scholarly journals Spherical-multipole analysis of an arbitrarily directed complex-source beam diffracted by an acoustically soft or hard circular cone

2015 ◽  
Vol 13 ◽  
pp. 57-61 ◽  
Author(s):  
A. Reinhardt ◽  
H. Bruens ◽  
L. Klinkenbusch ◽  
M. Katsav ◽  
E. Heyman
Keyword(s):  
Numerical Analysis ◽  
Boundary Value Problem ◽  
Analytical Approach ◽  
Circular Cone ◽  
Legendre Functions ◽  
Soft Or ◽  
Associated Legendre Functions ◽  
Multipole Analysis ◽  
Complex Source ◽  
Complex Valued

Abstract. An analytical approach to analyze the diffraction of an arbitrarily directed complex-source beam (CSB) by an acoustically soft or hard semi-infinite circular cone is presented. The beam is generated by assigning a complex-valued location to a point source; its waist and direction are defined by the real and imaginary parts of the source coordinate, respectively. The corresponding scalar boundary-value problem is solved by a spherical-multipole analysis. The solution requires the calculation of associated Legendre functions of the first kind for complex-valued arguments which turns out to be a non-trivial task. Beside a numerical analysis of the corresponding algorithms we present numerical results for the total near- and scattered far-fields.

scholarly journals Acoustic scattering of a complex-source beam by the edge of a plane angular sector

2014 ◽  
Vol 12 ◽  
pp. 179-186 ◽  
Author(s):  
H. Brüns ◽  
L. Klinkenbusch
Keyword(s):  
Plane Wave ◽  
Acoustic Scattering ◽  
Classical Case ◽  
Soft Or ◽  
New Approach ◽  
Angular Sector ◽  
Multipole Analysis ◽  
Incident Plane ◽  
Complex Source ◽  
Complex Valued

Abstract. The scattering and diffraction of a complex-source beam (CSB) by an acoustically soft or hard plane angular sector is treated by a rigorous spherical-multipole analysis in sphero-conal coordinates. By assigning a complex-valued radial source coordinate to the corresponding Green's function, the CSB is directed exactly towards the corner of the sector. Since the CSB can be interpreted as a localized plane wave, its interaction with the corner in the presence of the semi-infinite structure can be analyzed in detail. In opposite to the classical case of a non-localized incident plane wave, the resulting multipole series is strongly convergent and no summation techniques are necessary to obtain meaningful results. The numerical results include a convergence analysis, total near fields as well as scattered far fields and prove the applicability of this new approach.

scholarly journals Electromagnetic diffraction and scattering of a complex-source beam by a semi-infinite circular cone

2013 ◽  
Vol 11 ◽  
pp. 31-36 ◽  
Author(s):  
H. Brüns ◽  
L. Klinkenbusch
Keyword(s):  
Plane Wave ◽  
Electromagnetic Scattering ◽  
Scattering Problem ◽  
Multipole Expansion ◽  
Circular Cone ◽  
Plane Wave Incident ◽  
Complex Source ◽  
Full Plane ◽  
Complex Valued ◽  
Electromagnetic Diffraction

Abstract. A complex-source beam (CSB) is used to investigate the electromagnetic scattering and diffraction by the tip of a perfectly conducting semi-infinite circular cone. The boundary value problem is defined by assigning a complex-valued source coordinate in the spherical-multipole expansion of the field due to a Hertzian dipole in the presence of the PEC circular cone. Since the incident CSB field can be interpreted as a localized plane wave illuminating the tip, the classical exact tip scattering problem can be analysed by an eigenfunction expansion without having the convergence problems in case of a full plane wave incident field. The numerical evaluation includes corresponding near- and far-fields.

scholarly journals Derivatives of addition theorems for Legendre functions

1995 ◽  
Vol 37 (2) ◽  
pp. 212-234 ◽  
Author(s):  
D.E. Winch ◽  
P.H. Roberts
Keyword(s):  
Spherical Harmonics ◽  
Chebyshev Polynomials ◽  
Legendre Polynomials ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Addition Theorems ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Derivatives Of

AbstractDifferentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

scholarly journals Spatially Restricted Integrals in Gradiometric Boundary Value Problems

2009 ◽  
Vol 44 (4) ◽  
pp. 131-148 ◽  
Author(s):  
M. Eshagh
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Legendre Functions ◽  
Gravity Gradiometry ◽  
Satellite Gravity Gradiometry ◽  
Satellite Gravity ◽  
Earth’S Gravity Field ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Slepian Functions

Spatially Restricted Integrals in Gradiometric Boundary Value ProblemsThe spherical Slepian functions can be used to localize the solutions of the gradiometric boundary value problems on a sphere. These functions involve spatially restricted integral products of scalar, vector and tensor spherical harmonics. This paper formulates these integrals in terms of combinations of the Gaunt coefficients and integrals of associated Legendre functions. The presented formulas for these integrals are useful in recovering the Earth's gravity field locally from the satellite gravity gradiometry data.

scholarly journals The Asymptotic Expansions of the Spherical Harmonics

1922 ◽  
Vol 41 ◽  
pp. 82-93
Author(s):  
T. M. MacRobert
Keyword(s):  
Real Axis ◽  
Spherical Harmonics ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
The Real ◽  
Image Position

Associated Legendre Functions as Integrals involving Bessel Functions. Let,where C denotes a contour which begins at −∞ on the real axis, passes positively round the origin, and returns to −∞, amp λ=−π initially, and R(z)>0, z being finite and ≠1. [If R(z)>0 and z is finite, then R(z±)>0.] Then if I−m (λ) be expanded in ascending powers of λ, and if the resulting expression be integrated term by term, it is found that

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