The zeros of the Hankel function as a function of its order

1960 ◽  
Vol 2 (1) ◽  
pp. 228-244 ◽  
Author(s):  
Wilhelm Magnus ◽  
Leon Kotin
Keyword(s):  
Hankel Function

scholarly journals Zeros of the Hankel function of real order out of the principal Riemann sheet

1991 ◽  
Vol 37 (1-3) ◽  
pp. 89-99 ◽  
Author(s):  
Andrés Cruz ◽  
Javier Esparza ◽  
Javier Sesma
Keyword(s):  
Riemann Sheet ◽  
Hankel Function

On an integral transform associated with a condition of radiation

1972 ◽  
Vol 71 (2) ◽  
pp. 369-379 ◽  
Author(s):  
D. Naylor
Keyword(s):  
Integral Transform ◽  
Hankel Function ◽  
Orthogonal Functions ◽  
Considerable Debate ◽  
Formal Expansion ◽  
Image Position

1. There has been considerable debate on the conditions under which a given function f(r) can be expanded as a series of the orthogonal functions where υ1, υ2, … are the zeros of the Hankel function regarded as a function of its order. Here k, a are positive constants and 0 < a ≤ r < ∞. The formal expansion can be written aswhere

On some Cauchy-separable integral equations

1986 ◽  
Vol 99 (3) ◽  
pp. 547-564 ◽  
Author(s):  
D. Porter
Keyword(s):  
Integral Equations ◽  
Wave Diffraction ◽  
Hankel Function ◽  
Subsequent Paper ◽  
Singular Equations ◽  
Volterra Operators

In a recent paper, Porter [9] devised two generalized Volterra operators which convert integral equations with the Hankel function kernel into Cauchy singular equations. The transformations were exploited in [9], and in a subsequent paper (Porter and Chu [10]), in relation to certain wave diffraction problems.

Sound Radiation by a Point Source Near a Surface of Aircraft

2020 ◽  
pp. 17-25
Author(s):  
Natalia K. Musatova ◽  
Mezhlum A. Sumbatyan
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Physical Theory ◽  
Acoustic Pressure ◽  
Sound Radiation ◽  
Hankel Function ◽  
Ray Theory ◽  
Linear Algebraic Equations ◽  
Scattering Field ◽  
Element Method

The problem of sound radiation by a source located in the tail of an aircraft is considered. Three methods of finding acoustic pressure are compared: the boundary element method, the Kirchhoff’s physical theory of diffraction and the ray theory. The simplest model in the form of two-dimensional problem and some thin long shape with acute angle is considered. The diffraction problem for an acoustically solid obstacle lay in the solving Fredholm’s integral equation of the second kind. Due to the boundary element method application, the equation along the entire region is reduced to the equation along the boundary. Discretization by grid nodes, selected on the boundary curve, using the collocation method is applied for numerical solution. A system of linear algebraic equations with real coefficients is formed, then the total acoustic pressure is found. The Kirchhoff’s physical theory of diffraction is based on the fact that on an arbitrary convex body in case of short-wave diffraction in the vicinity of each boundary point in the zone of light the boundary value of pressure is equal to twice pressures in the incident field. By the ray theory the modulus of the acoustic pressure in the scattering field is described by the Hankel function. Argument of this function is equal to the length of full path of the beam when it is reflected once from the border. In conclusion, the pressure in cases, when in the sharp edge there is a split node and when there isn’t, are compared. Also a scattering field calculated by three theories and scattering field in the far receiving point are built.

scholarly journals Zeros of the Hankel function of real order and of its derivative

1982 ◽  
Vol 39 (160) ◽  
pp. 639-639 ◽  
Author(s):  
Andr{és Cruz ◽  
Javier Sesma
Keyword(s):  
Hankel Function

Buffon's Needle on Caustics and Torus Quantization

2003 ◽  
Vol 10 (01) ◽  
pp. 51-64 ◽  
Author(s):  
B. H. Lavenda
Keyword(s):  
Continuum Limit ◽  
Principal Term ◽  
Small Distance ◽  
Hankel Function ◽  
Shadow Zone ◽  
Parallel Lines ◽  
Uniform Asymptotic Expansion ◽  
Sine Law ◽  
Method Of Steepest Descent ◽  
The Continuum

The probability of n + 1 intersections of a long needle in the Buffon problem is the eikonal of a Hankel function which is the principal term in the uniform asymptotic expansion in powers of the small distance between the parallel lines. Evaluating this probability using the torus quantization conditions shows that in the physically meaningful region, where a closed convergence of rays covers the caustic circle, the probability is greater than unity. In addition, the method of steepest descent shows that the caustic and reflection indices are more general than the ones given by torus quantization. The distance from the light source to the center of caustic circle corresponds to the length of the needle, and n times the distance between the parallel lines is the radius of the caustic. Unlike diffraction problems, the solution cannot be extended to the shadow zone since the angles become imaginary. In the continuum limit where the distance between the parallels tends to zero, the number of intersections is governed by an arc sine law in which maximum number of intersections, or the maximum chord length in a circle of a given radius, is most probable.

Zeros of the modified Hankel function

1970 ◽  
Vol 16 (3) ◽  
pp. 278-284 ◽  
Author(s):  
Erasmo M. Ferreira ◽  
Javier Sesma
Keyword(s):  
Hankel Function

scholarly journals Method of Singular Integral Equations for Analysis of Strip Structures and Experimental Confirmation

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 140
Author(s):  
Liudmila Nickelson ◽  
Raimondas Pomarnacki ◽  
Tomyslav Sledevič ◽  
Darius Plonis
Keyword(s):  
Differential Equations ◽  
Integral Representation ◽  
Boundary Conditions ◽  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Integral Form ◽  
Hankel Function ◽  
Cauchy Type ◽  
Constitutive Parameters

This paper presents a rigorous solution of the Helmholtz equation for regular waveguide structures with the finite sizes of all cross-section elements that may have an arbitrary shape. The solution is based on the theory of Singular Integral Equations (SIE). The SIE method proposed here is used to find a solution to differential equations with a point source. This fundamental solution of the equations is then applied in an integral representation of the general solution for our boundary problem. The integral representation always satisfies the differential equations derived from the Maxwell’s ones and has unknown functions μe and μh that are determined by the implementation of appropriate boundary conditions. The waveguide structures under consideration may contain homogeneous isotropic materials such as dielectrics, semiconductors, metals, and so forth. The proposed algorithm based on the SIE method also allows us to compute waveguide structures containing materials with high losses. The proposed solution allows us to satisfy all boundary conditions on the contour separating materials with different constitutive parameters and the condition at infinity for open structures as well as the wave equation. In our solution, the longitudinal components of the electric and magnetic fields are expressed in the integral form with the kernel consisting of an unknown function μe or μh and the Hankel function of the second kind. It is important to note that the above-mentioned integral representation is transformed into the Cauchy type integrals with the density function μe or μh at certain singular points of the contour of integration. The properties and values of these integrals are known under certain conditions. Contours that limit different materials of waveguide elements are divided into small segments. The number of segments can determine the accuracy of the solution of a problem. We assume for simplicity that the unknown functions μe and μh, which we are looking for, are located in the middle of each segment. After writing down the boundary conditions for the central point of every segment of all contours, we receive a well-conditioned algebraic system of linear equations, by solving which we will define functions μe and μh that correspond to these central points. Knowing the densities μe, μh, it is easy to calculate the dispersion characteristics of the structure as well as the electromagnetic (EM) field distributions inside and outside the structure. The comparison of our calculations by the SIE method with experimental data is also presented in this paper.

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