Wave transformation by axisymmetric three-dimensional bathymetric anomalies with gradual transitions in depth

Coastal Engineering 52 (2005) 331 – 351 www.elsevier.com/locate/coastaleng

Wave transformation by axisymmetric three-dimensional bathymetric anomalies with gradual transitions in depth Christopher J. Bendera,T, Robert G. Deanb b

a Taylor Engineering, 9000 Cypress Green Dr #200, Jacksonville, FL 32256, United States Department of Civil and Coastal Engineering, University of Florida, 365 Weil Hall, P.O. Box 116590, Gainesville, FL 32611-6590, United States

Received 2 September 2003; received in revised form 10 December 2004; accepted 17 December 2004 Available online 26 February 2005

Abstract The development of an analytic model (Axisymmetric 3-D Step Model) for the propagation of linear water waves over an axisymmetric bathymetric anomaly in arbitrary water depth is presented. The Axisymmetric 3-D Step Model is valid in a region of uniform depth containing an axisymmetric bathymetric anomaly with gradual transitions in depth allowed as a series of steps approximating arbitrary slopes. The velocity potential is calculated by applying matching conditions at the interface between regions of constant depth. The velocity potential obtained determines the wave field in the domain for monochromatic incident waves of linear form. A second analytic model (3-D Shallow Water Exact Model) is developed for comparison within the shallow water limit. The Axisymmetric 3-D Step Model determines the wave transformation caused by the processes of wave refraction, diffraction and reflection. Wave transformation is demonstrated in plots of the relative amplitude for bathymetric anomalies in the form of pit or a shoal, highlighting areas of wave sheltering and wave focusing. Anomalies of constant volume, but variable cross-section are employed to isolate the effect of the transition slope on the wave transformation. Comparisons to a shallow water model, numerical models, and experimental data verify the results of the Axisymmetric 3-D Step Model for several bathymetries including both pits and shoals. Also included are estimates of the energy reflection induced by an axisymmetric depth anomaly. The 3-D Axisymmetric Step Model has been applied previously to account for nearshore transformation (sloping bathymetry) and associated shoreline changes [C.J. Bender, R.G. Dean, Coastal Engineering 51 (2004) 1143]. D 2005 Elsevier B.V. All rights reserved. Keywords: Wave transformation; Bathymetric anomaly; Sloped transition; Reflection; Coefficient; Transmission coefficient

1. Introduction

T Corresponding author. Fax: +1 904 731 9847. E-mail address: [email protected] (C.J. Bender). 0378-3839/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2004.12.005

Irregular and unexpected shoreline planforms adjacent to nearshore borrow areas have increased awareness of the wave field modification caused by

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bathymetric anomalies such as borrow pits or shoals. When a wave field in a region of generally uniform depth encounters a bathymetric anomaly, the wave field is modified through wave refraction, wave diffraction, wave reflection and wave dissipation, with the first three known collectively as scattering. These four wave transformation processes result in a modified wave field that propagates towards, and eventually impacts, the shoreline. Case studies such as the beach nourishment projects at Grand Isle, Louisiana, and Anna Maria Key, Florida, have shown the possible effects of a nearshore borrow area on the adjacent shoreline planform. The ability to predict, and possibly design for, the equilibrium planform in the vicinity of a bathymetric anomaly requires a better understanding of both the wave and sediment processes near the anomaly. The focus here is to better understand the wave field modifications caused by bathymetric anomalies of axisymmetric three-dimensional form, with results providing the foundation for future work on potential shoreline changes induced by altering the nearshore bathymetry. Several studies encompassing field and laboratory scales have investigated this issue. These studies examined the wave transformation over a bathymetric anomaly with the shoreline changes caused by the altered wave field. Earlier, dating back to the early 1900s, research focused on the modification of a wave train encountering a change in bathymetry by employing analytic methods. This early research included development of analytical solutions for bathymetric changes in the form of a step, or a pit, first of infinite length (in one horizontal dimension; 2-D models) and, more recently, of finite dimensions (in two horizontal dimensions; 3-D models). The complexity of the 3-D models has advanced from a pit/shoal with vertical sidewalls and uniform depth surrounded by water of uniform depth, which are solved analytically, to domains with arbitrary bathymetry solved with complex numerical schemes. Some models combine the calculation of the wave transformation and resulting shoreline change, whereas others perform the wave calculations separately and rely on a different program for shoreline evolution. The intent of the analytic model developed within this study is not to replace or supercede the more complex numerical schemes that allow arbitrary bathymetry and more complex wave-

related processes (such as nonlinearities), but to develop an alternative model that provides an analytic solution. A better understanding of the wave field near bathymetric anomalies is obtained through models that more accurately represent the bathymetric geometries and the local wave transformation processes. The analytic models developed in this study extend previous analytic methods to better approximate the natural domain. Previous analytic 3-D models have domains containing abrupt transitions in depth (vertical sidewalls) for the bathymetric anomaly (Black and Mei, 1970; Williams, 1990; Williams and Vasquez, 1991; McDougal et al., 1996; Bender, 2001). A more realistic representation of natural bathymetric anomalies should allow for gradual transitions (sloped sidewalls). Bender and Dean (2003a,b) provide an extensive review of recent work on wave field modification by bathymetric anomalies and shoreline response including a brief review of several other numerical nearshore wave models currently in use. The focus of the present study is an analytic solution of the propagation of water waves over an axisymmetric 3-D (pit or shoal) bathymetric anomaly of more realistic geometry and the wave transformation induced. For an axisymmetric 3-D domain, an analytic solution to the wave field modification caused by bathymetric anomalies with sloped transitions in depth is developed. This solution is an extension of previous work for 2-D anomalies with abrupt depth transitions in regions of otherwise uniform depth employing steps to approximate a gradual transition in depth. A shallow water analytic solution is also developed, which is valid for specific sidewall slope and pit size combinations. Employing the laboratory data of Chawla and Kirby (1996), and the numerical models REF/DIF-1 (Kirby and Dalrymple, 1994) and FUNWAVE 1.0 [2-D] (Kirby et al., 1998) validates the 3-D models. The application of the different models to realworld problems depends on the situation of interest. The wave transformation and energy reflection (current article), and longshore transport and shoreline evolution (future article) by an axisymmetric 3-D bathymetric anomaly with gradual transitions in depth are investigated through the methods developed in this study.

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333

(r, θ)

2. Step method theory and formulation r

This paper investigates the three-dimensional transformation of monochromatic, small-amplitude water waves in an inviscid and irrotational fluid of arbitrary depth. The waves propagate in an infinitely long, uniform depth domain containing a threedimensional axisymmetric anomaly (pit or shoal) of finite extent. The addition of the second horizontal dimension provides many new, and more practical, possibilities for study compared to model domains excluding longshore variation (2-D) (Bender and Dean, 2003a,b, 2004). The developments of two different models are presented for 3-D domains of uniform depth containing transitions in depth. The analytic step method (Axisymmetric 3-D Step Model) extends Bender (2001) by determining the wave transformation in arbitrary water depth for domains with gradual transitions in depth approximated as a series of steps of uniform depth. The 3-D Shallow Water Exact Model solves the wave transformation in shallow water for specific bathymetries reducing the governing equations to forms for which exact solutions exist. 2.1. Axisymmetric 3-D Step Model: formulation and solution The step method for a three-dimensional domain is an extension of the Bender (2001) formulation for the propagation of waves past a circular anomaly with abrupt transitions. This method allowed oblique wave incidence—an advancement over two-dimensional models, but was limited to the shallow water region. The governing equations for Axisymmetric 3-D Step Model are developed following Bender (2001) with significant changes in notation. The domain is divided into regions with the bathymetric anomaly and its projection comprising Regions 2YN s+1 where N s is the number of steps approximating the depth transition slope and with the rest of the domain, of depth h 1, in Region 1, see Fig. 1. For the case of an abrupt transition in depth the bathymetric anomaly occurs in Region 2 of uniform depth h 2, where abrupt is defined as one step either down or up. For the case of a gradual depth transition the bathymetric anomaly is divided into subregions

φ S2

φI φ S1

φ T2

θ

φ T3 Region 3

Region 2

Region 1

Z R

r

h1

r1 r2

h2 h3

Fig. 1. Definition sketch for boundaries of gradual depth transitions.

with the depth in each subregion equal to h j for each step j=2YN s+1. The solution starts with the definition of a velocity potential in cylindrical coordinates that is valid in each Region j:
 Uj ¼ Re /j ðr; h; zÞeixt

ð j ¼ 1YNs Þ

ð2  1Þ

where x is the wave frequency. Linear wave theory is employed and Laplace’s solution in cylindrical coordinates is valid: r2 / ¼

B2 / 1 B/ B2 / B2 / þ þ þ ¼0 Br2 r Br Bz2 r2 Bh2

ð2  2Þ

where the free surface boundary condition is B/ r2  / ¼ 0; Bz g

z¼0

ð2  3Þ

and the bottom boundary condition is taken as B/ ¼0 Bz at z=h 1 in Region 1 or z=h j in Region 2.

ð2  4Þ

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At each interface between changes in depth, matching conditions are applied to ensure continuity of pressure: /j ¼ /jþ1




 r ¼ rj ; ð j ¼ 1YNs Þ

ð2  5Þ

The solution starts with the definition of an incident velocity potential in the form of a plane progressive wave: " # l X cm cosðmhÞJm ðk1 rÞ /I ðr; h; z; t Þ ¼ MI m¼0

and continuity of horizontal velocity normal to the vertical boundaries: B/j B/jþ1 ¼ Brj Brj




 r ¼ rj ; ð j ¼ 1YNs Þ

ð2  6Þ

The matching conditions are applied over the vertical plane between the two regions: (h j VzV0) if h j bh j+1 or (h j+1VzV0) if h j Nh j+1. At the interface between two regions a no flow condition is mandated along the deeper side of the step: B/jþ1 ¼0 Brj




 r ¼ rj ð j ¼ 1YNs Þ

ð2  7Þ

The solution must also meet the radiation condition for large r:  
 pffiffi B  ik1 /1  /I1 limrYl r Br



ð2  9Þ

where c m =1 for m=0 and 2i m otherwise, M I=igH/2x, r is the radial distance from the center of the bathymetric anomaly to the point in the fluid domain, H is the incident wave height, x is the angular wave frequency, and h is the angle to the point measured counterclockwise from the h origin as shown in Fig. 1. The scattered velocity potential is defined at each boundary (R j ) and consists of a plane progressive wave and evanescent modes for ( j=1YN s+1): " # l X
 1 Aj;m cosðmhÞHm kj r /S; j ðr; h; z; t Þ ¼ m¼0


 cosh tkj hj þ z b ixt
 e  cosh kj hj ( " l l X X
 þ aj;m;n Km jj;n r

ð2  8Þ

where / 1I is the specified incident velocity potential. The step method extends the work of Bender (2001) by allowing a domain of arbitrary depth that contains a depth anomaly with gradual transitions (sloped sidewalls) between regions. Instead of having a bstep downQ from the upwave direction for a pit or a bstep upQ for the case of a shoal, in the step method a series of steps either up or down, of uniform depth, are connected by a uniform depth region for rbr N s. A sketch of a domain with a stepped pit was shown in Fig. 1 indicating the location and definition of the velocity potentials and boundaries for a pit with N s=2. Each region has a specified depth and each interface between regions has a specified radius where the matching conditions are applied. At each boundary the matching conditions depend on whether the boundary is a bstep upQ or a bstep down.Q With the incident wave specified, a set of equations with 2N sM(N e+1) unknown coefficients is formed where M is the number of Bessel function modes included in the solution and N e is the number of evanescent (nonpropagating) modes.

cosh½k1 ðh1 þ zÞ ixt e coshðk1 h1 Þ

m¼1

n¼1

)


 # cos jj;n hj þ z
  cosðmhÞ eixt cos jj;n hj ð2  10Þ Inside the pit the transmitted velocity potential is given by the form " /T; j ðr; h; z; t Þ ¼

l X


 Bj;m cosðmhÞJm kj r

#

m¼0


 cosh tkj hj þ z b ixt
 e  cosh kj hj ( " l l X X
 þ bj;m;n Im jj;n r m¼1






n¼1

cos jj;n hj þ z
  cos jj;n hj

 #

) cosðmhÞ eixt ð2  11Þ

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A m , a m,n , B m and b m,n are unknown amplitude coefficients that are determined for each Bessel function mode included in the solution. The solution is obtained by applying the matching conditions at each interface between regions for two truncated series (m=1YM) and (n=1YN e). Solving the matching conditions , integrating over depth and applying the orthogonality properties of the problem gives the resulting integral equations of the form (Takano, 1960): for j=1YN s. If (h j Nh j+1) at r =Rj the depth anomaly is a pit and the boundary is a bstep downQ and the matching conditions are Z

0

hj





 /j Rj ; z cosh kj hj þ z dz

¼

Z

0

hj

Z

0

hj





 /jþ1 Rj ; z cosh kj hj þ z dz



 /j Rj ; z cos jj;n hj þ z dz ¼

Z

0

hj


 /jþ1 Rj ; z


 cos jj;n hj þ z dz ðn ¼ 1YNe Þ Z

0

hj

ð2  12Þ

ð2  13Þ




 B/j
Rj ; z cosh kjþ1 hjþ1 þ z dz Bx Z 0 


 B/jþ1
Rj ; z cosh kjþ1 hjþ1 þ z dz ¼ Bx hjþ1

335

the cases presented—to ensure convergence of the solution. Appendix C details the solution method for the unknown coefficients matrix. 2.2. 3-D Shallow Water Exact Model: formulation and solution The exact shallow water solution method solves the long wave transformation by an axisymmetric pit or shoal that reduces the governing equation to an equation with a known solution. Therefore this method is only valid within the shallow water limit for certain bathymetries containing a region where the depth is a function of the radius that connects two uniform depth regions. The governing equation is developed with the continuity equation in cylindrical coordinates and the inviscid equations of motion averaged over depth and linearized. The governing equation has the form: r2 dhðrÞ FVðrÞ hðrÞ dr   þ r 2 k ðr Þ2  m 2 F ðr Þ ¼ 0

r2 FWðrÞ þ rFVðrÞ þ

ð2  16Þ

The solution starts by specifying an incident wave form: " # l X cm cosðmhÞJm ðk1 rÞ eixt /I ðr; h; t Þ ¼ m¼0

ð2  17Þ

ð2  14Þ Z




 B/j
Rj ; z cos jjþ1;n hjþ1 þ z dz hj Bx Z 0 


 B/jþ1
¼ Rj ; z cos jjþ1;n hjþ1 þ z dz hjþ1 Bx 0

ðn ¼ 1YNe Þ

ð2  15Þ

which is the shallow water form of Eq. (2-9). The velocity potential of the scattered wave and the wave inside the uniform depth region of the depth anomaly are given by the forms " # l X /S ðr; h; t Þ¼ Am cosðmhÞHm1 ðk1 rÞ eixt m¼0

Solving the bstep upQ case requires a change in the limits of integration to satisfy the matching conditions with depth. At each boundary (R j ) the appropriate evanescent mode contributions from the adjacent boundaries (R j1, R j+1) are taken into account. The resulting set of simultaneous equations is solved as a linear matrix equation with the values of M, N e and N s sufficiently large—approximately 60, 6 and 10 (linear slope) for

ð2  18Þ " /ins ðr; h; t Þ¼

l X

# Bm cosðmhÞJm ðk2 rÞ eixt

m¼0

ð2  19Þ Fig. 2 presents a definition sketch for the boundaries of the exact solution method. As in the step method,

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ηI ηS

η (r,n)+ η (r,n)-

η ins

Z 0

h(r)

r

h2

h1 R1

R2

Fig. 2. Definition sketch for boundaries of exact shallow water solution method.

matching conditions are applied at transitions where the depths are described by different functions. The matching conditions ensure continuity of the water surface and continuity of the gradient of the water surface (equivalent to continuity of discharge). Only specific bathymetries will allow the exact solution of Eq. (2-16). Table 4-1 contains two bathymetries, one pit and one shoal that solve the governing equation exactly. The resulting truncated set of simultaneous equations is solved as a linear matrix equation with the number of Bessel function modes included, M, large enough to ensure convergence of the solution.

3. Results and comparisons Solution of the axisymmetric models establishes the complex velocity potential anywhere in the fluid domain. The complex velocity potential allows calculation of the wave height and direction. These

quantities demonstrate the wave field modifications caused by axisymmetric bathymetric anomalies. The effect of the depth transition slope on the wave field modification was investigated by applying the Axisymmetric 3-D Step Model. Comparisons between the Axisymmetric 3-D Step Model and the exact analytic model are made in shallow water conditions for two bathymetries. The Axisymmetric 3-D Step Model is also validated through comparison with the laboratory data of Chawla and Kirby (1996), and the numerical models REF/DIF-1 (Kirby and Dalrymple, 1994) and FUNWAVE 1.0 [2-D] (Kirby et al., 1998). 3.1. Wave height modification Applying the Axisymmetric 3-D Step Model yields the wave field caused by the three wave transformation processes identified that act on a planar wave that encounters a bathymetric anomaly. Fig. 3 presents a contour plot of the wave amplitude field normalized by the incident wave amplitude for a circular shoal with sloped sidewalls (truncated cone). In this, and remaining figures, the incident waves propagate from left to right. A cross-section of the shoal taken through the centerline is shown in the inset diagram for one-half of the shoal. Ten equally sized steps approximate a transition slope of 0.1. The convergence of the wave field caused by the shoal is shown clearly in Fig. 3 with a large area of wave focusing evident directly shoreward of the shoal. Two bands of relative amplitude less than one, caused by the diverging wave field at these locations, flank this area. The relative amplitude values are symmetric about Y=0. The wave field modification is viewed in another way by taking transects in the cross-shore and

Table 4-1 Specifications for two bathymetries for exact shallow water solution method h(r)

C

r1 [m]

r 2 [m]

Solution~g(r, n)F

C/r (pit)

10

10

5

aðr; mÞ pffiffi !½ J ðn; fÞFiY ðn; fÞ

r

10

5

pffiffi aðr; mÞ! r !½ J ðn; fÞFiY ðn; fÞ

C*r (shoal)

0.2

n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4m2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4m2 3

f 4p T

rffiffiffiffiffiffiffi r gC

4p 3T

sffiffiffiffiffiffiffi 1 1:5 r gC

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Fig. 3. Contour plot of relative amplitude for shoal with k 1h 1=0.29 and transition slope=0.1; cross-section of shoal bathymetry through centerline included. Note distorted scale.

longshore directions. Fig. 4 shows the relative amplitude along transects taken in the cross-shore direction at Y=0 for a pit. In the figure a cross-section of the pit taken through the centerline is included in an inset diagram and the extent of the pit shown by the dotted line in the main plot. The cross-shore transect of the relative amplitude highlights the sheltering caused by pit with the sheltering effect diminishing with distance shoreward of the pit. A partial standing wave pattern is evident in the upwave direction indicating the reflection caused by the pit. A focal point of the current study was to investigate the effect of the transition slope on wave field modification. Different truncated cones of constant volume were created in an attempt to isolate the effect of the sidewall slope. Starting from a pit of uniform depth with abrupt depth transitions (cylinder), three other pits (truncated cones) were created with the same volume and depth by decreasing the transition slope. A cross-section of these bsame depthQ pits (slope=abrupt, 1, 0.2 and 0.07) taken through the centerline is shown in the inset diagram of Fig. 5. The main plot shows the relative amplitude along a cross-shore transect through the center of each pit for k 1h 1=0.3 with

the extent of the abrupt pit indicated by the dotted line. In this figure the greatest reflection upwave of the pit is associated with transition slopes of 1 and 0.2 with minimal reflection for a slope of 0.07. The sheltering downwave of the pit increases as the transition slope decreases. Although not presented in graphical form here, for k 1h 1=0.15 the upwave reflection diminishes and the downwave sheltering increases as the transition slope becomes more gradual. The effect of the transition slope for the case of a shoal is viewed in Fig. 6. A cross-section of the shoal configurations taken through the centerlines is shown in the inset diagram with each shoal in the form of a truncated cone with constant volume. The slopes for the transitions are abrupt, 1, 0.2 and 0.05. Fig. 6 shows the relative amplitude for a transect in the cross-shore direction through the center of the shoal for k 1h 1=0.15 with the extent of the shoal indicated by the dashed line. The features of the contour plot of relative amplitude for a shoal (Fig. 3) are shown with an increase in the relative amplitude over the shoal and shoreward of the shoal where wave focusing occurs. The shoal with a transition slope of 0.05 is seen to cause less focusing shoreward of the shoal,

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Fig. 4. Relative amplitude for cross-shore transect at Y=0 with k 1h 1=0.24 for pit with transition slope=0.1; cross-section of pit bathymetry through centerline included. Note small reflection.

Fig. 5. Relative amplitude for cross-shore transect at Y=0 for same depth pits for k 1h 1=0.3; cross-section of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2 and 0.07.

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339

Fig. 6. Relative amplitude for cross-shore transect located at Y=0 for same depth shoals with k 1h 1=0.15; cross-section of shoal bathymetries through centerline included with slopes abrupt, 1, 0.2 and 0.05.

Fig. 7. Wave direction for alongshore transect at X=300 m for same depth pits for k 1h 1=0.15; cross-section of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2 and 0.07. Negative directions indicate divergence of wave rays.

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significantly smaller relative amplitudes over the shoal, and a reflected wave that is out of phase, but similar in magnitude to the other transitions slopes.

tions in the longshore. Greater refraction occurs as the transition slope becomes more gradual and the planform (bfootprintQ) of the pit is increased.

3.2. Modification of wave direction

3.3. Comparison of Axisymmetric 3-D Step Model and Analytic Shallow Water Exact Model

Previous plots of the relative amplitude in a domain with a bathymetric anomaly have demonstrated the divergence and convergence of the wave field near the anomaly due to wave sheltering and focusing, respectively. The divergence and convergence of the wave field is also viewed by examining the wave direction modification near a bathymetric anomaly. The wave direction is defined as that associated with the vectorial energy flux—Appendix A (Bender, 2001). The effect of the transition slope on the wave direction was investigated using the constant volume, constant depth pits first shown in Fig. 4. The wave direction for a longshore transect located 300 m from the center of the pits is shown in Fig. 7 for k 1h 1=0.15. Decreasing the transition slope is seen to increase slightly the magnitude of the wave direction oscilla-

For certain shallow water bathymetries the Axisymmetric 3-D Step Model is compared to the 3-D Shallow Water Exact Model described in Section 2.2. Table 4-1 presents the characteristics of two bathymetric forms that satisfy the requirements of the exact analytic model. To specify a bathymetry for the exact analytic model the values of C, h 1, h 2, R 1, R 2 and T (wave period) are selected such that shallow water conditions are maintained. Fig. 8 shows the relative amplitude in the crossshore direction through the center of the pit for the Axisymmetric 3-D Step Model (N s=10 steps) and the Shallow Water Exact Model. The inset diagram shows a cross-section of the pit bathymetry through the centerline for the dimensions indicated in the figure, which satisfy the exact analytic model for the first case

Fig. 8. Relative amplitude for cross-shore transect at Y=0 for pit with for h=C/r in region of transition slope; cross-section of pit bathymetry through centerline included.

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341

Three-dimensional numerical models for nearshore wave transformation are employed to validate the Axisymmetric 3-D Step Model for similar bathymetries. As in comparing any models, one must ensure the models are representing comparable processes and within their limits. The widely applied numerical model REF/DIF-1 (Kirby and Dalrymple, 1994) was employed to assess the results of the Axisymmetric 3D Step Model for a uniform domain containing a bathymetric anomaly. REF/DIF-1 was compared to laboratory data from Berkhoff et al. (1982) for a submerged shoal on a sloped, plane beach. The composite nonlinear wave theory of REF/DIF-1 was quite accurate in predicting the wave field. Thus, while the Axisymmetric 3-D Step Model is compared here to the laboratory data from a single experiment (Chawla and Kirby, 1996) for the wave field modification by a bathymetric anomaly with gradual transitions in an otherwise uniform domain, comparing the Axisymmetric 3-D Step Model to numerical models can provide a certain level of validation for many domains as the numerical models have been compared successfully to other laboratory data.

upwave scattering (reflection) caused by a bathymetric anomaly is not included in REF/DIF-1. Fig. 9 presents a comparison of the Axisymmetric 3-D Step Model and REF/DIF-1 results for a crossshore transect of the relative amplitude. A crosssection of the bathymetric anomaly used in the Axisymmetric 3-D Step Model taken through the centerline for this comparison is shown in the inset diagram where 23 steps approximate the profile. A grid spacing of 5 m was used in the x and y directions for REF/DIF-1, which resulted in a coarser resolution for the anomaly than in the Axisymmetric 3-D Step Model. The solid line in the main plot shows the relative amplitude for the Axisymmetric 3-D Step Model and indicates slight reflection upwave of the pit and significant wave sheltering shoreward of and diminishing with distance from the pit. The circles show the REF/DIF-1 results for the case of linear wave theory, and generally indicate good agreement with the Axisymmetric 3-D Step Model results; however, upwave of the pit the REF/DIF-1 results identically equal 1 as reflection is not considered. The dashed line indicates results from REF/DIF-1 using composite nonlinear wave theory (Kirby and Dalrymple, 1986) for an incident wave height of 0.2 m. The nonlinear results follow the linear results over the pit and at small (b50 m) downwave distances. At large downwave distances the nonlinear results diverge significantly from the linear results and indicate much less wave sheltering due to the pit. Although not presented graphically here, the nonlinear REF/DIF-1 results were found to approach the linear results as the wave height was decreased to very small values (waves become more linear).

3.4.1. Comparison of Axisymmetric 3-D Step Model with REF/DIF-1 REF/DIF-1 (Kirby and Dalrymple, 1994) is a parabolic model for monochromatic water waves that is weakly nonlinear and accounts for wave refraction, diffraction, shoaling, breaking, and bottom friction. REF/DIF-1 assumes a mild bottom slope and is found to remain accurate for slopes up to 1 on 3 (Booij, 1983). REF/DIF-1 was run with no wave energy dissipation and with linear wave theory in order to most closely match the limitations of the Axisymmetric 3-D Step Model. One difference between REF/ DIF-1 and the Axisymmetric 3-D Step Model is that

3.4.2. Comparison with laboratory data of Chawla and Kirby (1996) In the Axisymmetric 3-D Step Model, the requirement of uniform depth outside of the bathymetric anomaly greatly limits the available laboratory data for comparison. The laboratory experiment of Chawla and Kirby (1996) is the only experiment known to the authors with wave measurements in a domain of uniform depth that contains a bathymetric anomaly. The bathymetric anomaly in Chawla and Kirby is a shoal of 2.57 m radius with the shape of the shoal being the upper portion of a sphere with a radius of 9.1 m. The dimensions for the basin used in the

in Table 4-1, h=C/r. Good agreement between the models is indicated in all regions (upwave, over the pit and downwave) providing verification for the Axisymmetric 3-D Step Model in shallow water. Although not presented here, a comparison in the longshore direction was performed for this bathymetry with good agreement between the models indicated. 3.4. Comparison of Axisymmetric 3-D Step Model to numerical models and laboratory data

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Fig. 9. Relative amplitude using Axisymmetric 3-D Step Model and REF/DIF-1 for cross-shore transect at Y=0 for Gaussian pit with k 1h 1=0.24; cross-section of pit bathymetry through centerline included.

Fig. 10. Relative amplitude using Axisymmetric 3-D Step Model, FUNWAVE 2-D and data from Test 4 of Chawla and Kirby (1996) for crossshore transect with k 1h 1=1.89; cross-section of shoal bathymetry through centerline included.

C.J. Bender, R.G. Dean / Coastal Engineering 52 (2005) 331–351

experiment were 18.2 m in the longshore direction and 20 m in the cross-shore direction. The data collected during the experiment of Chawla and Kirby that fit within the constraints of the Step Method are comprised of three different tests (Tests 1, 2 and 4) for monochromatic, non-breaking, incident waves with wave heights collected along seven transects. Figs. 10–12 compare the Step Method with results from Test 4 of Chawla and Kirby (1996). Chawla and Kirby (1996) present further details of the experiment and the collected data. The relative amplitudes along a cross-shore transect taken over the center of the shoal for the analytic model and the Chawla and Kirby (1996) data are shown in Fig. 10. An inset diagram is including showing the bathymetry with the shoal being approximated by 15 steps in the Axisymmetric 3-D Step Model. Also included in the main plot is the relative amplitude from FUNWAVE 1.0 [2-D] (Kirby et al., 1998). FUNWAVE 2-D is a two-dimensional fully nonlinear Boussinesq model based on the equations of Wei et al. (1995), which has been successfully

343

compared with the data of Chawla and Kirby (1986). The specifications of the inputs for FUNWAVE 2-D necessary to model the experiment of Chawla and Kirby (1986) are included in the freely distributed code for the numerical model and were used to generate the FUNWAVE 2-D results shown here. The incident wave height and period are 0.0233 m and 1 s, respectively. Good agreement is shown in Fig. 10 between the Axisymmetric 3-D Step Model and the data of Chawla and Kirby. FUNWAVE 2-D is found to agree well with the data up to X=5 m, where oscillations in the relative amplitude commence. Nonlinear effects could cause the difference in the maximum relative amplitude values over the shoal for the Step Model and those of the data and FUNWAVE 2-D. The maximum value of the Ursell parameter for the FUNWAVE 2-D results over the shoal is 83.13, while the maximum value upwave of the shoal is 0.60. Both the Axisymmetric 3-D Step Model and FUNWAVE 2D indicate smaller relative amplitude values than the data shoreward of the shoal (4bXb6 m).

Fig. 11. Relative amplitude using Axisymmetric 3-D Step Model, FUNWAVE 2-D and data from Test 4 of Chawla and Kirby (1996) for longshore transect located 1.35 m shoreward of the center of the shoal with k 1h 1=1.89; cross-section of shoal bathymetry through centerline included.

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C.J. Bender, R.G. Dean / Coastal Engineering 52 (2005) 331–351

Fig. 12. Relative amplitude using Axisymmetric 3-D Step Model, FUNWAVE 2-D and data from Test 4 of Chawla and Kirby (1996) for longshore transect located 2.995 m shoreward of the center of the shoal with k 1h 1=1.89; cross-section of shoal bathymetry through centerline included.

The relative amplitude values for a longshore transect located 1.35 m shoreward of the shoal center are found in Fig. 11. This transect lies over the shoreward side of the shoal. For each data point, the Axisymmetric 3-D Step Model results show good agreement with the data of Chawla and Kirby (1996). The FUNWAVE 2-D results nearly match those of the Axisymmetric 3-D Step Model with a smaller peak value at Y=0 and a noticeable asymmetry in the values due to a slight asymmetry in the experimental setup (sidewall locations), and therefore the FUNWAVE 2D domain. The relative amplitudes for a longshore transect located 2.995 m shoreward of the center of the shoal are shown in Fig. 12. The Axisymmetric 3-D Step Model shows comparable relative amplitude values to the data of Chawla and Kirby (1996) and FUNWAVE 2-D. In Fig. 12 the data of Chawla and Kirby (1996) show large (N1.4) values directly shoreward of the shoal where the Axisymmetric 3-D Step Model and FUNWAVE 2-D results indicate values around 1.1; a result first shown in Fig. 10. Overall the agreement

between the Analytic Step Method and experimental data of Chawla and Kirby (1996) is quite good. 3.5. Energy reflection Unlike the two-dimensional case of waves over a trench or shoal of infinite length, determining the energy reflection from a bathymetric anomaly in three dimensions is not trivial. To determine the energy reflection caused by a three-dimensional bathymetric anomaly a method was developed using far-field approximations of the Bessel functions along a halfcircular arc bounding the region of reflected energy— Appendix B (Bender, 2003). Summing the reflected energy through this arc and comparing it to the amount of energy incident on the anomaly gives a reflection coefficient. 3.5.1. Comparison to previous results The far-field energy reflection method will first be compared to a shallow water method that sums the energy flux through a transect. Bender (2001) de-

C.J. Bender, R.G. Dean / Coastal Engineering 52 (2005) 331–351

veloped a method to calculate the energy reflection caused by bathymetric anomalies in shallow water. This method used a time-averaged energy flux approach in which a transect was created outward from the center of the anomaly in the longshore direction and parallel to the incident wave fronts. At each location in the transect the energy flux was determined, which allowed for the total energy flux through the transect to be summed and compared with the energy flux through the transect if no pit were present, resulting in a reflection coefficient. Fig. 13 shows the reflection coefficient, K R, versus pit diameter divided by the wavelength outside the pit for the far-field approximation method and the shallow water transect method. In order to maintain shallow water conditions the transect method requires increasing the pit radius to obtain K R for larger values of the non-dimensional diameter; therefore, four different pit radii are included for the transect method results. Two different pit radii were used for the far-field approximation method with little difference in the values indicated in the results. The transect method

345

values for each of the four radii used show good agreement with the results of the far-field approximation method with the largest difference at a nondimensional diameter of 0.9. 3.5.2. Effect of transition slope on reflection The effect of the transition slope on reflection was investigated using the step method and the far-field approximation for the reflection coefficient. Bathymetric anomalies of constant volume were created either by keeping the depth or the bottom radius constant in an attempt to isolate the effect of the transition slope. Fig. 14 shows the reflection coefficient versus k 1h 1 for four pits in the shape of truncated cones with constant volume and equal depth; these are the same pit configurations used in Fig. 5. The gradual transition slopes are seen to greatly change the reflection properties of the pits. The K R values for each pit oscillate with k 1h 1 with the trend of K R decreasing as k 1h 1 increases. For the more gradual transition slopes the oscillations are almost completely damped for the larger k 1h 1 values. The only occurrence of complete transmission (K R=0) is

Fig. 13. Reflection coefficient versus non-dimensional diameter; comparison between shallow water transect method and far-field approximation method.

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C.J. Bender, R.G. Dean / Coastal Engineering 52 (2005) 331–351

Fig. 14. Reflection coefficient versus k 1h 1 based on far-field approximation and constant volume and depth pits; cross-section of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2 and 0.07.

the trivial solution of k 1h 1=0. This differs from the result for symmetric two-dimensional trenches of infinite length with the same depth on both sides of the trench, where for each oscillation there is an instance of complete transmission. The reflection coefficients for four shoals of equal volume and depth are shown in Fig. 15. The same shoal configurations were used as in Fig. 6. The more gradual transition slopes are seen to produce less reflection than the more abrupt slopes, as seen previously. The reflection coefficient oscillates with k 1h 1 with the general trend of the K R values increasing with k 1h 1 for all slopes except the most gradual; a result that differs from the case for a pit.

4. Conclusions and directions for further study Two methods (3-D Analytic Step Method and 3-D Shallow Water Analytic Exact Solution Method) developed in this study demonstrate the wave field transformation by axisymmetric bathymetric anoma-

lies with good agreement shown between the methods for shallow water conditions. The 3-D Step Method compares well with the laboratory data of Chawla and Kirby (1996) as well as the numerical models REF/DIF-1 (Kirby and Dalrymple, 1994) and FUNWAVE 1.0 [2-D] (Kirby et al., 1998) for several bathymetries and linear incident waves. Past and future numerical models can employ the Axisymmetric 3-D Step Model to their benefit, with the analytic model providing verification of the numerical scheme contained within the model. Pits or shoals (in the form of truncated cones) of constant volume illustrate the impact of the configuration of the bathymetric anomaly on wave transformation. Gradual transition slopes caused greater wave sheltering shoreward of a pit, with the degree of upwave reflection dependent on incident wave and pit characteristics. Recent interest in extracting large volumes of nearshore sediment for beach nourishment and construction purposes has increased the need for reliable predictions of wave transformation and associated shoreline changes caused by the resulting bathymetric

C.J. Bender, R.G. Dean / Coastal Engineering 52 (2005) 331–351

347

Fig. 15. Reflection coefficient versus k 1h 1 based on far-field approximation and constant volume and depth shoals; cross-section of shoal bathymetry through centerline included with slopes of abrupt, 1, 0.2 and 0.07.

anomalies. This predictive capacity would assist the designer of such projects in minimizing undesirable shoreline changes. The available laboratory and field data suggest that the effect of wave transformation by an offshore pit can result in substantial shoreward salients and associated erosional areas. Of the four wave transformation processes caused by a bathymetric anomaly, a significant number of wave models include only the effects of wave refraction and diffraction, with few models incorporating wave reflection and/or dissipation over a soft medium in the pit. Computational results incorporating only refraction and diffraction and accepted values of sediment transport coefficients appear incapable of predicting the observed salients landward of borrow pits. Therefore, improved capabilities to predict wave transformation and shoreline response to constructed borrow pits will likely require improvements in both: (1) wave modeling, particularly in representing wave reflection and dissipation, and (2) longshore sediment transport particularly by the breaking wave direction and wave height gradient terms.

Acknowledgements An Alumni Fellowship granted by the University of Florida sponsored this study with partial support from the Bureau of Beaches and Wetland Resources of the State of Florida. The authors would like to thank Professor Ulrich H. Kurzweg of the University of Florida for his constructive comments and efforts in the development of the 3-D Shallow Water Exact Model.

Appendix A. Analytic wave angle calculation The wave angles are calculated by a procedure outlined in Bender (2001) using the time-averaged energy flux based on the total velocity potential outside the bathymetric anomaly. The time averaging is obtained by taking the conjugate of one of the complex variables:   pT u4T P P EFluxX T ¼ pT uT ¼ Real 2 ð4 means conjugateÞ

ðA  1Þ

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C.J. Bender, R.G. Dean / Coastal Engineering 52 (2005) 331–351

 4 pT v T P P EFluxY T ¼ pT vT ¼ Real 2 ð4 means conjugateÞ

where p is the pressure and velr is the velocity in the radial direction: ðA  2Þ

p¼ q

where p is the pressure and u and v are the velocities in the cross-shore and longshore directions, respectively: B/ pT ¼  q T Bt

ðA  3Þ

uT ¼ velrT cosðhÞ  velhT sinðhÞ

ðA  4Þ

vT ¼ velrT sinðhÞ þ velhT cosðhÞ

ðA  5Þ

where velrT=B/ T/Br and velhT=(1/r)(B/ T/Bh). The total potential outside of the anomaly is defined as the sum of the incident potential and the scattered potential due to the change in depth: /T ¼ /I þ /S

ðA  6Þ

Therefore the equation to determine the time averaged energy flux, in the x direction, at a single point due to the incident potential and the reflected potential is P EFluxX T ¼ pT u4T ¼ ðpI þ pS ÞðuI þ uS Þ4 ¼ pI u4I þ pI u4S þ pS u4I þ pS u4S

ðA  7Þ

velr ¼

Z

3p=2

P EFlux S ¼ pI vel4rS þ pS vel4rI þ pS vel4rS

ðB  1Þ




M ¼1

Z

3p=2

p=2




"

ðB  4Þ

N ¼1

l   ipgH X pI vel4rS dh ¼ 4p M ¼1

l h  X p p cM A4N fQM N gcos kr  M  2 4 N¼1 # i p p  eiðkrN 2  4 Þ ðB  5Þ



3p=2

p=2

The far-field approximation for the energy reflection starts by constructing a half circle of large radius in the upwave region of the anomaly that captures all the reflected energy flux. The depth integrated, timeaveraged energy flux is calculated where total reflected energy flux at any location is

ðB  3Þ

 qx pS vel4rS dh ¼ 2p p=2 " # l l h i X X  AM A4N fQM N geiðN M Þp=2

Z

Appendix B. Analytic far-field approximation of energy reflection

B/ Br

ðB  2Þ

Large value approximations are taken for the appropriate Bessel functions (Abramowitz  and R 3p=2
Stegun, 1964) and the operation p=2 pvelr dh is performed to quantify the energy flux around the entire arc. After simplifying the resulting equations, the forms for the components of Eq. (B.1) are

The wave angle at each point is then solved employing P 1 0 real EFluxY T P  A þ aI a ¼ tan1 @ ðA  8Þ real EFluxX T where a I is the incident wave direction.

B/ Bt

"




l   ipgH X p4S velrI dh ¼ 4p M ¼1

l h  X p p A4N cM fQM N gisin kr  M  2 4 N¼1 # i p p  eiðkrN 2  4 Þ ðB  6Þ



where after the substitution   p p p p cos kr  M  þ isin kr  M  2 4 2 4 p p ¼ eiðkrM 2  4 Þ ðB  7Þ

C.J. Bender, R.G. Dean / Coastal Engineering 52 (2005) 331–351

which allows Eqs. (B-5) and (B-6) to be combined l X

 ipgH pI vel4r S þ p4S velr I ¼ 4p M ¼1 " # l h i X 4 ið NM Þp=2  cM AN fQM N ge

φI φ S1

φ T2

φ S2 R1

349

φ T3 φ S3 R2

φ T4 Z R3 0

r

ðB  8Þ

N ¼1

h1

h2

h3

h4

In Eqs. (B-4)–(B-8), A N is the unknown reflected wave amplitude coefficient for each Bessel function mode, c is defined in Eq. (2.8), and M and N are the Bessel function modes uses in the solution, where if M=N: QM N ¼ p

ðB  9Þ

and if MpN: QM N ¼

Fig. C-1. Definition sketch for pit with three steps through centerline.

sinð M þ N Þ3p=2  sinð M þ N Þp=2 ðM þ N Þ þ

sinð M  N Þ3p=2  sinð M  N Þp=2 ðM  N Þ ðB  10Þ

Using Eqs. (B-4)–(B-10) the energy reflected by a bathymetric anomaly can be approximated in the farfield using the amplitude coefficients determined using the Axisymmetric 3-D Step Model. The reflection coefficient is calculated after dividing the reflected energy flux in the far-field by the energy flux incident on the bathymetric anomaly and taking the square root.

Appendix C. Solution method for unknown coefficient matrix This section demonstrates the solution method for the case of an axisymmetric pit with three steps. Fig. C-1 presents a sketch–through the center line of the pit–indicating the four regions and associated velocity potentials within each region for this case—three transitions in depth. The incident (U I), scattered (U S) and transmitted (U T) velocity potentials correspond to Eqs. (2-9)–(2-11). Table C-1 presents an unknown coefficient matrix to illustrate the solution method for the case presented in Fig. C-1. The matrix applies to a case with three steps (N s) and two evanescent modes (N e) for a single Bessel function mode (M). The parameters require the

solution of 18 unknown coefficients [2N sM(N e+1)] for each Bessel function mode. The method requires a new matrix solution for any additional Bessel function modes included in the solution. The matrix separates the propagating and non-propagating (evanescent) mode components for the scattered and transmitted waves. The solution procedure requires a known incident velocity potential. The top row of the matrix identifies the unknown coefficient corresponding to that column. The left-hand column of the matrix presents the matching condition applied at the interface between regions of different depth for that row. The next column contains the appropriate eigenfunction for that row. The last column contains the incident potential values for the appropriate matching conditions; these incident potential values are specified and represent the solution component of the equation. The interior of the matrix indicates the components applied in the solution of each matching condition as developed in Eqs. (2-12)–(215) with the notation of the integrals over depth specified as:



 sinh 2kj hj I1ð j; intÞ ¼ cosh kj hj þ z dz ¼ 4kj hint



 sinh 2kj hj  hint hj  hint hj þ   2 4kj 2 Z

0

2

ðC  1Þ

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Table C-1 Matrix solution for unknown coefficients; case of pit with three steps and two evanescent modes Matching condition

Eigenfunction U scat1

U1=U2 B/1/Br=B/2/Br U1=U2 U1=U2 B/1/Br=B/2/Br B/1/Br=B/2/Br U2=U3 B/2/Br=B/3/Br U2=U3 U2=U3 B/2/Br=B/3/Br B/2/Br=B/3/Br U3=U4 B/3/Br=B/4/Br U3=U4 U3=U4 B/3/Br=B/4/Br B/3/Br=B/4/Br

cosh(k 1h 1) cosh(k 2h 2) cos(k 1,1h 1) cos(k 1,2h 1) cos(k 2,1h 2) cos(k 2,2h 2) cosh(k 2h 2) cosh(k 3h 3) cos(k 1,2h 2) cos(k 2,2h 2) cos(k 3,1h 3) cos(k 3,2h 3) cosh(k 3h 3) cosh(k 4h 4) cos(k 3,1h 3) cos(k 3,2h 3) cos(k 4,1h 4) cos(k 4,2h 4)

I2ð j; n; intÞ¼

Z

U trans2

U scatEV1,1

I2(1,2,1) I1(2,2) I3(1,2,1,1) I3(1,2,1,1) I4(1,1,1) I3(1,2,2,1) I3(2,1,1,1) I5(1,2,1,1,1) I3(2,1,2,1) I5(1,2,2,1,1) I1(2,2) I2(3,2,2) I1(1,1) I2(2,1,1)

I3(3,2,1,2) I3(3,2,2,2)

U scatEV1,2

I3(2,1,2,1)

U trans3

U scatEV2,1 I3(2,1,1,1) I5(1,2,1,1,1) I5(1,2,2,1,1) I4(2,1,2)

I2(2,3,2) I1(3,3) I3(2,3,1,2) I3(2,3,1,2) I4(2,1,2) I3(2,3,2,2) I5(2,3,1,1,2) I5(2,3,2,1,2) I1(3,3) I2(4,3,3)

I3(4,3,1,3) I3(4,3,2,3)


 1  sinh kj hj þ kn hn ¼
2 kj þ kn


   sinh  hj kj þ kn þ kj hj þ kn hn 
 1  sinh kj hj  kn hn þ
2 kj  kn


   sinh  hj kj  kn þ kj hj  kn hn 0

I3(2,1,1,1)

U scat2

I2(1,2,1) I1(2,2) I5(1,2,1,1,1) I5(1,2,1,2,1) I3(1,2,1,1) I4(1,2,1) I5(1,2,2,1,1) I5(1,2,2,2,1) I3(1,2,2,1) I5(1,2,1,2,1) I4(2,1,2) I5(1,2,2,2,1) I4(2,2,2) I1(2,2) I3(2,3,1,2) I3(2,3,2,2) I2(3,2,2) I4(2,1,2) I4(2,2,2) I5(2,3,1,1,2) I5(2,3,1,2,2) I3(3,2,1,2) I5(2,3,2,1,2) I5(2,3,2,2,2) I3(3,2,2,2)

hint

Z I3ð j; n; e; intÞ¼

U transEV2,2

I3(1,2,2,1)




 cosh kj hj þ z coshðkn ½hn þ z Þdz

0

U transEV2,1








ðC  2Þ

cos kej;e hj þz coshðkn ½hn þ z Þdz

I5ð j; n; e; f ; intÞ Z 0



 cos kej;e hj þ z cos ken; f ½hn þ z dz ¼ hint


 1  sin kej;e hj þ ken; f hn ¼
2 kej;e þ ken; f


   sin  hj kej;e þ ken; f þ kej;e hj þ ken; f hn 
 1  sin kej;e hj  ken; f hn þ
2 kej;e  ken; f


   sin  hj kej;e  ken; f þ kej;e hj  ken; f hn ðC  5Þ

hint


 1  sin kej;e hj þ ikn hn ¼
2 kej;e þ ikn


  sin  hj kej;e þ ikn þ kej;e hj 1  þ ikn hn g þ
2 kej;e  ikn
  fsin kej;e hj  ikn hn


  sin  hj kej;e  ikn þ kej;e hj ðC  3Þ  ikn hn g Z 0


 I4ð j; e; intÞ ¼ cos2 kej;e hj þ z dz hint

 sin 2kej;e hj hj sin 2kej;e thj  hint b þ  ¼ 4kej;e 4kej;e 2
 hj  hint ðC  4Þ  2

References Abramowitz, M., Stegun, I.A. (Eds.), 1964. Handbook of Mathematical Functions. National Bureau of Standards, Applied Mathematics Series, vol. 55. U.S. Government Printing Office, Washington, DC. Bender, C.J., 2001. Wave field modifications and shoreline response due to offshore borrow areas. Master’s thesis. Department of Civil and Coastal Engineering, University of Florida. Bender, C.J,. 2003. Wave transformation by bathymetric anomalies with gradual transitions in depth and resulting shoreline response. Doctoral dissertation. Department of Civil and Coastal Engineering, University of Florida. Bender, C.J., Dean, R.G., 2003a. Wave field modification by bathymetric anomalies and resulting shoreline changes: a review with recent results. Coastal Engineering 49 (1–2), 125 – 153. Bender, C.J., Dean, R.G., 2003b. Wave transformation by twodimensional bathymetric anomalies with sloped transitions. Coastal Engineering 50, 61 – 84.

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351

Table C-1 (continued) U scatEV2,2

U transEV3,1

U transEV3,2

U scat3

U trans4

U scat EV3,1

U scatEV3,2

U transEV4,1

U transEV4,2

I3(2,1,2,1)

U inc I1(1,1) I2(2,1,1)

I5(1,2,1,2,1) I5(1,2,2,2,1) I3(2,1,1,1) I3(2,1,2,1)

I4(2,2,2) I3(3,2,1,2)

I3(3,2,2,2)

I5(2,3,1,1,2) I5(2,3,2,1,2) I4(3,1,3)

I5(2,3,1,2,2) I5(2,3,2,2,2)

I3(2,3,2,2) I4(2,2,2) I5(2,3,1,2,2) I5(2,3,2,2,2)

I2(2,2,2) I1(3,3) I3(2,3,1,2) I3(2,3,2,2)

I3(3,2,1,2)

I3(3,2,2,2)

I5(2,3,1,1,2) I5(2,3,2,1,2) I4(3,1,3)

I5(2,3,1,2,2) I5(2,3,2,2,2)

I4(3,2,3) I3(3,4,1,3) I4(3,1,3) I5(3,4,1,1,3) I5(3,4,2,1,3)

I3(3,4,2,3) I4(3,2,3) I5(3,4,1,2,3) I5(3,4,2,2,3)

I4(3,2,3) I1(3,3) I2(4,3,3)

I2(3,4,3) I1(4,4) I3(3,4,1,3) I3(3,4,2,3)

I3(4,3,1,3) I3(4,3,2,3)

Bender, C.J., Dean, R.G., 2004. Potential shoreline changes induced by 3-dimensional bathymetric anomalies with gradual transitions in depth. Coastal Engineering 51, 1143 – 1161. Berkhoff, J.C.W., Booij, N., Radder, A.C., 1982. Verification of numerical wave propagation models for simple harmonic linear water waves. Coastal Engineering 6, 255 – 279. Black, J.L., Mei, C.C., 1970. Scattering and radiation of water waves. Technical Report No. 121. Water Resources and Hydrodynamics Laboratory, Massachusetts Institute of Technology. Booij, N., 1983. A note on the accuracy of the mild-slope equation. Coastal Engineering 7, 191 – 203. Chawla, A., Kirby, J.T., 1996. Wave transformation over a submerged shoal. Report No. CACR-96-03. Department of Civil Engineering, University of Delaware. Kirby, J.T., Dalrymple, R.A., 1986. An approximate model for nonlinear dispersion in monochromatic wave propagation models. Coastal Engineering 9, 545 – 561. Kirby, J.T., Dalrymple, R.A., 1994. User’s Manual, Combined Refraction/Diffraction Model, REF/DIF-1, Version 2.5. Center for Applied Coastal Research, Department of Civil Engineering, University of Delaware. Newark, Delaware.

I3(3,4,1,3) I4(3,1,3) I5(3,4,1,1,3) I5(3,4,2,1,3)

I3(4,3,1,3)

I3(4,3,2,3)

I5(3,4,1,1,3) I5(3,4,2,1,3) I4(4,1,4)

I5(3,4,1,2,3) I5(3,4,2,2,3)

I3(3,4,2,3) I4(3,2,3) I5(3,4,1,2,3) I5(3,4,2,2,3)

I4(4,2,4)

Kirby, J.T., Wei, G., Chen, Q., Kennedy, A.B., Dalrymple, R.A., 1998. FUNWAVE 1.0, Fully nonlinear Boussinesq wave model. Documentation and user’s manual. Report CACR-98-06. Center for Applied Coastal Research, Department of Civil and Environmental Engineering, University of Delaware. McDougal, W.G., Williams, A.N., Furukawa, K., 1996. Multiple pit breakwaters. Journal of Waterway, Port, Coastal, and Ocean Engineering 122, 27 – 33. Takano, K., 1960. Effets d’un obstacle parale´lle´le´pipe´dique sur la propagation de la houle. Houille Blanche 15, 247. Wei, G., Kirby, J.T., Grilli, S.T., Subramanya, R., 1995. A fully nonlinear Boussinesq model for surface waves. I. highly nonlinear, unsteady waves. Journal of Fluid Mechanics 294, 71 – 92. Williams, A.N., 1990. Diffraction of long waves by a rectangular pit. Journal of Waterway, Port, Coastal, and Ocean Engineering 116, 459 – 469. Williams, A.N., Vasquez, J.H., 1991. Wave interaction with a rectangular pit. Journal of Offshore Mechanics and Arctic Engineering 113, 193 – 198.