Modeling a Wave on Mild Sloping Bottom Topography and Its Dispersion Relation Approximation
DOI:
https://doi.org/10.15575/kubik.v7i1.18419Keywords:
linear wave, mild sloping seabed, dispersion relation approximation, velocity potentialAbstract
Linear wave theory is a simple theory that researchers and engineers often use to study a wave in deep, intermediate, and shallow water regions. Many researchers mostly used it over the horizontal flat seabed, but in actual conditions, sloping seabed always exists, although mild. In this research, we try to model a wave over a mild sloping seabed by linear wave theory and analyze the influence of the seabed’s slope on the solution of the model. The model is constructed from Laplace and Bernoulli equations together with kinematic and dynamic boundary conditions. We used the result of the analytical solution to find the relation between propagation speed, wavelength, and bed slope through the dispersion relation. Because of the difference in fluid dispersive character for each water region, we also determined dispersion relation approximation by modifying the hyperbolic tangent form into hyperbolic sine-cosine and exponential form, then approximated it with Padé approximant. As the final result, exponential form modification with Padé approximant had the best agreement to exact dispersion relation equation then direct hyperbolic tangent form.
References
R. R. Rahmawati, A. H. S. Putro, and J. L. Lee, “Analysis of Long-Term Shoreline Observations in the Vicinity of Coastal Structures: A Case Study of South Bali Beaches,†Water, vol. 13, no. 24, p. 3527, 2021.
K. He and J. Ye, “Physical modeling of the dynamics of a revetment breakwater built on reclaimed coral calcareous sand foundation in the South China Sea—tsunami wave,†Bull. Eng. Geol. Environ., vol. 80, no. 4, pp. 3315–3330, 2021.
F. S. Fathonah, D. Zulkarnaen, and E. Sukaesih, “Pencarian Solusi Persamaan Diferensial Parsial Non Linier menggunakan Metode Transformasi Pertubasi Homotopi dan Metode Dekomposisi Adomian,†Kubik J. Publ. Ilm. Mat., vol. 2, no. 1, pp. 35–42, May 2017, doi: 10.15575/kubik.v2i1.1472.
P. Guven Geredeli, “A time domain approach for the exponential stability of a linearized compressible flowâ€structure PDE system,†Math. Methods Appl. Sci., vol. 44, no. 2, pp. 1326–1342, Jan. 2021, doi: 10.1002/mma.6833.
C.-S. Liu, E. R. El-Zahar, and Y.-W. Chen, “Solving nonlinear elliptic equations in arbitrary plane domains by using a new splitting and linearization technique,†Eng. Anal. Bound. Elem., vol. 125, pp. 124–134, Apr. 2021, doi: 10.1016/j.enganabound.2021.01.012.
A. A. Aderogba and A. R. Appadu, “Classical and multisymplectic schemes for linearized kdv equation: Numerical results and dispersion analysis,†Fluids, vol. 6, no. 6, pp. 1–29, 2021, doi: 10.3390/fluids6060214.
L. H. Holthuijsen, Waves in oceanic and coastal waters. Cambridge university press, 2010. doi: 10.1017/cbo9780511618536.
G. Guannel and H. T. Özkan-Haller, “Formulation of the undertow using linear wave theory,†Phys. Fluids, vol. 26, no. 5, p. 056604, May 2014, doi: 10.1063/1.4872160.
T. Skrivan, A. Soderstrom, J. Johansson, C. Sprenger, K. Museth, and C. Wojtan, “Wave curves,†ACM Trans. Graph., vol. 39, no. 4, Aug. 2020, doi: 10.1145/3386569.3392466.
S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, “Propagation Dynamics of Airy Water-Wave Pulses,†Phys. Rev. Lett., vol. 115, no. 3, p. 034501, Jul. 2015, doi: 10.1103/PhysRevLett.115.034501.
G. J. Schiereck, Introduction to bed, bank and shore protection. CRC Press, 2017.
Z. You, “Discussion of ‘Simple and explicit evolution to the wave dispersion equation’ [Coastal Engineering 45 (2002) 71–74],†Coast. Eng., vol. 48, no. 2, pp. 133–135, Apr. 2003, doi: 10.1016/S0378-3839(02)00170-9.
W. Tarantino and S. Di Sabatino, “Diagonal Padé approximant of the one-body Green’s function: A study on Hubbard rings,†Phys. Rev. B, vol. 99, no. 7, p. 075149, Feb. 2019, doi: 10.1103/PhysRevB.99.075149.
S. Gluzman, Padé and post-padé approximations for critical phenomena, †Symmetry (Basel)., vol. 12, no. 10. 2020. doi: 10.3390/sym12101600.
B. Méhauté, “An Introduction to Water Waves,†in An Introduction to Hydrodynamics and Water Waves, Berlin, Heidelberg: Springer Berlin Heidelberg, 1976, pp. 197–211. doi: 10.1007/978-3-642-85567-2_15.
A. Abanov, T. Can, and S. Ganeshan, “Odd surface waves in two-dimensional incompressible fluids,†SciPost Phys., vol. 5, no. 1, p. 010, Jul. 2018, doi: 10.21468/SciPostPhys.5.1.010.
Downloads
Published
Issue
Section
License
Authors who publish in KUBIK: Jurnal Publikasi Ilmiah Matematika agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Attribution-ShareAlike 4.0 International (CC BY-SA 4.0) License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
Â
