## 1. Introduction

Scattering of tidal frequency waves around a small island may result in the local modification of their amplitude and phase (Proudman 1914), as described well by Defant (1961, Figs. 149 and 150) theoretically. For real tides, Larsen (1977) showed that scattering alters the time of high water around Hawaii by as much as an hour. Also, Kim and Lee (1986) and Lee and Kim (1988) found that amplitudes of semidiurnal and diurnal tides vary with geographical location around Cheju Island (Fig. 1): amplitudes of the *M*_{2} and *O*_{1} tides are larger than 75 and 18 cm at site 7 on the southwestern side of this island and as low as 61.2 and 14.3 cm at site 3 on the northeastern corner (Table 1). For the *K*_{1} tide, Lee and Kim (1988) found a way to utilize short records of sea level data to remove the contaminating effects of *P*_{1} on *K*_{1} and successfully showed that the amplitudes along the southern coast are significantly larger than those along the northern coast.

The amplitude–phase relationships shown in Fig. 2 are useful to examine how the characteristics of the *M*_{2} and *O*_{1} tides are changed as they propagate along the coast of Cheju Island. The amplitudes are noticeably larger on the left-hand side of the island for an observer looking down the direction of wave propagation and smaller on the right-hand side. The phases of the *M*_{2} and *O*_{1} tides increase in an orderly manner along the both sides of the island. The maximum phase difference across the island is about 39 degrees for the *M*_{2} tide and 16 degrees for the *O*_{1} tide, corresponding to 1.34 and 1.15 h, respectively. If tidal waves propagate around Cheju Island as a Kelvin wave for a mean water depth of 75 m, the phase difference across the island due to the incident wave would be about 15 degrees for the *M*_{2} tide and about 8 degrees for the *O*_{1} tide. These differences are about one-half of the observed differences, implying that propagation of tides around Cheju Island is not as simple as a Kelvin wave propagation. Therefore, the amplitude–phase relationship in Fig. 2 deserves a careful analysis and a physical explanation.

Ogura (1933), Choi (1980), Larsen et al. (1985), and Kang et al. (1991) showed that the tides propagate from the East China Sea to the Yellow Sea along the southern and western coast of Korea (Fig. 1) and that amplitudes of the semidiurnal and diurnal tides increase generally toward the southern coast of Korea from the East China Sea. However, the difference between the amplitudes along the southern and northern coasts of Cheju Island is opposite to this general trend. For tides in the Yellow Sea, Kang (1980) and Kang (1995) solved Taylor’s problem, assuming that the *M*_{2} tide in the Yellow and East China Sea comprises Kelvin and Poincare waves. But, Fang et al. (1991) supposed that the *M*_{2} tide around Cheju Island partially has Sverdrup wave characteristics because of the clockwise rotation of tidal currents. Hence, previous studies considered all tide-related waves that have different characteristics of currents.

Larsen (1977) applied an approximate solution by Proudman (1914) for the scattering of Sverdrup waves for tides around the Hawaiian Islands with periods shorter than the local inertial period. Following Larsen, Kim and Lee (1986) explained the amplitude variation of the semidiurnal tides around Cheju Island successfully, as their periods are shorter than the local inertial period. However, the periods of the *O*_{1} and *K*_{1} tides are longer than the local inertial period around Cheju Island (33°20′N), and the amplitude variation of the diurnal tides around Cheju Island remains to be explained since the propagation of Sverdrup and Poincare waves in an inviscid fluid is not supported at subinertial frequencies (Pedlosky 1979). Lee and Kim (1993) obtained the scattering solution for Kelvin waves and showed that Kelvin wave scattering in an inviscid fluid results in a uniform amplitude at the circumference of the island and the scattering with linear bottom friction produces a slight amplitude decrease along the direction of incident wave propagation due to frictional damping. Lee and Kim concluded that it would be inappropriate to explain the amplitude variation of tides around Cheju Island in terms of the Kelvin wave scattering.

Unlike tidal propagation around the Hawaiian Islands where the water depth is about 5000 m, tides around Cheju Island are propagating in shallow water where the depth is about 100 m. In this shelf sea, bottom friction plays an important role in dissipating tidal wave energy (Suh 1996). If bottom friction is neglected, the Sverdrup wave with subinertial frequency cannot propagate (Leblond and Mysak 1978; Lee 1989). With the bottom friction, the low-frequency cutoff is removed and the dispersion curves for Sverdrup and Poincare waves extend below the inertial frequency although the waves are evanescent due to large damping (Lee 1989;Kang 1995). Thus, bottom friction allows us to extend a scattering theory to the propagation of subinertial tide-related waves.

Larsen (1977) derived the solution for Sverdrup wave scattering without bottom friction by replacing the Helmholtz wave equation by Laplace’s equation and by approximating the incident wave directly into a linear form, which gave the same results as Proudman’s (1914) solution. It should be noted, however, that Poincare wave scattering may not be solved by Larsen’s (1977) approximation because the incident Poincare wave is not directly approximated in a linear form. Obtaining an approximate solution for the disturbance of an inertial wave without bottom friction around a small island, Longuet-Higgins (1970) also replaced the Helmholtz equation by Laplace’s equation. But Longuet-Higgins used the particle velocity of the inertial wave at a large distance from the island to get the spatial function of the inertial wave when the island is absent. In this paper, we followed the Longuet-Higgins approach to find an approximate solution of the incident wave field as a linear form.

The objective of this paper is to derive the scattering solution that 1) is widely applicable for Sverdrup, Poincare, and Kelvin waves propagating around a small island, 2) is valid for all frequency ranges, and 3) includes the bottom friction. We replaced the particle velocity of the incident wave around the island by that at the center point of the island in order to find a linear function for the incident wave field. This method allows us to minimize the inhomogeneity in the velocity field around the island due to friction and to take into account effects of bottom friction on the incident wave propagation. Examples of the amplitude and phase fields modified by the scattering are described in detail, and the model amplitude–phase variations are applied to tides around Cheju Island.

## 2. Theory

### a. Wave equations

*ζ,*

**u**) = Re[(

*Z,*

**U**) exp(−

*iωt*)] at a given wave frequency

*ω*> 0, where

**u**is the vertically averaged velocity and

*ζ*is the surface elevation, the long-wave equations in water of a uniform depth take the form (Leblond and Mysak 1978)

*σ*=

*γ*−

*iω, G*= −

*g*/(

*σ*

^{2}+

*f*

^{2}) and

*K*

^{2}

*iω*

*σ*

^{2}

*f*

^{2}

*σgh*

*h*is the water depth,

**f**is a constant Coriolis parameter vector in the vertical direction,

**is the horizontal gradient operator, and**

**∇***γ*=

*γ*′/

*h*where

*γ*′ is the linear bottom friction coefficient. The dispersion relation (3) for this system was presented by Rienecker and Teubner (1980) and Buchwald (1980).

### b. Incident waves

*a*and the origin of coordinate system be at the center of the island (Fig. 3). Consider an incident wave propagating westward (in a negative

*x*direction). If we assume no lateral boundary away from the island, we can take a possible incident wave as a Sverdrup plane wave that takes the form

*Z*

_{i}=

*A*

_{s}exp(

*ikx*). The complex wavenumber

*k*can be defined as

*k*=

*K*=

*α*

_{s}+

*iβ*

_{s}(see the appendix). The velocity components of the incident wave with (2) are given in the

*x*and

*y*directions by

*U, V*

*σ,*

*f*

*ikA*

_{s}

*G*

*ikx*

*y*=

*b,*a possible incident wave is a Poincare wave. A westward incident wave having a real amplitude

*A*

_{p}at (

*x, y*) = (0,

*b*) is the same as the right-bounded Poincare wave by Platzman (1971) and may take the form

*Z*

_{i}=

*A*

_{c}exp(

*ilx*), where

*A*

_{c}=

*A*

_{p}[cos(

*my*−

*mb*) + (

*ilf*/

*σm*) sin(

*my*−

*mb*)] is a complex amplitude, and

*l*and

*m*are the alongshore and offshore wavenumbers having the relation

*K*

^{2}=

*l*

^{2}+

*m*

^{2}. It should be noticed that the complex nature of

*A*

_{c}produces the wave phase

*φ*

^{c}= arg(

*A*

_{c}), which is a function of

*y.*As shown by Mofjeld (1980) and Brink and Allen (1978), the cophase lines of an incident wave with friction slant relative to the direction perpendicular to the straight coast due to this

*φ*

^{c}. The alongshore complex wavenumber

*l*can be defined as

*l*=

*α*

_{p}+

*iβ*

_{p}(see the appendix), and the wavenumber

*m*can be defined without channel width because there is only one lateral boundary (Platzman 1971). The velocity components of the incident wave with (2) are given in the

*x*and

*y*directions by

*A*

_{u}=

*A*

_{p}

*G*/

*σm.*

*y*=

*b*may also be a Kelvin wave that has the form

*Z*

_{i}=

*A*

_{k}exp(

*ik*

_{x}

*x*+

*ik*

_{y}

*y*). Complex alongshore and offshore wavenumbers may be defined as

*k*

_{x}=

*α*

_{k}+

*iβ*

_{k}and

*k*

_{y}=

*k*

_{x}

*f*/

*σ*=

*μ*+

*iν.*The resulting dispersion curves (see Mofjeld 1980) show that the Kelvin wave can propagate for all frequencies independent of bottom friction. The velocity components are given by

*U, V*

*ik*

_{x}

*g*

*σ*

*A*

_{k}

*ik*

_{x}

*x*

*ik*

_{y}

*y*

### c. Wave scattering

*a*is much smaller than the wavelength of the incident wave. Since we are interested in the scattering around a small island, (1) can be reduced to Laplace’s equation near the island, where |

*Kr*| ≪ 1,

^{2}

*Z*

^{2}≡ ∂

^{2}/∂

*r*

^{2}+ ∂/

*r*∂

*r*+ ∂

^{2}/

*r*

^{2}∂

*θ*

^{2}in polar coordinates as shown in Fig. 3. With this approximation, we can assume that

**∇***Z*for incident wave becomes constant within |

*Kr*| ≪ 1 so that the particle velocity in (2) becomes spatially uniform for a given

*σ.*Thus, the particle velocities for the several incident waves in (4), (5), and (6) within |

*Kr*| ≪ 1 can be replaced by (

*U*

_{0},

*V*

_{0}), which are the velocity components of each wave at the origin of the coordinate system (

*r*= 0) when the island is absent; that is, (

*U, V*) ≈ (

*U*

_{0},

*V*

_{0}). In our polar coordinates, the radial and tangential velocity components for each incident wave are then given by

*Z*

_{i}=

*A*

_{0}at

*r*= 0, it follows that the incident wave

*Z*

_{i}within |

*Kr*| ≪ 1 in the absence of the island is given uniquely by

*x*and

*y.*Next, to take into account the presence of an island, we add a scattered wave to (10). If the wave energy of the scattered wave radiating from the island is assumed to be constant within the circle of radius

*r*from the island, the amplitudes of the scattered wave will be inversely proportional to the distance

*r.*With this approximation, the scattered wave

*Z*

_{s}near the island can be represented by

*Z*

_{s}

*B*

*θ*

*C*

*θ*

*r,*

*Z*=

*Z*

_{i}+

*Z*

_{s}.

*U*

^{r}

*G*

*σ*

*r*

*f*

*r*

*θ*

*Z*

*r*

*a.*

*B*and

*C*in (11) can be determined by applying (12) since

*Z*=

*Z*

_{i}+

*Z*

_{s}must satisfy this boundary condition for any

*θ.*We can obtain

*B*= −

*a*

^{2}[

*σU*

_{0}+

*fV*

_{0}]/

*r*and

*C*= −

*a*

^{2}[

*σV*

_{0}−

*fU*

_{0}]/

*r*after some manipulations. Using (10) and (11) with

*B*and

*C,*the total wave solution within |

*Kr*| ≪ 1 takes the following form:

*U*

^{r}

*U*

^{θ}

*a*

^{2}

*r*

^{2}

*U*

^{r}

_{0}

*a*

^{2}

*r*

^{2}

*U*

^{θ}

_{0}

*U*

_{0}and

*V*

_{0}in (13) are replaced by the relevant velocity components in (4), (5), and (6) at

*r*= 0 for each incident wave type.

*r*= 0 in (4), (5), and (6) may be obtained as

*A*

_{0}=

*A*

_{s}, the incident Sverdrup wave in (10) may be written as

*Z*

_{i}=

*A*

_{s}(1 +

*ikr*cos

*θ*), which is the same form as the incident wave approximated by Larsen (1977). With (17) and

*A*

_{0}=

*A*

_{k}, the incident Kelvin wave in (10) may be written by

*Z*

_{i}=

*A*

_{k}(1 +

*ik*

_{x}

*r*cos

*θ*+

*ik*

_{y}

*r*sin

*θ*), which is also the same form as Lee and Kim’s (1993) approximation. However, the complex amplitude of incident Poincare wave

*A*

_{0}=

*A*

_{c}at

*r*= 0 can be written by

*A*

_{0}= (

*P*

^{2}

_{r}

*P*

^{2}

_{i}

^{1/2}exp(

*iφ*

^{c}

_{0}

*P*

_{r}=

*A*

_{p}{cos(

*mb*) + [

*f*(

*α*

_{p}

*ω*+

*β*

_{p}

*γ*)/(

*γ*

^{2}+

*ω*

^{2})

*m*] sin(

*mb*)} and

*P*

_{i}=

*A*

_{p}[

*f*(

*β*

_{p}

*ω*−

*α*

_{p}

*γ*)/(

*γ*

^{2}+

*ω*

^{2})

*m*] sin(

*mb*) are the real and the imaginary part of

*A*

_{c}at

*r*= 0 and the wave phase

*φ*

^{c}

_{0}

^{−1}(

*P*

_{i}/

*P*

_{r}). Thus,

*A*

_{0}of incident Poincare wave varies with

*mb.*With this

*A*

_{0}and (16), the incident Poincare wave can be obtained from (10).

The scattering solution of (13) is, in general, represented by *ζ* = *A* cos(*ωt* − *ϕ*), where *A* = mod(*Z*) and *ϕ* = arg(*Z*). However, since the imaginary part with incident Sverdrup and Kelvin waves, having the order of *Kr* in (13), is much smaller than the real part near the island (Proudman 1914; Lee and Kim 1993), the scattering solution to the order of our approximation can be written by *ζ* = Re(*Z*) cos(*ωt* − *ϕ*), where *ϕ* = Im(*Z*). We assume that the straight coast at *y* = *b* is located far enough from the island that the reflection of *Z*_{s} at that coast is negligible. Thus, (12) is a sufficient condition to solve the Poincare and Kelvin wave scattering around the island. Though Lee and Kim (1993) explained the amplitude modification in Kelvin wave scattering, we describe the scattering results briefly. When there is no friction, total wave amplitude becomes uniform with *A*_{k} at *r* = *a,* implying that the amplitude variation of incident wave along the coast of the island is compensated by the scattered wave amplitude. Total wave phase varies along the coast of the island as a function of cos*θ,* whether friction is included or not, and the maximum phase difference over the island becomes |4*aα*_{k}|, which is exactly twice of phase difference by the incident wave. Meanwhile, friction causes the total amplitude to vary along the coast of the island as much as ±2*a*|*β*_{k}|.

## 3. Examples of wave scattering

The preceding theoretical results are applied to a cylindrical island with radius *a* = 25 km. We take *f* = 0.8012 × 10^{−4} s^{−1} (33°20′N) and *h* = 75 m, for which the Rossby deformation radius (*R*_{d}) is about 340 km, and *m* = −1/*R*_{d} for the Poincare wave propagation. When friction is absent, *α*_{p} with this *m* has a cutoff frequency near 1.42*f.* The superinertial incident wave is taken as the *M*_{2} tide (*ω*/*f* = 1.75), and the subinertial incident wave as the *O*_{1} tide (*ω*/*f* = 0.84). These tidal frequencies are above and below the cutoff frequency for the Poincare wave. The bottom friction coefficient is assumed to be *γ* = 1.4147 × 10^{−5} s^{−1} (*γ*/*f* = 0.176), which produces the same frictional force as the quadratic form (*C*_{d}*u*^{2}/*h*) with a velocity of 50 cm s^{−1} and a drag coefficient of 0.0025. We calculate the wave amplitude and phase field within *r* = 150 km from the center of the island for the Sverdrup wave scattering with *A*_{s} = 1 m and within *r* = 100 km for the Poincare wave scattering with *A*_{p} = 1 m. Because Lee and Kim (1993) showed examples of Kelvin wave scattering, we will not repeat them in this paper.

### a. Sverdrup wave

An example of the wave field without bottom friction is drawn for the total amplitude and phase of the *M*_{2} tide (Fig. 4a). Cophase lines of the total wave around the island show that the wave travels much faster around the frontal and leeward coast of the island than elsewhere. Near the island, particularly around the northern (southern) sides of the island, the total amplitude is reduced (strengthened). Because the incident wave amplitude without bottom friction is spatially uniform and the cophase lines are parallel to each other, we can see that the phase of the scattered waves contributes to the more rapid travel of the total wave along both coasts of the island and that the scattered wave amplitude with opposite sign contributes to the amplitude modification around the island. With bottom friction, the total wave field of the *M*_{2} tide (Fig. 4b) shows that the phase and amplitude distributions near the coast of the island are very similar to those without friction. Coamplitude lines far from the island are different from those without friction, indicating that total amplitude distribution in the far field reflects the attenuation of the incident wave amplitude due to bottom friction. The cophase lines in Figs. 4a and 4b are very similar, implying that the wave phase is little affected by bottom friction.

The total wave field of the *O*_{1} tide with bottom friction (Fig. 4c) shows significant amplitude attenuation from east to west. Note that the maximum amplitude occurs on the southwestern side of the island. The location of the maximum amplitude is moved clockwise, compared to that shown in Fig. 4b, indicating that the scattered wave amplitude is positive off the west coast of the island. The phase on the southeastern coast of the island is lowest as the high water occurs first about 60 degrees clockwise from the east. Therefore, the propagating direction of total wave around the island does not coincide with the direction of incident wave propagation. This result indicates that locations of maximum amplitude and minimum phase of total wave around the island are mainly determined by the scattered wave fields. Amplitude damping of the incident wave due to generic effects on subinertial frequency contributes significantly to reduce the total amplitude differences around the island.

### b. Poincare wave

We locate the straight coast at a distance *b* = 100 km from the island such that *mb* = −1/3.4. Total wave fields of the *M*_{2} and *O*_{1} tides with bottom friction are presented in Fig. 5. Cophase lines farther outside from the island show a slight forward slanting for the *M*_{2} frequency wave. But cophase lines show a large forward slanting for the *O*_{1} frequency wave, implying that the incident wave propagates toward the straight coast. As discussed earlier, slanting of cophase lines is produced by the wave phase *φ*^{c}. Both tides show that coamplitude lines farther outside are not parallel to the straight coast due to frictional damping along the straight coast and the change of amplitude gradient along the direction perpendicular to the straight coast by the bottom friction.

Near the island, the phase differences across the island for each tide are very small, compared to Sverdrup wave case, and minimum phase of the *M*_{2} tide around the island occurs on the east coast and that of the *O*_{1} tide on the southwest coast. From the phase modification around the island, we can deduce that for an observer looking down the direction of incident wave propagation, the phase of scattered wave for the *M*_{2} tide is negative (positive) in the front (leeward) coast and that of the *O*_{1} tide is positive (negative) in the front (leeward) coast. This means that for the *M*_{2} tide the direction of scattered wave propagation is close to the direction of incident wave propagation, but for the *O*_{1} tide it is nearly opposite to the direction of the incident wave propagation. Thus, the scattered wave for the *M*_{2} (*O*_{1}) tide contributes to increase (decrease) the total phase difference across the island.

Amplitudes of both tides are larger on the northeast coast of the island than along other parts of the coast. From the amplitude modification near the island, we can also deduce that the amplitude of scattered waves for both tides is positive (negative) on the northeast (southwest) coast of the island, resulting in an increase of the amplitude difference of total wave across the island for both tides. However, amplitude difference for the *O*_{1} tide is much larger than that for the *M*_{2} tide because of the large gradient of incident amplitude of the *O*_{1} tide due to friction.

### c. Bottom friction effects on amplitude and phase differences

To examine the effects of bottom friction on Sverdrup and Poincare wave scattering, we estimate the maximum difference of the amplitude and phase across the island (Fig. 6) as a function of frequency (*ω*/*f*) for different values of the bottom friction coefficient (*γ*/*f*). Radius of island, water depth, and Coriolis parameter are the same as before. The amplitude difference is nondimensionalized by the incident wave amplitude |*A*_{0}|.

For the Sverdrup wave, the amplitude difference increases from 0.3*A*_{s} as the frequency decreases when friction is neglected (solid lines in Fig. 6a) and reaches infinity near the inertial frequency due to singularity effects. The phase difference also increases infinitely at the inertial frequency. However, the incident wave with the inertial frequency has to become an inertial wave, which is a special case, having *A*_{s} = *A*_{0} = 0. In our approach for the scattering solution, the incident wave with no friction at inertial frequency is not defined because its current becomes an indefinite value in (15) (see *U*_{1}, *V*_{2} in the appendix), which is unrealistic. As for this special case, Longuet-Higgins (1969, 1970) already derived the solution for a finite inertial current and showed that all amplitude and phase variation is caused by the scattered wave, representing a mode trapped exponentially at large distances from the island.

When bottom friction is present, the transitions of amplitude and phase differences near the inertial frequency become smoother with larger bottom friction coefficients, indicating that friction effectively prevents any singularity. As the frequency decreases below the inertial frequency, the differences for a given friction coefficient decrease monotonically. This can be explained as follows. Because the wavelength of the incident wave increases with the decrease of the frequency, the island dimension becomes relatively smaller in comparison to the wavelength, resulting in reducing scattering effects. In addition to this, the different variations of incident and scattered wave amplitude and phase contribute to the decrease of total amplitude and phase difference (see Fig. 4c). For low-frequency waves *ω*/*f* < 0.5, amplitude and phase differences are larger for larger friction coefficients as frictional effects on the incident wave dominate the amplitude and phase variations along the coast of the island.

Amplitude and phase differences for an incident Poincare wave are presented in Fig. 6b, when we take *b* = 100 km. When friction is present, the curves for amplitude and phase difference are extended below the cutoff frequency of *ω*/*f* ≈ 1.42 and singularity effects near inertial frequency are absent. For a given wave frequency, the larger friction coefficient makes the larger amplitude difference, but the friction effects on the phase difference are complicated with the friction coefficient below the cutoff frequency. Large differences of total amplitude below the cutoff frequency are mainly due to the generic effects of large amplitude damping of the incident wave and the additive effects of the scattered wave amplitude variation on this incident wave amplitude variation (see Fig. 5b). Note that the frequency where the phase difference becomes a minimum decreases from the cutoff frequency as the frictional coefficient increases. This minimum phase difference implies that high and low water for this frequency wave occurs concurrently along the circumference of the island.

## 4. Application to Cheju Island

To apply the theoretical results to tides around Cheju Island, we analyze first the observed coastal tides shown in Fig. 2 in detail. The maximum amplitude difference for the *M*_{2} tide is 16.2 cm and for the *O*_{1} tide it is 4.1 cm (Table 1). If the average value of the observed amplitude is assumed to be the incident wave amplitude *A*_{0} at the island center, *A*_{0} is about 70 cm for the *M*_{2} tide and 17 cm for the *O*_{1} tide. Then, the maximum amplitude differences are about 0.23*A*_{0} and 0.24*A*_{0} for the observed *M*_{2} and *O*_{1} tides. It should be noted that the minimum phase for the *O*_{1} tide occurs at site 6 and for the *M*_{2} tide it occurs at site 5, indicating that the phase variation along the coast of Cheju is different for various tidal constituents.

Proudman (1914) showed that the scale of the elliptic island in his scattering solution, which was given by *ρ* exp(*ξ*), where *ξ* is an elliptic coordinate representing the shore line of the island and *ρ* is the focus distance, determines the amplitude and phase differences by the scattering. Our scattering solution in (13) also indicates that the amplitude and phase difference due to scattering depends on the circular island scale 2*a.* These solutions show that the amplitude and phase differences depend on the island scale, while variations of amplitude and phase along the coast depend on the island shape and incident angle of the wave. Our circular model island has a scale corresponding with an elliptic-shaped Cheju Island, that is, for Cheju Island, *ρ* exp(*ξ*) = 52.6 km (Kim and Lee 1986), which is close to the scale of our model island 2*a* = 50 km. The water depth of 75 m around our model island is comparable with the mean water depth around Cheju Island.

Based upon these variables, theoretical amplitude and phase differences are calculated for *M*_{2} and *O*_{1} frequency Sverdrup and Kelvin waves, propagating, toward the west, subject to variable bottom friction in order to examine what is the best estimate for the friction coefficient (Table 2). We may do a least squares fit of theoretical results to Fig. 2 to find the offshore wave field, which is tedious work converting our solutions into an elliptic coordinate. With small friction coefficient *γ*/*f* < 0.3 in Table 2, the amplitude differences in Sverdrup wave scattering for both tides are significantly larger than the observed differences. On the other hand, the theoretical differences in Kelvin wave scattering are very much smaller than observed ones. However, the observed phase differences in Table 1 are close to the theoretical phase differences of Sverdrup wave scattering for the *M*_{2} tide when *γ*/*f* ⩽ 0.1 and for the *O*_{1} tide when *γ*/*f* > 0.3 as shown in Table 2. The phase differences of Kelvin wave scattering are less sensitive to the friction coefficient, though theoretical differences for the *M*_{2} tide are slightly smaller than the observed difference. Based upon these comparisons, we construct amplitude–phase diagrams with friction coefficients *γ*/*f* = 0.1 and 0.4 for both tides.

For the incident Sverdrup waves we take *A*_{s} = 70 and 17 cm for the *M*_{2} and *O*_{1} tides (Fig. 7a). Amplitude–phase relationships are elliptic and similar to the observed relationships shown in Fig. 2. Note that amplitude axes decrease upward to make Figs. 2 and 7 compatible. Amplitude–phase diagrams for the *O*_{1} tide with *γ*/*f* = 0.4 show a significantly reduced ellipse, implying that the *O*_{1} tide is more sensitive to friction coefficients than the *M*_{2} tide. It should be noted that the scattering of the *O*_{1} tide subject to bottom friction has the minimum phase located on the southeast coast of the island although the *O*_{1} tide has the same incident direction as the *M*_{2} tide. The position of maximum amplitude of the *O*_{1} tide is moved from the west coast to the southern coast as the friction coefficient *γ*/*f* changes from 0.1 to 0.4. In the case of Kelvin wave scattering, the amplitude–phase relationships are linear (Fig. 7b) as amplitudes decrease from the front coast to the leeward coast irrespective of north and south coasts.

Variation of the friction coefficient in Table 2 does not improve the comparison of amplitude difference between theory and observation against our expectation, indicating that the observed diagrams cannot be explained by a single incident wave scattering. When we consider the real *M*_{2} tide as a linear combination of Kelvin and Sverdrup waves, the amplitudes of incident Sverdrup wave around Cheju Island should be less than 70 cm. We apply this assumption to estimating the amplitude *A*_{s} because the scattering of Kelvin waves produces only small amplitude differences along the wave propagation direction in Fig. 7. The amplitude *A*_{s} of the *M*_{2} tide with *γ* = 0.1*f* should be about 66% of the observed mean amplitude *A*_{0} to fit the theoretical difference of 0.35*A*_{s} to the observed difference of 16.2 cm. This makes *A*_{s} equal to 46 cm, and we assume *A*_{k} = 24 cm. With this estimate, however, it turns out that the incident *O*_{1} tide for *γ* = 0.4*f* has entirely Sverdrup wave characteristics around Cheju Island because theoretical amplitude difference of 0.22*A*_{s} is smaller than the observed difference of 0.24*A*_{0}. Then, no energy of the *O*_{1} tide is carried by Kelvin waves, which is unrealistic. When we apply the friction coefficient of *γ* = 0.3*f* for the *O*_{1} tide, the amplitude difference becomes 0.25*A*_{s}, though the phase difference of 20.2 degrees is slightly larger than the observed values (Table 2), and then 96% of the mean amplitude *A*_{0} of the observed *O*_{1} tide will be carried by Sverdrup wave. We suspect that our observed amplitudes and phases of the *O*_{1} tide can be contaminated by other constituents due to one-month data analysis, which may result in the large amplitude and phase differences in the observed diagram. We need more long observations along the south coast of Cheju Island to explain the amplitude and phase differences of the *O*_{1} tide.

## 5. Discussion and conclusions

Our approach produces the same results as Larsen’s (1977) solution on Sverdrup wave scattering and as Lee and Kim’s (1993) solution on Kelvin wave scattering if bottom friction is neglected. An important merit of our approach is that we can solve the scattering of Poincare waves in linear form and our scattering solution is valid for all tide-related waves. Next, by including linear bottom friction in the wave equation, we can remove the frequency limitation on the application of the scattering theory for propagating Sverdrup and Poincare waves.

In our scattering solution, if the island were much smaller than the incident wavelength, it would scatter the wave with much smaller amplitude because the scattered wave amplitude in (13) is of the order |*Ka*|. In this case, the scattering effect could be easily buried in the incident wave field. For a very small island having a topographic skirt, of which the diameter is the order of one tidal excursion, Pingree and Maddock (1979, Fig. 2) showed that two amphidromic points with the reduced amplitudes are formed near the island when Coriolis effect is neglected, indicating that the different tide modification is produced by the inertial and frictional forces associated with high curvature and shallow water depth due to a sloping topographic skirt.

If the island is much larger than our model island, the scattering solution will not be applicable because the basic approximation in (7) is not valid anymore. For the effects of big islands on the tidal wave propagation, Platzman (1984) showed that the New Zealand Kelvin waves having the large energy in the frequencies of 13.8 and 10.8 h propagate counterclockwise around the island. However, this Kelvin wave is a trapped wave, analyzed from the normal mode experiments, and is entirely different from our scattering phenomena. Another near-inertial wave phenomenon around an island is subinertial wave trapping. Longuet-Higgins (1969) showed theoretically that trapping of a subinertial wave is always possible for any island when the azimuthal wavenumber is one, while a superinertial wave cannot be perfectly trapped. Following the Longuet-Higgins theory, trapping of the *O*_{1} frequency forced wave can take place around our model island only if the water depth is about 2 m. This is unrealistically shallow, implying that the scattering of a subinertial tide can be separated from the trapping phenomena.

In the application of our theoretical scattering solution to Cheju Island, we did not include Poincare wave scattering. Theoretical phase differences due to Poincare wave scattering with *γ*/*f* = 0.15 are about 11 degrees for the *M*_{2} tide, generally much smaller than those in Fig. 2, and the phase increasing direction along the coast of the island is different for the *M*_{2} and *O*_{1} tides (see Fig. 5). Moreover, maximum amplitudes in these theoretical examples occur at the northeast coasts of the island. Kang (1995) showed that energies of Poincare modes in numerical solutions of the Taylor problem are large near the northern part of the Yellow Sea and significantly decrease to the south due to bottom friction. Observed diagrams in Fig. 2, our theoretical results, and Kang’s (1995) results suggest that Poincare modes included in tides near Cheju Island are negligible.

The amplitude–phase relationships between the observed and model tides agree well for only Sverdrup incident waves at both superinertial and subinertial frequencies, though some parts of both tidal constituents are carried by Kelvin-type waves. This consistency implies also that the clockwise rotary currents off Cheju Island are changed to rectilinear currents near the coast of this island, producing the amplitude differences between the south and the north coasts by Coriolis effect (Kim and Lee 1986). In particular, it should be noticed in Fig. 7a that the location of minimum phase (the first high water on the island) for the *O*_{1} tide occurs slightly on the left-hand side of the incident wave angle, whereas it is just ahead for the *M*_{2} tide. This basic pattern also agrees well with the observed amplitude–phase diagram shown in Fig. 2.

In summary, the theoretical analysis shows that the amplitude and phase differences caused by scattering around the island depend on the incident wave type. The singularity effects near the inertial frequency for Sverdrup wave scattering produce large amplitude and phase variations along the coast of the island, while the singularity does not happen in the Poincare and Kelvin wave scattering. Bottom friction removes the singularity in Sverdrup wave scattering. For Sverdrup wave scattering, amplitude–phase relationships along the coast of the island show that the maximum amplitude appears on the left-hand side of the island for an observer looking down the direction of wave propagation around the island. For Poincare wave scattering with a fixed value of *mb* = −1/3.4, amplitude–phase relationships along the coast of the island show that, for the same observer as in Sverdrup wave scattering, the maximum amplitude at frequencies higher than the cutoff frequency occurs on the right-hand side and at the subinertial frequencies it occurs on the leeward coast. For Kelvin wave scattering, the amplitude–phase relationships along the coast of the island become linear along the direction of wave propagation. With these theoretical results, the amplitude and phase variations of the *M*_{2} and *O*_{1} tides around Cheju Island off the south coast of Korea are interpreted by the scattering of those tides that mainly comprise Sverdrup and Kelvin waves having superinertial and subinertial frequencies.

## Acknowledgments

The authors wish to thank R. Beardsley of the Woods Hole Oceanographic Institution and M. G. G. Foreman of the Institute of Ocean Sciences, British Columbia for their critical reviews on our initial manuscript. Anonymous reviewers’ comments were helpful to improve our paper structure and figures. Preparation of the final manuscript of this work was supported by the Basic Science Research Institute Program by the Ministry of Education of Korea under BSRI 97-5409 and 97-5406.

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## APPENDIX

### Wavenumbers and Current Components

#### The complex components of each incident wavenumber

*E*

_{s}=

*ω*

^{2}(

*ω*

^{2}+

*γ*

^{2}−

*f*

^{2}),

*F*

_{s}=

*ω*

^{4}(

*ω*

^{2}+

*γ*

^{2}−

*f*

^{2})

^{2}+

*γ*

^{2}

*ω*

^{2}(

*ω*

^{2}+

*γ*

^{2}+

*f*

^{2})

^{2}with positive sign for

*F*

^{1/2}

_{s}

*E*

_{p}=

*ω*

^{2}(

*ω*

^{2}+

*γ*

^{2}−

*f*

^{2}) − (

*γ*

^{2}+

*ω*

^{2})

*ghm*

^{2},

*F*

_{p}= [

*ω*

^{2}(

*ω*

^{2}+

*γ*

^{2}−

*f*

^{2}) − (

*γ*

^{2}+

*ω*

^{2})

*ghm*

^{2})]

^{2}+

*γ*

^{2}

*ω*

^{2}(

*ω*

^{2}+

*γ*

^{2}+

*f*

^{2})

^{2}with positive sign for

*F*

^{1/2}

_{p}

#### The complex components of the current for U_{0} = U_{1} + iU_{2} and V_{0} = V_{1} + iV_{2}

Amplitude–phase diagrams of the *M*_{2} and *O*_{1} tides using the observed values at the circumference of Cheju Island (see Table 1 and Fig. 1). Minimum phases of the *M*_{2} and *O*_{1} tides occur at sites 5 and 6.

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Amplitude–phase diagrams of the *M*_{2} and *O*_{1} tides using the observed values at the circumference of Cheju Island (see Table 1 and Fig. 1). Minimum phases of the *M*_{2} and *O*_{1} tides occur at sites 5 and 6.

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Amplitude–phase diagrams of the *M*_{2} and *O*_{1} tides using the observed values at the circumference of Cheju Island (see Table 1 and Fig. 1). Minimum phases of the *M*_{2} and *O*_{1} tides occur at sites 5 and 6.

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Coordinates system (right hand) around a cylindrical island with radius *a*.

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Coordinates system (right hand) around a cylindrical island with radius *a*.

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Coordinates system (right hand) around a cylindrical island with radius *a*.

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Distributions of total amplitude and phase for the *M*_{2} tidal frequency Sverdrup wave with (a) no friction, (b) bottom friction *γ*/*f* = 0.176, and (c) for *O*_{1} tidal frequency with the same bottom friction as in (b).

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Distributions of total amplitude and phase for the *M*_{2} tidal frequency Sverdrup wave with (a) no friction, (b) bottom friction *γ*/*f* = 0.176, and (c) for *O*_{1} tidal frequency with the same bottom friction as in (b).

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Distributions of total amplitude and phase for the *M*_{2} tidal frequency Sverdrup wave with (a) no friction, (b) bottom friction *γ*/*f* = 0.176, and (c) for *O*_{1} tidal frequency with the same bottom friction as in (b).

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Distributions of amplitude and phase of Poincare wave for (a) *M*_{2} and (b) *O*_{1} frequency in the case of *mb* = −1/3.4 when there is friction with *γ*/*f* = 0.176.

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Distributions of amplitude and phase of Poincare wave for (a) *M*_{2} and (b) *O*_{1} frequency in the case of *mb* = −1/3.4 when there is friction with *γ*/*f* = 0.176.

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Distributions of amplitude and phase of Poincare wave for (a) *M*_{2} and (b) *O*_{1} frequency in the case of *mb* = −1/3.4 when there is friction with *γ*/*f* = 0.176.

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Maximum amplitude and phase differences for (a) Sverdrup and (b) Poincare wave scattering across the island. The amplitude difference nondimensionalized by incident wave amplitude *A*_{0} and the phase differences are shown as a function of nondimensionalized frequency (*ω*/*f*). Various nondimensionalized friction coefficients are used; *γ*/*f* = 0 (solid lines), 0.1 (circles), 0.3 (triangles), and 0.6 (cross).

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Maximum amplitude and phase differences for (a) Sverdrup and (b) Poincare wave scattering across the island. The amplitude difference nondimensionalized by incident wave amplitude *A*_{0} and the phase differences are shown as a function of nondimensionalized frequency (*ω*/*f*). Various nondimensionalized friction coefficients are used; *γ*/*f* = 0 (solid lines), 0.1 (circles), 0.3 (triangles), and 0.6 (cross).

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Maximum amplitude and phase differences for (a) Sverdrup and (b) Poincare wave scattering across the island. The amplitude difference nondimensionalized by incident wave amplitude *A*_{0} and the phase differences are shown as a function of nondimensionalized frequency (*ω*/*f*). Various nondimensionalized friction coefficients are used; *γ*/*f* = 0 (solid lines), 0.1 (circles), 0.3 (triangles), and 0.6 (cross).

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Amplitude–phase diagrams of (a) Sverdrup and (b) Kelvin waves for the model island. The friction coefficients of *γ*/*f* = 0.1 (solid line) and 0.4 (dashed line) are used for the *M*_{2} and *O*_{1} tides. The phase axis is given for waves propagating to the left passing around the model island. Cross denotes the location *θ* = 0° on the island coast that points to the incident wave, open diamond *θ* = 90°, closed circle *θ* = 180°, and open triangle *θ* = 270°. Observed mean amplitudes around Cheju Island are used for each incident wave amplitude *A*_{0}.

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Amplitude–phase diagrams of (a) Sverdrup and (b) Kelvin waves for the model island. The friction coefficients of *γ*/*f* = 0.1 (solid line) and 0.4 (dashed line) are used for the *M*_{2} and *O*_{1} tides. The phase axis is given for waves propagating to the left passing around the model island. Cross denotes the location *θ* = 0° on the island coast that points to the incident wave, open diamond *θ* = 90°, closed circle *θ* = 180°, and open triangle *θ* = 270°. Observed mean amplitudes around Cheju Island are used for each incident wave amplitude *A*_{0}.

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Amplitude–phase diagrams of (a) Sverdrup and (b) Kelvin waves for the model island. The friction coefficients of *γ*/*f* = 0.1 (solid line) and 0.4 (dashed line) are used for the *M*_{2} and *O*_{1} tides. The phase axis is given for waves propagating to the left passing around the model island. Cross denotes the location *θ* = 0° on the island coast that points to the incident wave, open diamond *θ* = 90°, closed circle *θ* = 180°, and open triangle *θ* = 270°. Observed mean amplitudes around Cheju Island are used for each incident wave amplitude *A*_{0}.

Citation: Journal of Physical Oceanography 29, 3; 10.1175/1520-0485(1999)029<0436:SOTFWA>2.0.CO;2

Tide observations and harmonic constants of the *M*_{2} and *O*_{1} tides around Cheju Island. See Fig. 1 for the tide stations. Phase *k* is referred to 135°E longitude. Harmonic constants at sites 1 and 6 are analyzed from a one-year tidal record. Under data source Y and OH indicate Yih (1992) and Office of Hydrographic Affairs of Korea (1964, 1985).

Amplitude and phase differences in Sverdrup and Kelvin wave scattering with *M*_{2} and *O*_{1} frequencies around the model island, subject to variable nondimensional bottom friction. Amplitude differences are nondimensionalized by the incident amplitude *A _{s}* and

*A*.

_{k}