A note on variational principles for surface-wave scattering

Complementary variational formulations are developed for the scattering of a gravity wave by a circular dock. These formulations, which are based on assumed distributions of the radial velocity and the potential, respectively, on the projection of the cylindrical boundary, yield lower and upper bounds to an impedance parameter that determines the difference between the scattered wave for the dock and the corresponding wave for a circular cylinder. Numerical results, using trial functions based on the incident wave, are compared with the results implied by a Galerkin solution (Garrett 1971). The maximum errors in the variational approximations to the total scattering cross-section are found to be of the order of 2% for a typical depth/radius ratio, draft/depth ratios of 0, ½ and 1, and all wavelengths. The axisymmetric component of the scattering cross-section is found to be very close to the value for scattering by a circular cylinder (dock extending to bottom). The intensity of the scattered wave on the forward axis for long wavelengths and a certain range of the geometric parameters is significantly less than that for a circular cylinder, and may vanish for critical combinations of these parameters.

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Effects of the Fractal Prefactor on the Optical Properties of Fractal Soot Aggregates

ASME 2009 Second International Conference on Micro/Nanoscale Heat and Mass Transfer, Volume 2 ◽

10.1115/mnhmt2009-18473 ◽

2009 ◽

Cited By ~ 3

Author(s):

Fengshan Liu ◽

David R. Snelling ◽

Gregory J. Smallwood

Keyword(s):

Optical Properties ◽

Fractal Dimension ◽

Cross Section ◽

Laser Wavelength ◽

Scattering Cross Section ◽

Differential Scattering Cross Section ◽

Scattering Cross ◽

Primary Particles ◽

Total Scattering Cross Section ◽

The Difference

The effects of prefactor on the optical properties of numerically generated fractal soot aggregates were investigated using the numerically exact generalized multi-sphere Mie-solution method (GMM) and the approximate Rayleigh-Debye-Gans (RDG) theory. The numerically generated fractal aggregates consist of 50 to 400 primary particles of 30 nm in diameter. The considered incident laser wavelength is 266 nm. Attention is paid to the effect of prefactor on the vertical-vertical differential scattering cross section, since such quantity has often been used to infer the fractal dimension and prefactor based on the RDG formulation. The fractal prefactor affects the optical properties of the numerically generated soot aggregates through its influence on the compactness of the structure. Using GMM to calculate the optical properties of the numerically generated aggregates results in a lower aggregate absorption cross section, but a higher total scattering cross section with increasing the prefactor. The difference between the RDG results and those of GMM is primarily caused by multiple scattering and such effect is found significant, especially for the higher value of prefactor considered. The fractal dimension derived from the GMM non-dimensional differential scattering cross section agrees well with the morphological value in the case of the lower prefactor of 1.3 considered; however, the derived fractal dimension is much higher than the morphological value for fractal soot aggregates with a prefactor of 2.3. The light scattering derived prefactor is in general lower than the morphological value, especially when the morphological prefactor is higher.

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The electrical and thermal conductivities of monovalent metals

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1954.0267 ◽

1954 ◽

Vol 226 (1167) ◽

pp. 436-454 ◽

Cited By ~ 75

Keyword(s):

Cross Section ◽

Scattering Cross Section ◽

Lattice Vibrations ◽

Phonon Interaction ◽

Electron Phonon Interaction ◽

Scattering Cross ◽

Alternative Description ◽

The Difference ◽

As If ◽

Thermal Conductivities

The scattering of electrons by the thermal lattice vibrations is considered. The original Bloch theory is qualitatively in agreement with experiment, but has been shown by various authors to be unrealistic in its detailed assumptions. In particular, it ignores Umklapp processes, neglects the difference between longitudinal and transverse phonon velocities, and treats electron-phonon interaction as if it were independent of the scattering angle. The Bloch theory is generalized to take account of these effects, and formulae are obtained for the transport properties. Numerical calculations for the case of sodium, using the Bardeen (1937) formula for the scattering cross-section and Blackman’s (1951) value for a ‘longitudinal Debye temperature’, agree better with observation than do the simple Bloch expressions, but there still remain discrepancies. An alternative description of these is to treat the differential scattering cross-section, which is not really known exactly, as if it were an unknown function to be determined. The observed electrical and thermal conductivities of sodium are consistent with a cross-section which has a narrower forward lobe than the Bardeen formula, but does not become so small for scattering in the backward direction.

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Mass Determination by Direct Measurement of Electron Scattering

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100114827 ◽

1975 ◽

Vol 33 ◽

pp. 240-241

Author(s):

M. K. Lamvik ◽

A. V. Crewe

Keyword(s):

Cross Section ◽

Electron Scattering ◽

Mass Density ◽

Variable Mass ◽

Scattering Cross Section ◽

Mass Determination ◽

Number Density ◽

Cross Sectional ◽

Thin Slice ◽

Scattering Cross

If a molecule or atom of material has molecular weight A, the number density of such units is given by n=Nρ/A, where N is Avogadro's number and ρ is the mass density of the material. The amount of scattering from each unit can be written by assigning an imaginary cross-sectional area σ to each unit. If the current I0 is incident on a thin slice of material of thickness z and the current I remains unscattered, then the scattering cross-section σ is defined by I=IOnσz. For a specimen that is not thin, the definition must be applied to each imaginary thin slice and the result I/I0 =exp(-nσz) is obtained by integrating over the whole thickness. It is useful to separate the variable mass-thickness w=ρz from the other factors to yield I/I0 =exp(-sw), where s=Nσ/A is the scattering cross-section per unit mass.

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