESSARYPenalty for Private Use, S300 IF MAIUED

BUSINESS REPLY CARD ___SAE

FIRST CLASS PERMIT NO. 12503 WASH D.C.

POSTAGE WILL BE PAID BY ADDRESSEE

COMMANDING OFFICERCODE L34560 LABORATORY DRIVENAVAL CIVIL ENGINEERING LABORATORYPORT HUENEME CA 93043-4328

NCEL DOCUMENT EVALUATION

You are number one with us; how do we rate with you?

We at NCEL want to provide you our customer the best possible reports but we need your help. Therefore, I ask youto please take the time from your busy schedule to fill out this questionnaire. Your response will assist us in providingthe best reports possible for our users. I wish to thank you in advance for your assistance. I assure you that theinformation you provide will help us to be more responsive to your future needs.

R. N. STORER, Ph.D. P.E.Technical Director

DOCUMENT NO. TITLE OF DOCUMENT:

Date: Respondent Organization:

Name: Activity Code.Phone: Grade/Rank:

Category (please check):

Sponsor _ User Proponent - Other (Specify)

Please answer on your behalf only; not on your organization's. Please check (use an X) only the block that most closelydescribes your attitude or feeling toward that statement:

SA Strongly Agree A Agree 0 Neutral D Disagree SD Strongly Disagree

SA A N D SD SA A N D SD

1. The technical quality of the report () () () () () 6. The conclusions and recommenda- () () () () ()is comparable to most of my other tions are clear and directly sup-sources of technical information, ported by the contents of the

report.2. The report will make significant () ( ) () ()

improvements in the cost and or 7. The graphics, tables, and photo- () () () ( )performance of my operation. graphs are well done.

3. The report acknowledges related 0 0 0 0 0work accomplished by others. Do you wish to continue getting r r IN

4. The report is well formatted. NCELreports? YES NO

Please add any comments (e.g.. in what ways can we5. The report is clearly written. () () () () () improve the quality of our reports?) on the back of this

form.

Comments:

Fold on line, staple, and drop in madl.

DEPARTMENT OF THE NAVYNaval CivuI Eng0mesnng Laboratory560 Laboratory Drive

Port Huenem CA 93043-4328

Offa Busness POSTAGEO~al usummNECESSARY

Penalty for Private Use. S300 | MAX"1 I NmTHEI

BUSINESS REPLY CARD UIE TJ

FIRST CLASS PERMIT NO. 12503 WASH D.C.

POSTAGE WILL BE PAID BY ADDRESSEE

COMMANDING OFFICERCODE L03560 LABORATORY DRIVENAVAL CIVIL ENGINEERING LABORATORYPORT HUENEME CA 93043-4328