TY - JOUR

T1 - Scattering of a Surface Wave in a Polygonal Domain with Impedance Boundary

AU - Лялинов, Михаил Анатольевич

AU - Zhu, Ning Yan

N1 - Publisher Copyright: © 2022 American Mathematical Society

PY - 2022/3/4

Y1 - 2022/3/4

N2 - The two-dimensional (2D) domain under study is bounded from below by two semi-infinite and, between them, two finite straight lines; on each of the straight lines (segments), a usually individual impedance boundary condition is imposed. An incident surface wave, propagating from infinity along one semi-infinite segment of the polygonal domain, excites outgoing surface waves both on the same segment (a reflected wave) and on the second semi-infinite segment (a transmitted wave); in addition, a circular (cylindrical) outgoing wave will be generated in the far field. The scattered wave field satisfies the Helmholtz equation and the Robin (in other words, impedance) boundary conditions as well as some special integral form of the Sommerfeld radiation conditions. It is shown that a classical solution of the problem is unique. By the use of some known extension of the Sommerfeld–Malyuzhinets technique, the problem is reduced to functional Malyuzhinets equations and then to a system of integral equations of the second kind with integral operator depending on a characteristic parameter. The Fredholm property of the equations is established, which also leads to the existence of the solution for noncharacteristic values of the parameter. From the Sommerfeld integral representation of the solution, the farfield asymptotics is developed. Numerical results for the scattering diagram are also presented.

AB - The two-dimensional (2D) domain under study is bounded from below by two semi-infinite and, between them, two finite straight lines; on each of the straight lines (segments), a usually individual impedance boundary condition is imposed. An incident surface wave, propagating from infinity along one semi-infinite segment of the polygonal domain, excites outgoing surface waves both on the same segment (a reflected wave) and on the second semi-infinite segment (a transmitted wave); in addition, a circular (cylindrical) outgoing wave will be generated in the far field. The scattered wave field satisfies the Helmholtz equation and the Robin (in other words, impedance) boundary conditions as well as some special integral form of the Sommerfeld radiation conditions. It is shown that a classical solution of the problem is unique. By the use of some known extension of the Sommerfeld–Malyuzhinets technique, the problem is reduced to functional Malyuzhinets equations and then to a system of integral equations of the second kind with integral operator depending on a characteristic parameter. The Fredholm property of the equations is established, which also leads to the existence of the solution for noncharacteristic values of the parameter. From the Sommerfeld integral representation of the solution, the farfield asymptotics is developed. Numerical results for the scattering diagram are also presented.

KW - Far-field asymptotics

KW - Fredholm integral equation

KW - Functional equations

KW - Impedance boundary of a polygon

KW - Numerical solution

KW - Surface waves

UR - http://www.scopus.com/inward/record.url?scp=85126691090&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/ffb1a1e3-726b-3d2b-ae2c-70d9cadc85fa/

U2 - 10.1090/spmj/1700

DO - 10.1090/spmj/1700

M3 - Article

VL - 33

SP - 255

EP - 282

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 2

ER -