Coastal Engineering, 12 (1988) 133-156 133 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

Combined Refract ion-Di f fract ion of Short-Waves in Large Coastal Regions

V.G. PANCHANG 1, B. CUSHMAN-ROISIN ~ and B.R. PEARCE 1

I Department of Civil Engineering, The University of Maine, Orono, ME (U.S.A.) ~Departrnent of Oceanography, Florida State University, Tallahassee, FL (U.S.A.)

(Received April 13, 1987; accepted December 8, 1987)

ABSTRACT

Panchang, V.G., Cushman-Roisin, B. and Pearce, B.R., 1988. Combined refraction-diffraction of short-waves in large coastal regions. Coastal Eng., 12: 133-156.

A method for solving the combined refraction-diffraction equation in large domains is de- scribed. This equation is modified to the reduced wave equation, and the elliptic, boundary-value problem is solved by the marching or "Error Vector Propagation" method. The solution method is direct, and eliminates the computer storage problems associated with large matrices obtained in standard methods. This efficiency is obtained at the cost of resolution of the shorter wave components in a direction normal to the incident wave direction. As such, the method is limited by the paraxial approximation encountered in the parabolic equation method of solving the ocean- wave refraction-diffraction problem. But it overcomes the other limitation of the parabolic ap- proximation, in that it allows backscattering and propagation in the - x direction. Therefore it is possible to accommodate reflecting structures such as seawalls along the downwave boundary of the domain, and its computational convenience allows it to be applied to large coastal regions to study wave refraction and diffraction. The method has been tested for several cases, and some results are presented. The model compares very well with other model solutions and observed data.

1. INTRODUCTION

A significant advance in the field of wave modelling was made by Berkhoff (1972, 1976), who derived the "combined refraction-diffraction" equation. By including the effects of diffraction, this equation eliminates the usual problem encountered in refraction studies (viz. caustics). It can be used for a wide range of ocean wave frequencies, since it passes, in the limit, to the deep and shallow water equations. The usefulness of Berkhoff's equation in yielding good sim- ulations of wave behavior in a wide variety of different situations has been demonstrated by many investigators. Jonsson et al. (1976), Nachbin and Wro- bel (1984), Tsay and Liu (1983), and Houston (1981) have used it to study long wave propagation in the vicinity of islands; Berkhoff, Booy and Radder

0378-3839/88/$03.50 © 1988 Elsevier Science Publishers B.V.

134

( 1982 ) have used it to study short wave propagation over arbitrary variations in bottom topography. Berkhoff (1976) and Rottmann-Sode and Zielke (1984) have used it to investigate wave propagation in harbors. Tsay and Liu (1983) have made avail of it to compute wave forces on floating docks, and Pos and Kilner ( 1987 ) have used it to develop breakwater diffraction diagrams.

In view of these studies, coastal engineering firms are now attempting to implement the combined refraction-diffraction equation to determine wave conditions in coastal areas. However, even with access to a large computer, this is a formidable task, particularly when tackling short-wave propagation in most coastal areas of significant dimensions. The reason is that the refraction-dif- fraction equation is an inseparable elliptic partial differential equation with complex variables, and the solution makes prohibitve demands on computer time and memory.

Consider, for example, the case of Swansea Bay, studied by Heathershaw et al. (1980), on a square domain 25 km × 25 km. Refraction analyses have been performed by them for several wave frequencies and directions, and as ex- pected, there are many regions where caustics appear. The water depths in this region are of the order of 5 meters, and the centre of the frequency spectrum corresponds to a wavelength of about 50 meters. In light of the caustics, if we attempt to solve the combined refraction-diffraction equation for this case, with even as few as five grid points per wavelength, we would have to solve 2500 × 2500 simultaneous equations in complex variables. In the finite-dif- ference method, moreover, the equations are usually not even diagonally dom- inant, thereby requiring a direct solution and the storage of a huge system matrix. Indeed, even if the domain were only 1 km × 1 km, we would still have a very difficult problem.

Copeland (1985) has suggested an alternative to the elliptic stationary re- fraction-diffraction equation by using the time-dependent form, and trans- forming the resulting wave equation into three coupled hyperbolic equations. Initial conditions for the time-integration are not known in this approach, and one has to start from an arbitrary state. For large regions with arbitrarily varyJ ing topography, integration over several wave periods with small time-steps (governed by the Courant condition) is required for model "spin-up". The treatment of open boundary conditions for the time-dependent wave equation is also difficult, and a matter of considerable research (e.g. Engquist and Majda, 1977).

In view of these difficulties, it has become necessary to use the parabolic approximation (Radder, 1979) of the combined refraction-diffraction equa- tion. By this method, a solution is rendered feasible in fairly large open coastal areas, but there are two limitations to this method: ( 1 ) the waves must have a principal propagation direction (say x), since diffraction effects are restricted to the y-direction only, and (2) backscattering and the reflected component of the wave potential in the - x direction should be negligibly small. [In certain

135

cases, Liu and Tsay (1983) have illustrated that it is possible to correct the parabolic approximation to some extent, although efforts to further improve the solution will cause their method to break down (McCacken, 1986) ]. The parabolic approximation is therefore inappropriate when the bathymetry or structures such as harbor walls, breakwaters, etc., reflect the energy in the - x direction. More recently, Ebersole (1985) has suggested a finite-difference so- lution to the elliptic combined refraction-diffraction equation, in a manner suitable for large coastal areas. But this method solves the elliptic equation essentially as an initial value problem (downwave propagation), so that re- flections are again required to be negligible.

The development of approximate methods, such as those discussed above, represent a very useful and important contribution to the coastal engineer, enabling him to quickly compute the general wave propagation characteristics in coastal regions. These approximate solutions are also often adequate, since wave propagation in these areas is usually influenced by dissipation, atmos- pheric input and other factors which are not accounted for even in the complete solution. In this paper, a finite-difference solution of the combined refraction- diffraction equation is given. The complete elliptic boundary value problem is solved following the Error Vector Propagation (EVP) or marching method (Roache, 1978a). The EVP method of solving elliptic problems has a very appealing solution algorithm, which does not require the storage of the system matrix. In fact, its demands on computer memory are extremely modest. How- ever, the EVP method is usually characterized by an inherent instability {in the sense of Hadamard), which renders it useless for most elliptic problems with even as few as 75 rows in the "march direction". It is shown in this paper that it is possible to eliminate this instability for the reduced wave equation under certain criteria. Satisfaction of these stability criteria permits a solution of the combined refraction-diffraction equation that is approximate, in that the method is still restricted by the first assumption of the parabolic equation method given in the preceding paragraph. But it is not limited by the second assumption, and as such allows backscattering and the presence of coastlines, harbor walls or other structures at the downwave end of the domain.

Section 2 of this paper gives a formulation of the problem and a description of the marching method as applied to this problem. In Section 3, the difficulties characteristic to this method are pointed out, and criteria are established to overcome them by performing a stability analysis. A discussion of these criteria is also given. The solutions obtained with the marching method are compared with laboratory data and finite-element and standard finite-difference solu- tions in Section 4. Section 5 summarizes the advantages and limitations of this method.

2. PROBLEM STATEMENT AND METHOD OF SOLUTION

The combined refraction-diffraction equation, derived by Berkhoff (1972, 1976) and by Smith and Sprinks (1975), to describe the propagation of peri-

136

odic, small-amplitude, surface gravity waves over an arbitrarily varying, mild- sloped sea-bed is:

V. ( CC~ VcI) ) + o)~cI)=O (2.1)

where O(x,y) is the complex wave potential function, which gives a measure of the wave height, o) the frequency ( = 2n/T) under consideration, C (x,y) the phase velocity ( = w / k ) , Cg(x,y) the group velocity (=Ow/c)k), k(x,y) the wavenumber, related to the depth h (x,y) through the dispersion relation

(oe = gk tanh ( kh ) . (2.2)

Instead of working with eqn. (2.1), we work with the reduced wave equation,

v~O+ K~(x,y)O=O (2.3)

obtained from eqn. (2.1) through the transformation suggested by Radder (1979):

O=¢(CCg) °'5 andK2=k 2 V2(CC,) °'~ ( CC, )o.~ (2.4)

A rectangular domain is chosen (the method can be extended to non-rectan- gular regions also; see Roache (1978a), although that case is not considered here), and the coordinate axes, incoming wave direction, etc., are shown in Fig. 1.

Boundary conditions have to be imposed along AB, BD, CD and AC, and mostly these are of the Robbins type. Along AB, the incident wave is ¢i ~-~ Aiexp (iKx), and Ai is specified as 1. There exists also a backscattered com- ponent, approximated in this study by 0r =B [exp ( - iKx)]; for more details, see Booij ( 1983 ). Since B is not necessarily known, we can differentiate,

0_~¢= ig(oi - - Or ) = iK(oi - (O- Oi ) ) Ox

or, since Oi-- 1 at x = O,

0~= iK(2 - O) (2.5) Ox

Equation (2.5) can be one form of the boundary condition along AB. Alter- natively, the wavemaker condition

00 Ox = a constant ( 2.6 )

can be used, e.g. see Berkhoff, Booij and Radder (1982). If wave conditions

137

I~

p = N+I

AX ~ A y ~ +

• . y

A p=1 B

INCIDENT WAVE

Fig. 1. Model grid.

along AB are known or calculated using some other coarser model, the Diri- chlet condition may be used. Along CD, if a vertical wall exists that perfectly reflects waves, then the no flux condition can be imposed. If CD is an open boundary, along which there are only outgoing waves, the following condition can be used:

~x- iKO=O (2.7)

If a coastline is present along the boundary CD, the mixed boundary condition

a¢ --~x ÷ af~ = O (2.8)

where a is a complex reflection coefficient, can be imposed. Similar mixed or full reflection or radiating boundary conditions are used for the lateral bound- aries BD and AC (e.g. see Dalrymple et al. (1984); Kirby (1986a)).

138

We now give a description of the EVP or marching method, as applied to eqn. (2.3). The domain is discretized into ( N × M ) cells of size ~x and Ay (x= (p--1)Ax and y= (q-1)z~y). For demonstration, we use eqns. (2.5) and (2.7) on the lower and upper boundaries, and full reflection on the lateral boundaries, i.e.

°0_0, ~yy- a longBD a n d A C (2.9)

Equation (2.3) is writ ten in finite-difference form as:

0 ( p - 1,q) - 20(p,q) + ¢ ( p + 1,q) (/~X) 2

(2.10)

f)(p,q- i ) - 2~ (p,q) + O(p,q+ 1 ) ( ~y ) 2 ~- K2 (p,q ) ~ (p,q ) = 0

An initial (arbitrary) estimate ¢~' (2,q) is first made for the values of ¢ along row p = 2. Obviously these are in error, by an amount given by the error vector e(2,q), i.e.

¢(2,q) = ~ ' (2,q) +e(2 ,q) for 2<=q<=M (2.11)

Estimates along row p = 1 now can be obtained from the finite-difference rep- resentat ion of the boundary condition, eqn. (2.5):

¢ ' (1,q) = (~' (2,q) - 2 i K A x ) / ( 1 - i K ~ x ) , for 2<=q<=M (2.12)

Est imates ~' can now be found for the entire domain by marching the solution up from p = 2 to p = N in the following equation:

¢~' ( p + l , q ) = [2--Ke(p,q)z~x2+2{z~x/Ay)2]~ ' (p,q)

--(Ax/Ay)2[O'(p,q--1)+(~'(p,q+l)]--¢) ' (p-- l ,q) (2.13)

Equation (2.13), which gives estimates 0' for rows p = 3 to p - - N + 1, can be applied for q = 2 to M. The finite-difference representation of the full reflection condition on the lateral boundaries, i.e.,

¢} ' (p,1) = ~ ' (p,2) and ~ ' ( p , M + l ) = ~ ' (p,M) (2.14)

give boundary estimates during the march. When ~' has been computed for all rows, it is obvious that the upper bound-

ary condition, eqn. (2.7), will not be satisfied by the ¢~' values. Equation (2.7) may be writ ten as

~(N+ l,q)-f)(N,q) iK(N,q) f) (N,q) = 0 for all q. (2.15)

/~x

Or, since @--@' + e

139

~ ' ( N + 1,q)-~) ' (N,q) iK(N,q)# (N,q) ~x

=--[ e(N+ l'q)-e(N'q)Ax iK(N,q)e(N,q) ] or A (q)=S(q) (2.16)

where A and B are two vectors, each containing M - 1 elements (2 <-q<=M). B (q) is the amount by which the boundary condition at CD is violated. Since the vector A can be determined from the computed ¢~' values, the vector B , containing information about the errors at the top of the grid, is known. This information about the errors at the top of the grid can be correlated (as shown below ) to the original error vector e (2,q) = E (q), say, where E (q) also contains M - 1 elements ( 2 __< q __< M). ( It is not necessary to include the boundary values at q= 1 and q=M+ 1 in the vectors A, B and E, since these are determined by the lateral boundary conditions. )

The error vector propagation equation is obtained by subtracting eqn. ( 2.13 ) from eqn. (2.10):

e(p+ l,q) = [2--K2(p,q)Ax2 + 2(Ax/Ay)2]e(p,q) --(Ax/Ay)2[e(p,q--1)+e(p,q+l)]--e(p--l,q) (2.17)

0e The lateral boundary conditions for the above equation are ~yy = 0, in view of

eqns. (2.9) and (2.14). Along AB, subtracting eqn. (2.12) from the finite-dif- ference analog of eqn. (2.5) gives the following boundary condition:

e(1,q) =e(2,q)/(1-iKAx) (2.18)

The error B (q) at the top of the grid, for each q, consists of a linear combi- nation of contributions to it from each initial error along row 2. This relation between the vectors B and E may be expressed as:

[BI = [Cl [El (2.19)

where [C] is the correlation matrix. It can be determined, independently of previous computations, using the error propagation eqn. (2.17) with eqn. (2.18), following the procedure described in Roache (1978a). Having computed [C], the actual vector of errors A (q) is substituted for B in the left hand side of eqn. (2.19), which is then inverted to give the initial error vector E (q), consisting of e(2,q). These are added to the estimates ¢)' (2,q) to give the solution along row 2. Values along row I can be calculated from the finite difference represen- tation of the boundary condition along AB. A second sweep of the grid with eqn. (2.10) yields the solution ¢~.

It is thus seen that the marching method is easy to implement, and gives a direct, non-iterative solution. All problems associated with computer storage

140

are completely eliminated, since the propagation equations, eqns. (2.10) and (2.17), involve only three rows at a time. Only the information at the bottom and top of the grid, i.e. vectors ~' ( 2,q ) and A (q), have to be stored, in addition to the ( M - 1) × ( M - 1 ) matrix C. Since the propagation equations for O', O and e are the same, only one variable name suffices for all three quantities in the solution algorithm. The inversion of eqn. (2.19) also usually presents no problems and standard IMSL routines are adequate.

3. STABILITY ANALYSIS

While the marching method of solving the refraction-diffraction equation, described in the preceding section, appears to have many advantages, the march equations, such as eqns. (2.10) or (2.17), are usually unconditionally unstable for most elliptic problems. Roache (1978a) shows how a Poisson system re- sponds to a unit forcing applied at the centre of row 2. This applied error grows to the maximum allowable number on the computer in only about 32 rows, for Ax/,~y = 1.

For the reduced wave equation, however, it is possible to stabilize the march equation, eqn. (2.10) or (2.17). Here a v o n Neumann stability analysis, e.g. Smith (1978), is performed in a manner rather similar to that for the explicit solution of a parabolic equation. Consider the equation

/ 2 ~ + K 2 g t = 0 (3.1)

which has a solution

g~ ~ exp ( imx)exp (iny) (3.2)

(for constant K), such that

m 2 + n e = K e . (3.3)

The solution (3.2) can be written as:

¢(p,q) = e x p ( i m ( p - 1 ) A x ) e x p ( i n ( q - 1 ) A y ) = o p e x p ( i n ( q - 1 ) z l y ) (3.4)

where

Cp = exp (im (p - 1 )Ax).

Substituting eqn. (3.4) in eqn. (2.10),

~p+~--2~v + ~ _ ~ ~_~I2 cos nay -2 - ] __~ _ 2 j+K

Using the amplification factor

G.~)p+ i/~)p in eqn. (3.6) gives

(3.5)

(3.6)

(3.7)

141

G~+BG+ 1 =0 (3.8)

where B=K~Ax ~ - (2(Ax/Ay) sin a ) e - 2 , and a=nAy/2 . When the amplifi- cation factor satisfies a quadratic equation such as eqn. (3.8), it is convenient to invoke the formulas derived in Cushman-Roisin (1984) for investigating the stability. For eqn. (3.8), these formulas [ Cushman-Roisin ( 1984 ), p. 237 ] yield the following condition for stability:

( /~x ) ~ or, -2<_K~/~x~-4 sin~a - 2 < 2

- ( ~ y ) 2 -

The right inequality is always satisfied if KAx < 2, or

Ax <= L/7~ (where L = 27r/K).

The left inequality is always satisfied if

(3.9)

If the marching method described in the preceding section is used with the stability criteria, eqns. (3.9) and (3.10), a stable scheme results, and the in- herent instability described earlier is removed. While the condition (3.9) is similar to commonly obtained stability criteria for many partial differential equations, the condition (3.10) is unusual, and it has both advantages and disadvantages. Most important, it explains why the marching method can be used for the reduced wave equation, but is inherently unstable for most other commonly occurring elliptic problems. For instance, for the Poisson or Laplace equations, where K2=O, or L - ~ c , it is obvious, from eqn. (3.10), that any finite grid size Ay will result in an unstable scheme. In addition to the advan- tage of making the marching method usable for the refraction-diffraction equation, the condition (3.10) also reduces the number of computational points required. A further advantage of the condition (3.10) is that it results in the matrix [C] being well-posed even if N is of the order of a few thousand. This is because the components of initial row of errors E (q), when computing [C l, have sufficient separation, so that the resulting error vectors B (q) at the top of the grid are sufficiently different from each other. (In other words, adjacent columns of [C] can distinguish between errors located at adjacent points on row 2.) In the absence of such separation, Roache (1978a) has reported that N > 100 results in the matrix C being ill-posed.

There are two disadvantages due to eqn. (3.10) as well: (1) the grids being spaced L/Tr meters apart, information about depth variations between grid col- umns, and hence K2(x,y), is not well represented in the model, and (2) the wave components in the y-direction are not well resolved. The first disadvan-

Ay>__L/7~ (3.10)

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tage is not a major drawback, as seen in one example in Section 4, since the refraction-diffraction equation is applicable to coastal regions of "mild slope", and we are dealing with short wave propagation in these areas. As such it is not expected that a small, rapid and local variation of the ba thymetry will influence the wave pat tern significantly. Moreover, if the real bot tom topog- raphy is chaotic, it is usually subjected to a nominal amount of smoothing, prior to being input to a wave model. This should diminish the impact of the first disadvantage.

The second disadvantage is one that makes the marching solution approxi- mate, in that, as in the parabolic approximation, the dominant wave direction is still x. If we substi tute exp (im3x) for the amplification factor (using eqns. (3.5) and (3.7) in eqn. (3.8), we have the following relation for m and n in the numerical solution:

• 4 s~n ~ - - j sin 2 (g.ll) (~x) ~ + ~

We note that as Ax~O and A y e 0 , eqn. (8.11) simply reduces to the analytical relation given by eqn. (8.8) (plotted in ~ig. 2(a) as a circle). However the stability criterion (&10) does not permit Ay<2/K. If we therefore let Ay be equal to its minimum value, 2/K, we have

K~+ sin ~ = 1 (&12)

Equation (&12) is also plotted in ~ig. 2(a) , from which one can deduce the effect of the condition (g.10). At large angles from the 2 x axis, eqn. (~.12) shows the greatest deviation from the analytical relation (~.g). When the wave vector is directed entirely along the y-axis (i.e. when m ~ = 0 ) , n/h is a maxi- mum, and consequently, the Fourier wave len~hs in the y-direction are the shortest (L and 2L/~, respectively, for the analytical and numerical cases, an error of about g6% ). These short waves are poorly resolved, due to the condi- tion (g.10). The situation improves as the wave vector approaches the 2 x directions (i.e., m~> 0), when smaller n/h values yield y-components which are longer, and hence well represented. Equation (~.12) shows little departure from the analytical relation g.g for a range of directions 2 gg ~ from the 2 x- axis. (The maximum error is less than 9 ~ at 0=~5 ~ ). Thus, good solutions are obtained if the wave propagation indicates an x-directional preference.

The above situation is similar to that encountered in the parabolic approx- imation approach to combined refraction-diffraction. F i b r e 2 (b) shows the relations between the wave vector components for two parabolic equations, discussed in Kirby (1986b), which are used to approximate the Helmholtz equation. As indicated by Kirby (1986b), the parabolic equation method is a good approximation for n/k << 1, and the error (prior to numerical discretiza-

(+x)

\ ~\\ iim / K // / . . . .

f r o m ~ ~ / ~ \ ~', / / ~ vertical//~,. \~ ",,, // / ' "

< i / I

55° [ ¸ ' ¸ ' ¸~ from ~ J vertical / / -i /

! ~ ~

(-x)

~ 55 ° froff vertlcal /

/E~. ~.~ ~Eq. I. 2

55 ° from vertical

143

{+x)

I~/I.

~ - ~ - - !!____~

/ / ~ /

~ /

/ / ~ /

/ , / / /' /

~ / - ~ ~ ~ ~ Z ~ -

/ / ~ ~

/ ~, ~ ~

/ / '~k a / ~

/ / ~x.~ ~ ~

~ ~ ~ . . . . ~__ ~

. / ~ / /

" \ [~< , /~ /<< Eq. 3 . 3 (aria y t i ca ] )

"" , Parabolic Approximations , "( < / ~i~b,,, ~ ~ , ~

,~ ,, ~ ~ ~ ~, '~ x

, ' n 1(

2 :' ~ / ',

/ ', / ~x, 'x

/ ', / ,~ k

~ , x x

l (-x}

Fig. 2 (a) . Comparison of analytical relation between wave vectors wi th the relation in numerical ( E V P ) model.

Fig. 2 (b). Comparison of analytical relation between wave vectors with the relations in two par- abolic models.

144

tion) is 5% at 0 = +42.6 ° from the + x direction for the lower-order approxi- mation, and at 0= +55.9 ° from the + x direction for the higher-order approximation. The error is 10% at 0 = + 49.3 ° and at 0 = + 61.5 ° from the + x direction for the two approximations. Dalrymple et al. (1984b) also indicate 101 < 45 ° to be appropriate for the parabolic equation method. The EVP model is thus seen to exhibit a range of validity similar to that of the parabolic ap- proxmiation in the + x direction. In the - x direction, however, the parabolic approximation breaks down, Fig. 2 (b), since reflections are ignored. From Fig- ure 2(a) we see that the EVP solution overcomes this limitation, i.e., back- scattering and propagation in the negative x direction are included. It is thus possible to account for the presence of sea walls, coastlines, etc. along the boundary CD (Fig. 1 ) of the solution domain.

4. NUMERICAL VERIFICATION

The model described in the preceding sections was used to simulate wave propagation in intermediate depths of water. Three configurations of bot tom topography were considered. The first is a paraboloidal shoal (shown in Fig. 3 ), surrounded by a region of constant water depths. The conventional refrac- tion analysis for this shoal results in an abundance of caustics, as demon- strated by Ito and Tanimoto ( 1972 ), who also collected data by simulating this

0

s ~

0 I 2 3 4 5 y /L~

f ~

/ \ ,- /

~ ! / J

~ incoming WOV(~

Fig. 3. Bottom topography for first example (after Ito and Tanimoto, 1972 ).

145

3-

• ~ 2 "~ ~r_ . . ~

a Y_ =3

L i

~ ,,INCID~/f WAVE

~ SHOAL ~ 1 ' ' ~ ' ~

2 3 4 5 6 7 8

x/ L i

2

1

b _x = 3

L i

~ • ~ • ~ • • •

I I I J i ~

1 2 3 4 5 6

y l L ~

0

0

c x = 2

ki • • • • • Lab Data ( Re f. 15 )

~ O . . . . . Model Results

• • • • • • •

I ~ I I I ~

i 2 3 4 5 6

Y/h i

Fig. 4 (a-c). Wave height comparisons for first example.

case in a hydraulic model. The incident wavelength here is 40 m, and Ay was chosen to be 14 m, in accordance with the stability condition (3.10). Wave heights computed with ~Ix = 1 m are compared with the hydraulic model data of Ito and Tanimoto (1972) in Fig. 4 (a -c ) . It is seen that the model simulates the observed data very well. Several runs were made for different values of ~Ix, and Fig. 5 shows a comparison of the numerical results along x / L = 3 for Ax = 1 m, ~Ix = 2 m and ~Ix = 4 m ( i.e. ~Ix = L / 40, L / 20, L /10 ). It appears that the smaller grid size results in smoother solution; however the solution with the larger grid

146

I

,*~X = I :11

.-"'. . AX = 2 m ...o .-2.. ° x--'~. /.'<

.. ........................ ~x = 4 i~l

<.,

~ ~ ,:o.. . ~. ;'2. ;. : -" ". .' "~ ." • " ~ ."" ,:'" ." ~. : x .: "

2 3 4 5 6 7 8

x/ L i

Fig. 5. Solutions along axis of symmetry for different grid sizes.

5,8 CM, 1

t- ~R: ~o c~, - - - ~

30 c~.

a b c d f g h i '~t~ 1 2 ~ :

' t No ~ ~ , 'S

/ / . ~ ~ ~

~ I -

d - - ~ - -

l INCIDENT WAVE DIRECTION

PERIOD = 0,63 SEC,

i k

Fig. 6. Bottom topography for second example (after Williams et al., 1980).

147

size is still an excellent indicator of the final solution. [Note: A reviewer rightly suggested that the amplitude modulations in Fig. 5 can be reduced by repre- senting the boundary condition 2.7 by second-order accurate finite-differ- ences, instead of the first-order eqn. (2.15).] Similar behavior was observed in the other tests also (discussed below). Results along the other two sections with larger Ax showed almost no difference from the solutions shown in Figs. 4 (a) and 4 (b). Choosing very small values for Ax, of course, presents no diffi- culty in the marching scheme, since one deals with only three rows at a given time, and moreover, there are relatively few computing points in the y-direction.

In the second example, the bathymetry was again represented by a circular shoal of radius R = 60 cm, shown in Fig. 6. The local wavelength on the shoal at a distance r from the center is given by:

r L 0.3~+0.7

where Li = 60 cm is the incident wavelength in deep water. Laboratory data for wave propagation over this shoal may be found in Williams et al. (1980). This example is interesting for two reasons. First, the shoal represents an isolated scatterer with a radius as small as the wavelength (R= 2Li in the previous example), and therefore, with Ay = 20 cm, contains extremely few grid points in the y-direction. The wavelengths change more rapidly than in the previous example (AL /R = ( L i - Lmin)/R : 0.3 in this case, A L / R = O. 125 in the first ex- ample). This rapidly changing information about K is thus represented by a maximum of only 7 grid points (along section 5, Fig. 6). We can thus see how this affects the quality of the EVP model results. Secondly, finite-difference solutions of eqn. (2.1) by conventional matrix methods are available for com- parison (Williams et al., 1980). The results along several sections are shown in Fig. 7. (The EVP model results are connected by straight lines, in all ex- amples. No smoothing is applied. ) It is seen that the model compares very well with observed data. There is a slight overestimation of the maximum wave- height in the model. The data indicates a maximum H / H o = 1.7 (at the inter- section of sections a and 8A) whereas the corresponding model value is 1.89 (at the intersection of sections a and 10). Overall, the comparison with data and the results of Williams et al. (1980) is good, and, moreover, the results of the present study appear to show a better correlation with the observations than the solutions of Williams et al. (1980), which consistently underesti- mated the wave heights.

The third bottom configuration, representing more complicated bathyme- try, is taken from Berkhoff, Booy and Radder (1982). The bathymetry, shown in Fig. 8, consists of a shoal situated on a bottom sloping at an angle to the incident wave direction. Berkhoff, Booy and Radder (1982) simulated wave propagation over this bathymetry in a hydraulic model. In addition, they con-

148

H

H o

2 ] G • EVP Model

| ~ -- o-- -- -- Finite-Differences (Ref. 32)

- - , - - Laboratory Data (Ref. 32)

i { : g ~ . ~ - .... - -- J ®

SECTION Q

b

H° I - ----o --o~ - "

2- C

~H 1 H o ~ ~ o _.-o .... o

\\ .~o

o_.o SECTION 8

0 o ~ ~' Ig ~I i

Ho

2 Cl = EVP Model

~ ---~- -- Finite-Differences (Ref. 32)

.~.~\ ------- Laboratory Data (Ref. 32)

%\

~ ~ ~ ~ •

o vl o .... o .... o

~ / z ~v o ~

"o" SECTION 9

Fig. 7 (a-h). Wave height comparisons for second example.

Ho

e

\ SECTION 11

~- "~v °,\ "~,',. /7" . -~ '- '~ ~ ~.~ ~ • ~ ~ " ~ / ~

~

149

1

Ho

f

~ SECTION 1_3

" "0,~ ~,0.,

• ~.0

-";~~ ~.~2..~ ~

~ I I

~ ; ~ ~ ', i ~

:~ g

0 0 0 0 • • • •

H

go

SECTION 8A

o,~ ~, ~ ; ~; ,~ ,~

EVP Model

Fin~te-Differences (Ref. 32)

Laboratory Data (Re[. 32)

H__ ] Ho

2- h

SECTION a

~ ~ ~ ~ ~ ~ ~ IB I' ]l~ ~

150

l o - h Oz/ } T = 1 SEC ~ .'3" ~

~ ~ - - ~ ~ = __~_~_~-~ :'~ ~ • - o--z ~ - ~ L ~ ' ~ - - i ~ ~ -~-, , ~ - L ~ ~

~ ~ ~ _ ~ c c ' ~

. ~ ~__~___~

-,o~ ~ ~ ~ ~

-10 -5 0 5 10 x i=z m

4 ~x' in meters. ~,y,~o~y~ ~ ~

" ~ o I~ %',~ k ~ ,

outside shoal ~ . 3 + 0 5j 1- ( ~ ) - ( ~

~ ~ ~.x.~x ~.x ~ x x.x ~.xh,.x x x ~.x~ ~ ~

-3 0 3 ~ y'inm

h=O.45m outside slope

h=O.4~m ~

t Fig. 8. Bottom topography for third example (after Berkhoff et al., 1982).

structed a finite-element model for eqn. (2.1). However, for their numerical calculations, the domain, which was approximately 16Li × 13Li (Li = incident wavelength) was reduced to approximately 13Li × 5Li, because the solution re- quired too many grid points. The model described here cannot, of course, be expected to be as accurate as the finite-element solution of eqn. (2.1), in view of the limitation discussed in the preceding section; but it can easily solve a much larger domain. Solutions along Sections 1-8 (Fig. 8), obtained with the marching method, are compared to the finite-element solution and hydraulic model data of Berkhoff, Booy and Radder (1982) in Figs. 9 (a-h) . All the es- sential features of the observed wave pattern are reproduced by the EVP model. The results of the model described by Ebersole ( 1985 ), which treats eqn. (2.1) as an initial value problem, are also shown in Fig. 9. It is seen that the EVP model results, when compared to data, show the same order of error as not only

1 -

O ' - 5

(] EVP Model

Ebersole model (Re ~. 19)

.............................. F E M ( R e f . B)

• • • • • Data {Ref. 3)

SECTION !

~ ;

151

2 -

0

- 5

b

, . . ~ ° ° ' • ~ ~ ~ •

b ~ X

2 - SECTmN 3 Z~: ~: ~ ~-: ~\

0 -~ (~ ~ X

d • EVP Model

/~. model {Ref. i0) 2" S[C'[[0~ 4 ~'/ ~ ................... EbersoleFEM (Ref. 3)

i/ I"~'~ ° ° ° ° Data (Ref" 3) I• : .,,%

. -

.~o -s 6 g

x

~ig. 0 (fl-h). W~ve heighg comparisons for third example.

152

~: 2.] e

"~ 1. .,~

0

/ ' ~

SECTION 5 / ~ :'.:" ..~

•

~ ~ x

, -

2 .

f

SECTION 6

~ • •

o ~ 6 Y

N

2 -

/... ~--- .. , . . . ~ . ~ .

~ - ~ ~

SECTION 7 o ~ 6

O

0

EVP Hode 1

h Eberso]e Model (Ref. 1O)

SEC/IC~ 8 ......................... win. {P, ef. 3)

• • • • • Data {Ref. 3)

~ • ......-....... . . . . :'... ~ . . . ./ . . . . .. ~,...~... ....

.. ...-'" ~ • •

~ 6 Y

153

other aproximate methods (Ebersole, 1985 ), but also as the complete solution (Berkhoff et al., 1982). The results imply that the condition (3.10) is not too severe a limitation for short wave propagation.

The test program, in which no attempt was made to minimize computer time, was able to solve a 100 × 250 grid (approximately 30Li × 12Li ) in 13 sec- onds on the CYBER 205 at Florida State University. The solution time in- creased roughly linearly as N increased. A more efficient version of the computer program is available and may be obtained from the authors. It is noted (with no intention to draw a comparison) that Liu and Tsay (1983) computed wave refraction-diffraction for a semicircular domain of radius Li in 25 seconds on an IBM 370/168 using the hybrid finite-element method; their other, iterative scheme to correct the parabolic approximation (assuming a weak interaction between the forward and backward wavefields) required 19 seconds on the same machine for a region approximately 2L~ × 6Li.

5. SUMMARY AND CONCLUDING REMARKS

The derivation of the combined refraction-diffraction equation, eqn. (2.1), by Berkhoff (1972, 1976) and by Smith and Sprinks (1975) has been shown by many investigators to be a significant advance in coastal engineering stud- ies, since it overcomes the limitations of conventional refraction analyses, most notably the occurrence of caustics, and also a paucity of wave rays in regions known to have appreciable wave energy (Booij, 1981 ). For short wave propa- gation the various methods of solving this equation may be summarized as follows: ( 1 ) Solution of eqn. (2.1), with no physical assumptions, using, say, the finite-

difference method (Williams et al., 1980) or the finite element method (e.g. Berkhoff (1976), Houston (1980), Rottmann-Sode and Zielke ( 1984); see also Mei (1978)). However, large areas are not amenable to these methods; Berkhoff, Booy and Radder (1982) limit the domain to approx- imately nL 2, where n ~ 0 ( 10 ).

(2) The parabolic approximation method, due to Radder (1979) and Tsay and Liu ( 1982 ). This method can be applied to much larger areas; however the method assumes that the main direction of wave propagation is x, and it cannot allow reflections by structures and backscattering in the - x direction.

(3) The method of Ebersole (1985), where eqn. (2.1) is essentially solved as an initial value problem. It is possible to apply the method to large areas, and the unidirectionality requirement of the parabolic approximation is removed. However this method also does not allow reflections.

In this paper the refraction-diffraction equation is solved using the march- ing method. The basic marching method was described in Section 2. For vari- ations of this scheme, see Roache (1978b). While this method is usually

154

inherently unstable for most elliptic problems, it is shown that it can be applied to the reduced wave equation, if certain stability criteria are met.

It can be said that the EVP method presented here reflects a compromise between lateral resolution (Ay sufficiently small) and computational stability (z]y sufficiently large). This situation is reminiscent of several other problems in physics, especially the inverse problem. There also, a compromise is to be reached between resolution and numerical stability: on one side, one wishes to cover the domain with a grid as fine as possible, but on the other side, such refinement provokes redundancy of information among adjacent rows and re- sults in ill-conditioned matrices. This is why inverse methods require that the grid size not be smaller than a threshold guaranteeing a well-posed mathemat- ical problem, in analogy with the method developed here.

The EVP method, used in conjunction with the stability conditions derived in Section 3, yields a solution, which requires, in a sense, the opposite of Eber- sole's physical criteria. As in the parabolic approximation, there is a preferred directional dominance, to within approximately _+ 55 ° around the x-axis. How- ever, the boundary value problem is solved, and thus backscattering and re- flections in the - x direction can be accounted for. As such it is possible to have any boundary condition (seawall, reflecting structures) at the downwave end of the domain.

The method is applied to three configurations of bottom topography, and the solutions are compared with hydraulic model data and the results of other models. It is found that the marching method competes well with other models. Moreover the solution is direct, and the algorithm is simple, using only ele- mentary principles of linear algebra. It does not require the storage of large coefficient matrices, and can therefore be used to investigate the wave climate in large areas with arbitrary bathymetry, where ray theory is problematic.

In conclusion, while the usefulness of the combined refraction-diffraction equation has been well established, approximate solutions are necessary to make it suitable for use in reasonably large coastal areas. The parabolic ap- proximation (Radder, 1979; Tsay and Liu, 1982), the method suggested by Ebersole (1985), and the scheme proposed in this paper address the problem of making the refraction-diffraction viable, in different circumstances.

ACKNOWLEDGEMENTS

The authors are grateful to Professor J.J. O'Brien for helpful discussions, and to the reviewers for their suggestions. This research was supported in part by NOAA, Department of Commerce, Grant NA86-D-SG-047, Project R/ EMP8/Maine-New Hampshire Sea-Grant Program (for VGP and BRP), by NASA Grant NAGW - - 985 (for VGP) and by ONR Contract N00014-82-C- 0404 (for BCR).

155

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