Integral Representations of Solution in the Problem on Skew Incidence of a Surface Wave on the Straight Shoreline Water Wedge

M. A. Lyalinov; Лялинов М. А.; S. V. Polyanskaya; Полянская С. В.

doi:10.31857/S0032823524030055

Integral Representations of Solution in the Problem on Skew Incidence of a Surface Wave on the Straight Shoreline Water Wedge

Authors: Lyalinov M.A.¹, Polyanskaya S.V.²
Affiliations:
1. Saint Petersburg State University
2. North-Western Institute of Management of the RANEPA
Issue: Vol 88, No 3 (2024)
Pages: 406-421
Section: Articles
URL: https://ruspoj.com/0032-8235/article/view/675053
DOI: https://doi.org/10.31857/S0032823524030055
EDN: https://elibrary.ru/ZAWQKP
ID: 675053

Cite item

Full Text

Open Access
Restricted Access

Access granted
Restricted Access

Subscription Access

Abstract

In the linear approximation of the surface gravitational waves of small amplitude a classical model problem about the incursion of a surface wave under some angle on the shoreline is solved. The problem is formulated for the harmonic potential of velocity of the fluid in the 3D water wedge with the Robin-Steklov boundary condition on the free surface and with the no-flow condition along the normal on the bed of the water domain. Some critical comments about a known in the literature solution having a “non-physical” singularity of the logarithmic type on the coastal line are given. The asymptotics with respect to distance from the shoreline of the obtained solution, bounded on the edge, is constructed. The reflection coefficient of the wave reflected from the shoreline is calculated.

Keywords

surface wave, integral representations, functional equations, asymptotics

Full Text

About the authors

M. A. Lyalinov

Saint Petersburg State University

Author for correspondence.
Email: lyalinov@yandex.ru
Russian Federation, Saint Petersburg

S. V. Polyanskaya

North-Western Institute of Management of the RANEPA

Email: polyanskaya-sv@ranepa.ru
Russian Federation, Saint Petersburg

References

Shrira V.I. et al. Can edge waves be generated by wind? // J. of Fluid Mech., 2022, vol. 934, pp. A16–136.
Ehrenmark U.T. Oblique wave incidence on a plane beach: The classical problem revisited // J. of Fluid Mech., 1998, vol. 368, pp. 291–319.
Isaacson E. Water waves over a sloping bottom // Commun. Pure Appl. Math., 1950, vol. 3, pp. 11–31.
Kuznetsov N., Mazya V., Vainberg B. Linear Water Waves. Cambridge: Univ. Press, 2002. 513 p.
Ursell F. Edge waves on a sloping beach // Proc. R. Soc. Lond. Ser. A, 1952, vol. 214, pp. 79–97.
Lyalinov M.A. Comment on the eigenfunctions and eigenvalues of the Laplace operator in an angle with Robin boundary conditions // Proc. Sci. Sem.r of the St. Petersburg Branch of the Steklov Institute of Mathematics of the RAS, 2019, vol. 483, no. 49, pp. 116–127.
Khalile M., Pankrashkin K. Eigenvalues of Robin Laplacians in infinite sectors // Math. Nachrichten, 2018, vol. 291, no. 5–6, pp. 928–965.
Lyalinov M.A. Eigenoscillations in an angular domain and spectral properties of functional equations // Europ. J. of Appl. Math., 2021, vol. 33, pp. 538–559.
Lyalinov M.A. On the eigenfunctions of the essential spectrum of a model problem for the Schrödinger operator with a singular potential // Matem. sb., 2023, vol. 214(10), pp. 3–29.
Babich V.M., Lyalinov M.A., Grikurov V.E. Diffraction Theory: The Sommerfeld–Malyuzhinets Technique. Oxford: Alpha Sci. Int., 2008. 215 p.
Malyuzhinetz G.D. Excitation, reflection and radiation of surface waves from a wedge with arbitrary surface impedances // Dokl. AN SSSR, 1985, vol. 3. pp. 752–755.
Gradshteyn I.S., Ryzhik M.I. Table of Integrals, Series and Products. N.Y.: Acad. Press, 2007. 1108 p.
Lyalinov M.A. Functional Difference Equations and their link with Perturbations of the Mehler operator // Rus. J. of Math. Phys., 2022, vol. 29, no. 3, pp. 379–397.