Functional vlues of function f are determined from the vlues F(s) of its Laplace transform at discrete points of s. Evaluation of F(s) t points given by s (8 -1 /),/c 0, 1, determine coefficients in an infinite series expansion of f(t) in terms of Jacobi polynomials. The vlues of and determine the position along the rel s-xis at which F(s) is evaluated. An approximation to f(t) is given by using finite number of terms of the infinite series expansion of f(t).Numerical examples are given and results are compared with some known rmmerical methods for pproximating f(t).Introduction. The problem of numericMly inverting the Laplace transform is known to mthemticins, physicists, nd engineers nd hs been

Beam patterns of arrays of sensors can now be optimized (low sidelobe level with minimum mainlobe width) by shading the array with amplitude weights that vary with incoming plane-wave angles. For the simple case of a one-dimensional array [x,f(x)], we demonstrate a method of shading for wide steering angles and give computer-simulated results. This method of shading is immediately applicable to two-dimensional arrays of the form [x,y,f(x)], and permits analysis of a restricted class of conformal arrays.

Common‐depth‐point seismic reflection data were generated on a computer using simple ray tracing and analyzed with processing techniques currently used on actual field recordings. Constant velocity layers with curved interfaces were used to simulate complex geologic shapes. Two models were chosen to illustrate problems caused by curved geologic interfaces, i.e., interfaces at depths which vary laterally in a nonlinear fashion and produce large spatial variations in the apparent stacking velocity. A three‐layer model with a deep structure and no weathering was used as a control model. For comparison, a low velocity weathering layer also of variable thickness was inserted near the surface of the control model. The low velocity layer was thicker than the ordinary thin weathering layers where state‐of‐the‐art static correction methods work well. Traveltime, moveout, apparent rms velocities, and interval velocities were calculated for both models. The weathering introduces errors into the rms velocities and traveltimes. A method is described to compensate for these errors. A static correction applied to the traveltimes reduced the fluctuation of apparent rms velocities. Values for the thick weathering layer model were “over corrected” so that synclines (anticlines) replaced false anticlines (synclines) for both near‐surface and deep zones. It is concluded that computer modeling is a useful tool for analyzing specific problems of processing CDP seismic data such as errors in velocity estimates produced by large lateral variations in overburden.

An analysis is made of a surface wave traveling axially along the boundary of an infinitely long cylindricalsurface that has small periodic variations superimposed upon it. By the application of Floquet's theorem, the acoustic pressure is expressed in terms of an infinite series. For waveguides that have acoustic reactances that are periodic with period L in the axial direction, an infinite system of equations is obtained. Unknown amplitude coefficients are determined by a method of infinite determinants. After the amplitude coefficients are known in terms of one coefficient, the phase variation for the surface wave along the cylindrical axis is calculated by •(z) --arctan (ImP/ReP). The general method is illustrated by considering a waveguide with a variable reactance given by x=x0+x cos(2•-z/L), where x<

Numerical predictions for the reflection of plane sound waves from a pressure-release sinusoidal surface of large slope are compared with experimental data. The approach introduced recently by Uretsky [J. Acoust. Soc. Am. 35, 1293-1294 (1963)] for the calculation of reflection coefficients was adopted with some modification owing to the geometrical parameters of the surface and the frequency of the incident sound. The geometry of the surface and the experiment were such that theory could be tested with three orders of the grating spectra.