Watson Method Research Articles

Observations and modeling of the backscattering of short tone bursts from a spherical shell: Lamb wave echoes, glory, and axial reverberations

Tone bursts having durations of 3 or 4 cycles were incident on an air-filled stainless steel shell in water. The resulting sequence of echoes included a specular reflection and echoes radiated by Lamb waves on the shell. Echo structure was studied for ka of 24 to 75, where a denotes the outer radius; b/a=0.838, where b denotes the inner radius. The amplitudes of Lamb wave echoes were modeled using an elastic generalization of the geometrical theory of diffraction (GTD) [P. L. Marston, J. Acoust. Soc. Am. 83, 25–37 (1988)]. The required Lamb wave parameters (the phase velocity cl and damping βl ) were found by the Sommerfeld–Watson method; an efficient numerical method for the computation of the required complex root νl is described. The echoes were identified by comparing arrival times with predictions; bursts reflected from a solid tungsten carbide sphere were used for a reference amplitude. Measurements with ka=24 of the largest Lamb wave echo (which was due to a flexural wave) were made at various backscattering angles γ. The echo was largest for small γ in agreement with predictions. This is a manifestation of the glory pertinent to other bistatic measurements. The ringing of longitudinal wave reverberations (which could be seen following the specular reflection) may be useful for determining the thickness of a shell.

Effective torsion-rotation Hamiltonian for methanol-type molecules

A fourth-order effective Hamiltonian has been derived for the torsion-rotation problem of a methanol-type molecule, i.e., for a C 3 v top attached to a C s frame. First, symmetry considerations based on a frame-fixed axis system are used to determine allowed terms in the Hamiltonian. These terms are then subjected to a contact transformation to remove the indeterminate ones. This procedure is essentially an extension of Watson's method for semirigid molecules to the torsion-rotation problem. It is demonstrated that the Hamiltonian derived in the present work is capable of improving the fittings of the millimeter and submillimeter absorption frequencies of CH 3OH and CH 3SH.

Green's function for the bending of a plate on an elastic half-space

Improper Woinowski-Krieger integrals /1/ expressing the deflections of an infinite plate and the contact reactions of an elastic half-space subjected to a unit normal force are considered. Elementary formulas to calculate the quantities mentioned in the neighbourhood of the point of application of the load are obtained from the power series expansion with a logarithm by Watson's method. The results of calculations using these formulas are in good agreement with the results of a numerical integration of quadratures /2/. The analytical representations obtained for Green's functions are convenient for utilization as kernels of the integral equations when solving contact problems for the interaction of bodies, one of which is reinforced by a thin covering.

The carbon dioxide molecule

Accurate ground state rotational constants and reliable vibration-rotation interaction constants are available for CO2 and its isotopomers. Therefore CO2 has been used as a test case to study the various internuclear distances. We found that the r 0 distances appear, for a given carbon isotope, as a linear function of the inverse of the reduced mass. The r 3 distances obtained, even from purely ‘theoretical’ parameters, still show a small isotopic dependence. Finally Watson's method seems to yield, in the present case, r m distances differing from r e by only 5 × 10-5 Å.

Is the Aliev and Watson method a suitable tool to relate DK0 of symmetric top molecules to excited state rovibrational parameters?

The M. R. Aliev and J. K. G. Watson sum rule [ J. Mol. Spectrosc. 75, 150–160 (1979)] has been tentatively applied to the following symmetric top molecules: H 3SiCl, H 3GeCl, H 3GeBr, H 3SiF, H 3CCl, D 3CCl, H 3CBr, D 3CBr, H 3CI, D 3CI, H 3CF, D 3CF, NH 3, PH 3, PD 3, AsH 3, H 3CD, H 3SiD, H 3GeD, HCF 3, H 2CCCH 2, and D 2CCCD 2. This sum rule relates η tJ and η tK to the quartic centrifugal distortions and should allow the determination of D K 0. Therefore we have collected the best available values of the experimental quantities η tJ and u t = D K 0 − 1 4 η tK for each perpendicular band. We find that the sum role or its simplified version usually provides estimates which are at least as good as force field predictions of D K 0 for which a direct determination is frequently precluded. Since this method is based on the rovibrational analysis of all perpendicular bands, it is also useful to detect perturbations not accounted for in the rovibrational analysis of these fundamentals. In such cases, D K 0 values from different sources turn out to be inconsistent. A critical evaluation of the signs and magnitudes of the η tJ and η tK values may be helpful to find the origin for such inconsistencies. Because of the greater difficulties met and correlation of parameters usually found in the rovibrational analysis of the light ( XY 3) and/or nearly spherical top molecules, the Aliev and Watson sum rule appears at the moment to be of greater value for molecules with a large A 0 B 0 ratio.

The Verification of the Moderator Response Function via Watson Transform

With the aid of residue evaluation, the timedependent reflection response G /SUB gr/ (t) of Shinkawa et al., which was recently used in a coupled-core stability analysis, is verified. The expansion series of the G /SUB gr/ (t) is summed with the Watson method, and its consistency with neutron conservation is checked. The result justifies the previous addition of -delta(t) to a reflection term, which was made in the stability analysis. The meaning of the response flux eigenfunction expansion for a moderator region is discussed.

On zeros of special functions

The zeros of special functions are an integral part of the theory of numerical integration and they arise in the computational problems associated with the inverse Laplace transform. A number of theorems are given which show that Watson's method applied to entire functions with only real positive zeros, provides estimates for both the upper‐ and lower‐bounds of these zeros, in terms of the coefficients in the Taylor's series expansions of the entire functions. Some numerical examples are included.

On the stability of plane Poiseuille flow to finite-amplitude disturbances, considering the higher-order Landau coefficients

In this work a study has been made of the Stuart (1960)–Watson (1960) formalism as applied to plane Poiseuille flow. In particular, the higher-order Landau coefficients have been calculated for the Reynolds & Potter (1967) method and for the Watson (1960) method. The results have been used to study the convergence of the Stuart–Landau series. A convergence curve in the (α, R)-plane has been obtained by using suitable Domb–Sykes plots. In the region of poor convergence of the series, and also in a part of the divergent region of the series, it has been found that the Shanks (1955) method, using the em1 transformation, serves as a very effective way of finding the proper sum of the series, or of finding the proper antilimit of the series. The results for the velocity calculations at R = 5000 are in very good agreement with Herbert's (1977) Fourier-truncation method using N = 4. The Watson method and the Reynolds & Potter method have also been compared inthe subcritical and supercritical regions. It is found in the supercritical region that there is not much difference in the results by the ‘true problem’ of Watson and the ‘false problem’ of Reynolds & Potter when the respective series in both methods are summed by the Shanks method. This fact could possibly be capitalized upon in the subcritical region, where the Watson method is difficult to apply.

On estimation of critical temperatures of fast reactor fuels

The applicability of Watson's method to estimate critical temperatures Tc of fast reactor fuels using their normal boiling points Tb, liquid density at Tb and molecular weight is reviewed in the context of the peculiar situation arising from the dissociation of those polyatomic nuclear materials into different gas-like species in the vapour phase. A simpler empirical relation proposed by Gates and Thodos for evaluating Tc directly from Tb appears to be more appropriate for such materials, with its predictions comparing favourably with more sophisticated approaches for both oxide and carbide fuels. Based on the consistency between these independent approaches, the most probable values of Tc are recommended.

Structure of dichlorine monoxide as studied by microwave spectroscopy. Determination of equilibrium structure by a modified mass dependence method

The microwave spectra of four isotopic species of dichlorine monoxide (OCl 2) have been observed, and the rotational constants have been obtained. The r m structure for each isotopic species has been determined by Watson's method. The equilibrium structure has been estimated by taking proper averages of r m structures to be r e (OCl) = 1.69587(7) A ̊ and ∠ e ClOCl = 110.886(6)°. The general applicability and the merit of the present method for estimating the equilibrium structure are pointed out.

Molecular structure of phosgene as studied by gas electron diffraction and microwave spectroscopy: The rs, rm, and re structures

The microwave spectra of eight isotopic species of COCl 2 have been observed, and the following rotational constants have been obtained: An analysis of the rotational constants has resulted in the r s and r m structures. The equilibrium structure, r e , has been estimated by combining the r m parameters derived according to Watson's method and the r e bond distances estimated in our recent electron-diffraction and spectroscopic studies to be r e (CO) = 1.1756 ± 0.0023 A ̊ , r e (CCl) = 1.7381 ± 0.0019 A ̊ , ∠ e ClCCl = 111.79 ± 0.24°.

Nonlinear Stability of Parallel Flows by High-Ordered Amplitude Expansions

* * The Landau-Stuart theory and its subsequent modifications suffer from some restrictions and from the nonuniqueness in determining higher-order terms of the amplitude expansions, which limit the range of applicability as well as the validity of the results. In the present paper, a well-defined amplitude is introduced a priori. In this way, uniqueness for terms of any order is achieved. Moreover, Watson's method is no longer restricted to almost neutral disturbances. This offers not only more accurate approximations and numerical studies on convergence but, as a consequence, a whole series of new applications. As a first example, the nonlinear equilibrium states of the plane Poiseuille flow are investigated. The numerical results are discussed in context with the author's solutions of the nonlinear equations and with special emphasis on the convergence of Landau's series. f

Application of Watson's method to a problem of the theory of waves on the surface of a heavy liquid

Application of Watson's method to a problem of the theory of waves on the surface of a heavy liquid

An Estimation of the Thermodynamic Properties of Organic Compounds in the Ideal Gas State. I. Acyclic Compounds and Cyclic Compounds with a Ring of Cyclopentane, Cyclohexane, Benzene, or Naphthalene

Abstract A new group-contribution method for estimating the standard enthalpy of formation, the standard absolute entropy, and the standard molar heat of organic compounds in the ideal gas state is proposed with tables of the contribution values. This method (the ABWY method) is a completely revised and enlarged version of the Anderson, Bayer, and Watson method, and it can be used with regard to a more diverse variety of compounds with a higher reliability. The neighboring effects in group-contribution methods are introduced to obtain better estimated values. The substitutions of various functional groups on aromatic rings are discriminated. Type-number corrections of the functional groups and multiple-substitution effects of halogens are also introduced; thus even heterofunctional compounds can be processed. A new concept, the quality of contribution values, is proposed to designate the qualitative reliability of the estimated physical properties. The manual procedures and tables of contribution values are given for estimating the thermodynamic properties of the compounds, as mentioned in the title, composed of C, H, O, halogens, S, and N. This method was critically compared with the Benson method in versatility and reliability.

Shortwave diffraction by an oblate spheroid

The diffraction of shortwaves emitted by a point source located on the equator on an oblate spheroid is investigated by a three-dimensional analog of Watson's method. The zone of deep shadow is considered under the assumption that the point of observation is near the source and is either on the surface of the spheroid or near this surface.

均一電着性の計算式の改善

Several equations to calculate throwing power index have been proposed to compare the throwing power for various plating solutions. In Watson's method (Hull Cell method), Field's equation is mostly preferred. Throwing power index, T(%), calculated from the equation does not necessarily indicate an essential strength of throwing power because any value can be obtained for a certain secondary current distribution, if measuring points on a Hull Cell panel are intentionally selected. Field's equation contains the terms of primary and secondary current density ratio (L and M) which are determined from the thicknesses at two points on the panel. If the measuring points are adjacent to each other and are on a high current density edge of the panel, L and M values gradually reach unity and T value zero. In order to improve this defect in Field's equation a new equation has been formulated on the basis of the slope of a secondary current distribution curve determined from the thicknesses measured at two points on the Hull Cell panel and from plating time.

Comparison of approximate methods for multiple scattering in high-energy collisions. II

The scattering in one dimension of a particle by a target of $N$ like particles in a bound state has been studied. The exact result for the transmission probability has been compared with the predictions of the Glauber theory, the Watson optical potential model, and the adiabatic (or fixed scatterer) approximation. The approximate methods work well only at energies well above the binding energy per target particle. The adiabatic approximation works best and the Watson optical potential model is second best. The Watson method is found to work better when the kinematics suggested by Foldy and Walecka are used rather than that suggested by Watson, that is to say, when the two-body $t$ matrix is calculated with the nucleon-target reduced mass instead of the nucleon-nucleon reduced mass.

Kinetics of the methane–steam reaction over nickel catalyst in a continuous stirred tank reactor

AbstractThe kinetics of steam‐methane reforming over a supported nickel catalyst have been investigated in a continuous stirred tank reactor, under isothermal conditions between 350 and 450 °C at 1–2 atm. The rate data obtained have been analysed by the Hougen–Watson method. Linear as well as non‐linear least squares analysis combined with other statistical considerations were used to discriminate among forty reaction rate models. The following Hougen–Watson type rate equation has been proposed: The constants in the rate equation have been expressed as functions of temperature.

Green's function for Laplace's equation in an infinite cylindrical cell

The Green's function for Laplace's equation in an infinite-length cylinder with a homogeneous mixed boundary condition is considered. Its eigenfunction expansion converges slowly when the axial separation between the source and observation points is small compared to the cylinder radius, and diverges when the axial separation is zero. Applying a modified form of a contour integral method of Watson to an integral representation of the Green's function, a more general expansion of the Green's function is derived. Watson's original method had previously been applied to the case when the source and observation points were both on the axis of the cylinder. The expansion contains a free parameter which may be adjusted to give rapid convergence for any axial separation. It fails, however, when the source and observation points are both near the surface of the cylinder. For two special values of the parameter, the general expansion reduces to the eigenfunction expansion or to the integral representation. The derivation is somewhat obscure, but the resulting formula has a simple interpretation as the superposition of the potential of two related boundary value problems in finite-length cylinders. Some numerical results are given in the spatial region which previously could not be calculated, for a boundary condition approaching a homogeneous Neumann condition, and for a homogeneous Dirichlet condition.

On gamma function inequalities

Watson's method [1] is used to find two convergent monotonically non-decreasing sequences whose upper bounds are equal to Γ(l)Γ(l∓2a)/Γ2(l∓a) ( = K say), provided l > max (0, - 2a). Boyd [2] showed that Gurland's inequality [3] for K corresponds to the first term of the first sequence; Raja Rao's inequality [4] corresponds to the second term of the first sequence. The first two terms of the second sequence also yield easily calculated inequalities.