A Bessel–Watson transform pair

1981 ◽  
Vol 22 (1) ◽  
pp. 39-41 ◽  
Author(s):  
S. N. Samaddar
Keyword(s):  
Watson Transform

Field Transforms for Multilayered Cylindrical and Spherical Structures of Finite Conductivity

1975 ◽  
Vol 53 (11) ◽  
pp. 1078-1087 ◽  
Author(s):  
E. Bahar
Keyword(s):  
Radius Of Curvature ◽  
Finite Conductivity ◽  
Electromagnetic Propagation ◽  
Wave Guides ◽  
Spherical Structures ◽  
Fourier Type ◽  
The Relationship ◽  
Conducting Cylinders ◽  
Dielectric Structures ◽  
Watson Transform

Problems of propagation around multilayered cylindrical or spherical structures possessing highly conducting cores have been analyzed in terms of discrete sets of modes. However, when propagation through the core of the structure is significant and of particular interest, the discrete set of modes is not suitable for the complete expansion of the electromagnetic fields.To provide a suitable basis for the expansion of the electromagnetic field in nonuniform multilayered, cylindrical or spheroidal, dielectric structures we derive expressions for the electric and magnetic field transforms consisting of both a discrete and a continuous spectrum of waves. The relationship between these transforms, the Kontorowich–Lebedev transform, and the discrete Watson transform is discussed. When the radius of curvature of the structure is infinite these transforms merge with generalized Fourier type transforms.The transforms can be used to investigate electromagnetic propagation through irregularly shaped dielectric wave guides, and through irregularly shaped spheroids. It can also be used to solve the problem of propagation in the interior (concave side) of irregularly shaped conducting cylinders or spheroids.

The Verification of the Moderator Response Function via Watson Transform

1984 ◽  
Vol 88 (1) ◽  
pp. 94-97
Author(s):  
Kojiro Nishina ◽  
Shoji Watanabe
Keyword(s):  
Response Function ◽  
Watson Transform

The S-Matrix and Its Analysis

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Acoustic Waves ◽  
Bound States ◽  
Scattering Matrix ◽  
Infinite Dimensional ◽  
Levinson’S Theorem ◽  
S Matrix ◽  
Radial Schrödinger Equation ◽  
Square Well ◽  
Watson Transform ◽  
Remote Past

This chapter focuses on the scattering matrix, or S-matrix, an infinite-dimensional matrix or operator that expresses the state of a scattering system consisting of waves or particles or both in the far future in terms of its state in the remote past. In the case of electromagnetic (or acoustic) waves, the S-matrix connects the intensity, phase, and polarization of the outgoing waves in the far field at various angles to the direction and polarization of the beam pointed toward an obstacle. The chapter first considers the problem of scattering by a square well, located symmetrically with respect to the origin, before discussing bound states and a heuristic derivation of the Breit-Wigner formula. It als describes the Watson transform and Regge poles before concluding with an analysis of the time-independent radial Schrödinger equation and Levinson's theorem.

scholarly journals An earth-flattening transformation for waves from a point source

1972 ◽  
Vol 62 (5) ◽  
pp. 1195-1210 ◽  
Author(s):  
David P. Hill
Keyword(s):  
Point Source ◽  
Continuous Wave ◽  
Body Wave ◽  
Helmholtz Equations ◽  
Vertical Heterogeneity ◽  
Spherical Earth ◽  
Field Form ◽  
Wave Problems ◽  
Watson Transform ◽  
Flat Geometry

Abstract An earth-flattening transformation is developed for wave-propagation problems that can be formulated in terms of uncoupled scalar Helmholtz equations. Through the transformation, wave problems in isotropic, spherically symmetric media with a specified radial heterogeneity can be expressed in terms of a flat geometry with a suitably vertical heterogeneity. The transformation is exact for homogeneous (no source) problems and is useful for normal mode studies. When a point source of waves is present, the earth-flattening transformation together with the Watson transform converts the reflected wave field from a sum over discrete, spherical eigenfunctions to an integral over continuous wave numbers in a flat geometry. The far-field form of this integral shares many properties with the Weyl integral and is useful for body-wave studies in a spherical earth.

Some Results Relating the Behaviour of Fourier Transforms Near the Origin and at Infinity

1969 ◽  
Vol 21 ◽  
pp. 942-950 ◽  
Author(s):  
C. Nasim
Keyword(s):  
Fourier Transforms ◽  
Fourier Inversion Formula ◽  
Fourier Cosine ◽  
Summation Formulae ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
Image Position ◽  
Watson Transform ◽  
Special Case

It is known that under special conditions, Fourier sine transforms and Fourier cosine transforms behave asymptotically like a power of x, either as x → 0 or as x → ∞ or both. For example (3),where f(x) = x–αϕ(x), 0 < α < 1, and ϕ(x) is of bounded variation in (0, ∞) and Fc(x) is the Fourier cosine transform of f(x). This suggests that other results connecting the behaviour of a function at infinity with the behaviour of its Fourier or Watson transform near the origin might exist. In this paper wre derive various such results. For example, a special case of these results iswhere f(x) is the Fourier sine transform of g(x). It should be noted that the Fourier inversion formula fails to give f(+0) directly in this case. Some applications of these results to show the relationships between various forms of known summation formulae are given.

Distributional Watson Transforms

1972 ◽  
Vol 24 (6) ◽  
pp. 1191-1197 ◽  
Author(s):  
Hsing-Yuan Hsu
Keyword(s):  
Hilbert Transform ◽  
Mellin Transform ◽  
Hankel Transform ◽  
Special Cases ◽  
Image Position ◽  
Watson Transform

All our notation is as denned in [2] with the restriction to n = 1. However, for our purposes, we introduce a sequence of norms byin It is not difficult to see that turns out to be a fundamental space.It is a well-known fact that the Watson transform and the Mellin transform are connected by the fact thatandif and only if K(s)K(l — s) = 1, where K(s) is the Mellin transform of k(x). Further, the Hankel transform and Hilbert transform can be considered as special cases of Watson transforms.

The Spectral resolution of Watson transforms

1958 ◽  
Vol 54 (3) ◽  
pp. 368-376
Author(s):  
Edwin J. Akutowicz
Keyword(s):  
Spectral Resolution ◽  
Image Position ◽  
Watson Transform

If φ belongs to L2(0, ∞), the Watson transform W φ of φ is defined bywhere k(u)/u belongs to L2(0, ∞) andThe operator W is unitary in L2(0, ∞), and its inverse is given by

Pseudo-differential operators involving Watson transform

2007 ◽  
Vol 86 (10) ◽  
pp. 1223-1236 ◽  
Author(s):  
Ram Shankar Pathak ◽  
Sadhana Tiwari
Keyword(s):  
Differential Operators ◽  
Pseudo Differential Operators ◽  
Watson Transform

Transmission of creeping-wave modes around a periodically modulated impedance cylinder

1976 ◽  
Vol 54 (18) ◽  
pp. 1839-1849 ◽  
Author(s):  
S. H. Cho ◽  
J. R. Wait
Keyword(s):  
Perturbation Methods ◽  
Diagnostic Value ◽  
Wave Mode ◽  
Spatial Period ◽  
Radio Wavelength ◽  
Oceanographic Data ◽  
Radio Oceanography ◽  
Watson Transform ◽  
Impedance Cylinder ◽  
Creeping Wave

Motivated by recent advances in radio oceanography, an idealized two-dimensional model of diffraction by a non-uniform cylinder is analyzed. The cylinder is characterized by a surface impedance that varies in a periodic fashion in the azimuthal direction. For excitation by a magnetic line source, a solution for the resulting field is obtained that satisfies the specified periodic boundary condition. This representation for the fields is converted, via a Watson transform procedure, to a form that is more convergent and useful for short wavelengths. It is shown that, for each creeping-wave mode, an infinite number of spatial harmonics is excited. These harmonics are a consequence of the periodic non-uniformity of the surface and they include the backward scattered waves. The latter becomes very strong in the region where the spatial period is half the radio wavelength in accordance with the well-known Bragg phenomenon. In this case, contrary to what is sometimes assumed, the attenuation of the forward creeping waves is much increased. This suggests that perturbation methods, such as used to interpret radio-oceanographic data, may be invalid just in the region when they appear to have the most diagnostic value.

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