The concept of the transmission line, which has proved so fruitful in electronics generally, is adapted for application to planar dielectric waveguiding structures. The possibility of such structures losing signal strength through radiation, and the difficulty of assessing this power loss, is well known in opto-electronics. Here, it is shown that the slab transmission line may be used to obtain exact solutions for open waveguiding structures that have hitherto been unapproachable by such simple means. A new class of waveguide intersections is discovered. The latter have the property that they consist entirely of guided waves, and there is no general radiation. These simple structures may be regarded as outer solutions, where the inner solution depends on the precise structure of the region of intersection. The new solutions predict the possibility of manufacturing new structures which in principle achieve total transmission around a corner at which a waveguide is reflected from a metal or a dielectric mirror. General applications, and applications to opto-electronics using the concept of the effective dielectric constant are discussed. The solutions also provide accurate controls for numerical methods of solution.
{"title":"Some exact slab transmission line solutions for planar dielectric waveguiding structures","authors":"R. C. Hewson-Browne, P. Kendall","doi":"10.1098/rspa.1990.0044","DOIUrl":"https://doi.org/10.1098/rspa.1990.0044","url":null,"abstract":"The concept of the transmission line, which has proved so fruitful in electronics generally, is adapted for application to planar dielectric waveguiding structures. The possibility of such structures losing signal strength through radiation, and the difficulty of assessing this power loss, is well known in opto-electronics. Here, it is shown that the slab transmission line may be used to obtain exact solutions for open waveguiding structures that have hitherto been unapproachable by such simple means. A new class of waveguide intersections is discovered. The latter have the property that they consist entirely of guided waves, and there is no general radiation. These simple structures may be regarded as outer solutions, where the inner solution depends on the precise structure of the region of intersection. The new solutions predict the possibility of manufacturing new structures which in principle achieve total transmission around a corner at which a waveguide is reflected from a metal or a dielectric mirror. General applications, and applications to opto-electronics using the concept of the effective dielectric constant are discussed. The solutions also provide accurate controls for numerical methods of solution.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"114 1","pages":"473 - 492"},"PeriodicalIF":0.0,"publicationDate":"1990-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79233687","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 beam of electrons, grazing the surface of a semiconductor or insulator, can cause transitions between bands of localized surface states. When the scattered beam is focused in an energy-resolving transmission electron microscope, an image of the surface is obtained. This paper uses a semiclassical model to analyse inelastic electron-surface scattering and predict the brightness and shape of the surface image. The range of validity of the semiclassical model depends on the symmetries of the initial and final surface wavefunctions in a direction perpendicular to the surface.
{"title":"Imaging surface states in the electron microscope: a semiclassical model","authors":"J. Bolton, L. M. Brown","doi":"10.1098/rspa.1990.0036","DOIUrl":"https://doi.org/10.1098/rspa.1990.0036","url":null,"abstract":"A beam of electrons, grazing the surface of a semiconductor or insulator, can cause transitions between bands of localized surface states. When the scattered beam is focused in an energy-resolving transmission electron microscope, an image of the surface is obtained. This paper uses a semiclassical model to analyse inelastic electron-surface scattering and predict the brightness and shape of the surface image. The range of validity of the semiclassical model depends on the symmetries of the initial and final surface wavefunctions in a direction perpendicular to the surface.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"5 1","pages":"291 - 305"},"PeriodicalIF":0.0,"publicationDate":"1990-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84176217","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 Copley Medal is awarded to Dr C. Milstein, F. R. S., in recognition of his outstanding contributions to immunology, in particular to the discovery of monoclonal antibodies and to the understanding of the role of somatic mutations in the maturation of the immune response. The thrust of Dr Milstein’s work has been to understand, at a molecular level, how the diversity of antibodies arises. With G. Brownlee he first proved that the variable and constant regions of antibody light chains are encoded in one messenger RNA molecule. His studies with G. Kohler on certain cell hybrids led to the immortalization of antibody secreting cells, and hence monoclonal antibodies, for which discovery they shared the Nobel Prize for Physiology and Medicine. He has applied these methods to the production of diagnostic reagents and the isolation of cell surface antigens, and has extended the technology to the production of bispecific antibodies, the first example of antibody engineering. Most recently he has developed an approach to studying steps in the antibody response, and has demonstrated the critical role of hypermutation in the maturation process. In all his work, Dr Milstein has shown an unusual gift in making major discoveries by unconventional routes, a gift that he continues to apply in his current work in immunology.
{"title":"The award of medals by the President, Sir George Porter, at the Anniversary Meeting, 30 November 1989","authors":"G. Porter","doi":"10.1098/rspa.1990.0033","DOIUrl":"https://doi.org/10.1098/rspa.1990.0033","url":null,"abstract":"The Copley Medal is awarded to Dr C. Milstein, F. R. S., in recognition of his outstanding contributions to immunology, in particular to the discovery of monoclonal antibodies and to the understanding of the role of somatic mutations in the maturation of the immune response. The thrust of Dr Milstein’s work has been to understand, at a molecular level, how the diversity of antibodies arises. With G. Brownlee he first proved that the variable and constant regions of antibody light chains are encoded in one messenger RNA molecule. His studies with G. Kohler on certain cell hybrids led to the immortalization of antibody secreting cells, and hence monoclonal antibodies, for which discovery they shared the Nobel Prize for Physiology and Medicine. He has applied these methods to the production of diagnostic reagents and the isolation of cell surface antigens, and has extended the technology to the production of bispecific antibodies, the first example of antibody engineering. Most recently he has developed an approach to studying steps in the antibody response, and has demonstrated the critical role of hypermutation in the maturation process. In all his work, Dr Milstein has shown an unusual gift in making major discoveries by unconventional routes, a gift that he continues to apply in his current work in immunology.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"26 1","pages":"251 - 254"},"PeriodicalIF":0.0,"publicationDate":"1990-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75234730","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 detailed description is given of a new method for calculating the high- velocity tail of the electron velocity distribution function in a plasma slab with large temperature and density gradients. Thermal electrons in a plasma are strongly coupled with each other and in a steady state their velocity distribution is always near to maxwellian. On the other hand, the collision frequency of an electron decreases rapidly with increasing speed (v ~ v-3), so that the coupling between the high-velocity electrons and the plasma is very weak. These electrons move almost freely through the plasma and an inhomogeneity can strongly affect the high-velocity part of the distribution function. In our method electrons are classified into two groups, depending on their velocity. The distribution function for the first group (thermal electrons) is accurately given by the Spitzer-Harm solution of the Landau-Fokker-Planck equation. For the second group (high-velocity electrons) the Spitzer-Harm solution is inaccurate and we calculate the distribution function as a solution to the high-velocity approximation of the Landau-Fokker-Planck equation (HVL). The two solutions are matched at a suitably chosen value of the normalized speed ξ. We solve the HVL equation numerically using an efficient method that we have developed. Application is made to the transition region of the quiet Sun using several data-sets for temperature and density gradients by different authors. The results exhibit large deviations from maxwellian throughout the transition region as well as a strongly anisotropic character of the high-velocity tail of the distribution function. The results are very sensitive to the gradients. Also, the non-local character of the formation of the velocity distribution is clearly seen.
{"title":"Calculation of the electron velocity distribution function in a plasma slab with large temperature and density gradients","authors":"N. Ljepojevic, A. Burgess","doi":"10.1098/rspa.1990.0026","DOIUrl":"https://doi.org/10.1098/rspa.1990.0026","url":null,"abstract":"A detailed description is given of a new method for calculating the high- velocity tail of the electron velocity distribution function in a plasma slab with large temperature and density gradients. Thermal electrons in a plasma are strongly coupled with each other and in a steady state their velocity distribution is always near to maxwellian. On the other hand, the collision frequency of an electron decreases rapidly with increasing speed (v ~ v-3), so that the coupling between the high-velocity electrons and the plasma is very weak. These electrons move almost freely through the plasma and an inhomogeneity can strongly affect the high-velocity part of the distribution function. In our method electrons are classified into two groups, depending on their velocity. The distribution function for the first group (thermal electrons) is accurately given by the Spitzer-Harm solution of the Landau-Fokker-Planck equation. For the second group (high-velocity electrons) the Spitzer-Harm solution is inaccurate and we calculate the distribution function as a solution to the high-velocity approximation of the Landau-Fokker-Planck equation (HVL). The two solutions are matched at a suitably chosen value of the normalized speed ξ. We solve the HVL equation numerically using an efficient method that we have developed. Application is made to the transition region of the quiet Sun using several data-sets for temperature and density gradients by different authors. The results exhibit large deviations from maxwellian throughout the transition region as well as a strongly anisotropic character of the high-velocity tail of the distribution function. The results are very sensitive to the gradients. Also, the non-local character of the formation of the velocity distribution is clearly seen.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"9 1","pages":"111 - 71"},"PeriodicalIF":0.0,"publicationDate":"1990-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91330068","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}
F. G. Leppington, K. Heron, E. Broadbent, S. M. Mead
A rectangular sandwich panel, consisting of two thin plane shells on either side of a cellular structure, has perforations in one of the bounding shells so that some or all of the cells may act as Helmholtz resonators. The compound panel is set in an otherwise rigid plane baffle and is irradiated by a plane wave travelling from the unperforated side of the system. An estimate is presented, in integral form, for the acoustic power transmitted through the panel, averaged with respect to time and with respect to all possible directions of incidence, in the limit of large values of ka and kb, where k is the acoustic wavenumber and a, b denote the plate dimensions. For operating frequencies below a certain critical coincidence value, the basic estimate is supplemented by an additional contribution from the resonantly driven but acoustically inefficient modes, which become significant at modest values of ka and kb if the mechanical loss factor ϵ is sufficiently small. There is good agreement between the numerical evaluation of the estimates and the available experimental data.
{"title":"The acoustic transmission properties of anisotropic sandwich panels with perforations","authors":"F. G. Leppington, K. Heron, E. Broadbent, S. M. Mead","doi":"10.1098/rspa.1990.0024","DOIUrl":"https://doi.org/10.1098/rspa.1990.0024","url":null,"abstract":"A rectangular sandwich panel, consisting of two thin plane shells on either side of a cellular structure, has perforations in one of the bounding shells so that some or all of the cells may act as Helmholtz resonators. The compound panel is set in an otherwise rigid plane baffle and is irradiated by a plane wave travelling from the unperforated side of the system. An estimate is presented, in integral form, for the acoustic power transmitted through the panel, averaged with respect to time and with respect to all possible directions of incidence, in the limit of large values of ka and kb, where k is the acoustic wavenumber and a, b denote the plate dimensions. For operating frequencies below a certain critical coincidence value, the basic estimate is supplemented by an additional contribution from the resonantly driven but acoustically inefficient modes, which become significant at modest values of ka and kb if the mechanical loss factor ϵ is sufficiently small. There is good agreement between the numerical evaluation of the estimates and the available experimental data.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"16 1","pages":"27 - 47"},"PeriodicalIF":0.0,"publicationDate":"1990-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86037639","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}
This paper concerns wave propagation in a discrete nonlinear system of equations proposed and studied by G. F. Newell as a model for car- following in traffic flow. In particular, Newell found exact solutions for shock waves and related phenomena. Here, exact solutions representing periodic waves and solitary waves are obtained. The method relates travelling wave solutions to the Toda and Kac-van-Moerbeke discrete systems. In this and other ways, much of the interest is in the general phenomena possible in discrete systems, here including also a time lag, rather than in just the specific traffic flow setting.
本文研究了由G. F. Newell提出并研究的一个离散非线性方程组作为交通流中车辆跟随模型中的波传播问题。特别是，纽厄尔找到了激波和相关现象的精确解。本文给出了周期波和孤立波的精确解。该方法将Toda离散系统和Kac-van-Moerbeke离散系统的行波解联系起来。在这方面和其他方面，大部分的兴趣是在离散系统中可能出现的一般现象，这里也包括时间滞后，而不仅仅是在特定的交通流设置中。
{"title":"Exact solutions for a discrete system arising in traffic flow","authors":"G. Whitham","doi":"10.1098/rspa.1990.0025","DOIUrl":"https://doi.org/10.1098/rspa.1990.0025","url":null,"abstract":"This paper concerns wave propagation in a discrete nonlinear system of equations proposed and studied by G. F. Newell as a model for car- following in traffic flow. In particular, Newell found exact solutions for shock waves and related phenomena. Here, exact solutions representing periodic waves and solitary waves are obtained. The method relates travelling wave solutions to the Toda and Kac-van-Moerbeke discrete systems. In this and other ways, much of the interest is in the general phenomena possible in discrete systems, here including also a time lag, rather than in just the specific traffic flow setting.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"19 1","pages":"49 - 69"},"PeriodicalIF":0.0,"publicationDate":"1990-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77324179","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 velocity potential of the Kelvin ship-wave source is fundamental in the mathematical theory of the wave resistance of ships, but is difficult to evaluate numerically. We shall be concerned with the integral term F(x, ρ, ∝) = ∫∞ -∞ exp {— 1/2ρ cosh (2u — i∝)} cos (x cosh u)du in the source potential, where x and ρ are positive and —1/2π ≼ ∝ ≼ 1/2π, which is difficult to evaluate when x and ρ are small. It will be shown here that F(x, ρ, ∝) = 1/2ƒ(x, ρ, ∝ ) + 1/2ƒ(x, ρ, ─∝) + 1/2ƒ(─x , ρ, ∝) + ½ƒ(─x, ρ, ─∝), where ƒ(x, ρ, ∝) = P0 (x, ρ e-i∝) Σgm (x, ρ ei∝) cm (x, ρ e-i∝) + P1 (x, ρ e-i∝) Σgm (x, ρ ei∝) bm (x, ρ e-i∝) + Σgm (x, ρ ei∝) am (x, ρ e-i∝) In this expression each of the functions gm(x, ρ ei∝), am (x, ρ e-i∝), bm (x, ρ e-i∝), cm(x, ρ e-i∝), satisfies a simple three-term recurrence relation and tends rapidly to 0 for small x and ρ when m → ∞, and the functions P 0 (x, ρ e-i∝) and P1(x, ρ ei∝) are simply related to the parabolic cylinder functions Dv(ζ) respectively, where ζ = — ix(2ρ)-1/2e1/2i∝.
{"title":"On the theory of the Kelvin ship-wave source: the near-field convergent expansion of an integral","authors":"Fritz Joseph Ursell","doi":"10.1098/rspa.1990.0023","DOIUrl":"https://doi.org/10.1098/rspa.1990.0023","url":null,"abstract":"The velocity potential of the Kelvin ship-wave source is fundamental in the mathematical theory of the wave resistance of ships, but is difficult to evaluate numerically. We shall be concerned with the integral term F(x, ρ, ∝) = ∫∞ -∞ exp {— 1/2ρ cosh (2u — i∝)} cos (x cosh u)du in the source potential, where x and ρ are positive and —1/2π ≼ ∝ ≼ 1/2π, which is difficult to evaluate when x and ρ are small. It will be shown here that F(x, ρ, ∝) = 1/2ƒ(x, ρ, ∝ ) + 1/2ƒ(x, ρ, ─∝) + 1/2ƒ(─x , ρ, ∝) + ½ƒ(─x, ρ, ─∝), where ƒ(x, ρ, ∝) = P0 (x, ρ e-i∝) Σgm (x, ρ ei∝) cm (x, ρ e-i∝) + P1 (x, ρ e-i∝) Σgm (x, ρ ei∝) bm (x, ρ e-i∝) + Σgm (x, ρ ei∝) am (x, ρ e-i∝) In this expression each of the functions gm(x, ρ ei∝), am (x, ρ e-i∝), bm (x, ρ e-i∝), cm(x, ρ e-i∝), satisfies a simple three-term recurrence relation and tends rapidly to 0 for small x and ρ when m → ∞, and the functions P 0 (x, ρ e-i∝) and P1(x, ρ ei∝) are simply related to the parabolic cylinder functions Dv(ζ) respectively, where ζ = — ix(2ρ)-1/2e1/2i∝.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"28 1","pages":"15 - 26"},"PeriodicalIF":0.0,"publicationDate":"1990-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84215963","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}
B. Figgis, J. Forsyth, E. Kucharski, P. A. Reynolds, F. Tasset
A polarized neutron diffraction experiment has been done on deuterated ammonium ferrous Tutton salt at 1.5 K, 4.6 T with orientations of the magnetic field along the crystal b and c* axes. The flipping ratios of 303 and 280 reflections respectively were used, after correction for extinction, to give 121 and 118 unique values of the magnetic structure factor FM, Z(hkl) (eff). Those values were used in refinements of models for a description of the magnetization density in the crystal. All models resulted in substantial (37° and 45°) canting of the magnetization direction in the paramagnet away from the magnetic field, to an almost constant direction with respect to the O6 ligand framework, indicating large magnetic anisotropy at the iron atom sites. There is delocalization of magnetization density away from the iron atom into the Fe-O overlap region ( — 4.5%) and onto the OD 2 ligands (6.5%), values comparable with the delocalization of spin from the metal atom in other Tutton salts studied. An earlier ligand field model for the electronic structure of the ion based upon spectroscopic and magnetic data is shown to be inadequate, because it is incompatible with the observed anisotropy in the magnetization around the iron atoms.
{"title":"The magnetization of [Fe(OD2)6]2+ in ammonium ferrous Tutton salt studied by polarized neutron diffraction - a canted paramagnet","authors":"B. Figgis, J. Forsyth, E. Kucharski, P. A. Reynolds, F. Tasset","doi":"10.1098/rspa.1990.0027","DOIUrl":"https://doi.org/10.1098/rspa.1990.0027","url":null,"abstract":"A polarized neutron diffraction experiment has been done on deuterated ammonium ferrous Tutton salt at 1.5 K, 4.6 T with orientations of the magnetic field along the crystal b and c* axes. The flipping ratios of 303 and 280 reflections respectively were used, after correction for extinction, to give 121 and 118 unique values of the magnetic structure factor FM, Z(hkl) (eff). Those values were used in refinements of models for a description of the magnetization density in the crystal. All models resulted in substantial (37° and 45°) canting of the magnetization direction in the paramagnet away from the magnetic field, to an almost constant direction with respect to the O6 ligand framework, indicating large magnetic anisotropy at the iron atom sites. There is delocalization of magnetization density away from the iron atom into the Fe-O overlap region ( — 4.5%) and onto the OD 2 ligands (6.5%), values comparable with the delocalization of spin from the metal atom in other Tutton salts studied. An earlier ligand field model for the electronic structure of the ion based upon spectroscopic and magnetic data is shown to be inadequate, because it is incompatible with the observed anisotropy in the magnetization around the iron atoms.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"5 1","pages":"113 - 127"},"PeriodicalIF":0.0,"publicationDate":"1990-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86685789","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}
Analytical solutions for various hydrodynamic problems are briefly reviewed. The case of a submerged sphere moving in a circular path at constant angular velocity is then analysed based on the linearized velocity potential theory. The potential is expressed by means of a Green function and a distribution of sources over the body surface, written in terms of Legendre functions. The coefficients in the series of the Legendre functions are obtained by imposing the body surface condition. Figures are provided showing the hydrodynamic forces on the sphere.
{"title":"The hydrodynamic forces on a submerged sphere moving in a circular path","authors":"G. Wu, R. E. Taylor","doi":"10.1098/rspa.1990.0031","DOIUrl":"https://doi.org/10.1098/rspa.1990.0031","url":null,"abstract":"Analytical solutions for various hydrodynamic problems are briefly reviewed. The case of a submerged sphere moving in a circular path at constant angular velocity is then analysed based on the linearized velocity potential theory. The potential is expressed by means of a Green function and a distribution of sources over the body surface, written in terms of Legendre functions. The coefficients in the series of the Legendre functions are obtained by imposing the body surface condition. Figures are provided showing the hydrodynamic forces on the sphere.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"18 1","pages":"215 - 227"},"PeriodicalIF":0.0,"publicationDate":"1990-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83749067","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}
This paper discusses the eigenvalue problem associated with the Laplace tidal wave equation (LTWE) given, for μϵ (—1,1), by 1−μ2μ2−τ2y′(μ)′+1μ2−τ2sτμ2+τ2μ2−τ2+s21+μ2y(μ)=λy(μ),(LTWE) where s and τ are parameters, with s an integer and 0 < τ < 1, and λ determines the eigenvalues. This ordinary differential equation is derived from a linear system of partial differential equations, which system serves as a mathematical model for the wave motion of a thin layer of fluid on a massive, rotating gravitational sphere. The problems raised by this differential equation are significant, for both the analytic and numerical studies of Sturm-Liouville equations, in respect of the interior singularities, at the points ± τ, and of the changes in sign of the leading coefficient (1 - μ2)/(μ2 - τ2) over the interval (-1, 1). Direct sum space methods, quasi-derivatives and transformation theory are used to determine three physically significant, well-posed boundary value problems from the Sturm-Liouville eigenvalue problem (LTWE), which has singular end-points ± 1 and, additionally, interior singularities at ± τ. Self-adjoint differential operators in appropriate Hilbert function spaces are constructed to represent each of the three well-posed boundary value problems derived from LTWE and it is shown that these three operators are unitarily equivalent. The qualitative nature of the common spectrum is discussed and finite energy properties of functions in the domains of the associated differential operators are studied. This work continues the studies of LTWE made by earlier workers, in particular Hough, Lamb, Longuet-Higgins and Lindzen.
{"title":"Boundary value problems for the Laplace tidal wave equation","authors":"M. Homer","doi":"10.1098/rspa.1990.0029","DOIUrl":"https://doi.org/10.1098/rspa.1990.0029","url":null,"abstract":"This paper discusses the eigenvalue problem associated with the Laplace tidal wave equation (LTWE) given, for μϵ (—1,1), by 1−μ2μ2−τ2y′(μ)′+1μ2−τ2sτμ2+τ2μ2−τ2+s21+μ2y(μ)=λy(μ),(LTWE) where s and τ are parameters, with s an integer and 0 < τ < 1, and λ determines the eigenvalues. This ordinary differential equation is derived from a linear system of partial differential equations, which system serves as a mathematical model for the wave motion of a thin layer of fluid on a massive, rotating gravitational sphere. The problems raised by this differential equation are significant, for both the analytic and numerical studies of Sturm-Liouville equations, in respect of the interior singularities, at the points ± τ, and of the changes in sign of the leading coefficient (1 - μ2)/(μ2 - τ2) over the interval (-1, 1). Direct sum space methods, quasi-derivatives and transformation theory are used to determine three physically significant, well-posed boundary value problems from the Sturm-Liouville eigenvalue problem (LTWE), which has singular end-points ± 1 and, additionally, interior singularities at ± τ. Self-adjoint differential operators in appropriate Hilbert function spaces are constructed to represent each of the three well-posed boundary value problems derived from LTWE and it is shown that these three operators are unitarily equivalent. The qualitative nature of the common spectrum is discussed and finite energy properties of functions in the domains of the associated differential operators are studied. This work continues the studies of LTWE made by earlier workers, in particular Hough, Lamb, Longuet-Higgins and Lindzen.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"92 1","pages":"157 - 180"},"PeriodicalIF":0.0,"publicationDate":"1990-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76392381","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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