Surface integral

Abstract: Three different integral formulations have been used as a basis for obtaining approximate solutions of the exterior steady‐state acoustic radiation problem for an arbitrary surface whose normal velocity is specified: (1) the simple‐source formulation, adapted from potential theory; (2) the surface Helmholtz integral formulation, based on the integral expression for pressure in the field in terms of surface pressure and normal velocity; and (3) the interior Helmholtz integral formulation, in which the surface pressure is determined by making a certain integral vanish for all points interior to the radiating surface. For certain characteristic wavenumbers, it is shown that no solution of the simple‐source formulation exists in general and that there is no unique solution of the surface Helmholtz integral formulation. The interior Helmholtz integral formulation is subject to similar difficulties and has undesirable computational characteristics. A Combined Helmholtz Integral Equation Formulation (CHIEF) that overcomes the deficiencies of the first two methods and the undesirable computational characteristics of the third, is described. The significant improvement over the previous three methods, which is accomplished through the use of CHIEF, is illustrated by numerical examples involving spheres, finite cylinders, cubes, and a steerable array mounted in two different boxlike structures.

Abstract: Computer migration of seismic data emerged in the late 1960s as a natural outgrowth of manual migration techniques based on wavefront charts and diffraction curves. Summation (integration) along a diffraction hyperbola was recognized as a way to automate the familiar point‐to‐point coordinate transformation performed by interpreters in mapping reflections from the x, t (traveltime) domain into the x, z (depth domain). We will discuss the mathematical formulation of migration as a solution to the scalar wave equation in which surface seismic observations are the known boundary values. Solution of this boundary value problem follows standard techniques, and the migrated image is expressed as a surface integral over the known seismic observations when areal or 3-D overage exists. If only 2-D seismic coverage is available, wave equation migration is still possible by assuming the subsurface and hence surface recorded data do not vary perpendicular to the seismic profile. With this assumption, the surface inte...

Abstract: 1. The Description of a Material.- 3. The Constitutive Laws. Case of No Constraint on the State Quantities or Their Velocities.- 5. The Constitutive Laws on a Discontinuity Surface.- 6. Deformable Solids with Interactions at a Distance.- 7. Deformable Solids Without Interaction at a Distance.- 8. Collision of Rigid Bodies. Multiple Collisions.- 9. Evolution of Two Deformable Solids with Collisions.- 10. Material with Volume Interactions at a Distance. Fibre Reinforced Material.- 11. Solid-Liquid Phase Change. Supercooling. Soil Freezing.- 12. Damage. Gradient of Damage.- 13. Shape Memory Alloys.- 14. Unilateral Contact. Contact with Adhesion.- A.1 Convex Functions.- A.1.1 Subgradient of a Convex Function. Subdifferential Set.- A.1.2 Indicator Function of a Set.- A.1.5 Indicator Function of the Segment [0, 1].- A.1.7 Indicator Function of a Triangle.- A.1.9 A Property of the Subdifferential Set.- A.1.10The Dual Function of a Convex Function.- A.2 Material Derivatives.- A.2.1 Material Derivative of a Function.- A.2.2 Material Derivative of a Surface Integral.- A.2.3 Material Derivative of a Double Surface Integral.- A.2.4 Mass Balance on a Surface.- A.2.5 Material Derivatives of Integrals of Mass Densities.- References.