THE PRESSURE PULSE PRODUCED BY A LARGE EXPLOSION IN THE ATMOSPHERE

1961 ◽  
Vol 39 (7) ◽  
pp. 993-1009 ◽  
Author(s):  
V. H. Weston
Keyword(s):  
Fourier Transform ◽  
Green’S Function ◽  
Green's Function ◽  
Pressure Pulse ◽  
Closed Surface ◽  
Excess Pressure ◽  
Normal Velocity ◽  
Harmonic Frequency ◽  
Large Explosion ◽  
Ring Source

The pressure pulse produced by a large explosion in the atmosphere is investigated. The explosion is represented in terms of the excess pressure and normal velocity on a closed surface, outside of which the hydrodynamical equations are linearized. The pulse is represented in terms of a Fourier transform of the associated harmonic frequency problem, for which a ring-source Green's function is obtained in terms of an expansion of the discrete modes. It is shown that the excess pressure may be represented in terms of an integral (containing the Green's function) over the surface surrounding the source. The gravity wave portion of the pressure pulse at the ground is computed for various ranges from the source, which is located at various altitudes, and for three models of the atmosphere. In calculating the head of the pulse a new asymptotic technique is introduced which gives very good results for intermediate and long ranges.

Radar signature of a 2.5-D tunnel

Geophysics ◽  
1993 ◽  
Vol 58 (11) ◽  
pp. 1573-1587 ◽  
Author(s):  
Mark L. Moran ◽  
Roy J. Greenfield
Keyword(s):  
Fourier Transform ◽  
Green’S Function ◽  
Green's Function ◽  
Electric Dipole ◽  
Short Pulse ◽  
Three Dimensional ◽  
Acute Angle ◽  
Peak Power Output ◽  
Early Arrival ◽  
Low Amplitude

The effects of an infinitely long cylindrical void on short‐pulse cross‐borehole radar waveforms are modeled and analyzed. Pulsed electromagnetic sensing system (PEMSS) data are of particular interest. The PEMSS system developed by the Southwest Research Institute uses a vertically oriented electric dipole that emits a short electromagnetic pulse with peak power output centered around 30 MHz, which gives wavelengths of roughly 1.5 cavity diameters. The transmitter and receiver are typically located in boreholes separated by approximately 30 m. The model is based on field solutions for a vertically oriented point‐source electric dipole. A three‐dimensional (3-D) analytical frequency domain derivation of the Green’s function is found using a spatial Fourier transform over the cylinder axis. The resulting wavenumber integral is evaluated by a numerical integration over wavenumber. Time‐domain waveforms are produced by applying a Fourier transform to a 7-80 MHz band of frequencies in the Green’s function spectrum. Model results agree well with PEMSS field data sets. Further modeling examines the effects on waveforms for a wide variety of cases in which the raypath is not orthogonal to the tunnel axis, including the effect of tunnel dip. An air‐filled tunnel with a radius greater than 1.0 m produces a low amplitude shadow zone along its entire length. A low amplitude early arrival is observed in simulations with air‐filled tunnels in which the source to receiver path forms an acute angle larger than 45 degrees with the tunnel axis. This arrival is interpreted as propagation through the tunnel. When this angle is smaller than 45 degree the tunnel is effectively an opaque object and only the energy diffracted around the cylindrical void is observed. Waveform behavior gradually transitions from propagation through the tunnel in the vicinity of 45 degrees.

scholarly journals Integral Equations and the Determination of Green's Functions in the Theory of Potential

1912 ◽  
Vol 31 ◽  
pp. 71-89 ◽  
Author(s):  
H. S. Carslaw
Keyword(s):  
Green’S Function ◽  
Integral Equations ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Closed Surface ◽  
Second Derivatives

In the Theory of Potential the term Green's Function, used in a slightly different sense by Maxwell, now denotes a function associated with a closed surface S, with the following properties:—(i) In the interior of S, it satisfies ∇2V = 0.(ii) At the boundary of S, it vanishes.(iii) In the interior of S, it is finite and continuous, as also its first and second derivatives, except at the point (x1, y1,z1).

The propagator matrix related to the Kirchhoff‐Helmholtz integral in inverse wavefield extrapolation

Geophysics ◽  
1994 ◽  
Vol 59 (12) ◽  
pp. 1902-1910 ◽  
Author(s):  
Lasse Amundsen
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Matrix Method ◽  
Pressure Field ◽  
Normal Derivative ◽  
Closed Surface ◽  
Observation Point ◽  
Propagator Matrix ◽  
Helmholtz Integral

The Kirchhoff‐Helmholtz formula for the wavefield inside a closed surface surrounding a volume is most commonly given as a surface integral over the field and its normal derivative, given the Green’s function of the problem. In this case the source point of the Green’s function, or the observation point, is located inside the volume enclosed by the surface. However, when locating the observation point outside the closed surface, the Kirchhoff‐Helmholtz formula constitutes a functional relationship between the field and its normal derivative on the surface, and thereby defines an integral equation for the fields. By dividing the closed surface into two parts, one being identical to the (infinite) data measurement surface and the other identical to the (infinite) surface onto which we want to extrapolate the data, the solution of the Kirchhoff‐Helmholtz integral equation mathematically gives exact inverse extrapolation of the field when constructing a Green’s function that generates either a null pressure field or a null normal gradient of the pressure field on the latter surface. In the case when the surfaces are plane and horizontal and the medium parameters are constant between the surfaces, analysis in the wavenumber domain shows that the Kirchhoff‐Helmholtz integral equation is equivalent to the Thomson‐Haskell acoustic propagator matrix method. When the medium parameters have smooth vertical gradients, the Kirchhoff‐Helmholtz integral equation in the high‐frequency approximation is equivalent to the WKBJ propagator matrix method, which also is equivalent to the extrapolation method denoted by extrapolation by analytic continuation.

Elastic Green's function for a damaged interface in anisotropic materials

1996 ◽  
Vol 11 (2) ◽  
pp. 537-544 ◽  
Author(s):  
J. R. Berger ◽  
V. K. Tewary
Keyword(s):  
Fourier Transform ◽  
Green’S Function ◽  
Plane Strain ◽  
Green's Function ◽  
Functional Form ◽  
Anisotropic Materials ◽  
Displacement Discontinuity ◽  
Interface Problems ◽  
Transform Method ◽  
Anisotropic Bimaterial

We present the derivation of the elastic Green's function for an anisotropic bimaterial in a state of plane strain. A Fourier transform method is used to calculate the Green's function. A discontinuity in displacement is permitted across the interface between the two solids. This provides a useful functional form for parameterizing damage along an interface. We show several examples for the form of the displacement discontinuity and calculate the displacement Green's function for each. The Green's function derived here is applicable to a variety of interface problems between two different anisotropic solids or for two similar solids at different orientations.

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