Field of a Uniformly Moving Charge in an Anisotropic Dispersive Medium

1973 ◽  
Vol 28 (6) ◽  
pp. 907-910
Author(s):  
S. Datta Majumdar ◽  
G. P. Sastry
Keyword(s):  
Electromagnetic Field ◽  
Fourier Transform ◽  
Point Charge ◽  
Rest Frame ◽  
Dispersive Medium ◽  
Scalar Potential ◽  
Fourier Integral ◽  
The Third ◽  
Dispersion Formula ◽  
The Fourier Transform

The electromagnetic field of a point charge moving uniformly in a uniaxial dispersive medium is studied in the rest frame of the charge. It is shown that the Fourier integral for the scalar potential breaks up into three integrals, two of which are formally identical to the isotropic integral and yield the ordinary and extraordinary cones. Using the convolution theorem of the Fourier transform, the third integral is reduced to an integral over the isotropic field. Dispersion is explicitly introduced into the problem and the isotropic field is evaluated on the basis of a simplified dispersion formula. The effect of dispersion on the field cone is studied as a function of the cut-off frequency.

On: “Gravity Interpretation Using the Fourier Integral”, by M. E. Odegard and J. W. Berg, Jr. (GEOPHYSICS, June 1965, p. 127–138)

Geophysics ◽  
1975 ◽  
Vol 40 (2) ◽  
pp. 356-357
Author(s):  
Jay Gopal Saha
Keyword(s):  
Fourier Transform ◽  
Gravity Anomaly ◽  
Fourier Integral ◽  
Two Dimensional ◽  
Transform Method ◽  
Reverse Problem ◽  
Vertical Fault ◽  
Vertical Step ◽  
The Fourier Transform ◽  
Gravity Interpretation

In their paper, Odegard and Berg claim that from the gravity anomaly due to a two‐dimensional vertical fault the density, the throw, and the depth can be determined uniquely by a Fourier transform method. It is true that the solution of the reverse problem for a two‐dimensional vertical step is theoretically unique. The derivation of the Fourier transform by the authors, however, is erroneous.

The Remaining Cases Where m = 2 and B = 3 or B = 4

2016 ◽  
Author(s):  
Isroil A. Ikromov ◽  
Detlef Müller
Keyword(s):  
Fourier Transform ◽  
Domain Decomposition ◽  
Frequency Domain ◽  
Fourier Integral ◽  
Basic Approach ◽  
Small Letter ◽  
The Fourier Transform ◽  
Frequency Domain Decomposition

This chapter discusses the remaining cases for l = 1. With the same basic approach as in Chapter 5, the chapter again performs an additional dyadic frequency domain decomposition related to the distance to a certain Airy cone. This is needed in order to control the integration with respect to the variable x₁ in the Fourier integral defining the Fourier transform of the complex measures ν‎subscript Greek small letter delta superscript Greek small letter lamda. It first applies a suitable translation in the x₁-coordinate before performing a more refined analysis of the phase Φ‎superscript Music sharp sign. The chapter then treats the case where λ‎ρ‎(̃‎δ‎) ≲ 1 and hereafter deals with the case where λ‎ρ‎(̃‎δ‎) ≲ 1 and B = 4. Finally, the chapter turns to the case where B = 3.

The Fourier transform of 1/r

Geophysics ◽  
1986 ◽  
Vol 51 (2) ◽  
pp. 417-418
Author(s):  
Maurice Craig
Keyword(s):  
Fourier Transform ◽  
Potential Theory ◽  
Bessel Functions ◽  
Integral Formula ◽  
Fourier Integral ◽  
Elementary Functions ◽  
The Fourier Transform

Potential theory employs the Fourier integral [Formula: see text] where [Formula: see text]. Bhattacharyya (1966) obtained the value [Formula: see text] with the use of Bessel functions; the same argument has been repeated often in the geophysical literature (see Courant, 1961; Fuller, 1971; Lourenço, 1972; Gunn, 1975; Weil, 1976; Nabighian, 1984). The following alternative, short derivation involves only elementary functions and may, therefore, be of interest.

Introduction

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Nuclear Magnetic Resonance ◽  
Fourier Transform ◽  
Magnetic Resonance ◽  
Fourier Analysis ◽  
Fourier Integral ◽  
Sampling Theorem ◽  
Two Dimensional ◽  
The Fourier Transform ◽  
Array Imaging ◽  
Nuclear Magnetic

Jean Baptiste Joseph Fourier’s powerful idea of decomposition of a signal into sinusoidal components has found application in almost every engineering and science field. An incomplete list includes acoustics [1497], array imaging [1304], audio [1290], biology [826], biomedical engineering [1109], chemistry [438, 925], chromatography [1481], communications engineering [968], control theory [764], crystallography [316, 498, 499, 716], electromagnetics [250], imaging [151], image processing [1239] including segmentation [1448], nuclear magnetic resonance (NMR) [436, 1009], optics [492, 514, 517, 1344], polymer characterization [647], physics [262], radar [154, 1510], remote sensing [84], signal processing [41, 154], structural analysis [384], spectroscopy [84, 267, 724, 1220, 1293, 1481, 1496], time series [124], velocity measurement [1448], tomography [93, 1241, 1242, 1327, 1330, 1325, 1331], weather analysis [456], and X-ray diffraction [1378], Jean Baptiste Joseph Fourier’s last name has become an adjective in the terms like Fourier series [395], Fourier transform [41, 51, 149, 154, 160, 437, 447, 926, 968, 1009, 1496], Fourier analysis [151, 379, 606, 796, 1472, 1591], Fourier theory [1485], the Fourier integral [395, 187, 1399], Fourier inversion [1325], Fourier descriptors [826], Fourier coefficients [134], Fourier spectra [624, 625] Fourier reconstruction [1330], Fourier spectrometry [84, 355], Fourier spectroscopy [1220, 1293, 1438], Fourier array imaging [1304], Fourier transform nuclear magnetic resonance (NMR) [429, 1004], Fourier vision [1448], Fourier optics [419, 517, 1343], and Fourier acoustics [1496]. Applied Fourier analysis is ubiquitous simply because of the utility of its descriptive power. It is second only to the differential equation in the modelling of physical phenomena. In contrast with other linear transforms, the Fourier transform has a number of physical manifestations. Here is a short list of everyday occurrences as seen through the lens of the Fourier paradigm. • Diffracting coherent waves in sonar and optics in the far field are given by the two dimensional Fourier transform of the diffracting aperture. Remarkably, in free space, the physics of spreading light naturally forms a two dimensional Fourier transform. • The sampling theorem, born of Fourier analysis, tells us how fast to sample an audio waveform to make a discrete time CD or an image to make a DVD.

Fundamentals of Fourier Analysis

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Fourier Transform ◽  
Discrete Time ◽  
Continuous Time ◽  
Fourier Integral ◽  
Time Signal ◽  
Continuous Sampling ◽  
Common Reference ◽  
Multidimensional Signals ◽  
Time Signals ◽  
The Fourier Transform

This chapter contains foundational material for modelling of signals and systems. Section 2.2 introduces classes of functions useful in signal processing and analysis. The Fourier transform, in Section 2.3, begins with the Fourier integral and develops the Fourier series, the discrete time Fourier transform and the discrete Fourier transform as special cases. The following material in this chapter can be skipped on a first reading. † denotes material relevant to multidimensional signals in Chapters 8 and 11. ‡ denotes material relevant to probability and stochastic processes in Chapter 4. ¶ denotes material used in continuous sampling in Chapter 10. There are a number of signal classes to which we will make common reference. Continuous time signals are denoted with their arguments in parentheses, e.g., x(t). Discrete time signals will be bracketed, e.g., x[n]. A continuous time signal, x(t), is periodic if there exists a T such that x(t) = x(t − T) for all t. The function x(t) = constant is periodic. A discrete time signal, x[n], is periodic if there exists a positive integer N such that x[n] = x[n − N] for all n. The function x[n] = constant is periodic.

scholarly journals Strain Sensitivity Enhancement of Broadband Ultrasonic Signals in Plates Using Spectral Phase Filtering

2021 ◽  
Vol 11 (6) ◽  
pp. 2582
Author(s):  
Lucas M. Martinho ◽  
Alan C. Kubrusly ◽  
Nicolás Pérez ◽  
Jean Pierre von der Weid
Keyword(s):  
Fourier Transform ◽  
Guided Waves ◽  
Strain Level ◽  
Energy Concentration ◽  
Short Time Fourier Transform ◽  
Time Frequency ◽  
Frequency Representation ◽  
The Fourier Transform ◽  
Short Time ◽  
Sensitivity Gain

The focused signal obtained by the time-reversal or the cross-correlation techniques of ultrasonic guided waves in plates changes when the medium is subject to strain, which can be used to monitor the medium strain level. In this paper, the sensitivity to strain of cross-correlated signals is enhanced by a post-processing filtering procedure aiming to preserve only strain-sensitive spectrum components. Two different strategies were adopted, based on the phase of either the Fourier transform or the short-time Fourier transform. Both use prior knowledge of the system impulse response at some strain level. The technique was evaluated in an aluminum plate, effectively providing up to twice higher sensitivity to strain. The sensitivity increase depends on a phase threshold parameter used in the filtering process. Its performance was assessed based on the sensitivity gain, the loss of energy concentration capability, and the value of the foreknown strain. Signals synthesized with the time–frequency representation, through the short-time Fourier transform, provided a better tradeoff between sensitivity gain and loss of energy concentration.

Determination of starch crystallinity with the Fourier-transform terahertz spectrometer

2021 ◽  
Vol 262 ◽  
pp. 117928
Author(s):  
Shusaku Nakajima ◽  
Shuhei Horiuchi ◽  
Akifumi Ikehata ◽  
Yuichi Ogawa
Keyword(s):  
Fourier Transform ◽  
The Fourier Transform ◽  
Starch Crystallinity
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