On reflection of a planar solitary wave at a vertical wall

The reflection of a planar solitary wave at a vertical wall is investigated by solving the Boussinesq equations analytically as well as numerically. The analytical solution is obtained by means of the inner-outer expansions technique, while the numerical solution is based on a finite-difference scheme. The maximum wave amplitude at the wall and the time at which this maximum amplitude is reached are presented. It is also found that the incident wave does not reflect immediately at the wall as predicted by the linear wave theory. Rather, the wave suffers a time delay, called the phase lag, during the reflection process. This phase lag is found to be inversely proportional to the square root of the initial wave amplitude. As the reflected wave eventually propagates away from the wall, it has a phase shift in comparison with that obtained by the linear wave theory. The analytical results obtained in this paper are in good agreement with the numerical results, and they also agree fairly well with the existing experimental data.     

C1-algebras generated by Hankel operators and Toeplitz operators

Results relating the spectra and essential spectra of Hankel operators to their symbols are obtained by various considerations of the C¹-algebras that they generate. Such considerations exploit the elementary relationships between Hankel operators and Toeplitz operators and established techniques in the theory of Toeplitz operators and their generated C¹-algebras.     

Refinement of the crystal and molecular structure of tetraacetylethane by neutron diffraction

The structure of tetraacetylethane has been determined by neutron diffraction. The compound exists in the dienolic form, the enolic hydrogen being intramolecularly bonded to a neighbouring oxygen in a clearly asymmetric mode. The acetylacetonate residues are essentially planar.     

Certain yawed magnetogasdynamic flows

Certain steady yawed magnetogasdynamic flows, in which the magnetic field is everywhere parallel to the velocity field, are related to certain reduced three-dimensional compressible gas flows having zero magnetic field. Under a restriction, the reduced flows are linked, by certain reciprocal relations, to a four parameter class of plane gas flows. In the instance of constant entropy an approximation method is suggested for obtaining magnetogasdynamic flows from the corresponding plane, irrotational gasdynamic flows and examples are given.

Nomenclature

magnetogasdynamic flow variables H magnetic intensity - q fluid velocity - fluid density - p pressure - s entropy - Q _t, H _t component of q, H in the x–y plane - w , h component of q, H perpendicular to the x–y plane reduced gasdynamic flow factor of proportionality - q* fluid velocity - * fluid density - p* pressure - Q _t ^* =u*î+v*, w* components of q* - l arbitrary constant - A _v Alfvén speed - Q _t, , p fluid velocity, density, pressure of the reciprocal gas dynamic flow - L, n, k, arbitrary constants - , velocity potential, stream function - angle made by Q _t, Q _t ^* , and V with the x-axis - adiabatic gas constant - a ²=(–1)/2 constant - M Mach number - W constant value of w* - E approximate constant value of g(p) - * modified potential function - modified velocity coordinate - +i - complex potential of the irrotational flow - B arbitrary constant - V incompressible flow velocity - V modified fluid velocity - X _p, Y _p points on the profile     

Certain two-dimensional solutions to Poisson's equation

Summary A method is presented which derives the solution to Poisson's equation in two dimensions, subject to boundary conditions of sufficient generality to be applicable to many branches of mathematical physics. In the first instance, a circular boundary is considered, and by means of conformal transformation an indication is given showing how the solution can be adapted to other boundary shapes.     

Application of a reciprocal property to subsonic flow

Summary A reciprocal property for the two-dimentional, irrotational flow of a compressible fluid is derived. From this certain flows can be obtained which possess the property of having simply-related velocity potentials and stream functions. To each flow pattern there corresponds a certain pressure density law. Using this property, an approximation method for subsonic flow is deduced, and an example is given which indicates that satisfactorily accurate results can be achieved.     

Relaxations and nano-phase-separation in ultraviscous heptanol-alkyl halide mixture

To gain insight into the effects of liquid-liquid phase separation on molecular relaxation behavior we have studied an apparently homogeneous mixture of 5-methyl-2-hexanol and isoamylbromide by dielectric spectroscopy over a broad temperature range. It shows two relaxation regions, widely separated in frequency and temperature, with the low-frequency relaxation due to the alcohol and the high-frequency relaxation due to the halide. In the mixture, the equilibrium dielectric permittivity epsilon(s) of the alcohol is 41% of the pure state at 155.7 K and epsilon(s) of isoamylbromide is approximately 86% of the pure state at 128.7 K. The difference decreases for the alcohol component with decreasing temperature and increases for the isoamylbromide component. The relaxation time tau of 5-methyl-2-hexanol in the mixture at 155.7 K is over five orders of magnitude less than in the pure state, and this difference increases with decreasing temperature, but tau of isoamylbromide in the mixture is marginally higher than in the pure liquid. This shows that the mixture would have two T(g)'s corresponding to its tau of 10(3) s, with values of approximately 121 K for its 5-methyl-2-hexanol component and approximately 108 K for its isoamylbromide component. It is concluded that the mixture phase separates in submicron or nanometer-size aggregates of the alcohol in isoamylbromide, without affecting the latter's relaxation kinetics, while its own epsilon(s) and tau decrease markedly.     

Synthesis and characterization of the homologous M-M bonded series Ar'MMAr' (M = Zn, Cd, or Hg; Ar' = C6H3-2,6-(C6H3-2,6-Pr(i)2)2) and related arylmetal halides and hydride species

The synthesis and structural characterization of the first homologous, molecular M-M bonded series for the group 12 metals are reported. The compounds Ar'MMAr' (M = Zn, Cd, or Hg; Ar' = C(6)H(3)-2,6-(C(6)H(3)-2,6-Pr(i)(2))(2)) were synthesized by reduction of the corresponding arylmetal halides by alkali metal/graphite (Zn or Hg) or sodium hydride (Cd). These compounds possess almost linear C-M-M-C core structures with two-coordinate metals. The observed M-M bonds distances were 2.3591(9), 2.6257(5), and 2.5738(3) A for the zinc, cadmium, and mercury species, respectively. The shorter Hg-Hg bond in comparison to that of Cd-Cd is consistent with DFT calculations which show that the strength of the Hg-Hg bond is greater. The arylmetal halides precursors (Ar'MI)(1 or 2), and the highly reactive hydrides (Ar'MH)(1 or 2), were also synthesized and fully characterized by X-ray crystallography (Zn and Cd) and multinuclear NMR spectroscopy. The arylzinc and arylcadmium iodides have iodide-bridged dimeric structures, whereas the arylmercury iodide, Ar'HgI, is monomeric. The arylzinc and arylcadmium hydrides have symmetric (Zn) or unsymmetric (Cd) mu-H-bridged structures. The Ar'HgH species was synthesized and characterized by spectroscopy, but a satisfactory refinement of the structure was precluded by the contamination of monomeric Ar'HgH by Ar'H. It was also shown that the decomposition of Ar'Cd(mu-H)(2)CdAr' at room temperature leads to the M-M bonded Ar'CdCdAr', thereby supporting the view that the reduction of the iodide proceeds via the hydride intermediate.