An Analytical Solution for Wave Transformation Over Axi-Symmetric Topography

The present study considers wave scattering phenomena around a cylindrical island mounted on a general axi-symmetric topography or a general submerged truncated axi-symmetric shoal based on the mild-slope equation. The method of separation of variables and Taylor series expansion are invoked to find the approximate solution to the variable water depth region which varies proportionally to an arbitrary power of radial distance. Validations against the solutions for the combined wave refraction and diffraction around a cylindrical island mounted on a paraboloidal shoal of Liu et al. in 2004 and the scattering and trapping of wave energy by a submerged truncated paraboloidal shoal of Lin and Liu in 2007 show excellent agreements as the power of radial distance being equal to two. For the solutions of wave refraction and diffraction around a cylindrical island mounted on a shoal with depth proportionally to an arbitrary power of radial distance, good agreements with Zhai et al.’s (2013) solutions are demonstrated. Since the robustness of the assumption of a general axi-symmetric geometry based on an arbitrary power variability of the radial distance, the present solution can be very conveniently employed to investigate the effects of bottom topography on wave scattering and trapping patterns.


Introduction
The complex dynamical behaviors of wave transformation around an axi-symmetric structure have received many research attempts during the last several decades. Homma (1950) is a pioneer to obtain an analytical solution for long wave propagation over a cylindrical island mounted on a paraboloidal shoal that later on the paradigm used in his study was named Homma's island. Since then, many studies have further extended Homma's work which have made certain contribution to coastal and marine engineering understanding. Several analytical solutions to the linear shallow-water equation for some axi-symmetrical topographies have been found, for example, a conical island (Zhang and Zhu, 1994), a circular cylindrical island mounted on a conical shoal (Zhu and Zhang, 1996), a circular bowl pit (Suh et al., 2005), a circular hump (Zhu and Harun, 2009), a circular island mounted on a general shoal (Liu et al., 2012), and a submerged cylinder in an axi-symmetric pit (Liu and Sun, 2014). Based on the linear shallow-water equation, Kânoǧlu and Synolakis (1998) developed an analytic solution for simple piecewise linear topographies. Yu and Zhang (2003) also derived a solution for the linear shallow-water equation for waves over circular shoals. However, all these analytical studies resorted to the supposition of long wave to simplify the problems concern; when starting with the more general equation, i.e, the mild-slope equation, the circumstance becomes much more challenging and sophisticated.
The mild-slope equation (MSE) was independently proposed by Berkhoff (1972Berkhoff ( , 1976 and Smith and Sprinks (1975) and was proved to be an advantageous model for a wide spectrum of water wave transformation problems over uneven bottom topography as it can resolve the combined effect of wave refraction and diffraction in two dimensions. The principal barrier to solve the MSE explicitly comes from the implicit dispersion relation. This makes approximate techniques as the only wise choice to solve the MSE analytically.
When examining combined wave refraction and diffraction around a cylindrical island mounted on a paraboloidal shoal, and by employing Hunt's (1979) approximation to the linear dispersion equation, Liu et al. (2004) successfully derived a solution to the MSE in terms of combined Fourier series and Taylor series for a wider range of wave spectrum. It was then further employed to study wave scattering patterns over axi-symmetrical topographies. Examples include waves incident to a circular island on an axi-symmetric shoal (Cheng, 2007), waves propagation over an axi-symmetric pit (Jung and Suh, 2007), wave energy trapped by a submerged truncated paraboloidal shoal , wave scattering by a cylinder mounted on a conical shoal , waves propagation over an axisymmetric pit (Jung and Suh, 2008), and waves evolution around a circular conical island (Hsiao et al., 2010). Cheng et al. (2012) also provided a series solution to the MSE for wave scattering by a circular island mounted on an axi-symmetrical shoal. Recently, Zhai et al. (2013) proposed an exact analytical solution in terms of Taylor series to solve explicitly the modified mild-slope equation (MMSE) for wave propagation over a general Homma island with the maximal bottom slope of 4.27:1 for whose additional bottom curvature and slope-squared terms are included to better resolve the bottom effect when compared with the MSE solutions.
The present study is motivated by Liu et al.'s (2004) work to further bridge up the gap of easing topographic presumption that the geometry considered could be an idealized island standing on a general axi-symmetric structure or a general submerged truncated axi-symmetric shape. By employing the method of separation of variables and Taylor series expansion, the solution is derived for the interior computational domain shifting proportionally to an arbitrary power of radial distance. The required conditions will then be imposed to establish appropriate initial and boundary conditions to verify the robustness of the present solution with the existed solutions of wave scattering around a circular island on a paraboloidal shoal (Liu et al., 2004) and wave energy trapped by a submerged truncated paraboloidal shoal . Fig. 1 depicts the computational domain in case of an idealized cylindrical island mounted on a paraboloidal shoal when placing circular wall vertically along the coastline and in case of a submerged and truncated paraboloidal shoal when applying horizontal cutting plane to the former case. Assume a long-crested incident wave propagates in the positive x-direction and the origin of the Cartesian coordinates (x, y) situates at the centre of the shoal on the quiescent water level. The radii of the shoal and cylindrical island (or truncated plane) are correspondingly expressed as b and a. The variable water depth profile is denoted as:

Analytic solution
h a h b α where , , and are the water depths along the coastline (or water depth in the center of the shoal), the constant water depth outside the shoal, and the positive real number which can be arbitrarily selected, respectively.
η(x, y) The water surface elevation is governed by the MSE defined as: in which is the horizontal gradient operator, is the angular frequency. In addition, the phrase velocity and group velocity are defined respectively as: and h (x, y) k (x, y) where g is the gravitational acceleration. The relation of water depth and wave number is determined through the dispersion equation of linear wave theory For simple harmonic incident wave with the amplitude A, resorting to the following dimensionless quantities , k ′ = ka, Then the governing Eq. (2) and the dispersion relation Eq. (5) become and Hereafter, all the primes will be dropped for notational convenience. By assuming the axi-symmetric topography of the physical problem concerned, rewrite Eq. (6) in the cylindrical coordinates with and as: Hunt's (1979) direct solution to the linear dispersion relation Eq. (7) takes the form (12) Substituting Eqs. (10)-(12) into Eq. (8) yields the following approximate form of the mild-slope equation By the method of separation of variable, the solution of Eq. (13) can be expressed as: is the i-th solution corresponding to the m-th angular mode that satisfies the following ordinary differential equation, for i=1 or 2 and m = 0, 1, 2, …, ∞. In addition, denotes the unknown coefficients which will later be determined by the connection conditions. By substituting into Eq.
(15) and noting that with , Eq. (15) be- The general solution of the resulting second-order ordinary differential equation can be directly found by constructing a Taylor series around an ordinary point of Eq. (16). For the sake of analyzing the convergence of Taylor series solution, the following mappings are employed. Then Eq. (16) becomes , for the i-th solution. Then the solutions are given by and in which, and correspond to the v-th order derivatives of A(t), B(t), C(t), D(t) at the expansion point . The solutions in the varying water domain ( ) are given by b/a ⩽ r In the exterior uniform water depth region ( ), the well-known analytical solution was derived by MacCamy and Fuchs (1954), and is given by where , k b is the wavenumber corresponding to the constant water depth h b /a, J m is the Bessel function of the first kind of order m, is the Hankel function of the first kind of order m, and is the Jacobi symbol defined by for m = 0 and for m > 0. r = b/a On the exterior circle , the dynamic and kinematic matching conditions require (29) For the problem of water wave scattering by a cylindrical island mounted on a paraboloidal shoal, on the interior circle r = 1, we have the condition Eqs. (28)-(30) can be used to solve the unknowns , , and and therefore the problem of water wave scattering by a cylindrical island mounted on a paraboloidal shoal is solved.
On the other hand, for the problem of water wave scattering and trapping by a submerged truncated paraboloidal shoal, the interior solution is given by where is the wavenumber corresponding to the constant water depth h a /a. Following Lin and Liu (2007), the matching conditions on the interior circle , are given by Eqs. (28), (29), (32), and (33) can be used to solve the unknowns , , and . These complete the solution procedures.

Results and discussion
First, the convergence of present study which greatly relies upon Fourier series and Taylor series is thoroughly examined. In our numerical manipulation, the terms in Taylor  series in Eq. (26) are summed till convergent results are obtained, the summation process is stopped when, is satisfied. As the incident wave becomes shorter, more terms N in Taylor and angular modes m in Fourier series are required to obtain the convergence. While the typical numbers for N and m are 10 and 60, respectively, for incident waves with periods larger than 120 s to calculate the wave amplification along the coastline for Homma's island, the figures for N and m increase sharply to 15 and 250 when the wave periods are less than 90 s. In addition, the farther the distance is from the island center, more angular modes m and numbers of Taylor series are needed. From  Fig. 2, 15 angular modes are sufficient to obtain convergent results in calculating wave amplification along the coastline (r=1); however, in calculating wave amplification along the shoreline with r =3, 25 angular modes are asserted to achieve the convergence.

Comparison with Liu et al.'s (2004) solution
Employing Hunt's (1979) direct solution to the linear dispersion relation, Liu et al. (2004) successfully derived an analytical solution for the study of combined wave refraction and diffraction by a cylindrical island mounted on a paraboloidal shoal. Although higher order approximation can be used to approximate the linear dispersion relation, when exceeding the 4th-order approximation as pointed out by Hsiao et al. (2010), the analytical solution becomes much more complicated in phase speed calculation. Therefore, the third order of Hunt's (1979) direct solution will be employed in the following studies.
It is supposed that the solutions of the considered problems reduce to Liu et al.'s (2004) solution for . In Fig. 5, good agreements can be observed between the present and Zhai et al.'s (2013) soluitons for different as for T = 480 s. This should have validated the present solution for . Some discussions on the advantages and weakness among the solutions can be found in the article of Zhai et al. (2013). Lin and Liu's (2007)

Comparison with
As the power of the radial distance equal to two, the present study approaches to the solution of Lin and Liu (2007) for the study of wave energy scattering and trapping by a submerged truncated paraboloidal shoal. To validate against Lin and Liu's (2007) solution, several typical parameters of a = 10 km, b = 30 km, h a = 444 m, h b = 4 km, T = 120 s, 180 s, 720 s, and 1440 s are chosen. Excellent agreements of the present solution for the case of with the analytical solution of Lin and Liu's (2007) are shown in Fig. 6. 3.4 Paraboloidal shoal α A more detailed study is provided for the bottom effect in addition to Section 3.2. Fig. 7 shows the relative wave height corresponding to different wave periods of 1440 s, 720 s, 240 s, and 120 s. Clearly, the wave amplification is stronger near the cylindrical island for larger value of . In Fig. 7c, a comparison with Zhai et al.'s (2013) solution is provided with good agreements.
To investigate the main pattern of wave transformation over the entire region of cylindrical island, the contour lines of relative wave height on a rectangular region of and for the Homma's island in case of s are plotted in Fig. 8. In the figure, the changes of wave patterns with respect to the parameter are significant.

Truncated shoal
In this subsection, all the parameters of a, b, h b , h b , and are kept the same as the previous subsection. However, the considered problem is changed from the wave scatterings by a cylindrical island mounted on a paraboloidal shoal to wave scattering and trapping by a submerged truncated paraboloidal shoal. By the settings, the solutions of the problems reduce to Lin and Liu's (2007) solution for . Fig. 9 shows the relative moving location of focal point corresponding to different incident wave periods of 1440 s, 720 s, 240 s and 120 s. It is seen that for a long wave (T= 1440 s), the shoal does not impose significant modification to waves above it. The maximal wave amplification occurs in front of the shoal center due to the formation of partial standing waves by the incident and reflected waves. As the wavelength decreases (T=720 s, 240 s and 120 s), the refraction effect becomes increasingly important and the focal effect become stronger as the wave can be amplified up to about 5−7 times its original amplitude some distances downstream the shoal center. In addition, a smaller value of results in a weaker focal effect as the shoal becomes smaller. α In order to identify the effects of power to the wave pattern around the truncated paraboloidal shoal, the related contour lines of wave amplification on a rectangular region of and for the Homma's island in case of s are plotted in Fig. 10. Finally, the effects of extent of shoal submergences on   h a wave energy trapping are examined. We discuss the results corresponding to various values of the power and radius of shoal submergence by changing while other parameters are fixed at s, km, km and from Eq. (35). Fig. 11 shows the variation of relative wave height regarding the extent of shoal submergences with dif-α ferent values of . As expected, it is seen that for a larger value of a/b or deep submergence, the wave amplification is weaker, as expected. As the increase of shoal height, the maximum value of amplification becomes larger and the focal point shifts gradually toward the center of the shoal from the rear part of the shoal.

Conclusion
An analytical solution of the mild-slope equation is obtained for wave propagating over axi-symmetric topography which could be idealized symmetric or symmetrical truncated shoal. The method of separation of variable and Hunt's (1979) approximate solution to the linear dispersion relation has been employed to transfer the implicit coefficients in the governing equation into explicit form. The solutions to the resulting ordinary differential equation are then given in terms of Fourier series and Taylor series. By introducing an arbitrary power of radial distance on variable water depth domain, the present solutions extend Liu et al.'s (2004) and Lin and Liu's (2007) solutions to a more general axi-symmetric geometries. The present study further investigates wave patterns by varying values of power of radial distance and more interesting reveals of wave transformation have been examined. For the problem of wave scattering and trapping by a submerged truncated paraboloidal shoal, the effect of shoal submergences is studied.