Amplitude and phase shift due to caustics

1981 ◽  
Vol 71 (5) ◽  
pp. 1445-1461
Author(s):  
A. P. Choi ◽  
F. Hron
Keyword(s):  
Phase Shift ◽  
Saddle Point ◽  
Airy Function ◽  
Turning Points ◽  
Integral Solution ◽  
Saddle Point Method ◽  
Third Order ◽  
Function Solution ◽  
Formal Integral ◽  
Inhomogeneous Elastic Medium

abstract The formal integral solution for an arbitrary ray in a plane parallel-layered, vertically inhomogeneous elastic medium is evaluated using a modified third-order saddle point method. The result, which reduces to the Airy function solution where the latter is valid, is shown to be more generally valid and just as simple to compute. In addition, it is shown that the phase shift due to caustics is intimately related to the occurrence of turning points along the ray. An expression is derived explicitly relating this phase shift to the number of turning points and to whether the ray is on a direct or reverse travel-time branch.

scholarly journals A Note on the Asymptotic Behavior of the Heights in $b$-Tries for $b$ Large

10.37236/1517 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Charles Knessl ◽  
Wojciech Szpankowski
Keyword(s):  
Asymptotic Behavior ◽  
Saddle Point ◽  
Generating Functions ◽  
Single Point ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Point Method ◽  
Saddle Point Method ◽  
Numerical Verification ◽  
Height Distribution

We study the limiting distribution of the height in a generalized trie in which external nodes are capable to store up to $b$ items (the so called $b$-tries). We assume that such a tree is built from $n$ random strings (items) generated by an unbiased memoryless source. In this paper, we discuss the case when $b$ and $n$ are both large. We shall identify five regions of the height distribution that should be compared to three regions obtained for fixed $b$. We prove that for most $n$, the limiting distribution is concentrated at the single point $k_1=\lfloor \log_2 (n/b)\rfloor +1$ as $n,b\to \infty$. We observe that this is quite different than the height distribution for fixed $b$, in which case the limiting distribution is of an extreme value type concentrated around $(1+1/b)\log_2 n$. We derive our results by analytic methods, namely generating functions and the saddle point method. We also present some numerical verification of our results.

scholarly journals The Saddle Point Method for the Integral of the Absolute Value of the Brownian Motion

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Leonid Tolmatz
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Saddle Point ◽  
Point Method ◽  
Saddle Point Method ◽  
Tail Asymptotics ◽  
Absolute Value ◽  
Exact Tail Asymptotics ◽  
International Audience ◽  
The Absolute

International audience The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takács in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.

The reflexion of a spherical acoustic pulse by an absorbent infinite plane and related problems

1952 ◽  
Vol 215 (1121) ◽  
pp. 233-254 ◽  
Keyword(s):  
Acoustic Waves ◽  
Acoustic Pulse ◽  
Practical Interest ◽  
Exact Result ◽  
Integral Solution ◽  
Blast Waves ◽  
Angle Of Incidence ◽  
Infinite Plane ◽  
Formal Integral ◽  
Wide Range

A formal integral solution is given for the problem of the reflexion of a spherical acoustic pulse by an infinite plane interface having an impedance of arbitrary dependence on frequency and angle of incidence. In many cases of practical interest the impedance may be assumed to be independent of angle of incidence, and under this assumption the integral solution is relatively easy to evaluate. A simple exact expression for the reflected pulse, in closed form, is obtained when the wall impedance is purely resistive (i.e. independent of frequency). This solution is a special case of a general type of solution of the wave equation when it is reduced to a rotationally symmetric Laplace’s equation in the ‘spherical polar’ co-ordinates [√{( ct / r ) 2 - sin 2 θ}, ( ct cos θ/ r )/ √{( ct / r ) 2 - sin 2 θ}]. To illustrate the relatively wide range of validity of the assumption of an impedance independent of angle of incidence, when applied to real materials, this exact result is compared with an approximate solution for the case where the interface separates two homogeneous isotropic lossless materials. The formal integral solution is evaluated approximately for wall impedances of the following types: (i) resistance and mass, (ii) resistance and stiffness, (iii) resistance, mass and stiffness. The solutions are compared with corresponding solutions for plane incident waves, and the behaviour of the scattered wave, distinguishing between the spherical and the plane wave, is discussed. Possible applications of the results for acoustic waves to problems in the reflexion of blast waves and of transient radiation by an electric dipole are indicated briefly.

scholarly journals Electromagnetically Induced Transparency in Media with Rydberg Excitons 2: Cross-Kerr Modulation

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 160 ◽  
Author(s):  
David Ziemkiewicz ◽  
Sylwia Zielińska - Raczyńska
Keyword(s):  
Quantum Information ◽  
Solid State ◽  
Phase Shift ◽  
Cuprous Oxide ◽  
Quantum Information Processing ◽  
Electromagnetically Induced Transparency ◽  
Optimum Conditions ◽  
Third Order ◽  
All Optical ◽  
Induced Transparency

By mapping photons into the sample of cuprous oxide with Rydberg excitons, it is possible to obtain a significant optical phase shift due to third-order cross-Kerr nonlinearities realized under the conditions of electromagnetically induced transparency. The optimum conditions for observation of the phase shift over π in Rydberg excitons media are examined. A discussion of the application of the cross-phase modulations in the field of all-optical quantum information processing in solid-state systems is presented.

scholarly journals Sommes friables d'exponentielles et applications

2015 ◽  
Vol 67 (3) ◽  
pp. 597-638 ◽  
Author(s):  
Sary Drappeau
Keyword(s):  
Asymptotic Formula ◽  
Saddle Point ◽  
Exponential Sums ◽  
Prime Factor ◽  
Character Sums ◽  
Contour Integration ◽  
Point Method ◽  
Saddle Point Method ◽  
Greatest Prime Factor

AbstractAn integer is said to be y–friable if its greatest prime factor is less than y. In this paper, we obtain estimates for exponential sums over y–friable numbers up to x which are non–trivial when y ≥ . As a consequence, we obtain an asymptotic formula for the number of y-friable solutions to the equation a + b = c which is valid unconditionally under the same assumption. We use a contour integration argument based on the saddle point method, as developped in the context of friable numbers by Hildebrand and Tenenbaum, and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.

scholarly journals Power partitions and saddle-point method

2019 ◽  
Vol 204 ◽  
pp. 435-445 ◽  
Author(s):  
Gérald Tenenbaum ◽  
Jie Wu ◽  
Ya-Li Li
Keyword(s):  
Saddle Point ◽  
Point Method ◽  
Saddle Point Method
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