scholarly journals The generalized Wiener–Hopf equations for wave motion in angular regions: electromagnetic application

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210040
Author(s):  
V. G. Daniele ◽  
G. Lombardi
Keyword(s):  
General Theory ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Current Literature ◽  
Functional Equations ◽  
Wave Motion ◽  
Scattering Problems ◽  
First Order ◽  
Homogeneous Media ◽  
General Method

In this work, we introduce a general method to deduce spectral functional equations and, thus, the generalized Wiener–Hopf equations (GWHEs) for wave motion in angular regions filled by arbitrary linear homogeneous media and illuminated by sources localized at infinity with application to electromagnetics. The functional equations are obtained by solving vector differential equations of first order that model the problem. The application of the boundary conditions to the functional equations yields GWHEs for practical problems. This paper shows the general theory and the validity of GWHEs in the context of electromagnetic applications with respect to the current literature. Extension to scattering problems by wedges in arbitrarily linear media in different physics will be presented in future works.

scholarly journals The generalized Wiener–Hopf equations for the elastic wave motion in angular regions

2022 ◽  
Vol 478 (2257) ◽  
Author(s):  
Vito G. Daniele ◽  
Guido Lombardi
Keyword(s):  
Differential Equation ◽  
Functional Equations ◽  
Wave Motion ◽  
Elastic Field ◽  
Angular Region ◽  
First Order ◽  
Vector Differential Equation ◽  
First Time ◽  
Homogeneous Media ◽  
General Method

In this work, we introduce a general method to deduce spectral functional equations in elasticity and thus, the generalized Wiener–Hopf equations (GWHEs), for the wave motion in angular regions filled by arbitrary linear homogeneous media and illuminated by sources localized at infinity. The work extends the methodology used in electromagnetic applications and proposes for the first time a complete theory to get the GWHEs in elasticity. In particular, we introduce a vector differential equation of first-order characterized by a matrix that depends on the medium filling the angular region. The functional equations are easily obtained by a projection of the reciprocal vectors of this matrix on the elastic field present on the faces of the angular region. The application of the boundary conditions to the functional equations yields GWHEs for practical problems. This paper extends and applies the general theory to the challenging canonical problem of elastic scattering in angular regions.

A Note on the Disadvantages Associated with the Treatment of an Eigenvalue as an Additional Dependent Variable in Differential Eigenvalue Problems

1968 ◽  
Vol 72 (696) ◽  
pp. 1068
Author(s):  
B. Dawson ◽  
M. Davies
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
The Other ◽  
Unknown Boundary ◽  
Boundary Values ◽  
First Order ◽  
Basic Set ◽  
Non Linear ◽  
First Order Differential Equations

A novel technique of dealing with differential eigenvalue problems has recently been introduced by Wadsworth and Wilde . The differential equation is expressed as a set of simultaneous first-order differential equations, the eigenvalueλbeing regarded as an additional variable by adding the equationto the basic set. The differential eigenvalue problem is thus reduced to a set of non-linear first-order differential equations with two-point boundary conditions. This treatment of the problem, although novel, suffers from two serious disadvantages. First, it introduces non-linearity into an otherwise linear set of equations. Thus, the solution can no longer be obtained by linear combinations of independent particular solutions. One method of solving the non-linear systems is by assigning arbitrary starting values at one boundary and performing a step-by-step integration to the other boundary where in general the boundary conditions are not satisfied. The problem can be solved by adjustment of the initial assigned arbitrary values until the given conditions at the other boundary are satisfied. A second method and the one used by Wadsworth and Wilde is to estimate the unknown boundary values at both boundaries and integrate inwards to a meeting point. Changes can then be made to the unknown boundary values to make the two branches of the curve fit together.

Exact Equations for the Large Inextensional Motion of Elastic Plates

1981 ◽  
Vol 48 (1) ◽  
pp. 109-112 ◽  
Author(s):  
J. G. Simmonds
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Space Variable ◽  
Second Order ◽  
Straight Edge ◽  
Elastic Plates ◽  
Natural Coordinates ◽  
Governing Equations ◽  
First Order ◽  
Second Order In Time

The governing equations for plates that twist as they deform are reduced to 14 differential equations, first-order in a single space variable and second-order in time. Many of the equations are the same as for statics. Nevertheless, the extension to dynamics is nontrivial because the natural coordinates to use to describe the deformed, developable midsurface are not Lagrangian. The plate is assumed to have two curved, stress-free edges, one built-in straight edge, and one free straight edge acted upon by a force and a couple. There are 7 boundary conditions at the built-in end and 7 at the free end.

Note on the Petzval optical condition

1926 ◽  
Vol 23 (4) ◽  
pp. 461-464
Author(s):  
G. C. Steward
Keyword(s):  
General Theory ◽  
Optical System ◽  
Geometrical Meaning ◽  
First Order ◽  
Diffraction Patterns ◽  
General Method ◽  
Usual Notation

As a preliminary to an investigation of certain diffraction patterns I was led to consider, in some detail, the geometrical aberrations of a symmetrical optical system; and it appeared convenient then to classify the aberrations in orders according as they depend upon various powers of certain small quantities and to exhibit them as coefficients in the expansion of an ‘ Aberration Function.’ If aberrations of the first order only are considered, it becomes evident that one of them stands, in some sense, apart from the rest; I refer to the so-called ‘Petzval’ condition for flatness of field. It is of interest to notice that this condition was known to Coddington and to Airy before the time of Petzval—known at least as far as concerns systems of thin lenses. In the usual notation the condition is ΣΚ/μμ′ = 0; it is therefore independent of the positions of the object and pupil planes and in this respect it stands alone among the first order aberrations. But an increasing number of similar aberrations of higher orders will be found and it is of interest to examine these and to investigate their geometrical meaning. In the following note is given a proof of the Petzval condition, differing from that usually given and falling more into line with the general theory, and indicating also a general method of examining aberrations of this peculiar type.

Decay of Turbulent Axisymmetrical Free Flows With Rotation

1967 ◽  
Vol 34 (4) ◽  
pp. 806-812 ◽  
Author(s):  
A. Chervinsky ◽  
D. Lorenz
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Algebraic Equations ◽  
Free Jets ◽  
Swirling Jets ◽  
First Order ◽  
Free Jet ◽  
Order Algebraic ◽  
First Order Differential Equations

Previous studies on turbulent swirling jets are extended to cover general axisymmetrical turbulent free flows with rotation, with particular considerations of free jets and wakes. The equations governing the flow are integrated subject to the pertaining boundary conditions making use of the usual boundary-layer approximations. The axial and tangential components of velocity are assumed to retain similar forms of radial distributions and a set of two ordinary first-order differential equations is derived, the solution of which describes the axial decay of the maximal axial and tangential velocities. Integration of the differential equations in the particular case of uniform external velocity subject to conditions at the orifice results in a set of two second-order algebraic equations which are readily solved. The theoretical solutions derived for a free jet are compared with available experimental results.

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