# Jackup rig

The physics of ultrasound is well known and widely described in many publications. Recording amplitudes from model reflectors at different depths by Dr. Josef Krautkramer in 1959 led to the DGS-diagram Echo amplitudes from disk shaped reflectors of different sizes were [c.812]

Another reason why mass-scaled coordinates are useful is that they simplify the transfomiation to the Jacobi coordinates that are associated with the products AB + C. If we define. S as the distance from C to the centre of mass of AB, and s as the AB distance, mass scaling is accomplished via [c.974]

A definition of catalysis similar to that given above was stated first in about 1895 by Wilhelm Ostwald, whose work on catalysis was recognized with a Nobel prize. Sixty years before, Jakob Berzelius had coined the tenn [c.2697]

Landau E M, Rummel G, Cowan-Jacob S W and Rosenbusch J P 1997 Crystallization of a polar protein and small molecules from the aqueous compartment of lipidic cubic phases J. Phys. Chem. B 101 1935-7 [c.2846]

The vector potential is derived in hyperspherical coordinates following the procedure in [54], where the connections between Jacobi and the hyperspherical coordinates have been considered as below (see [67]) [c.87]

The gradient of v l with respect to Jacobi coordinates (the vector potential) considering the physical region of the conical intersection, is obtained by using Eqs. (C.6-C.8) and after some simplification ( ) we get, [c.89]

Variationally deriving with respect to a leads to the Hamilton-Jacobi equation [c.160]

In Eqs. (5) and (6), M is the total mass of the nuclei and is the mass of one electron. By using Eq. (2), the system s internal kinetic energy operator is given in terms of the mass-scaled Jacobi vectors by [c.183]

Figure 1. Jacobi vectors for a three-nuclei, four-electron system. The nuclei are Pi, P2, P3, and the electrons are ei, 02, 63, 64, |

The H -b H2 H2 + H hydrogen atom exchange reaction is the simplest atom-molecule scattering system. Molecules and atoms colliding is a basic step in chemical reactivity, and much work has been made to understand this system in all its details [111,112]. As well as experimental work, extensive calculations have been made using both a time-independent framework [113] and wavepacket methods [114-116] to obtain fully state resolved cross-sections for the reaction. This system is best described by Jacobi coordinates, shown in Figure la, and the reaction is dominated by the colinear configuration. The PES for this configuration (i.e., a cut with 0 = 0°) has a C-shaped minimum energy channel, with a saddle point as a transition region at the apex. This is shown in Figure 2. [c.260]

Using Jacobi coordinates and reduced masses, the Hydrogen-Chlorine interaction is modeled quantum mechanically whereas the Ar-HCl interaction classically. The potentials used, initial data and additional computational parameters are listed in detail in [16]. [c.406]

[c.553]

We have expressed P in tenns of Jacobi coordinates as this is the coordmate system in which the vibrations and translations are separable. The separation does not occur in hyperspherical coordinates except at p = oq, so it is necessary to interrelate coordinate systems to complete the calculations. There are several approaches for doing this. One way is to project the hyperspherical solution onto Jacobi s before perfonning the asymptotic analysis, i.e. [c.977]

The classical action, or solution of the Flamilton-Jacobi equation V S (/ ) = k (/ ), for relative motion under [c.2053]

It would be convenient for obtaining the expressions of the gradient of the hyperangle

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). [c.101]

The aim of this section is to show how the modulus-phase formulation, which is the keytone of our chapter, leads very directly to the equation of continuity and to the Hamilton-Jacobi equation. These equations have formed the basic building blocks in Bohm s formulation of non-relativistic quantum mechanics [318]. We begin with the nonrelativistic case, for which the simplicity of the derivation has [c.158]

Again, the summation convention is used, unless we state otherwise. As will appear below, the same strategy can be used upon tbe Dirac Lagrangean density to obtain the continuity equation and Hamilton-Jacobi equation in the modulus-phase representation. [c.159]

The result of interest in the expressions shown in Eqs. (160) and (162) is that, although one has obtained expressions that include corrections to the nonrelativistic case, given in Eqs. (141) and (142), still both the continuity equations and the Hamilton-Jacobi equations involve each spinor component separately. To the present approximation, there is no mixing between the components. [c.164]

The terms before the square brackets give the nonrelativistic part of the Hamilton-Jacobi equation and the continuity equation shown in Eqs. (142) and (141), while the term with the squaie brackets contribute relativistic corrections. All terms from are of the nonmixing type between components. There are further relativistic terms, to which we now turn. [c.165]

Consider a polyatomic system consisting of N nuclei (where > 3) and elecbons. In the absence of any external fields, we can rigorously separate the motion of the center of mass G of the whole system as its potential energy function V is independent of the position vector of G (rg) in a laboratory-fixed frame with origin O. This separation introduces, besides rg, the Jacobi vectors R = (R , , R , .. , Rxk -1) = (fi I "21 I fvji) fot nuclei and electrons, [c.182]

[c.261]

Caustics The above formulae can only be valid as long as Eq. (9) describes a unique map in position space. Indeed, the underlying Hamilton-Jacobi theory is only valid for the time interval [0,T] if at all instances t [0, T] the map (QOi4o) —> Q t, qo,qo) is one-to-one, [6, 19, 1], i.e., as long as trajectories with different initial data do not cross each other in position space (cf. Fig. 1). Consequently, the detection of any caustics in a numerical simulation is only possible if we propagate a trajectory bundle with different initial values. Thus, in pure QCMD, Eq. (11), caustics cannot be detected. [c.384]

See pages that mention the term

**Jackup rig**:

**[c.374] [c.499] [c.426] [c.712] [c.1459] [c.1499] [c.2051] [c.2051] [c.2055] [c.2293] [c.87] [c.153] [c.166] [c.168] [c.182] [c.182] [c.182] [c.183] [c.608] [c.620] [c.89] [c.407] [c.146]**

Standard Handbook of Petroleum and Natural Gas Engineering Volume 1 (1996) -- [ c.1363 ]