Analytical Solution for Long-Wave Scattering by a CircularIsland Mounted on a General Shoal

Huan-Wen Liu1; Jian-Jian Xie2; and Zhang-Hua Luo3

Abstract: In this paper, an analytical solution is given in the form of a Taylor series for long-wave scattering by a cylindrical islandmounted ona general shoal, where the water depth in the shoal region can be a real constant plus a power function of the radial distance. In other words, thecrest of the shoal is not restricted to be on the still water level. The distribution of singular points becomes complicated; therefore, the solutiontechnique needs to be more sophisticated. Because of the generality of the location of the shoal crest, the analytical solutions find several clas-sical analytical solutions to be its special cases, which include long-wave scattering by a cylindrical islandmounted on an idealized paraboloidalshoal, a cylindrical island mounted on an idealized conical shoal, a cylindrical island mounted on an idealized shoal, and a cylindrical islandlocated on a flat bottom. The present analytical solution covers a muchwider range of problems, and therefore, is muchmore useful. Finally, theeffect of the shoal size to the wave-scattering pattern is investigated using the present analytical solution. DOI: 10.1061/(ASCE)WW.1943-5460.0000149. © 2012 American Society of Civil Engineers.

CE Database subject headings: Analytical techniques; Shoaling; Long waves.

Author keywords: Analytical solution; Long-wave equation; Wave scattering; General shoal; Deviation of shoal crest.

Introduction

Because of the practical importance of understanding the phenom-enon of wave transformation in a coastal region caused by reflection,refraction, and diffraction, many numerical models have been de-veloped to predict the transformation of waves. Because they in-volve approximation error, numerical models must be validatedagainst field data (Massel 1996, p. 398), laboratory experimentalresults (Lie and Tørum 1991), or analytical solutions. In comparisonwith both field data and laboratory experimental results, analyticalsolutions are more favorable because they not only greatly save costin time, labor, and experimental equipment, but also have high ac-curacy because they are only limited by the underlying assumptions.Moreover, analytical solutions are more efficient than experimentaldata because they can reveal unknown phenomena.

However, analytical solutions are generally obtainable only forsimple bottom geometries or simple wave equations. In linear wavetheory, the mild-slope equation proposed by Berkhoff (1972, 1976)and the modified mild-slope equation (Chamberlain and Porter1995) have proved to be powerful models for water-wave problemsin the complete wave spectrum. However, exact analytical solutionsto them are quite difficult because the linear dispersion equation isimplicit, although some approximate analytical solutions have beenrecently presented based on Hunt’s (1979) approximate solution to

the implicit dispersion relation (Liu et al. 2004; Cheng 2007; Jungand Suh 2007; Lin and Liu 2007; Liu and Lin 2007; Jung and Suh2008; Hsiao et al. 2010) and based on function fitting (Cheng 2011).

If this study is restricted to the linear long-wave equation, theproblem becomes simpler, and exact analytical solutions are pos-sible. In one dimension, Lin and Liu (2005) and Liu and Lin (2005)recently presented a closed-form analytical solution for wave in-teraction with a trench and an obstacle of general trapezoidal shapethat includes several well-known analytical solutions as specialcases, such as Lamb’s (1932) classical solution for waves passingover an infinitely long step, Mei’s (1989) solution for waves pasta rectangular obstacle, and Dean’s (1964) solution for an infinitelylong shelf located behind a linear slope. Recently, Jung et al. (2008)and Jung and Cho (2009) also presented closed-form analyticalsolutions for long-wave interaction with trenches of various shapesand arbitrarily varying topography. Xie et al. (2011) presenteda closed-form analytical solution for long-wave reflection bya rectangular obstacle with two scour trenches.

In two dimensions, several analytical solutions to the long-waveequation have been found for some axisymmetrical bottom geom-etries, such as a circular cylinder mounted on a paraboloidal shoal(Homma 1950), a circular cylinder mounted on a conical shoal(Zhu and Zhang 1996), a circular cylinder mounted on a generalidealized shoal (Yu and Zhang 2003), a circular bowl pit (Suh et al.2005), and a submerged circular truncated shoal (Liu and Li 2007).However, it is worthwhile to note that all these analytically solvedproblems are restricted to idealized bottom topographies, that is, theshoal crest is assumed to be on the still water level. It is no doubt thatthis restriction will narrow down the range of application of theseanalytical solutions.

Only a few long-wave analytical solutions have been found forgeneral bottom topographies, such as a conical island and a sub-merged paraboloidal shoal (Zhang and Zhu 1994), a circular pa-raboloidal hump (Zhu and Harun 2009), a general circular island(Jung et al. 2010), a general hump (Niu and Yu 2011a; Liu and Xie2011), a vertical cylinder with a scour pit (Niu and Yu 2011b), anda dredge excavation pit (Niu and Yu 2011c).

1Professor, School of Sciences, Guangxi Univ. for Nationalities,Nanning, Guangxi 530006, P.R. China (corresponding author). E-mail:mengtian29@163.com

2Research Assistant, School of Sciences, Guangxi Univ. for National-ities, Nanning, Guangxi 530006, P.R. China.

3Associate Professor, School of Sciences, Guangxi Univ. for National-ities, Nanning, Guangxi 530006, P.R. China.

Note. This manuscript was submitted on January 4, 2011; approved onMarch 1, 2012; published online on March 3, 2011. Discussion period openuntil April 1, 2013; separate discussions must be submitted for individualpapers. This paper is part of the Journal of Waterway, Port, Coastal, andOcean Engineering, Vol. 138, No. 6, November 1, 2012. ©ASCE, ISSN0733-950X/2012/6-425e434/$25.00.

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In this paper, long-wave scattering by a circular cylindermounted on a general axisymmetrical shoal is considered. Becauseof the generality of the shoal, the problem of solving the long-waveequation becomes complicated. By using variable separation, vari-able transformation, Fourier-cosine series expansion, and Taylorseries expansion, an analytical solution is constructed.

Development of Analytical Techniques

Long-wave scattering by a circular cylinder mounted on a generalshoal is shown in Fig. 1, with the water depth being a real number h0plus a power function of radial distancewith an arbitrary value of thepower exponent, that is,

hðx; yÞ ¼ hðrÞ ¼�brm þ h0; ra # r, rbhb; r$ rb

ð1Þ

where b5 ðhb 2 h0Þ=rmb , the exponent m is a positive real number,and h0 is the deviation of the shoal crest from the still water level.When h0 5 0, the shoal degenerates into an idealized shoal; whenh0 . 0, the shoal crest is below the still water level, andwhen h0 , 0,the shoal crest is above the still water level. Let ha be the water depthalong the coastline, which can be expressed as

ha ¼ brma þ h0 ð2Þ

In this paper, always assume that ha $ 0.According to the linear long-wave theory, the water surface

elevation hðx; yÞ satisfies the linear long-wave equation as follows:

= × ðh=hÞ þ v2

gh ¼ 0 ð3Þ

in which =5 ð∂=∂x; ∂=∂yÞ; v 5 angular frequency; and g 5 grav-itational acceleration.

Usingcylindrical coordinates ðr;uÞwith x5r cos u and y5r sin u,Eq. (3) becomes

h∂2h∂r2

þ dhdr

∂h∂r

þ hr∂h∂r

þ hr2

∂2h∂2u

þ v2

gh ¼ 0 ð4Þ

In the outer regionwith constantwater depth hb, the water surfaceelevation h0 can be written as (MacCamy and Fuchs 1954)

h0ðr; uÞ ¼ P‘n¼0

hin«n JnðkbrÞ þ Að0Þ

n Hð1Þn ðkbrÞ

icos nu;

 ðr$ rb; 0 # u , 2pÞð5Þ

in which JnðkbrÞ 5 Bessel function of the first kind of order n;Hð1Þ

n ðkbrÞ5 Hankel function of the first kind of order n; kb 5 wavenumber with respect to the constant water depth hb; and the Jacobisymbol «n 5 1 for n5 0 and «n 5 2 for n. 0, respectively. Thecomplex constants Að0Þ

n are to be determined.In the inner region with variable water depth ra # r, rb, because

the seabed is axisymmetrical, a solution to Eq. (4) may be written asa Fourier-cosine series

h1ðr; uÞ ¼ P‘n¼0

RnðrÞcos nu;  ðra # r, rb; 0 # u, 2pÞ ð6Þ

inwhich the integern corresponds to the nth angular mode and RnðrÞis the corresponding coefficient that varies in r direction.

ByputtingEq. (6) into Eq. (4) and using the technique of variableseparation, the following ordinary differential equation is obtained:

r2h d2Rn

dr2þ

�hr þ r2dh

dr

�dRn

drþ

�v2

gr22 n2h

�Rn ¼ 0 ð7Þ

for n5 0; 1; 2; . . . ;‘. Further, substituting hðrÞ5brm 1 h0 intoEq. (7) yields

AðrÞ d2Rn

dr2þ BðrÞ dRn

drþ CðrÞRn ¼ 0 ð8Þ

for n5 0; 1; 2; . . . ;‘, where

Fig. 1. A definition sketch of a circular cylinder mounted on a general shoal with m5 2=3, rb 5 3ra: (a) h0 . 0; (b) h0 , 0

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AðrÞ ¼ brm12 þ h0r2

BðrÞ ¼ ðm þ 1Þbrm11 þ h0r

CðrÞ ¼ v2r2

g2 n2brm 2 n2h0

All of the coefficientsAðrÞ, BðrÞ, andCðrÞ are analytical functions inthe whole inner region with variable water depth.

When the deviation h0 5 0, Eq. (8) can be simplified into

rm12 d2Rn

dr2þ ðm þ 1Þrm11 dRn

drþ �

k2brmb r

2 2 n2rm�Rn ¼ 0

ð9Þ

and the general topography degenerates into idealized islands, ofwhich the paraboloidal case m5 2 was studied by Homma (1950),the conical case m5 1 was studied by Zhu and Zhang (1996), thegeneral case with m being an arbitrary real number was studied byYu and Zhang (2003), and the flat bottom case with m approachingzero was studied by MacCamy and Fuchs (1954).

Because of the idealization of the islands, Eq. (9) is either an Eulerequation (Homma 1950) or a Bessel equation (MacCamy and Fuchs1954), or can be transformed into Bessel equations after some variabletransformation (e.g., Zhu and Zhang 1996; Yu and Zhang 2003);therefore, analytical solutions in closed form have been obtained.

There is a unique singularity r1 5 0 in Eq. (9) that is alwaysoutside the complex disk: jr2 ðra 1 rbÞ=2j# ðrb 2 raÞ=2. However,when the deviation h0 � 0, besides the singularity r1 5 0, there areother singularities in Eq. (8) as follows:

rl ¼

�2h0

ha 2 h0

�1m

raei2ðl22Þp

m ; l ¼ 2; . . . ;m þ 1;

if m is an integer;�2h0

ha2 h0

�1m

raei2ðl22Þp

q ; l ¼ 2; . . . ; q þ 1;

if m ¼ qpðq $ 1; p $ 2Þ

8>>>>>>>>><>>>>>>>>>:

ð10Þ

If the deviation h0 , 0, then all the singular points are still outside thecomplex disk; if the deviation h0 . 0, the singular point r1 5 0 is stilloutside the physical domain, whereas other singular points may dropinto the complex disk. Therefore, different ways need to be used toconstruct series solutions toEq. (8) forh0 # 0 andh0 . 0, respectively.

Case I

For h0 # 0, no singularity of Eq. (8) drops into the complex diskjr2 ðra 1 rbÞ=2j# ðrb 2 raÞ=2; therefore, the solution RnðrÞ can beexpanded to Eq. (8) into a Taylor series at the ordinary point r5 rj 5ðra 1 rbÞ=2, which converges in the whole inner region. Using thetechnique proposed by Liu et al. (2004), the general solution RnðrÞ toEq. (8) can be written as

RnðrÞ ¼ Að1Þn Rð1Þ

n ðrÞ þ Að2Þn Rð2Þ

n ðrÞ ð11Þ

in which

Rð1Þn ðrÞ ¼ P‘

s¼0an;s

�r2 rj

�s;   Rð2Þ

n ðrÞ ¼ P‘l¼0

bn;l

�r2 rj

�l ð12Þ

where

an;0 ¼ 1 ð13Þ

an;1¼ 0 ð14Þ

an;2 ¼ 2C�rj�

2!A�rj� ð15Þ

an;s ¼ 2 1s!A

�rj� Ps22

n¼1

�s2 2

n

�ðs2 nÞ!An

�rj�an;s2n

þ Ps22

n¼0

�s2 2

n

�ðs2 n2 1Þ!Bn

�rj�an;s2n21

þ Ps22

n¼0

�s2 2

n

�ðs2 n2 2Þ!Cn

�rj�an;s2n22

;   s$ 3 ð16Þ

and

bn;0 ¼ 0 ð17Þ

bn;1 ¼ 1 ð18Þ

bn;2 ¼ 2B�rj�

2!A�rj� ð19Þ

bn;l ¼ 2 1l!A

�rj� Pl22

n¼1

�l2 2

n

�ðl2 nÞ!An

�rj�bn;l2n

þ Pl22

n¼0

�l2 2

n

�ðl2 n2 1Þ!Bn

�rj�bn;l2n21

þ Pl22

n¼0

�l2 2

n

�ðl2 n2 2Þ!Cn

�rj�bn;l2n22

;   l$ 3

ð20Þ

in which, AnðrjÞ;BnðrjÞ, and CnðrjÞ denote the nth order derivativesofAðrÞ,BðrÞ, andCðrÞ at the expansion point r5 rj, respectively. It isobvious thatAnðrjÞ;BnðrjÞ, andCnðrjÞ can be evaluated analytically.

Case II

For h0 . 0, the singular point r1 5 0 is still outside the complexdisk jr2 ðra 1 rbÞ=2j# ðrb 2 raÞ=2, whereas other singular pointsmay drop into the complex disk. In this case, if the precedingmethod is still used, to guarantee the convergence of the twosolutions of Eq. (12) in the whole inner region, the singular pointsr5 rlðl$ 2Þmust be required to be outside the complex disk, thatis, jrlj, ra or jrlj . rb, which implies that�����

�2h0

ha 2 h0

�1m

�����, 1 ð21Þ

or ������

2h0hb 2 h0

�1m

�����. 1 ð22Þ

that is,h0 , ha=2 or h0 . hb=2. Thismeans that the solution obtainedin the case h0 . 0 is valid onlywhen the shoal crest is slightly shiftedor sufficiently shifted from the still water level. This condition is

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similar to the restriction on the shoal submergence given by Niu andYu (2011a). The convergence of the analytical solution given byZhu and Harun (2009) should be also restricted on the shoalsubmergence, although they claimed that their solution was un-conditionally convergent in the whole inner region. In Zhu andHarun (2009, lines 4e5 on p. 318), the condition for a. b should beequivalent to h0 :h1 . 1=2, which is actually the same as the re-striction given by Niu and Yu (2011a). Recently, this restriction hasbeen totally removed by Liu and Xie (2011).

To overcome this restriction on the shoal submergence, thefollowing alternate transforms (Liu and Xie 2011) are used:

t ¼ brm

h0 þ brmð23Þ

RnðrÞ ¼ ~RnðtÞ ¼ ~Rn

�brm

h0 þ brm

�ð24Þ

which give

dRn

dr¼ m

�b

h0

�1m

t121mð12 tÞ111

md~Rn

dt

d2Rn

dr2¼ m2

�b

h0

�2m

t222mð12 tÞ212

md2~Rn

dt2

þ mðm2 2mt2 1Þ�

b

h0

�2m

t122mð12 tÞ112

md~Rn

dt

and Eq. (8) can be rewritten into

~AðtÞ d2~Rn

dt2þ ~BðtÞ d~Rn

dtþ ~CðtÞ~Rn ¼ 0 ð25Þ

with

~AðtÞ ¼ t2ð12 tÞ212m

~BðtÞ ¼ tð12 tÞ212m

~CðtÞ ¼ v2h2m2 10

m2gb2m

t2mð12 tÞ2 n2

m2ð12 tÞ2m

Under the variable transformEq. (23), the inner region ra # r, rbhas been mapped onto 12 ðh0=haÞ # t, 12 ðh0=hbÞ. It is clearthat there are two singular points, t5 0 and t5 1, in Eq. (25). Ifthe solution ~RnðtÞ is expanded into a Taylor series at the ordinarypoint t5 1=2, then the series solution converges in the complexdisk: jt2 1=2j, 1=2, which includes the whole inner region as itssubdomain.

By using the technique proposed by Liu et al. (2004), the generalsolution ~RnðtÞ to Eq. (25) can be written as

~RnðtÞ ¼ Að1Þn

~Rð1Þn ðtÞ þ Að2Þ

n~Rð2Þn ðtÞ ð26Þ

in which

~Rð1Þn ðtÞ ¼ P‘

p¼0gn;p

�t2 1

2

�p;   ~R

ð2Þn ðtÞ ¼ P‘

q¼0dn;q

�t2 1

2

�qð27Þ

where

gn;0 ¼ 1 ð28Þ

gn;1 ¼ 0 ð29Þ

gn;2 ¼ 2~C�12

�2! ~A

�12

� ð30Þ

gn;p ¼ 2 1

p! ~A�12

� Pp22

n¼1

�p2 2

n

�ðp2 nÞ! ~An

�12

�gn;p2n

þ Pp22

n¼0

�p2 2

n

�ðp2 n2 1Þ! ~Bn

�12

�gn;p2n21

þ Pp22

n¼0

�p2 2

n

�ðp2 n2 2Þ! ~Cn

�12

�gn;p2n22

;  p$ 3

ð31Þ

and

dn;0 ¼ 0 ð32Þ

dn;1 ¼ 1 ð33Þ

dn;2 ¼ 2~B�12

�2! ~A

�12

� ð34Þ

dn;q ¼ 2 1

q!~A�12

� Pq22

n¼ 1

�q2 2

n

�ðq2 nÞ! ~An

�12

�dn;q2n

þ Pq22

n¼ 0

�q2 2

n

�ðq2 n2 1Þ! ~Bn

�12

�dn;q2n21

þ Pq22

n¼ 0

�q2 2

n

�ðq2 n2 2Þ! ~Cn

�12

�dn;q2n22

;  q$ 3

ð35Þ

Therefore, the water surface elevation in the inner region can beexpressed as

h1ðr; uÞ ¼

(P‘n¼ 0

hAð1Þn Rð1Þ

n ðrÞ þ Að2Þn Rð2Þ

n ðrÞicos nu; h0 # 0

P‘n¼ 0

Að1Þn ~R

ð1Þn

�brm

h0 þ brm

�

þAð2Þn ~R

ð2Þn

�brm

h0 þ brm

�cos nu; h0 . 0

ð36Þ

the constants Að1Þn and Að2Þ

n are yet to be determined.The zero-flux condition along the coastline r5 ra requires

dh1ðr; uÞdr

����r5ra

¼ 0 ð37Þ

and the continuity of wave-surface elevations and flow fluxes acrossthe common boundary r5 rb between the inner region and the outerregion requires

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h0ðr; uÞjr5rb ¼ h1ðr; uÞjr5rb ;  ∂h0ðr; uÞ

∂r

����r5rb

¼ ∂h1ðr; uÞ∂r

����r5rb

ð38Þ

Eqs. (37) and (38) are equivalent to

2664

0 Rð1Þ9n ðraÞ Rð2Þ9

n ðraÞ2Hð1Þ

n ðkbrbÞ Rð1Þn ðrbÞ Rð2Þ

n ðrbÞ2kbH

ð1Þ9n ðkbrbÞ Rð1Þ9

n ðrbÞ Rð2Þ9n ðrbÞ

3775

Að0Þn

Að1Þn

Að2Þn

2664

3775

¼24 0

inenJnðkbrbÞinenkbJ9nðkbrbÞ

35 ð39Þ

for h0 # 0, and

26664

0 ~Rð1Þ9n ðlaÞ ~R

ð2Þ9n ðlaÞ

2Hð1Þn ðkbrbÞ ~R

ð1Þn ðlbÞ ~R

ð2Þn ðlbÞ

2kbHð1Þ9n ðkbrbÞ s~R

ð1Þ9n ðlbÞ s~R

ð2Þ9n ðlbÞ

37775

Að0Þn

Að1Þn

Að2Þn

2664

3775

�24 0

inenJnðkbrbÞinenkbJ9nðkbrbÞ

35 ð40Þ

for h0 . 0, where la 5 12 h0=ha; lb 5 12 h0=hb; and s5mh0ðhb 2 h0Þ=h2brb.

By solving the systemEq. (39) and the system Eq. (40), and usingthe Wronskian identity

JnðxÞHð1Þ9n ðxÞ2 J9nðxÞHð1Þ

n ðxÞ ¼ 2ipx

ð41Þ

all the coefficients Að0Þn , Að1Þ

n and Að2Þn can be determined as follows:

AðiÞn ¼ Di=D;   i ¼ 0; 1; 2 ð42Þ

where Di and D are given as follows.When h0 # 0

D0 ¼ inen

��������0 Rð1Þ9

n ðraÞ Rð2Þ9n ðraÞ

JnðkbrbÞ Rð1Þn ðrbÞ Rð2Þ

n ðrbÞkbJn9ðkbrbÞ Rð1Þ9

n ðrbÞ Rð2Þ9n ðrbÞ

��������ð43Þ

D1 ¼ 2in11enprb

Rð2Þ9n ðraÞ ð44Þ

D2 ¼ 22in11enprb

Rð1Þ9n ðraÞ ð45Þ

D ¼ 2

��������0 Rð1Þ9

n ðraÞ Rð2Þ9n ðraÞ

Hð1Þn ðkbrbÞ Rð1Þ

n ðrbÞ Rð2Þn ðrbÞ

kbHð1Þ9n ðkbrbÞ Rð1Þ9

n ðrbÞ Rð2Þ9n ðrbÞ

��������ð46Þ

When h0 . 0

D0 ¼ inen

���������0 ~R

ð1Þ9n ðlaÞ ~R

ð2Þ9n ðlaÞ

JnðkbrbÞ ~Rð1Þn ðlbÞ ~R

ð2Þn ðlbÞ

kbJn9ðkbrbÞ s~Rð1Þ9n ðlbÞ s~R

ð2Þ9n ðlbÞ

���������ð47Þ

D1 ¼ 2in11enplb

~Rð2Þ9n ðlaÞ ð48Þ

D2 ¼ 22in11enprb

~Rð1Þ9n ðlaÞ ð49Þ

D ¼ 2

���������0 Rð1Þ9

n ðlaÞ ~Rð2Þ9n ðlaÞ

Hð1Þn ðkbrbÞ ~R

ð1Þn ðlbÞ ~R

ð2Þn ðlbÞ

kbHð1Þ9n ðkbrbÞ s~R

ð1Þ9n ðlbÞ s~R

ð2Þ9n ðlbÞ

���������ð50Þ

Recently, the same scattering problem was also considered byJung and Lee (2012), where a variable transform different from thetransform Eq. (23) was employed, and then an analytical solution inthe form of a Frobenius series was obtained. However, their solutionis valid only when the shoal crest is located under the still water level(i.e., the deviation h0 must be positive) (Jung and Lee 2012, p.154).Otherwise, their solution may diverge in the physical domain of theproblem. There is no doubt that the present analytical solutionwithout any restriction on h0 is more general.

Results and Discussion

In this section, the present analytical solution for general islandsshall be validated against four classical analytical solutions foridealized islands. Further, the present analytical solution is used toinvestigate the influence of the submergence of the island, the powerorder m, and the shoal size to the wave amplification along thecoastline.

Validation against Homma’s Solution and Discussion

Homma (1950) designed an island system in which a circular cy-lindrical island is mounted on an idealized paraboloidal shoal andthen presented an analytical solution of the long-wave equation forthe case of ra 5 10 km, rb 5 30 km, and hb 5 4 km. BecauseHomma’s analytical solution was found, it has been employed toverify various numerical models based on the long-wave equation(Vastano and Reid 1967; Bettess and Zienkiewicz 1977). Becauseanalytical solution is very rare, Homma’s long-wave analyticalsolution has also been used by some researchers to check their moregeneral numericalmodels based on themild-slope equation (Jonssonet al. 1976; Houston 1981; Tsay and Liu 1983; Zhu 1993; Zhu et al.2000; Hsiao et al. 2009) and so on.

Clearly, whenm5 2, ra 5 10 km, rb 5 30 km, hb 5 4 km, andh0 5 0 m, the present general topography actually degenerates intoa circular cylindrical island mounted on an idealized paraboloidalshoal, and Homma’s (1950) analytical solution is applicable if thewater depth hb is kept shallow enough.

The general analytical model herein is first validated againstHomma’s analytical solution. For T5 480 s, h0 5 2200,2100, 0,100, 200, 300, and 400m, respectively, inwhich the case h0 5 0 m isthe case studied by Homma (1950). It is easy to see that the cor-responding value of kbhb 5 0:2645, which is in the long-wave range.As shown in Fig. 2, as h0 approaches zero, the normalized waveamplitudes along the coastline calculated by the general analytical

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model also approach Homma’s (1950) analytical solution. Onceh0 5 0, the present analytical solution is exactly the same asHomma’s (1950) analytical solution.

Validation against Zhu and Zhang’s Solution andDiscussion

Secondly, the present analytical solution for general islands withm5 1 is validated against the analytical solution for an idealizedisland presented by Zhu and Zhang (1996). The parameters ra 5 10km, rb 5 30 km, hb 5 4 km,m5 1, and T 5 720 s are chosen. Thecorresponding value of kbhb 5 0:1763, which is in the long-waverange. Clearly, when h0 5 0, the general island topographydegenerates into a circular cylindrical island mounted on an ide-alized conical shoal, to which Zhu and Zhang’s (1996) analyticalsolution is applicable.

The normalized wave amplitudes along the coastline are calcu-lated using the present general analytical model for the water depthh0 in the island center ranging from 21,800 to 2,000 m. The sol-utions for all these cases are presented in Fig. 3 together with theanalytical solution presented by Zhu and Zhang (1996). It can beseen that when h0 5 0, the present general analytical solution agreesperfectly with Zhu and Zhang’s (1996) analytical solution. In ad-dition, Fig. 3 exhibits the interaction between the diffraction andrefraction effects. When the total water depth hb as well as theincident wave period T are held constant, the diffraction effects aresignificantly enhanced as the water depth h0 in the island centerdecreases, which leads to the increase of the shoal size. When theshoal size is small (i.e., h0 5 2;000 m), the refraction is weak andthe maximum normalized wave amplitude is only about 1.5. As h0decreases to21,800m, the size of the shoal becomes larger; not onlydoes the maximum normalized wave amplitude increase to about2.5, but also the variation of normalized wave amplitude along thecoastline is significantly enhanced.

Validation against Yu and Zhang’s Solution andDiscussion

Thirdly, the present general analytical solution for general islands isvalidated against the analytical solution for an idealized islandpresented byYu andZhang (2003). In Yu and Zhang (2003), a resultof normalized wave amplitude along the coastline was displayed forlinear long waves with period being T5 410 s scattered by a circularcylindrical island mounted on an idealized shoal with m54, ra510 km, hb54 km, ha5444 m, and h050 m. After some simplecalculation, rb517:32484 km. To validate the general analyticalmodel against this analytical solution, m54, ra510 km, rb517:32484 km, hb54 km, and ha5444 m are chosen, but let thewater depth h0 in the island center vary from 2400 to 700 m.

The solutions for all these cases are plotted in Fig. 4 together withthe analytical solution presented byYuandZhang (2003).When h0 50 m, the agreement between thepresent general analytical solution andYu and Zhang’s (2003) analytical solution is excellent. Because all theislands have a relatively small shoal size with rb 5 17:32484 km, thenormalized wave amplitudes along the coastline are not significant.

Validation against Maccamy and Fuchs’ Solution andDiscussion

Finally, the present analytical model for general islands is validatedagainst the analytical solution presented by MacCamy and Fuchs(1954) for a circular cylindrical island. When the parameters ra, rb,h0, hb are fixed and power order m decreases to zero, a series ofgeneral islands are obtained that will approach a circular cylindricalisland with the radius being ra, which is the case in whichMacCamyand Fuchs’ (1954) analytical solution becomes applicable.

At first, for ra510 km, rb530 km, hb54 km, h05500 m, andT5720 s, m5 1=2, 1/22, 1/24 and 1/28, respectively. The relatednormalizedwave amplitudes along the coastline are calculated usingthe present general analytical model, and the solutions for all these

Fig. 2. The present analytical solutions for general islands are validated against Homma’s (1950) analytical solution for an idealized island with ra 510 km, rb 5 30 km, hb 5 4 km, T 5 480 s, m5 2

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four cases are presented in Figs. 5(a and b) together with the ana-lytical solution presented by MacCamy and Fuchs (1954). It can beseen that as m decreases to zero, the present analytical solutionsapproach MacCamy and Fuchs’ analytical solution for a cylindricalisland. When m51=28, MacCamy and Fuchs’ analytical solutiontotally overlaps the present general analytical solution. It can also be

seen that, because of the lack of refraction effect, the maximumnormalized wave amplitude in MacCamy and Fuchs’ analyticalsolution is only about 1.3, which is the smallest among all five cases.

Secondly, for ra510 km, rb530 km, hb54 km, h052500 m,and T5480 s, m5 1=2, 1/22, 1/24, and 1/28, respectively. Similarverification can be seen in Figs. 6(a and b).

Fig. 3. The present general analytical solutions with T5 720 s for general islands are validated against Zhu and Zhang’s (1996) analytical solution foran idealized conical island with ra 5 10 km, rb 5 30 km, hb 5 4 km, and m5 1

Fig. 4. The present general analytical solutions with T 5 410 s for general islands are validated against Yu and Zhang’s (2003) analytical solution foran idealized island with ra 5 10 km, rb 5 17:32484 km, hb 5 4 km, Ha 5 444 m, and m5 4

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Shoal Size Effect on the Wave-Scattering Pattern

With the previous validation of the present general analytical model,how the shoal size would affect the wave-scattering pattern is in-vestigated. Forh0 5 0 m and h0 5 500 m, the following parametersare chosen: ra 5 10 km, hb 5 4 km, m5 1, and T 5 720 s. Thenormalized wave amplitudes along the coastline are calculated for sixcases,with the toe size rb of the shoal ranging from ra to 6ra; the resultsare displayed in Figs. 7(a and b), where two cases for rb 5 ra are thecase that a cylindrical island stand on the flat sea bottom, for whichMacCamy and Fuchs’ (1954) analytical solution is applicable.

For each individual case of h0 5 0 m and h0 5 500 m, it can beseen from Figs. 7(a and b) that when rb increases, not only does the

maximumnormalized wave amplitude increase, but also the variationof normalized wave amplitude along the coastline is significantlyenhanced. It can also be seen that, without the shoal effect for the casewith rb 5 ra, the associated normalized wave amplitude along thecoastline is the smallest and shows the flattest variation. Further, bycomparing Figs. 7(a and b), the normalized wave amplitudes for thecase h0 5 0 m is globally more significant than that for the case h0 5500 m because all the shoals are flatter for the latter case.

Conclusion

In this paper, a general analytical solutionwas derived for long-wavescattering by a circular cylindrical island mounted on a general

Fig. 5. Comparison between the present analytical solutions for general islands with different power m and MacCamy and Fuchs’ (1954) analyticalsolution for a cylindrical island with T 5 720 s and h0 5 500 m: (a) the topographies; (b) normalized wave amplitudes

Fig. 6. Comparison between the present analytical solutions for general islands with different power m and MacCamy and Fuchs’ (1954) analyticalsolution for a cylindrical island with T 5 480 s and h0 5 2500 m: (a) the topographies; (b) normalized wave amplitudes

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shoal, that is, the shoal crest is not restricted to be on the still waterlevel, which leads to the variable water depth being a generalpolynomial function of the radial distance. Because of this gener-ality, the singularity of the long-wave equation becomes compli-cated, and the solution technique needs to be more skillful.

Also, because of the generality, the general analytical solutionobtained in this study finds three well-known analytical solutions ofthe long-wave equation to be its special cases. These include wavescattering by an idealized paraboloidal island (Homma 1950), by an

idealized conical island (Zhu and Zhang 1996), and by an idealizedarbitrary island (Yu and Zhang 2003). It is also shown that the well-known case of wave diffraction by a circular cylinder standing ina flat sea bottom studied byMacCamy and Fuchs (1954) is a limitingcase of wave scattering by the general islands considered in thispaper when m approach zero. In addition, the present solution issuperior to Jung and Lee’s (2012) analytical solution for the samescattering problem, because there is a restriction in the latter that theshoal crest must be located under the still water level. Naturally, this

Fig. 7. Variation of normalized wave amplitudes along the coastline for different shoal sizes rb: (a) h0 5 0 m; (b) h0 5 500 m

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analytical solution covers a much wider range of problems and maybe very useful in testing numerical models.

Acknowledgments

The first writer is supported by the Natural Science Foundation ofChina (10962001, 51149007), Guangxi Natural Science Foundation(2010GXNSFA013115, 2011GXNSFD018006) and Scientific Re-search Foundation of Guangxi Universities (201102ZD014). Allthe writers would like to gratefully acknowledge some very usefulsuggestions from three anonymous referees.

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