Diffraction of a plane wave by a half plane under mixed boundary conditions

1981 ◽  
Vol 59 (8) ◽  
pp. 974-984 ◽  
Author(s):  
T. C. Kaladhar Rao
Keyword(s):  
Plane Wave ◽  
Boundary Conditions ◽  
Half Plane ◽  
Infinite Series ◽  
Series Solution ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Condition ◽  
Neumann Condition ◽  
Transform Method

The problem of diffraction of a plane wave by a semi-infinite half plane with mixed boundary conditions (Dirichlet condition on one face and Neumann condition on the other) is solved by a direct and rather straightforward method. The infinite series solution and the far field are in agreement with the previous solutions obtained by the Lebedev–Kontorovich transform method as expected, as the two methods are basically equivalent. An alternate representation of the infinite series solution is presented which is valid for any type of incident field including cylindrical and spherical fields. This representation facilitates easy analysis of transient problems and the special case of an incident plane unit step function on the half plane is given as an example.

scholarly journals Optimal Design Problems for the First p-Fractional Eigenvalue with Mixed Boundary Conditions

2018 ◽  
Vol 18 (2) ◽  
pp. 323-335 ◽  
Author(s):  
Julian Fernández Bonder ◽  
Julio D. Rossi ◽  
Juan F. Spedaletti
Keyword(s):  
Optimal Design ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Condition ◽  
Shape Design ◽  
First Eigenvalue ◽  
Design Problems ◽  
Asymptotic Bounds ◽  
Optimization Variable

AbstractIn this paper, we study an optimal shape design problem for the first eigenvalue of the fractionalp-Laplacian with mixed boundary conditions. The optimization variable is the set where the Dirichlet condition is imposed (that is restricted to have measure equal to a prescribed quantity α). We show existence of an optimal design and analyze the asymptotic behavior when the fractional parametersconverges to 1, and thus obtain asymptotic bounds that are independent of α.

scholarly journals The solution of arbitrary smoothness of the one-dimensional wave equation for the problem with mixed conditions

2021 ◽  
Vol 57 (3) ◽  
pp. 286-235
Author(s):  
V. I. Korzyuk ◽  
I. S. Kozlovskaya ◽  
V. Y. Sokolovich ◽  
V. A. Sevastyuk
Keyword(s):  
Wave Equation ◽  
Classical Solution ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Analytical Form ◽  
Dirichlet Condition ◽  
Half Line ◽  
Neumann Condition ◽  
The One ◽  
Necessary And Sufficient

In this paper, we represented an analytical form of a classical solution of the wave equation in the class of continuously differentiable functions of arbitrary order with mixed boundary conditions in a quarter of the plane. The boundary of the area consists of two perpendicular half-lines. On one of them, the Cauchy conditions are specified. The second half-line is separated into two parts, namely, the limited segment and the remaining part in the form of a half-line. The Dirichlet condition is specified on the segment, as well as the Neumann condition is fulfilled on the second part in the form of a half-line. In a quarter of the plane, the classical solution of the problem under consideration is determined. To construct this solution, a particular solution of the original wave equation is established. For the given functions of the problem, the concordance conditions are written, which are necessary and sufficient for the solution of the problem to be classical of high order of smoothness and unique.

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