Abstract
This study examines oblique wave interaction with a submerged perforated semicircular breakwater based on the linear potential theory. The fluid domain is divided into inner and outer regions by a perforated semicircular arc. The velocity potentials in the inner and outer regions are constructed by eigenfunction expansions and multipole expansions, respectively. The unknown constants in the velocity potentials are determined by matching the boundary conditions on the perforated arc. The convergence of the solution method we have developed is rapid. The reflection and transmission coefficients and the wave forces acting on the breakwater are calculated and examined. Some useful results are presented for practical engineering.
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Acknowledgments
We wish to thank Prof. Guo-Xiong Wu at UCL for his valuable private notes on this work. We would like to thank the four reviewers and the associate editor Dr Andrew Hogg for their valuable comments, which greatly enhanced the quality of this paper. We also wish to thank Sarah Wood for proofreading this paper. This work was supported by the National Natural Science Foundation of China (Grants 51010009 and 50909086).
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Appendix: Multipoles with singularities on the water bottom for oblique wave
Appendix: Multipoles with singularities on the water bottom for oblique wave
Multipoles with singularities on the water bottom were given by Chapman [9]. Here the symmetric and antisymmetric multipoles, which have singularities at the semicircular breakwater centre \(r\) \(=\) 0 and satisfy Eqs. (3)–(6), are briefly introduced for the completeness of the presentation.
Solutions to Eqs. (3) and (4), which are singular at \(r\) \(=\) 0, can be written respectively as
where \(K_{n}\) is the modified Bessel function of the second kind of order \(n\). The preceding solutions can be represented as [15, Eqs. (B.101) and (B.103)]
where \(\nu =k_y \cosh \mu \).
The symmetric and antisymmetric multipoles, \(\varphi _n^+ \) and \(\varphi _n^- \), may have the following forms:
Obviously, Eqs. (33) and (34) satisfy the field equation in Eq. (3) and the water bottom condition in Eq. (4). The unknowns \(f^{+}(\mu )\) and \(f^{-}(\mu )\) can be determined using the free surface condition in Eq. (5). Then the symmetric and antisymmetric multipoles for the present problem can be written as
where the path of the integration passes below the pole at \(\mu =\kappa =\ln \left( {1/{\sin \beta }+\sqrt{1/{\sin ^{2}\beta }-1}} \right) \) to satisfy the radiation conditions in Eq. (6) [34, 35].
As \(x\rightarrow \pm \infty \), we have
where \(N_0^2 =\left[ {1+{\sinh \! \left( {2kh} \right) }/{\left( {2kh} \right) }} \right] /2\).
The multipoles can be written as power series expansions [29, Eq. A.4]:
where \(I_{m}\) is the modified Bessel function of the first kind of order \(m\) and
in which the path of the integration passes below the pole at \(\mu =\kappa \), and \(\varepsilon _0 =1\) and \(\varepsilon _m =2\) (\(m\ge 1)\).
Computational methods for such types of integrals in Eqs. (41) and (42) can be found in Linton [36, Eqs. (3.7) and (4.11)]. For example, the integral in Eq. (41) can be rewritten as
where PV denotes the principal value integral; the function \(S\left( {x,y} \right) \) is defined as \(S\left( {x,y} \right) =(K+x)\cosh \! \left( {2ny} \right) \cosh \left( {2my} \right) /{\left( {{\text{ e }}^{2xh}+1} \right) }\); the symbol \(b\) denotes the positive real root of the relationship
using the following integral:
in which \(\tilde{\nu }=\nu \;\tanh \! \left( {\nu h} \right) -k_y \tanh \! \left( {k_y h} \right) \). A similar expression can be obtained for Eq. (42).
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Liu, Y., Li, H.J. Analysis of oblique wave interaction with a submerged perforated semicircular breakwater. J Eng Math 83, 23–36 (2013). https://doi.org/10.1007/s10665-013-9625-x
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DOI: https://doi.org/10.1007/s10665-013-9625-x