Transient Compressional Waves in an Infinite Elastic Plate With a Circular Cylindrical Cavity

1964 ◽  
Vol 31 (4) ◽  
pp. 627-634 ◽  
Author(s):  
R. A. Scott ◽  
Julius Miklowitz
Keyword(s):  
Stationary Phase ◽  
Cylindrical Cavity ◽  
Integral Transform ◽  
Formal Solution ◽  
Linear Equations ◽  
Normal Displacement ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Hole Radius ◽  
Circular Cylindrical Cavity

The problem treated is that of an infinite free plate with a circular cylindrical cavity subjected to a step normal displacement. The linear equations of elasticity are employed and the formal solution is obtained using a multi-integral transform technique, necessitating the introduction of extended Hankel transforms, and residue theory. Some properties of the Rayleigh-Lamb frequency equation, pertinent to the inversion process, are derived. Numerical information for the far field, showing the effect of the hole radius on the displacements, is obtained using the stationary phase method and, in the case of the radial displacement, the solution is compared with the corresponding slab solution. The results show that at a given station the plate-cavity solution approaches that for the slab, as the hole radius decreases. The head of the pulse and stationary-phase approximations to the corresponding horizontal slab displacement are also compared, and some discrepancies between the two are found in the vicinity of the wave-front arrival time.

Transient Compressional Waves in an Infinite Elastic Plate or Elastic Layer Overlying a Rigid Half-Space

1962 ◽  
Vol 29 (1) ◽  
pp. 53-60 ◽  
Author(s):  
Julius Miklowitz
Keyword(s):  
Half Space ◽  
Elastic Layer ◽  
Integral Transform ◽  
Formal Solution ◽  
Equations Of Motion ◽  
Low Frequency ◽  
Frequency Equation ◽  
Point Load ◽  
Phase Method ◽  
Stationary Phase Method

The problem treated is that of an infinite free plate excited symmetrically by two equal and normally opposed step point-loads on its faces. The problem is equivalent to that of the surface normal point-load excitation of an infinite elastic layer, half the thickness of the plate, overlying a rigid half-space with lubricated contact. The formal solution is obtained from the equations of motion in linear elasticity with the aid of a double integral transform technique and residue theory. The stationary phase method, and known characteristics of the governing Rayleigh-Lamb frequency equation, are used to analyze and evaluate numerically the far field displacements. It is shown that the head of the disturbance is composed predominantly of the low-frequency long waves from the lowest mode of wave transmission.

scholarly journals Dispersive Estimates for Full Dispersion KP Equations

2021 ◽  
Vol 23 (1) ◽  
Author(s):  
Didier Pilod ◽  
Jean-Claude Saut ◽  
Sigmund Selberg ◽  
Achenef Tesfahun
Keyword(s):  
Stationary Phase ◽  
Initial Value Problem ◽  
Bessel Functions ◽  
Linear Part ◽  
Independent Interest ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Initial Value ◽  
Kp Equations ◽  
Well Posed

AbstractWe prove several dispersive estimates for the linear part of the Full Dispersion Kadomtsev–Petviashvili introduced by David Lannes to overcome some shortcomings of the classical Kadomtsev–Petviashvili equations. The proof of these estimates combines the stationary phase method with sharp asymptotics on asymmetric Bessel functions, which may be of independent interest. As a consequence, we prove that the initial value problem associated to the Full Dispersion Kadomtsev–Petviashvili is locally well-posed in $$H^s(\mathbb R^2)$$ H s ( R 2 ) , for $$s>\frac{7}{4}$$ s > 7 4 , in the capillary-gravity setting.

scholarly journals Focusing properties of an axicon pair

1993 ◽  
Vol 71 (1-2) ◽  
pp. 70-78 ◽  
Author(s):  
Marc Couture ◽  
Michel Piché
Keyword(s):  
Stationary Phase ◽  
Numerical Method ◽  
Gaussian Beam ◽  
Fresnel Diffraction ◽  
Optical Systems ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Diffraction Integral ◽  
Peak Intensity ◽  
Focusing Properties

The focusing properties of a so-called reflaxicon (a combination of a diverging and a converging axicon) are studied both theoretically and experimentally. Calculations of intensity distributions produced by this system are made by evaluating the Kirchhoff–Fresnel diffraction integral, first by means of an approximate technique, the stationary phase method, then by a more exact numerical method. The calculations are presented for various planes along the axis of the axicons. The effects of the presence of the supporting mount of the axicons and of some important misalignments of the system on the distributions is also investigated. Experimental results of actual intensity distributions produced by focusing a near-fundamental Gaussian beam by such a system are also presented and are seen to be in fair agreement with numerical calculations. Such calculations would be valuable in many applications for predicting important characteristics (e.g., peak intensity, length of the focal line, degree of asymmetry) of the intensity distributions formed by optical systems containing an axicon pair as the focusing component.

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