Waves In Open Channel Research Articles

Calculation of intermittent waves in open channels

Calculation of intermittent waves in open channels

Propagation of Flood Waves in Open Channels

Using the method of characteristics, the equations of motion and continuity were numerically solved for the case of a sinusoidal flood wave in a long rectangular open channel. The effects of various parameters, such as the friction coefficient, slope, distance of propagation, initial amplitude of the flood wave, etc., on the damping of the flood wave were studied. It was revealed that the maximum dimensionless discharge and depth are exponential functions of the distance traveled. An empirical formula relating the damping coefficient and other flow parameters was derived. A slight nonlinear relationship between the initial flood amplitude and that at a downstream section was noted.

Self-Induced Waves in open Channels

When water is led from a reservoir through an open smooth contraction into an open horizontal channel fitted with a weir at outlet, a stream is produced which is apparently uniform. However, if the velocity is increased so that the Froude number is raised above about 1/2, a stationary wave train forms on the surface. In an attempt to overcome this disturbance, experiments were made with a tilting channel with and without a weir. To obtain a waveless stream, it was found essential to raise the Froude number somewhat above unity.

Theory of Low-Energy Scattering of Electrons by Hydrogen Atoms

The theory of low-energy electron-hydrogen atom scattering is reformulated to include, in any truncated coupled-channel calculation, the contribution from continuum states due to exchange effects. The existence of such a contribution has been explicitly demonstrated by Castillejo, Percival, and Seaton, although the standard treatment does not include it. Our formulation of the problem employs the expansion of the exact wave function ${\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}$ in terms of the complete set of hydrogenic states ${{\ensuremath{\phi}}_{\ensuremath{\alpha}}}$ in the form ${\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}=\ensuremath{\Sigma}{\ensuremath{\alpha}}^{}{\ensuremath{\phi}}_{\ensuremath{\alpha}}(1){{\ensuremath{\psi}}_{\ensuremath{\alpha}}}^{\ifmmode\pm\else\textpm\fi{}}(2)$. Although this expansion is not termwise symmetric or antisymmetric, the entire sum is made so by use of the integral equation for ${\ensuremath{\Psi}}^{\ifmmode\pm\else\textpm\fi{}}$. The boundary conditions satisfied by the scattering coefficients ${{\ensuremath{\psi}}_{\ensuremath{\alpha}}}^{\ifmmode\pm\else\textpm\fi{}}$ are automatically specified in this approach. Furthermore, it is now clear that the entire, exact elastic or inelastic amplitude is contained in the scattering wave function ${{\ensuremath{\psi}}_{\ensuremath{\alpha}}}^{\ifmmode\pm\else\textpm\fi{}}$. The elastic wave function ${{\ensuremath{\psi}}_{0}}^{\ifmmode\pm\else\textpm\fi{}}$ (and also the set of open-channel wave functions) is shown to obey both homogeneous (optical potential) and inhomogeneous equations. The equivalence of these equations with each other and with the equation found using Feshbach's projection-operator method is demonstrated. A set of homogeneous equations are found for the case where only certain chosen channels are kept and the rest are truncated. This latter set of equations is identical to those obtained by Hahn, O'Malley, and Spruch as an adjunct to calculations establishing bounds on scattering parameters. Both the exact and the approximate sets of equations for the ${{\ensuremath{\psi}}_{\ensuremath{\alpha}}}^{\ifmmode\pm\else\textpm\fi{}}$ differ from the corresponding equations used in all previous coupled-channel calculations. Using our results, the relation between the Born-Oppenheimer amplitude and first-order exchange amplitude of Bell and Moiseiwitsch is discussed. Each is shown to be a different first-order approximation to the exact amplitude, although the Born-Oppenheimer amplitude is seen to be an approximation to the amplitude derived from the inhomogeneous equation. Some simple single-channel calculations of phase shifts and scattering lengths have been carried out. The triplet phase shifts and scattering length derived from the approximate homogeneous equation are identical to those of the standard static exchange approximation, a result expected on the basis of the antisymmetry of the ${\ensuremath{\Psi}}^{\ensuremath{-}}$. However, the singlet phase shifts and scattering length differ from those of the standard static exchange approximation. We find that for ${k}^{2}\ensuremath{\le}0.09$, the singlet phase shifts are larger than those calculated in the $1s\ensuremath{-}2s$ strong-coupling approximation, and that the singlet scattering length is 7.85, where quantities are given in atomic units. For ${k}^{2}>0.09$, the singlet phase shifts are less than those of the $1s\ensuremath{-}2s$ case and approach those of the static exchange calculation. Reasons for this behavior are discussed.

The neutron strength function and the shell model

The projection operator-equivalent potential method is employed for the study of neutron widths and the neutron strength function. Assuming that the residual potential, V R, consists of a sum of two particle potentials it can be shown with the aid of another plausible assumption that the width can be written as a product of two factors. One of these is the square of the matrix element of V R between the open channel wave function and a “shell model” two particle-one hole state. This factor which plays a crucial role measures the probability that in the first scattering of the incident particle a transition to the two particle-one hole state occurs. It contains the fluctuations in the width which accompany changes in the target nucleus. The second factor, giving the probability that the compound nucleus wave function contains the two particle-one hole state, decreases as the difference between the energy of the compound nucleus and the energy of the two particle-one hole state increases. This factor has a Lorentzian form with a half-width Δ corresponding to the lifetime of the two particle-one hole state. These results are generalized to include the case where the energies of several two particle-one hole states fall close to each other. Here it is found that the giant resonance for the widths will now have a substructure which should be observable. The strength function is found to depend upon the average of the transition probabilities to the two particle-one hole states, the number of these states, and Δ. The number is sharply limited by the requirements that angular momentum and parity be conserved in the transition and that energy be conserved to within an error equal to Δ. If the residual potential is expanded in multipoles, keeping terms up to and including the quadripole, it is found that for the most part in the region 40 ≦ A ≦ 64 the only possible transitions involve the monopole; between A = 68 and 85, both dipole and quadripole terms are effective as well, while between 85 and 130 only the quadripole term can induce transitions. This behavior enables one to obtain a fit to the data by means of three parameters: the average matrix elements squared for the monopole, dipole, and quadripole parts of the residual potential. The resulting fit is excellent providing a description not only of the deviations of the strength function from the optical model result but also of the fluctuations in the strength function as the target nucleus is changed. Predictions of the dependence of the strength function for various isotopes become possible. The success of these calculations demonstrates the essential importance of the “first” collision which the incident particle can undergo.

MATHEMATICAL STUDY OF THE MOTION OF INTUMESCENCES IN OPEN CHANNELS OF UNIFORM SLOPE

This paper is the first of a series dealing with the motion of translation waves in open channels. The case treated is that of an arbitrary intumescence with a small height in a uniform channel with a rectangular cross section containing water originally moving with a uniform velocity. Approximate formulas are derived which give the deformation and motion of the intumescence. The case of the intumescence of the form of a rectangular type is also dealt with and for that the form of the rigorous solution is shown. Special emphasis is laid on the dispersive property of intumescences. A method of illustration of the dispersion is presented and both the propagation speed and the dispersion of intumescences of any arbitrary shape are illustrated. The results are all derived with the method of operational calculus on the plane of the complex variables.