Analytical solutions for long waves over a circular island

Analytical solutions for long waves over a circular island

Coastal Engineering 57 (2010) 440–446 Contents lists available at ScienceDirect Coastal Engineering j o u r n a l h o m e p a g e : w w w. e l s ev ...

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Coastal Engineering 57 (2010) 440–446

Contents lists available at ScienceDirect

Coastal Engineering j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o a s t a l e n g

Analytical solutions for long waves over a circular island Tae-Hwa Jung a, Changhoon Lee b,⁎, Yong-Sik Cho c a b c

Department of Civil Engineering, Hanbat National University, San 16-1, Duckmyoung-dong, Yuseong-gu, Daejeon 305-719, South Korea Department of Civil and Environmental Engineering, Sejong University, 98 Kunja-dong, Kwangjin-gu, Seoul 143-747, South Korea Department of Civil and Environmental Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, South Korea

a r t i c l e

i n f o

Article history: Received 1 December 2008 Received in revised form 12 November 2009 Accepted 23 November 2009 Available online 16 December 2009 Keywords: Long wave Circular island Analytical solution Arbitrary bottom slope

a b s t r a c t In this study, we derive an analytical solution for long waves over a circular island which is mounted on a flat bottom. The water depth on the island varies in proportion to an arbitrary power, γ, of the radial distance. Separation of variables, Taylor series expansion, and Frobenius series are used to find the solutions, which are then validated by comparing them with previously developed analytical solutions. We also investigate how different wave periods, radii of the island toe, and γ values affect the solutions. For a circular island with a small value of γ (e.g. γ = 2/3, as in the equilibrium beach (Bruun, 1954)), the wave rays approaching near the island center reach the coastline, whereas the rays approaching away from the center bend away from the coastline, leading to smaller wave amplitudes along the coast. However, for a circular island with a large value of γ, e.g. γ = 2, all the rays on the island reach the coast, giving large coastline wave amplitudes. If the island domain is small compared to the wavelength, the wave amplitudes on the coastline do not increase significantly; however, when the island domain is not small, the wave amplitudes increase significantly. If γ is also large, the amplitudes can be so large as to cause a disaster on the island. © 2009 Elsevier B.V. All rights reserved.

1. Introduction There are several methods used to study the dynamic behavior of water waves, including physical experimentation, numerical simulation, and the finding of analytical solutions. Before the development of mathematical formulae to describe dynamic behavior, physical experiments were the preferred method of investigation. However, due to high cost and their time consuming nature, experiments were usually conducted under very limited conditions, often leading to distorted information due to the small model scale. Since the advent of computers, numerical simulations (finite difference methods, finite element methods and so on) have become increasingly popular, because one is able to investigate more varied conditions with lower costs and in less time. However, numerical simulations may still produce erroneous solutions if the boundary conditions are not properly specified. For example, an open boundary condition at the offshore boundary requires the outgoing waves to pass through the boundary without any reflection, a condition which has not been perfectly specified in two dimensions until now. An approximate way to accomplish this is to put sponge layers at the offshore boundaries and internally generate waves on an arc inside the computational domain (Lee and Suh, 1998; Lee and Yoon, 2007). The final course of investigation, searching for analytical solutions, is used today mainly to verify numerical solutions and compare experimental data. ⁎ Corresponding author. Tel: + 82 2 3408 3294; fax: + 82 2 3408 4332. E-mail addresses: [email protected] (T.-H. Jung), [email protected] (C. Lee), [email protected] (Y.-S. Cho). 0378-3839/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2009.11.010

Analytical solutions on simple bottom topographies have been developed in only a limited manner. Several researchers have obtained analytical solutions for long waves on axis-symmetric topographies, such as a circular cylindrical island mounted on a paraboloidal shoal (Homma, 1950; Vastano and Reid, 1967; Jonsson et al., 1976), a conical island (Zhang and Zhu, 1994), a circular cylindrical island mounted on a conical shoal (Zhu and Zhang, 1996), a circular paraboloidal pit (Suh et al., 2005), and a circular hump (Zhu and Harun, 2009). Recently, solutions that can be applied to deeper water depths were developed by Liu et al. (2004), Lin and Liu (2007), and Jung and Suh (2007). However, most solutions are limited to cases in which the water depth on a varying topography is proportional to the first or second power of the radial distance; in real bottom topography, however, the water depth may be proportional to a power of the radial distance that is less than unity. In particular, Bruun (1954) argues that, in an equilibrium beach, the water depth is actually proportional to twothirds the power of the offshore distance from the shoreline. In this study, we develop analytical solutions for long waves over a circular island. The present solution does not have any limitation on the value of the exponent relating the water depth to the radial distance, whereas Zhang and Zhu's (1994) previous solutions were limited only to conical islands. In Section 2, we develop analytical solutions using separation of variables, Taylor series expansion, and Frobenius series. In Section 3, we verify the developed solutions by comparing them with previous solutions. We also investigate the solutions with different incident wave periods, island toe radii, and powers of the radial distance. Finally, we summarize our work in Section 4.

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the island toe. The vertex level, hδ, is determined from r0, r1, h1 and γ by

2. Development of analytical solutions For linear long waves of incompressible and inviscid water, the mild-slope equation of Berkhoff (1972) can be written in polar coordinates (r, θ) as ! 2 2 2 ∂ η 1 ∂η 1∂ η dh ∂η ω η=0 + h + 2 2 + + 2 r ∂r dr ∂r g ∂r r ∂θ

ð1Þ

where h is the still water depth, η is the water surface elevation, ω is the angular frequency, and g is the gravitational acceleration. By ∞

separating the variable η as ηðr; θÞ = ∑ Rn ðrÞΘn ðθÞ, Eq. (1) can be n=0

split into the following two equations: 2

d R h 2n + dr

  h dh dRn + + r dr dr

! 2 2 ω n h − 2 Rn = 0 g r

Θn ðθÞ = C1n cos nθ + C2n sin nθ

hδ =

h1 ½ðr1 =r0 Þγ −1

ð2Þ

ð3Þ

The general solution of Eq. (2) can be obtained in terms of the Frobenius series, which requires the physical domain of interest to be located within a convergent circle of the series solution. To ensure the domain of interest lies within the convergent region, we employ the following mapping: r γ t = 1− 0 r

2

tð1−tÞ

ð6Þ

"  2 # d 2 Rn ω2 r02 n 2 dRn −γ2 +1 + ð1−tÞ ð1−tÞ −t Rn = 0 + γ dt dt 2 ghδ γ2



Rn = ∑ αm;n t

ð4Þ

where, r0 and r1 are the radial distances from the center to the coastline and the island toe, respectively, h1 is the water depth at

m+c

ð8Þ

m=0

Using a Taylor series expansion, the first term in the square brackets of Eq. (7) can be expressed in polynomial form as −γ2 + 1

ω r0 ð1−tÞ g

r≥r1

ð7Þ

The inner region of varying water depths (r0 ≤ r ≤ r1) is mapped onto 0 ≤ t ≤ 1 − (r0/r1)γ b 1, which is within the convergent region. From Frobenius' theory, we obtain the following series solution to Eq. (7):

2 2

r0 ≤rbr1

ð5Þ

Under this mapping, Eq. (2) can be written as

where Rn(r) is the function of r corresponding to each eigenvalue n, and C1n and C2n are constants. We consider for waves over a circular island on a flat bottom. The water depth on the island varies in proportion to an arbitrary power, γ, of the radial distance. Fig. 1 shows the computational domain for waves over the circular island with γ = 2/3. The water depth may be expressed as 8  γ  r > : h1 ;

441



= ∑ βl t

l

ð9Þ

l=0

where β0 = ω 2r02/g. Substituting Eqs. (8) and (9) into Eq. (7) and collecting terms with the same power of t gives, at the lowest-order, the solution of c as c = 0 ðdouble rootÞ

ð10Þ

and, further, two linearly independent solutions as ∞

Rn = ∑ αm;n t

m

ð11Þ

m=0



Rn = ln t ∑ αm;n t

m

ð12Þ

m=0

where αm,n is a constant to be determined. The condition that the water surface elevation must be finite at the coastline (t = 0) excludes the solution given by Eq. (12). Substituting Eq. (11) into Eq. (7) and collecting the terms of the same order of t give following recursive relations: α1;n = −β0 α0;n

ð13Þ m

2ðm−1Þ2 αm−1;n −½ðm−2Þ2 −ðn= γÞ2 αm−2;n − ∑ βk αm−k−1;n αm;n =

k=0

m2

ðm≥2Þ

ð14Þ Since the values of αi,n with i ≥ 1 can be determined in terms of α0,n from Eq. (7), the solution of Eq. (11) can be expressed as

Fig. 1. Computational domain for waves over a circular island with γ = 2/3.

Rn = α0;n R0;n ðtÞ = α0;n 1 +

! α1;n α2;n 2 t+ t +⋯ α0;n α0;n

ð15Þ

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Finally, the water surface elevation of long waves on a varying water depth can be expressed as ∞

ηi = ∑ ½α0;n R0;n ðtÞ cos nθ

ð16Þ

n=0

where the subscript i denotes the inner region with varying water depths, and the terms with sin nθ have been dropped due to symmetry about the x-axis. The water surface elevation ηo in the outer region with constant water depth can be represented by I

S

ηo = η + η

ð17Þ

where ηI and ηS are the incident and scattered waves, respectively. These can be expressed as (see Mei (1989) for details) I



n

η = ∑ i εn Jn ðkrÞ cos nθ

ð18Þ

n=0

S



ð2Þ

η = ∑ Dn Hn ðkrÞ cos nθ

ð19Þ

n=0

where Jn is the Bessel function of the first kind of order n, H(2) n is the Hankel function of the second kind of order n, εn is the Jacobi symbol (i.e., εn = 1 for n = 0 and εn = 2 for n ≥ 1), and Dn is a set of complex constants to be determined. To determine the unknowns α0,n and Dn, the following matching conditions are used: ηi = ηo ;

r = r1

ð20Þ

∂ηi ∂ηo = ; ∂r ∂r

r = r1

ð21Þ

which mean the continuities of pressure and radial velocity at r = ri, respectively. Thus, we get the unknowns as Jn ðkt1 ÞHnð2Þ ′ðkt1 Þ−Jn′ ðkt1 ÞHnð2Þ ðkt1 Þ

n

α0;n = ki εn

n

Dn = i εn

ð2Þ

ð2Þ

kR0;n ðt1 ÞHn ′ðkt1 Þ−R′0;n ðt1 ÞHn ðkt1 Þ

kJn′ ðkt1 ÞR0;n ðt1 Þ−Jn ðkt1 ÞR′0;n ðt1 Þ ð2Þ ð2Þ Hn ðkt1 ÞR′0;n ðt1 Þ−kHn ′ðkt1 ÞR0;n ðt1 Þ

ð22Þ

ð23Þ

where t1 = 1 − (r0 / r1)γ. When the unknowns are determined, the following boundary condition at the coastline (Mei, 1989) is automatically satisfied, which means that the normal flux at the coastline is zero: lim h

r→r0

∂η =0 ∂r

ð24Þ

3. Investigation of analytical solutions Our analytical solutions were verified by comparing them with previous solutions for waves on a conical island, and the solutions were further investigated by varying the radius of the island toe, wave period, and γ value. The effects of wave diffraction and refraction were also investigated in more detail. 3.1. Verification and application For verification, we compared the present solution to that of Zhang and Zhu (1994) for waves on a conical island with γ = 1. The conditions of bottom topography used were r0 = 10 km, r1 = 30 km, and h1 = 4 km. Fig. 2 shows the variation of the normalized wave amplitudes along the coastline with incident wave periods of

T = 12 min and T = 24 min. The present solution is the same as the solution found by Zhang and Zhu. Zhu and Zhang (1996) showed that, for long waves around a cylindrical island mounted on a conical shoal, the wave amplitude and phase depend on a dimensionless geometric parameter r1 / r0 (r0 and r1 are the radii of coastline and shoal toe, respectively) and a dimensionless wavelength L1 / r1 (L1 is the wavelength on a constant water depth). Zhang and Zhu (1994) argued that the solution for waves on a conical island with γ = 1 depends on the values of r1 / r0 (in this case, r1 is the radius of island toe), L1 / r1, and combinations of the water depth h1 and the wave period T. For waves over a circular island with an arbitrary value of γ, we also add γ as an additional parameter. For the bottom topography tested previously, the dimensionless geometric parameter was r1 / r0 = 3 and the dimensionless wavelengths were L1 / r1 = 4.7 and L1 / r1 = 9.5 for wave periods of T = 12 min and T = 24 min, respectively. When the wave period was 24 min, the wave amplitude variation was not significant. When the incident wave period was reduced to 12 min, however, the variation became significant, because the refractive focusing of wave energies over the island occurred more prominently for waves with a shorter period. The largest wave amplitudes were observed at the front face (i.e., θ = 0°) because of refractive focusing of the wave energy and, to a minor degree, wave reflection. At the rear side, around θ = 120°, the wave amplitudes were minimal because the wave energy was reduced due to diffraction in the shadow zone. Where θ N 120°, the wave amplitudes increased with increasing θ because the refracting and diffracting wave rays intersected from opposite directions, implying that the leeside of an island is not safe when a tsunami approaches. Pocinki (1950) investigated refraction of waves on a circular island where the water depths were almost constant near the coastline of r = r0 and increased gradually to the island toe of r = r1. He showed that, when ln(r1 / r0) b π / 2, no wave rays could reach the lee side of the island. This happened because there were almost no wave deflections near the coastline and the island was not so large to allow deflection of wave rays all around the island. Mei (1989) showed that, when ln(r1 / r0) N π / 2, the island was large enough for some wave rays to reach the lee side and further intersect from opposite directions as seen in the present case. For the sensitivity analysis, we first varied the radial distance of the island toe r1 to study the effects of the dimensionless geometric parameter r1 / r0. Second, we varied the wave period to study the effects of the dimensionless wavelength L1 / r1. For each case, we tested the solutions with γ = 2/3, 1, and 2 in order to study the effects of water depth as a power of radial distance. The bottom topography with γ = 2/3 is the same as the equilibrium beach profile of Bruun (1954). Fig. 3 shows the variation of normalized wave amplitudes along the coastline for different island toe radii r1. When the island toe radius was r1 = 3r0, all three cases (γ = 2/3, 1, and 2) showed a similar trend in that the amplitudes were maximized at the front face, decreased to a minimal value around θ = 120°, and then increased up to θ = 180°. The difference in wave amplitudes for different values of γ was not significant in this case because the domain was too small for the waves to feel the bottom topography. However, wave diffraction did dominantly affect the solution. As the radius of the island toe r1 increased to r1 = 9r0, the wave energies were more focused at the coastline due to refraction. Also, the number of maximal and minimal amplitudes increased with an increase in γ. The wave amplitude variation along the coastline was more significant with a larger γ, regardless of the island toe radius. Fig. 4 shows the variation of the normalized wave amplitudes along the coastline for the wave periods T = 12 min, 24 min. The corresponding dimensionless wavelengths were L1 / r1 = 4.7, 9.5. The shortest wave period satisfying the shallow-water wave condition was 6.8 min. When the wavelength was long as compared to the

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solution was dominated by the diffraction effect whereas, when γ = 2, refraction dominated instead. 3.2. Analysis of the diffraction and refraction effects

Fig. 2. Variation of normalized wave amplitudes along the coastline of a conical island with γ = 1: T = 12 min, r0 = 10 km, r1 = 30 km, h1 = 4 km.

island toe radius (as at T = 24 min), the wave amplitude variation was not significant, nor did it depend on γ. This is because the domain of varying water depths was so small that the refraction effect was minor and the diffraction effect was dominant. As the wavelength became shorter (as at T = 12 min), however, the value of γ did have a significant effect on the solution. In particular, when γ = 2/3, the

Fig. 3. Variation of normalized wave amplitudes along the coastline of a circular island with different radii of island toe: T = 12 min, r0 = 10 km, h1 = 4 km: (a) r1 = 3r0; (b) r1 = 9r0.

In order to analyze the diffraction effect, we tested waves around a cylindrical island on a flat bottom. The island radius was 10 km (=r0), and the water depth was set to either 4 km (=h1), 2 km(=h1 / 2), or 1 km(=h1 / 4). The wave period was T = 12 min. Fig. 5 shows the analytical solutions of the normalized wave amplitudes along the island wall. With a 4 km water depth, the wave amplitude was maximal at the front of the island and decreased monotonically up to θ = 180°. This happened because the island radius was small compared to the wavelength (i.e., L / r0 = 14.2), and so the wave rays originating from opposite sides of the x-axis did not intersect on the leeside of the island. When the water depth was 2 km, the amplitude decreased to a minimum value around θ = 140° and then increased up to θ = 180°. Here, the island radius was comparable to the wavelength (i.e., L / r0=10.1), and so the wave rays did intersect on the leeside. When the water depth was 1 km, even more wave rays intersected on the leeside. The trends observed for the 2 km and 1 km water depths were similar to those seen for waves on a circular island with a smaller domain (r 1 = 3r 0 ) and longer wave periods (T = 12 min and T = 24 min) (see Figs. 2, 3a, and 4a–b). Therefore, we may conclude that, for longer period waves on a small circular island, wave

Fig. 4. Variation of normalized wave amplitudes along the coastline of a circular island with different wave periods: r0 = 10 km, r1 = 30 km, h1 = 4 km: (a) T = 12 min; (b) T = 24 min.

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T.-H. Jung et al. / Coastal Engineering 57 (2010) 440–446

Fig. 5. Variation of normalized wave amplitudes along the wall of a cylindrical island on a flat bottom: r0 = 10 km, T = 12 min.

Fig. 6. Variation of relative water depths kr with radial distance of a circular island with r1 = 9r0: T = 12 min, r0 = 10 km, h1 = 4 km.

amplitudes will decrease on the rear side of the coastline (due to diffraction) and will increase on the leeside (due to the intersection of wave rays). In order to analyze the refraction effect, we investigated how wave rays deflected over circular islands with γ = 2/3 and γ = 2. The wave period was T = 12 min and the bottom topography was expressed by r0 = 10 km, h1 = 4 km. The island toe radius was r1 = 9r0. When water depth depends only on the radial distance, the deflection of wave rays depends on the relative wave number kr (Mei, 1989), and the direction of the wave ray θ is determined by

θ = 970° for γ = 2 and γ = 2/3, respectively. This means that, for all cases, wave rays from opposite sides of the x-axis intersected on the leeside of the island and the wave amplitudes increased on the side of intersection. We also found that the variation of wave amplitudes on the island was more significant for r1 = 9r0 compared to the case with r1 = 3r0 due to more refraction and intersections. It should be noted that the ray approach shows the refraction effect not the diffraction.

dθ jκj =F 2 2 dr rðk r −κ2 Þ

ð25Þ

or, in integral form, θ−θ1 = Fj κj ∫

r r1

dr rðk2 r 2 −κ2 Þ1 = 2

ð26Þ

where κ = kr sin θ(=k1r1 sin θ1 = constant) is determined by the initial position and direction of the ray at r = r1. From Eq. (26), the rays exist only in the region where kr N κ. At the critical radius r = r⁎, we have kr = κ and the corresponding wave angle θ = θ⁎ is determined by r

θ⁎ −θ1 = Fjκ j ∫

*

r1

dr rðk2 r 2 −κ2 Þ1 = 2

ð27Þ

From Eq. (25), dr/dθ = 0 at (r⁎, θ⁎); that is, the ray is either closest to or farthest from the island center. Fig. 6 shows the variation of the relative wave numbers kr with the radius r for r1 = 9r0. When γ = 2, the value of kr increased monotonically as the radius decreased from the island toe to the coastline. When γ = 2/3, as the radius decreased, the value of kr also decreased to a minimum value of kr = 2.09 (at r⁎ = 0.20r1), and then increased to infinity at the coastline. We found that for the cases with γ ≤ 1.97, the value of kr first decreased to a minimum and then increased with a decrease in radius. Fig. 7b shows the rays when γ = 2 with r1 = 9r0. In the figure, the wave rays started at the up-wave boundary of x/r1 = 2 with an interval of Δy = 0.1r1. All the rays on the island bent toward and reached the coastline. Fig. 7a shows the rays when γ = 2/3 with r1 = 9r0. Here, the wave rays approaching near the island center reached the coastline, whereas the rays approaching away from the center first bent toward, and then away from, the center after the minimum r*. The rays reached the coastline up to θ = 1129° and

Fig. 7. Wave rays for a circular island with r1 = 9r0: T = 12 min, r0 = 10 km, h1 = 4 km: (a) γ = 2/3; (b) γ = 2.

T.-H. Jung et al. / Coastal Engineering 57 (2010) 440–446

Thus, the resulting rays would be bent more than the analytical solutions which include the diffraction effect as well. It should be noted also that the values of bent angle qualitatively show the refraction effect because the bent angles depend on the ray interval Δy. In order to analyze the effects of refraction and intersection in more detail, we investigated the contours of normalized wave amplitudes for r1 = 9r0, as shown in Fig. 8. When γ = 2/3, the wave rays approaching the island center bent towards the coastline, whereas the wave rays approaching away from the center bent away from it, meaning that the superposition of wave rays was not significant near the coastline. However, when γ = 2, all of the rays

Fig. 8. Contour of normalized wave amplitudes for a circular island with r1 = 9r0: T = 12 min, r0 = 10 km, h1 = 4 km: (a) γ = 2/3; (b) γ = 2.

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approaching the island reached the coastline. Furthermore, some rays deflected more than 360° before reaching the coastline. Thus, the superposition of wave rays occurred several times near the coastline, and the amplitudes were more than 40 times the incident amplitude at some positions. In Fig. 8, the dash–dotted line represents the radial line crossing the node or anti-node of the coastline (see Fig. 3b). Fig. 9 shows the variation of wave amplitudes along the radial line (i.e., the dash–dotted line in Fig. 8), which crosses the node or antinode at the coastline for r1 = 9r0. When γ = 2/3, the variation of wave amplitudes was not significant and there were a few nodes or antinodes along the radial line. When γ = 2, however, the variation of

Fig. 9. Variation of normalized wave amplitudes along radial line which crosses the node or anti-node at the coastline of a circular island with r1 = 9r0: T = 12 min, r0 = 10 km, h1 = 4 km: (a) γ = 2/3; (b) γ = 2.

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amplitudes was significant, especially near the coast (r = r0), and there were several nodes and anti-nodes along the radial line due to the superposition of wave rays. 4. Conclusions In this study, we derived an analytical solution for long waves over a circular island on a flat bottom. The water depth on the island varies in proportion to an arbitrary power, γ, of the radial distance. The method of the separation of variables was used to transform the partial differential equation into an ordinary differential equation and, prior to the use of the Frobenius series solution, the transformation of variables was performed to ensure that the variable water depths were within a convergent region. For waves over a conical island with γ = 1, excellent agreement was found between the present solution and that of Zhang and Zhu (1994). The analytical solutions were further investigated by varying the island toe radius r1 or wave period T in order to study the effects of the dimensionless geometric parameter r1 / r0 (where r0 is the radius of coastline) and the dimensionless wavelength L1 / r1 (where L1 is the wavelength on the island toe), respectively. Also, the solutions with different γ values were compared for each case to study the effect of water depth as a power of the radius. The effect of wave diffraction was investigated in more detail by testing waves around a cylindrical island on a flat bottom with similar geometric and wave conditions, and wave refraction was qualitatively investigated by tracing wave rays on the circular island. For waves with longer periods over a circular island with a smaller domain, wave amplitudes were found to decrease on the rear side due to diffraction and increase on the leeside due to the intersection of wave rays. For a circular island with a small value of γ, such as γ = 2/3 as in the equilibrium beach (Bruun, 1954), the wave rays approaching near the island center reached the coastline, whereas the rays approaching away from the center bent away from the coastline, meaning that the wave amplitudes were not very large along the coast. However, for a circular island with a large value of γ, like γ = 2, all the rays on the island reached the coast, leading to large coastline wave amplitudes. Furthermore, if the island domain was large enough (r1 = 9r0), some rays deflected more than 360° before reaching the coastline. Thus, superposition of wave rays occurred several times in a large area, and the amplitudes were more than 40 times the incident amplitude at some positions along the coastline.

Conclusively, the propagation of long waves over a circular island depends on the geometry of the island and the wave period. If the island domain is small compared to the wavelength, wave amplitudes on the coastline do not significantly increase. However, when the island domain is not as small compared to the wavelength, wave amplitudes significantly increase and, if γ is also large, the wave amplitudes can become so great as to cause a disaster on the island. Acknowledgements This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no.: 2009-0080576). References Berkhoff, J.C.W., 1972. Computation of combined refraction–diffraction. Proceedings of the 13th Coastal Engineering Conference. ASCE, New York, pp. 471–490. Bruun, P., 1954. Coast erosion and the development of beach profiles. Technical Memo, 44, US Army Corps of Engineering, Beach Erosion Board, Washington, DC. Homma, S., 1950. On the behavior of seismic sea waves around circular island. Geophysical Magazine XXI, 199–209. Jonsson, I.G., Skovgaard, O., Brink-Kjaer, O., 1976. Diffraction and refraction calculations for waves incident on an island. Journal of Marine Research 34, 469–496. Jung, T.H., Suh, K.D., 2007. An analytic solution to the mild slope equation for waves propagating over an axi-symmetric pit. Coastal Engineering 54, 865–877. Lee, C., Suh, K.D., 1998. Internal generation of waves for time-dependent mild-slope equations. Coastal Engineering 34, 35–57. Lee, C., Yoon, S.B., 2007. Internal generation of waves on an arc in a rectangular grid system. Coastal Engineering 54, 357–368. Lin, P., Liu, H.-W., 2007. Scattering and trapping of wave energy by a submerged truncated paraboloidal shoal. Journal of Waterway, Port, Coastal, and Ocean Engineering 133, 94–103. Liu, H.-W., Lin, P., Shankar, N.J., 2004. An analytical solution of the mild-slope equation for waves around a circular island on a paraboloidal shoal. Coastal Engineering 51, 421–437. Mei, C.C., 1989. The Applied Dynamics of Ocean Surface Waves. World Scientific. Pocinki, L.S., 1950. The application of conformal transformations to ocean wave refraction problems. Transactions of American Geophysical Union 31, 856–860. Suh, K.D., Jung, T.H., Haller, M.C., 2005. Long waves propagating over a circular bowl pit. Wave Motion 42, 143–154. Vastano, A.C., Reid, R.O., 1967. Tsunami response for island: verification of a numerical procedure. Journal of Marine Research 25, 129–139. Zhang, Y.L., Zhu, S.P., 1994. New solutions for the propagation of long water waves over variable depth. Journal of Fluid Mechanics 278, 391–406. Zhu, S.P., Harun, F.N., 2009. An analytical solution for long wave refraction over a circular hump. Journal of Applied Mathematics and Computing 30, 315–333. Zhu, S.P., Zhang, Y.L., 1996. Scattering of long waves around a circular island mounted on a conical shoal. Wave Motion 23, 353–362.