Stress-Intensification Near a Semi-Infinite Rigid-Smooth Strip Due to Diffraction of Elastic Waves

1967 ◽  
Vol 34 (1) ◽  
pp. 119-126 ◽  
Author(s):  
S. A. Thau ◽  
Yih-Hsing Pao
Keyword(s):  
Radial Distance ◽  
Scattered Wave ◽  
Wave Number ◽  
Parabolic Cylinder ◽  
Principal Stresses ◽  
Plane Barrier ◽  
Diffraction Of Elastic Waves ◽  
Parabolic Cylinder Functions ◽  
P And Sv Waves ◽  
Rigid Smooth

Scattering of plane harmonic compressional and shear waves (P and SV-waves) by a semi-infinite rigid-smooth strip or ribbon which is a plane barrier with its top and bottom surfaces being confined normally, but free in lateral directions, is treated. Under the condition that displacements must be regular, exact solutions for the combined incident and scattered-wave fields are obtained in terms of Weber’s parabolic cylinder functions. Principal stresses are calculated on both sides of the strip and the stresses are shown to be singular of the order (kr)−1/2, where k is the incident wave number and r the radial distance from the tip.

Asymptotic expansion of the Mathieu and prolate spheroidal eigenvalues for large parameter c

1999 ◽  
Vol 77 (8) ◽  
pp. 635-652 ◽  
Author(s):  
T Do-Nhat
Keyword(s):  
Wave Number ◽  
Parabolic Cylinder ◽  
Prolate Spheroidal ◽  
Spheroidal Wave Functions ◽  
Prolate Spheroidal Wave Functions ◽  
Asymptotic Eigenvalue ◽  
Parabolic Cylinder Functions ◽  
Expansion Coefficients ◽  
Analytical Expressions ◽  
Operating Wave

The asymptotic expansions of the Mathieu eigenfunctions and the prolate spheroidal wave functions can be expanded in terms of the parabolic cylinder functions, from which their asymptotic eigenvalue can be expressed in an inverse power series of c, where the parameter c is proportional to the operating wave number. Analytical expressions of the eigenvalues, as well as those of the expansion coefficients of the eigenfunctions, are derived and verified with numerical results.PACS. No.: 2.30 MV

Saturation mechanism and generated viscosity in gravito-turbulent accretion disks

2020 ◽  
Vol 640 ◽  
pp. A53
Author(s):  
L. Löhnert ◽  
S. Krätschmer ◽  
A. G. Peeters
Keyword(s):  
Growth Rate ◽  
Numerical Simulations ◽  
Accretion Disks ◽  
Gravitational Instability ◽  
Turbulent Viscosity ◽  
Radial Distance ◽  
Maximum Growth ◽  
Wave Number ◽  
Radiative Cooling ◽  
Mixing Length

Here, we address the turbulent dynamics of the gravitational instability in accretion disks, retaining both radiative cooling and irradiation. Due to radiative cooling, the disk is unstable for all values of the Toomre parameter, and an accurate estimate of the maximum growth rate is derived analytically. A detailed study of the turbulent spectra shows a rapid decay with an azimuthal wave number stronger than ky−3, whereas the spectrum is more broad in the radial direction and shows a scaling in the range kx−3 to kx−2. The radial component of the radial velocity profile consists of a superposition of shocks of different heights, and is similar to that found in Burgers’ turbulence. Assuming saturation occurs through nonlinear wave steepening leading to shock formation, we developed a mixing-length model in which the typical length scale is related to the average radial distance between shocks. Furthermore, since the numerical simulations show that linear drive is necessary in order to sustain turbulence, we used the growth rate of the most unstable mode to estimate the typical timescale. The mixing-length model that was obtained agrees well with numerical simulations. The model gives an analytic expression for the turbulent viscosity as a function of the Toomre parameter and cooling time. It predicts that relevant values of α = 10−3 can be obtained in disks that have a Toomre parameter as high as Q ≈ 10.

scholarly journals On certain infinite integrals involving Struve functions and parabolic cylinder functions

1946 ◽  
Vol 7 (4) ◽  
pp. 171-173 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Present Note ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Struve Functions ◽  
Image Position

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved thatFrom (1)follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.

An infinite integral involving Bessel functions and parabolic cylinder functions

1937 ◽  
Vol 33 (2) ◽  
pp. 210-211 ◽  
Author(s):  
R. S. Varma
Keyword(s):  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Infinite Integral

The object of this paper is to evaluate an infinite integral involving Bessel functions and parabolic cylinder functions. The following two lemmas are required:Lemma 1. provided that R(m) > 0.

scholarly journals Distributions of the Ratio and Product of Two Independent Weibull and Lindley Random Variables

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
N. J. Hassan ◽  
A. Hawad Nasar ◽  
J. Mahdi Hadad
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Random Variables ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Parabolic Cylinder ◽  
Generalized Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Parabolic Cylinder Functions ◽  
The Moment

In this paper, we derive the cumulative distribution functions (CDF) and probability density functions (PDF) of the ratio and product of two independent Weibull and Lindley random variables. The moment generating functions (MGF) and the k-moment are driven from the ratio and product cases. In these derivations, we use some special functions, for instance, generalized hypergeometric functions, confluent hypergeometric functions, and the parabolic cylinder functions. Finally, we draw the PDF and CDF in many values of the parameters.

scholarly journals On the squares of Weber's parabolic cylinder functions and certain integrals connected with them

1934 ◽  
Vol 4 (1) ◽  
pp. 27-32 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Image Position

The parabolic cylinder functions Dn(x) and D−(n+1) (± ix) are defined byfor all values of n and x.

Tables of Weber Parabolic Cylinder Functions

1957 ◽  
Vol 41 (336) ◽  
pp. 144
Author(s):  
A. Fletcher ◽  
J. C. P. Miller
Keyword(s):  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions
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