The propagation of modes in a smoothly non-uniform Timoshenko beam is considered. We present an asymptotic theory of interaction of two modes whose phase velocities intersect at a single point and are non-tangent there. The high-frequency regime in the time-harmonic case and propagation of discontinuities are both considered.
{"title":"An asymptotic theory of resonance interaction of shear and bending modes in a non-uniform Timoshenko beam","authors":"M. Perel, I. Fialkovsky, A.P. Kiaelev","doi":"10.1109/DD.2000.902364","DOIUrl":"https://doi.org/10.1109/DD.2000.902364","url":null,"abstract":"The propagation of modes in a smoothly non-uniform Timoshenko beam is considered. We present an asymptotic theory of interaction of two modes whose phase velocities intersect at a single point and are non-tangent there. The high-frequency regime in the time-harmonic case and propagation of discontinuities are both considered.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"11 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":"127820477","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 following problem is studied. Consider two linear homogeneous second-order ordinary differential equations of the form ry''+r'y'=fy (eqn.1) and ru''+r'u'=gu (eqn.2). These equations are chosen to be formally self-adjoint. The function /spl upsi/(z) is defined as a product of the arbitrary solutions y(z) and g(z) of these equations. /spl upsi/:=yu. It is assumed that the functions r(z), f(z), and g(z) are analytical functions. Moreover, if applications to special functions are studied then r(z) may be taken a polynomial, and f(z) g(z) are fractions of two polynomials. The question arises: what is the differential equation for which the function /spl upsi/(z) is a solution? A more sophisticated question is: is there a differential equation for which singularities are located only at the points where singularities of eqs. 1 and 2 are? These are discussed.
{"title":"The equation for a product of solutions of two second-order linear ODEs","authors":"S. Slavyanov","doi":"10.1109/DD.2000.902370","DOIUrl":"https://doi.org/10.1109/DD.2000.902370","url":null,"abstract":"The following problem is studied. Consider two linear homogeneous second-order ordinary differential equations of the form ry''+r'y'=fy (eqn.1) and ru''+r'u'=gu (eqn.2). These equations are chosen to be formally self-adjoint. The function /spl upsi/(z) is defined as a product of the arbitrary solutions y(z) and g(z) of these equations. /spl upsi/:=yu. It is assumed that the functions r(z), f(z), and g(z) are analytical functions. Moreover, if applications to special functions are studied then r(z) may be taken a polynomial, and f(z) g(z) are fractions of two polynomials. The question arises: what is the differential equation for which the function /spl upsi/(z) is a solution? A more sophisticated question is: is there a differential equation for which singularities are located only at the points where singularities of eqs. 1 and 2 are? These are discussed.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"32 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":"121531950","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 diffraction of shortwave radiation by a convex body with the boundary having a jump of curvature. In cross-section the boundary consists of two parts: convex and planar, smoothly joined. A special case of diffraction by the curve with the curvature jump is under consideration: the jump point is situated in the penumbra region. Using asymptotic methods we obtain new formulae for the wave field in the main approximation in problems with Dirichlet, Neumann and impedance boundary conditions.
{"title":"Diffraction by a line of jump of curvature (a special case)","authors":"A. S. Kirpichnikova, V. Philippov","doi":"10.1109/DD.2000.902357","DOIUrl":"https://doi.org/10.1109/DD.2000.902357","url":null,"abstract":"We consider the diffraction of shortwave radiation by a convex body with the boundary having a jump of curvature. In cross-section the boundary consists of two parts: convex and planar, smoothly joined. A special case of diffraction by the curve with the curvature jump is under consideration: the jump point is situated in the penumbra region. Using asymptotic methods we obtain new formulae for the wave field in the main approximation in problems with Dirichlet, Neumann and impedance boundary conditions.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"604 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":"123235366","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 question of uniqueness for problems describing the interaction of submerged bodies with an ideal unbound fluid is far from resolution. In the present work a new criterion of uniqueness is suggested based on Green's integral identity and the maximum principle for elliptic differential equations. The criterion is formulated as an inequality involving integrals of the Green's function over bodies' wetted contours, and when being satisfied guarantees uniqueness of the problem. This criterion is quite general and applicable for any number of bodies of arbitrary shape (satisfying the exterior sphere condition) and in any dimension.
{"title":"A uniqueness criterion for linear problems of wave-body interaction","authors":"O. Motygin, P. Mciver","doi":"10.1109/DD.2000.902363","DOIUrl":"https://doi.org/10.1109/DD.2000.902363","url":null,"abstract":"The question of uniqueness for problems describing the interaction of submerged bodies with an ideal unbound fluid is far from resolution. In the present work a new criterion of uniqueness is suggested based on Green's integral identity and the maximum principle for elliptic differential equations. The criterion is formulated as an inequality involving integrals of the Green's function over bodies' wetted contours, and when being satisfied guarantees uniqueness of the problem. This criterion is quite general and applicable for any number of bodies of arbitrary shape (satisfying the exterior sphere condition) and in any dimension.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"421 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":"133453481","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 Gegenbauer functions of the first and the second kind are introduced, which generalize the Gegenbauer polynomials. The addition theorem for these functions is obtained by using the 'interpolation of dimensions' technique.
{"title":"Addition theorem for Gegenbauer functions","authors":"E. Tropp, L. Bakaleinikov","doi":"10.1109/DD.2000.902371","DOIUrl":"https://doi.org/10.1109/DD.2000.902371","url":null,"abstract":"The Gegenbauer functions of the first and the second kind are introduced, which generalize the Gegenbauer polynomials. The addition theorem for these functions is obtained by using the 'interpolation of dimensions' technique.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"30 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":"116744204","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 problem of describing a negative spectrum of the periodic self-adjoint Schroedinger operator is very popular in quantum mechanics and it is called the tight-binding approximation. Our aim is to show that the main aspects of the theory are illustrated by a very simple one-dimensional example of minus the second derivative with arbitrary boundary conditions at the vertices of the lattice. We consider one-dimensional Schroedinger equation.
{"title":"Tight-binding investigation of the generalized Dirac comb","authors":"A. B. Mikhaylova","doi":"10.1109/DD.2000.902362","DOIUrl":"https://doi.org/10.1109/DD.2000.902362","url":null,"abstract":"The problem of describing a negative spectrum of the periodic self-adjoint Schroedinger operator is very popular in quantum mechanics and it is called the tight-binding approximation. Our aim is to show that the main aspects of the theory are illustrated by a very simple one-dimensional example of minus the second derivative with arbitrary boundary conditions at the vertices of the lattice. We consider one-dimensional Schroedinger equation.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"53 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":"126646224","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 ordinary differential equation of the second order x/spl uml/+/spl psi/(/spl epsi/t) sin(x-/spl phi/(/spl epsi/t))=0 with the coefficients /spl psi/ and /spl phi/ depending slowly on time. By using a Wentzel-Kramers-Brillouin (WKB)-like method we construct two asymptotic series for a general solution of the equation in the limit /spl epsi//spl rarr/0 (adiabatic limit). One of them is true when the variable t is far from the zeroes of the coefficient /spl psi/ and the other one is valid in the neighborhoods of these these zeroes.
{"title":"WKB-like method for the adiabatic limit of a pendulum type equation","authors":"Andrey V. Ivanov","doi":"10.1109/DD.2000.902355","DOIUrl":"https://doi.org/10.1109/DD.2000.902355","url":null,"abstract":"We consider the ordinary differential equation of the second order x/spl uml/+/spl psi/(/spl epsi/t) sin(x-/spl phi/(/spl epsi/t))=0 with the coefficients /spl psi/ and /spl phi/ depending slowly on time. By using a Wentzel-Kramers-Brillouin (WKB)-like method we construct two asymptotic series for a general solution of the equation in the limit /spl epsi//spl rarr/0 (adiabatic limit). One of them is true when the variable t is far from the zeroes of the coefficient /spl psi/ and the other one is valid in the neighborhoods of these these zeroes.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"28 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":"133070311","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 solution of the Schro/spl uml/dinger equation with potential sin /spl phi//r ((r,/spl phi/)-polar coordinates) presents serious mathematical difficulties which so far have prevented the reliable calculation of the electrical resistivity of edge dislocations in metals. The nature of the difficulties is analyzed by studying related, explicitly solvable problems and by physical reasoning. It is argued that a complete solution will show a logarithmic dependence on an external cut-off radius and that this may account for the experimental results. A recursion-formula approach based on Hankel transformations and a mathematical technique originally developed for Mathieu functions promise to permit a full solution of the scattering problem.
{"title":"Scattering by the two-dimensional potential sin /spl phi//r","authors":"A. Seeger","doi":"10.1109/DD.2000.902366","DOIUrl":"https://doi.org/10.1109/DD.2000.902366","url":null,"abstract":"The solution of the Schro/spl uml/dinger equation with potential sin /spl phi//r ((r,/spl phi/)-polar coordinates) presents serious mathematical difficulties which so far have prevented the reliable calculation of the electrical resistivity of edge dislocations in metals. The nature of the difficulties is analyzed by studying related, explicitly solvable problems and by physical reasoning. It is argued that a complete solution will show a logarithmic dependence on an external cut-off radius and that this may account for the experimental results. A recursion-formula approach based on Hankel transformations and a mathematical technique originally developed for Mathieu functions promise to permit a full solution of the scattering problem.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"110 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":"130367887","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 explicit solutions of a wave equation in 3 dimensions found by Moses and Prosser (1990) and called 'bullets' are studied in detail. These solutions behave for t/spl rarr/+/spl infin/ as the characteristic function of an intersection of the annulus ct+a
{"title":"On the solution of the wave equation asymptotically localized at infinity","authors":"A.S. Blagovestchenskii, A. A. Novitskaya","doi":"10.1109/DD.2000.902354","DOIUrl":"https://doi.org/10.1109/DD.2000.902354","url":null,"abstract":"The explicit solutions of a wave equation in 3 dimensions found by Moses and Prosser (1990) and called 'bullets' are studied in detail. These solutions behave for t/spl rarr/+/spl infin/ as the characteristic function of an intersection of the annulus ct+a<r<ct+b and the circular cone divided by r, where r is the spherical radius, c is the wave speed, a, b=const, a<b. We establish that such a solution tends to a plane wave when the cone becomes narrow.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"10 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":"122460290","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}
A two-dimensional body moves forward with constant velocity in an inviscid incompressible fluid under gravity. The fluid consists of two layers having different densities, and the body intersection interface between the layers. The boundary value problem for the velocity potential is considered in the framework of linearized water-wave theory. The problem is augmented by a pair of physically justified supplementary conditions at points where the body intersects the interface. The extended problem is reduced to an integro-algebraic system. The solvability of the system is proved.
{"title":"The integral equation method for the Neumann-Kelvin problem for an interface-intersecting body in a two-layer fluid","authors":"A. Klimenko","doi":"10.1109/DD.2000.902358","DOIUrl":"https://doi.org/10.1109/DD.2000.902358","url":null,"abstract":"A two-dimensional body moves forward with constant velocity in an inviscid incompressible fluid under gravity. The fluid consists of two layers having different densities, and the body intersection interface between the layers. The boundary value problem for the velocity potential is considered in the framework of linearized water-wave theory. The problem is augmented by a pair of physically justified supplementary conditions at points where the body intersects the interface. The extended problem is reduced to an integro-algebraic system. The solvability of the system is proved.","PeriodicalId":184684,"journal":{"name":"International Seminar Day on Diffraction Millennium Workshop (IEEE Cat. No.00EX450)","volume":"72 1 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":"132500575","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：