Viscoelastic Love numbers and long period geophysical effects

2021 ◽  
Author(s):  
A Michel ◽  
J-P Boy
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Earth Tides ◽  
Rheological Parameters ◽  
Earth Model ◽  
Ice Melting ◽  
Long Period ◽  
Love Numbers ◽  
The Laplace Transform ◽  
The Fourier Transform

Summary Long term deformations strongly depend on the Earth model and its rheological parameters, and in particular its viscosity. We give the general theory and the numerical scheme to compute them for any spherically non rotating isotropic Earth model with linear rheology, either elastic or viscoelastic. Although the Laplace transform is classically used to compute viscoelastic deformation, we choose here instead, to implement the integration with the Fourier transform in order to take advantage of the Fast Fourier Transform algorithm and avoid some of the Laplace transform mathematical difficulties. We describe the methodology to calculate deformations induced by several geophysical signals regardless of whether they are periodic or not, especially by choosing an adapted time sampling for the Fourier transform. As examples, we investigate the sensitivity of the displacements due to long period solid Earth tides, Glacial Isostatic Adjustment (GIA), and present-day ice melting, to anelastic parameters of the mantle. We find that the effects of anelasticity are important for long period deformation and relatively low values of viscosities for both Maxwell and Burgers models. We show that slight modifications in the rheological models could significantly change the amplitude of deformation but also affect the spatial and temporal pattern of the signal to a lesser extent. Especially, we highlight the importance of the mantle anelasticity in the low degrees deformation due to present-day ice melting and encourage its inclusion in future models.

New Laplace, z and Fourier-related transforms

2007 ◽  
Vol 463 (2081) ◽  
pp. 1179-1198 ◽  
Author(s):  
Michael J Corinthios
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Large Class ◽  
Large Family ◽  
Step Function ◽  
Dirac Delta ◽  
Mellin Transforms ◽  
Generalized Distributions ◽  
The Laplace Transform ◽  
The Fourier Transform

In this paper, the author uses his recently proposed complex variable generalized distribution theory to expand the domains of existence of bilateral Laplace and z transforms, as well as a whole new class of related transforms. A vast expansion of the domains of existence of bilateral Laplace and z transforms and continuous-time and discrete-time Hilbert, Hartley and Mellin transforms, as well as transforms of multidimensional functions and sequences are obtained. It is noted that the Fourier transform and its applications have advanced by leaps and bounds during the last century, thanks to the introduction of the theory of distributions and, in particular, the concept of the Dirac-delta impulse. Meanwhile, however, the truly two-sided ‘bilateral’ Laplace and z transforms, which are more general than Fourier, remained at a standstill incapable of transforming the most basic of functions. In fact, they were reduced by half to one-sided transforms and received no more than a passing reference in the literature. It is shown that the newly proposed generalized distributions expand the domains of existence and application of Laplace and z transforms similar to and even more extensively than the expansion of the domain of Fourier transform that resulted from the introduction, nearly a century ago, of the theory of distributions and the Dirac-delta impulse. It is also shown that the new generalized distributions put an end to an anomaly that still exists today, which meant that for a large class of basic functions, the Fourier transform exists while the more general Laplace and z transforms do not. The anomaly further manifests itself in the fact that even for the one-sided causal functions, such as the Heaviside unit step function u ( t ) and the sinusoid sin βtu ( t ), the Laplace transform does not exist on the j ω -axis, and the Fourier transform which does exist cannot be deduced thereof by the substitution s =j ω in the Laplace transform, which by definition it should. The extended generalized transforms are well defined for a large class of functions ranging from the most basic to highly complex fast-rising exponential ones that have so far had no transform. Among basic applications, the solution of partial differential equations using the extended generalized transforms is provided. This paper clearly presents and articulates the significant impact of extending the domains of Laplace and z transforms on a large family of related transforms, after nearly a century during which bilateral Laplace and z transforms of even the most basic of functions were undefined, and the domains of definition of related transforms such as Hilbert, Hartley and Mellin transforms were confined to a fraction of the space they can now occupy.

Inversion of potential field data in Fourier transform domain

Geophysics ◽  
1985 ◽  
Vol 50 (4) ◽  
pp. 685-691 ◽  
Author(s):  
J. C. Mareschal
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Potential Field ◽  
Homogeneous Problem ◽  
Source Distribution ◽  
Flow Profile ◽  
Surface Source ◽  
Potential Field Data ◽  
The Laplace Transform ◽  
The Fourier Transform

A relationship is derived between the Fourier transform of a potential field at the Earth’s surface and the transform of the inducing source distribution. The Fourier transform of the field is the Laplace transform of the source distribution spectrum when the Laplace transform variable p is equal to the wavenumber. This relationship can be used to determine all possible source distributions compatible with the data. The solution is the superposition of a particular solution to an inhomogeneous problem and of the general solution to the homogeneous problem (i.e., for which the field vanishes at the surface). Source distribution can be expanded into a set of known functions; coefficients of the expansion are determined by solving a system of linear equations. Physical constraints can be introduced to restrict the variation range of the coefficients of expansion. Two examples are presented to illustrate the method: a synthetic gravity profile and a heat flow profile are inverted to determine density or heat source distributions compatible with the data.

scholarly journals Analytical solutions of some general classes of differential and integral equations by using the Laplace and Fourier transforms

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 2869-2876
Author(s):  
H.M. Srivastava ◽  
Mohammad Masjed-Jamei ◽  
Rabia Aktaş
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Integral Equations ◽  
Analytical Solutions ◽  
General Class ◽  
Fourier Transforms ◽  
Laplace And Fourier Transforms ◽  
The Laplace Transform ◽  
The Fourier Transform ◽  
Differential And Integral Equations

This article deals with a general class of differential equations and two general classes of integral equations. By using the Laplace transform and the Fourier transform, analytical solutions are derived for each of these classes of differential and integral equations. Some illustrative examples and particular cases are also considered. The various analytical solutions presented in this article are potentially useful in solving the corresponding simpler differential and integral equations.

Asymptotic solution of the Vlasov and Poisson equations for an inhomogeneous plasma

1991 ◽  
Vol 45 (1) ◽  
pp. 59-70 ◽  
Author(s):  
Riccardo Croci
Keyword(s):  
Magnetic Field ◽  
Laplace Transform ◽  
Integral Equations ◽  
Initial Conditions ◽  
Inhomogeneous Plasma ◽  
Algebraic Equations ◽  
Poisson Equations ◽  
The Laplace Transform ◽  
Eigenvalue Parameter ◽  
The Fourier Transform

The purpose of this paper is to derive the asymptotic solutions to a class of inhomogeneous integral equations that reduce to algebraic equations when a parameter ε goes to zero (the kernel becoming proportional to a Dirac δ function). This class includes the integral equations obtained from the system of Vlasov and Poisson equations for the Fourier transform in space and the Laplace transform in time of the electrostatic potential, when the equilibrium magnetic field is uniform and the equilibrium plasma density depends on εx, with the co-ordinate z being the direction of the magnetic field. In this case the inhomogeneous term is given by the initial conditions and possibly by sources, and the Laplace-transform variable ω is the eigenvalue parameter.

scholarly journals On solutions of local fractional Schrodinger equation

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 1929-1934
Author(s):  
Resat Yilmazer ◽  
Neslihan Demirel
Keyword(s):  
Fourier Transform ◽  
Schrödinger Equation ◽  
Laplace Transform ◽  
Closed Form ◽  
Schrodinger Equation ◽  
Fractional Schrödinger Equation ◽  
The Laplace Transform ◽  
Fractional Schrodinger Equation

In this study, we obtain the solution of a local fractional Schrodinger equation. The solution is obtained by the implementation of the Laplace transform and Fourier transform in closed form in terms of the Mittag-Leffler function.

scholarly journals Solution of Schrödinger equation by Laplace transform

1968 ◽  
Vol 8 (3) ◽  
pp. 557-567 ◽  
Author(s):  
M. J. Englefield
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Laplace Transform ◽  
Schrodinger Equation ◽  
Bound State ◽  
Transform Method ◽  
The Laplace Transform ◽  
Scattering Phase ◽  
Straightforward Solution ◽  
The Fourier Transform

Laplace transform techniques for solving differential equations do not seem to have been directly applied to the Schrödinger equation in quantum mechanics. This may be because the Laplace transform of a wave function, in contrast to the Fourier transform, has no direct physical significance. However, this paper will show that scattering phase shifts and bound state energies can be determined from the singularities of the Laplace transform of the wave function. The Laplace transform method can thereby simplify calculations if the potential allows a straightforward solution of the transformed Schrödinger equation. Suitable cases are the Coulomb, oscillator and exponential potentials and the Yamaguchi separable non-local potential.

A note on holomorphic functions and the Fourier-Laplace transform

2017 ◽  
Vol 120 (2) ◽  
pp. 225 ◽  
Author(s):  
Marcus Carlsson ◽  
Jens Wittsten
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Half Plane ◽  
Classical Problem ◽  
Holomorphic Functions ◽  
Function Spaces ◽  
Classical Function ◽  
Upper Half Plane ◽  
The Fourier Transform ◽  
Half Axis

We revisit the classical problem of when a given function, which is analytic in the upper half plane $\mathbb{C} _+$, can be written as the Fourier transform of a function or distribution with support on a half axis $(-\infty ,b]$, $b\in \mathbb{R} $. We derive slight improvements of the classical Paley-Wiener-Schwartz Theorem, as well as softer conditions for verifying membership in classical function spaces such as $H^p(\mathbb{C} _+)$.

Dynamics of the liquid core of the Earth

1972 ◽  
Vol 273 (1233) ◽  
pp. 237-260 ◽  
Keyword(s):  
Boundary Layer ◽  
Inner Core ◽  
Liquid Core ◽  
Stable Stratification ◽  
Earth Model ◽  
Unstable Stratification ◽  
The Core ◽  
Long Period ◽  
Love Numbers ◽  
Neutral Equilibrium

An asymptotic theory is developed for the long-period bodily tides in an Earth model having a liquid core. The yielding inside the core is found to be different in the case of a stable density stratification from the case of an unstable stratification. In the latter case, a boundary layer is formed in which the stress decreases exponentially with depth below the core surface, the scale length of the exponent being proportional to the frequency. In the limit of vanishing frequency the stress tends to zero through most of the liquid core, except near the boundary layer at the surface, where it grows to a finite value. In case of a stable stratification, the stress oscillates with depth below the surface of the core with a wavelength which is proportional to frequency. An infinite number of 'core oscillations' with indefinitely increasing periods exist in a liquid core with stable stratification, but in the case of an unstable stratification, none exist above the fundamental spheroidal oscillation (53.7 min) for n — 2. The assertion made that a liquid core must be in neutral equilibrium is not true. The displacements and stresses within a liquid core in long-period tidal yielding are determinate, even in the static limit, and are not arbitrary. Love numbers are derived for uniformly stable, neutral, and unstable liquid cores, as well as for a model with a rigid inner core.

RADIATION FROM A FINITE CYLINDRICAL EXPLOSIVE SOURCE

Geophysics ◽  
1977 ◽  
Vol 42 (7) ◽  
pp. 1384-1393 ◽  
Author(s):  
Anas M. Abo‐Zena
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Fourier Transforms ◽  
Inverse Fourier Transform ◽  
Applied Pressure ◽  
Inverse Laplace Transform ◽  
Laplace And Fourier Transforms ◽  
Explosion Pressure ◽  
Explosive Source ◽  
The Laplace Transform

For an elastic material with an infinite circular cylindrical hole, the exact solution due to a pressure on a finite length of the cylinder is obtained as a function of the Laplace transform parameter on time and Fourier transform parameter on the z-coordinate (the axis of the cylinder). The applied pressure is a function of the time and the position z. Numerical inversion of the Laplace and Fourier transforms are required to determine the field quantities in the time and space parameters. In the far field, the inverse Fourier transform can be obtained by an asymptotic expansion. It remains to obtain the inverse Laplace transform numerically. We have found that for cylinders whose radius is small compared with the smallest wavelength of interest, an analytical solution can be obtained. Graphical results for the cases of instantaneous explosion and progression of the detonation with constant velocity are given. In both cases an exponential decay of the explosion pressure is assumed.

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