DIFFRACTION BY A NARROW SLIT IN THE INTERFACE BETWEEN TWO DIFFERENT MEDIA

1967 ◽  
Vol 45 (1) ◽  
pp. 57-81 ◽  
Author(s):  
Karen Houlberg
Keyword(s):  
Integral Equation ◽  
Plane Electromagnetic Wave ◽  
Perturbation Technique ◽  
Narrow Slit ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Special Values ◽  
Differential Integral Equation ◽  
Conducting Screen ◽  
Formal Solutions

The two-dimensional problem of diffraction of a plane electromagnetic wave by a narrow slit in an infinitely thin, perfectly conducting screen between two different media is studied. The cases of both E and H polarization are considered. In the case of E polarization, a differential-integral equation is obtained for the unknown function in the slit; in the H polarized case, a pure integral equation is obtained for the unknown. These integral equations are solved by a perturbation technique and formal solutions are given in the form of series in powers of the ratio of slit width to wavelength, the coefficients of which depend on the logarithm of this ratio. Expressions are found for the transmission and back-scatter coefficients and some numerical results are given for special values of the parameters. Errors in earlier work (Barakat 1963; Stöckel 1964) are noted.

A NOTE ON DIFFRACTION BY AN INFINITE SLIT

1960 ◽  
Vol 38 (1) ◽  
pp. 38-47 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Integral Equation ◽  
Normal Derivative ◽  
Cylindrical Wave ◽  
Narrow Slit ◽  
Angle Of Incidence ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Boundary Values ◽  
Transmission Coefficients ◽  
Differential Integral Equation

The two-dimensional problem of diffraction of a plane wave by a narrow slit is considered. The assumed boundary values on the screen are the vanishing of either the total wave function or its normal derivative. In the former case, a differential–integral equation is obtained for the unknown function in the slit; in the latter, a pure integral equation is found. Solutions to these equations are given in the form of series in powers of ε (where ε/π is the ratio of slit width to wavelength), the coefficients of which depend on log ε. Expressions are found for the transmission coefficients as functions of ε and the angle of incidence; these are compared with previous determinations of other authors.A brief outline is given for the treatment of diffraction of a cylindrical wave by the slit.

Integral equation methods in potential theory. II

1963 ◽  
Vol 275 (1360) ◽  
pp. 33-46 ◽  
Keyword(s):  
Integral Equation ◽  
Potential Theory ◽  
Digital Computer ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Integral Equation Methods ◽  
Representative Selection ◽  
Selection Of

The boundary integral equations of potential theory can be solved to a tolerable accuracy without undue labour by digital computer techniques, and the computed datagenerate numerical values of the potential field wherever required. Tests have been made with a representative selection of two-dimensional problem s, some of which would not be amenable to any other treatment.

Interaction Between a Rigid Indenter and a Near-Surface Void or Inclusion

1983 ◽  
Vol 50 (3) ◽  
pp. 615-620 ◽  
Author(s):  
G. R. Miller ◽  
L. M. Keer
Keyword(s):  
Integral Equation ◽  
Stress Field ◽  
Size Effects ◽  
Maximum Shear Stress ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Near Surface ◽  
Subsurface Maximum ◽  
Contact Stress Distribution ◽  
Shear Stress Field

A solution is presented to the two-dimensional problem of a rigid indenter sliding with friction on a half plane containing a near-surface imperfection in the form of a circular void or rigid inclusion. The complex variable formulation of Muskhelishivili is used to reduce the problem to a Fredholm integral equation of the second kind. This integral equation is solved numerically thus enabling the numerical calculation of the stress field. The behavior of the stress field is depicted in plots of the contact stress distribution and the subsurface maximum shear stress field. Results are presented showing location and size effects in the case of an inclusion, and finally, comparisons are made between the disturbances due to inclusions and voids.

The pitching motion of a circular disk

1963 ◽  
Vol 17 (4) ◽  
pp. 607-629 ◽  
Author(s):  
W. D. Kim
Keyword(s):  
Integral Equation ◽  
Three Dimensional ◽  
Circular Disk ◽  
Moment Of Inertia ◽  
Damping Coefficient ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Pitching Motion ◽  
Surrounding Fluid

The interaction of a pitching circular disk with the motion induced by the disk in the surrounding fluid is investigated in this paper. MacCamy's (1961) method of simplifying the three-dimensional problem of a circular disk to the two-dimensional problem is found to apply in the present analysis. The integral equation is solved numerically to determine the dependence of pressure, added moment of inertia, and damping coefficient on the frequency of the oscillation.

Diffraction of Pulses by Cylindrical Obstacles of Arbitrary Cross Section

1962 ◽  
Vol 29 (1) ◽  
pp. 40-46 ◽  
Author(s):  
M. B. Friedman ◽  
R. Shaw
Keyword(s):  
Shock Wave ◽  
Integral Equation ◽  
Cross Section ◽  
Analytical Solutions ◽  
Arbitrary Cross Section ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Dimensional Problem ◽  
Acoustic Shock Wave ◽  
Square Box

The two-dimensional problem of the diffraction of a plane acoustic shock wave by a cylindrical obstacle of arbitrary cross section is considered. An integral equation for the surface values of the pressure is formulated. A major portion of the solution is shown to be contributed by terms in the integral equation which can be evaluated explicitly for a given cross section. The remaining contribution is approximated by a set of successive, nonsimultaneous algebraic equations which are easily solved for a given geometry. The case of a square box with rigid boundaries is solved in this manner for a period of one transit time. The accuracy achieved by the method is indicated by comparison with known analytical solutions for certain special geometries.

Modifications of the Godunov method for computing disturbance propagation in an elastic body

2016 ◽  
Vol 11 (1) ◽  
pp. 119-126 ◽  
Author(s):  
A.A. Aganin ◽  
N.A. Khismatullina
Keyword(s):  
Elastic Body ◽  
Godunov Method ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Order Accuracy ◽  
One Dimensional ◽  
Disturbance Propagation ◽  
Linear Waves ◽  
Longitudinal And Shear Waves ◽  
Contact Discontinuities

Numerical investigation of efficiency of UNO- and TVD-modifications of the Godunov method of the second order accuracy for computation of linear waves in an elastic body in comparison with the classical Godunov method is carried out. To this end, one-dimensional cylindrical Riemann problems are considered. It is shown that the both modifications are considerably more accurate in describing radially converging as well as diverging longitudinal and shear waves and contact discontinuities both in one- and two-dimensional problem statements. At that the UNO-modification is more preferable than the TVD-modification because exact implementation of the TVD property in the TVD-modification is reached at the expense of “cutting” solution extrema.

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

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