Ray tomography widely applied in the global seismology, exploration geophysics and seismic engineering usually uses body waves. Surface waves are employed rarely, partly because of their more complicated cinematic and dynamical properties. However, it is possible to point out problems whose effective solution can be gained only by surface wave ray tomography. In the paper we consider, by means of numerical modeling, the time inversion of creeping spiral waves. These waves are able to transit along the curved interface for a long distance reaching the shadow. The essential feature of the problem is in the existence of the velocity dependence on the local interface curvature. As it result, an effect just the same as anisotropy, appears. Thus, we are facing a 2D inversion problem for the anisotropic medium. The short wave approximation for the spiral wave velocities, as functions of the interface curvature and frequency, were given in Krauklis (1974). We use this approximation to consider both the direct and the inverse problem. For the time inversion, the method based upon the Tichonov regularisation (Ryzhikov and Troyan 1994) is applied here. From our point of view it affords to take the a priori information into consideration in the most natural way. To avoid the computational difficulties caused by a singularity in the solution, we use a suboptimal procedure.
射线层析成像在全球地震学、勘探地球物理和地震工程中广泛应用，通常采用体波。表面波很少被使用，部分原因是它们更复杂的电影和动力特性。然而，有可能指出只有表面波射线层析成像才能得到有效解决的问题。本文用数值模拟的方法研究了蠕变螺旋波的时间反演问题。这些波能够沿着弯曲的界面传播很长一段距离到达阴影。该问题的本质特征是速度依赖于局部界面曲率的存在。结果，出现了一种与各向异性相同的效应。因此，我们面临的是各向异性介质的二维反演问题。Krauklis(1974)给出了螺旋波速度作为界面曲率和频率函数的短波近似。我们用这个近似来考虑正问题和逆问题。对于时间反演，本文采用了基于Tichonov正则化(Ryzhikov and Troyan 1994)的方法。从我们的观点来看，它提供了以最自然的方式考虑先验信息。为了避免由于解中存在奇异点而造成的计算困难，我们使用了次优过程。
{"title":"Direct and inverse problems of the ray tomography on the creeping waves","authors":"A. Krauklis, A. I. Malik, V. Troyan","doi":"10.1109/DD.2000.902360","DOIUrl":"https://doi.org/10.1109/DD.2000.902360","url":null,"abstract":"Ray tomography widely applied in the global seismology, exploration geophysics and seismic engineering usually uses body waves. Surface waves are employed rarely, partly because of their more complicated cinematic and dynamical properties. However, it is possible to point out problems whose effective solution can be gained only by surface wave ray tomography. In the paper we consider, by means of numerical modeling, the time inversion of creeping spiral waves. These waves are able to transit along the curved interface for a long distance reaching the shadow. The essential feature of the problem is in the existence of the velocity dependence on the local interface curvature. As it result, an effect just the same as anisotropy, appears. Thus, we are facing a 2D inversion problem for the anisotropic medium. The short wave approximation for the spiral wave velocities, as functions of the interface curvature and frequency, were given in Krauklis (1974). We use this approximation to consider both the direct and the inverse problem. For the time inversion, the method based upon the Tichonov regularisation (Ryzhikov and Troyan 1994) is applied here. From our point of view it affords to take the a priori information into consideration in the most natural way. To avoid the computational difficulties caused by a singularity in the solution, we use a suboptimal procedure.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116087207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
N. Kirpichnikova, V. Philippov, A. S. Kirpichnikova
The problem of diffraction of an electromagnetic plane wave on the impedance interface between two media is investigated. The impedance differs from constant on a segment of the interface, where the impedance is described by piecewise linear, quadratic or step functions.
{"title":"Diffraction of electromagnetic waves from the impedance with different perturbations","authors":"N. Kirpichnikova, V. Philippov, A. S. Kirpichnikova","doi":"10.1109/DD.2000.902356","DOIUrl":"https://doi.org/10.1109/DD.2000.902356","url":null,"abstract":"The problem of diffraction of an electromagnetic plane wave on the impedance interface between two media is investigated. The impedance differs from constant on a segment of the interface, where the impedance is described by piecewise linear, quadratic or step functions.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115541309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Explicit solutions of the inhomogeneous Maxwell equations are obtained. The source is a moving pulsing radial current. The general case of finite pulse duration and radiator length is considered.
{"title":"On transient electromagnetic waves produced by the pulsed radial current moving with the velocity of light","authors":"I. Simonenko","doi":"10.1109/DD.2000.902368","DOIUrl":"https://doi.org/10.1109/DD.2000.902368","url":null,"abstract":"Explicit solutions of the inhomogeneous Maxwell equations are obtained. The source is a moving pulsing radial current. The general case of finite pulse duration and radiator length is considered.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122674603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The diffraction of a plane hydroacoustic wave by a short joint of two fluid loaded semi-infinite thin elastic plates is studied. A generalized point model of the joint is suggested. This model reproduces not only the principal order term in the asymptotics of the far field amplitude by the width of the joint, but also the logarithmically smaller corrections.
{"title":"Diffraction of hydroacoustic wave by a short joint of two semi-infinite elastic plates","authors":"I. Andronov","doi":"10.1109/DD.2000.902352","DOIUrl":"https://doi.org/10.1109/DD.2000.902352","url":null,"abstract":"The diffraction of a plane hydroacoustic wave by a short joint of two fluid loaded semi-infinite thin elastic plates is studied. A generalized point model of the joint is suggested. This model reproduces not only the principal order term in the asymptotics of the far field amplitude by the width of the joint, but also the logarithmically smaller corrections.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125088595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Most problems of seismic prospecting deal with structures containing interfaces where different materials adhere to one another. It is usual practice to assume these interfaces to have welded contact across which both stresses and displacements are continuous. In engineering geology there exists the concept of a loosely bound interface at surfaces along which the rigid connection between adjacent parts of the medium is broken, which in some cases results in the arising of an emergency situation. In carrying out mining operations, the diagnostics and localization of surfaces with unwelded contacts by acoustic and seismic methods is an important practical task. Earlier it was shown that in a layer sandwiched between two elastic halfspaces and having an unwelded contact an unusual wave propagates. This wave has a number of interesting properties: the wave spectrum has resonance frequencies; wave group velocity is equal to an intermediate value between the shear (V/sub s/) and longitudinal (V/sub p/) velocities; attenuation of the wave increases when the frequency varies from the resonance frequencies. The important feature of this wave is the enrichment of its spectrum by the resonance frequencies during the wave propagation. Such behavior of the wave can be explained if we consider the problem of reflection and transmission of SV waves incident on the boundary of two media with sliding contact. For incidence of the SV wave at angle /spl phi/=arcsin(V/sub s//V/sub p/) total reflection occurs. Here we show that an interference wave with such properties exists in a circular cylindrical body which has a sliding contact with the surrounding elastic medium.
{"title":"Unexpected wave in a cylindrical rod placed in the elastic medium","authors":"P.V. Krauklis, L.A. Krauklis","doi":"10.1109/DD.2000.902359","DOIUrl":"https://doi.org/10.1109/DD.2000.902359","url":null,"abstract":"Most problems of seismic prospecting deal with structures containing interfaces where different materials adhere to one another. It is usual practice to assume these interfaces to have welded contact across which both stresses and displacements are continuous. In engineering geology there exists the concept of a loosely bound interface at surfaces along which the rigid connection between adjacent parts of the medium is broken, which in some cases results in the arising of an emergency situation. In carrying out mining operations, the diagnostics and localization of surfaces with unwelded contacts by acoustic and seismic methods is an important practical task. Earlier it was shown that in a layer sandwiched between two elastic halfspaces and having an unwelded contact an unusual wave propagates. This wave has a number of interesting properties: the wave spectrum has resonance frequencies; wave group velocity is equal to an intermediate value between the shear (V/sub s/) and longitudinal (V/sub p/) velocities; attenuation of the wave increases when the frequency varies from the resonance frequencies. The important feature of this wave is the enrichment of its spectrum by the resonance frequencies during the wave propagation. Such behavior of the wave can be explained if we consider the problem of reflection and transmission of SV waves incident on the boundary of two media with sliding contact. For incidence of the SV wave at angle /spl phi/=arcsin(V/sub s//V/sub p/) total reflection occurs. Here we show that an interference wave with such properties exists in a circular cylindrical body which has a sliding contact with the surrounding elastic medium.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125342352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze the Maxwell equations for light propagation in a cylindrical waveguide acousto-optically modulated by plane elastic wave. Using the small parameter of acousto-optic interaction we obtain the dispersion equation explicitly through the matrix determinant. Numeric analysis of this determinant reveals the band-gap structure of the spectrum. We also discuss the regime where light propagating in a fibre radiates into the cladding.
{"title":"Acousto-optic scattering in a single-mode optic fibre","authors":"A.V. Badanin, B. Pavlov, A.A. Pokrovski","doi":"10.1109/DD.2000.902353","DOIUrl":"https://doi.org/10.1109/DD.2000.902353","url":null,"abstract":"We analyze the Maxwell equations for light propagation in a cylindrical waveguide acousto-optically modulated by plane elastic wave. Using the small parameter of acousto-optic interaction we obtain the dispersion equation explicitly through the matrix determinant. Numeric analysis of this determinant reveals the band-gap structure of the spectrum. We also discuss the regime where light propagating in a fibre radiates into the cladding.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121265808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the diffraction series (Schwarzschild's series) to solve the problem of diffraction at a slit with ideal boundary conditions. Using this series we derive the representation obtained by Williams (1982) and the differential equations for the unknown functions.
{"title":"To the problem of diffraction on a slit: some properties of Schwarzschild's series","authors":"A. Shanin","doi":"10.1109/DD.2000.902367","DOIUrl":"https://doi.org/10.1109/DD.2000.902367","url":null,"abstract":"We study the diffraction series (Schwarzschild's series) to solve the problem of diffraction at a slit with ideal boundary conditions. Using this series we derive the representation obtained by Williams (1982) and the differential equations for the unknown functions.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114062621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Polynomial solutions of the hypergeometric equation-Jacobi polynomials constitute an infinite set of orthogonal functions and coincide with eigenfunctions of a singular Sturm-Liouville problem with endpoints of the corresponding interval being regular singularities of the equation (Fuchsian second-order equations with three regular singularities). Among others there are two simple ways of generating these polynomials: i) one way is by using three-term recurrence relations and ii) the other way is by using the Rodrigues formula. The question arises whether it is possible to construct polynomial solutions for the third-order Fuchsian equation with four singularities. These solutions are supposed to be bound at three regular singularities. Taken in general, this problem leads to the necessity to solve algebraic equations of an arbitrary order. However, in particular cases explicit expressions with a generalization of the Rodrigues formula exist. Our starting point is a particular Fuchsian third-order equation with four regular singularities.
{"title":"Polynomial solutions of the third-order Fuchsian linear ODE","authors":"A. Melezhik","doi":"10.1109/DD.2000.902361","DOIUrl":"https://doi.org/10.1109/DD.2000.902361","url":null,"abstract":"Polynomial solutions of the hypergeometric equation-Jacobi polynomials constitute an infinite set of orthogonal functions and coincide with eigenfunctions of a singular Sturm-Liouville problem with endpoints of the corresponding interval being regular singularities of the equation (Fuchsian second-order equations with three regular singularities). Among others there are two simple ways of generating these polynomials: i) one way is by using three-term recurrence relations and ii) the other way is by using the Rodrigues formula. The question arises whether it is possible to construct polynomial solutions for the third-order Fuchsian equation with four singularities. These solutions are supposed to be bound at three regular singularities. Taken in general, this problem leads to the necessity to solve algebraic equations of an arbitrary order. However, in particular cases explicit expressions with a generalization of the Rodrigues formula exist. Our starting point is a particular Fuchsian third-order equation with four regular singularities.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124086086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the problem of constructing spectral series of a three-dimensional Schrodinger operator. Our results deal with the properties (like stability, reducibility, etc.) of the family of Hill equations (ordinary differential equations of second order with periodic coefficients). These properties are investigated by analytical and numeric methods and the curious structure of the mentioned spectral series and related quasimodes is described.
{"title":"Spectral series of the three-dimensional quantum anharmonic oscillator","authors":"M. Poteryakhin","doi":"10.1109/DD.2000.902365","DOIUrl":"https://doi.org/10.1109/DD.2000.902365","url":null,"abstract":"We consider the problem of constructing spectral series of a three-dimensional Schrodinger operator. Our results deal with the properties (like stability, reducibility, etc.) of the family of Hill equations (ordinary differential equations of second order with periodic coefficients). These properties are investigated by analytical and numeric methods and the curious structure of the mentioned spectral series and related quasimodes is described.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130544451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a novel theoretical approach which allows the analytical reconstruction of profile functions associated with thin films as observed by array diffraction. Our concept is based on the representation of the profile function by appropriate linear differential equations with polynomial coefficients that have straight forward Fourier transforms. Several theoretical examples of practical importance are discussed.
{"title":"Rigorous mathematical models for the reconstruction of thin films profiles from X-ray intensities","authors":"S. Slavyanov, C. Ern, H. Dosch","doi":"10.1109/DD.2000.902369","DOIUrl":"https://doi.org/10.1109/DD.2000.902369","url":null,"abstract":"We present a novel theoretical approach which allows the analytical reconstruction of profile functions associated with thin films as observed by array diffraction. Our concept is based on the representation of the profile function by appropriate linear differential equations with polynomial coefficients that have straight forward Fourier transforms. Several theoretical examples of practical importance are discussed.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"5 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120935851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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