The U Wave

Classically the electrocardiogram (ECG) was believed to be a graphic tracing of the electric current produced by the myoneural activity associated with heart muscle excitation. The normal ECG, it was believed, showed deflections resulting from atrial and ventricular activity. The first signal, P, is due to atrial excitation with the QRS deflections arising from ventricular activity. T waves are believed to be due to ventricular recovery (repolarization) while the U waves are seen in the normal ECG and are accentuated in hypokalaemia (low potassium levels in the blood). 

U waves are not always seen on the ECG (Fig. 2.16). Opinion is divided over what exactly the U wave represents. Many consider the U wave to represent repolarisation of the ventricular septum or Purkinje fibres. These waves can be difficult to see and are usually smaller than P waves. 

One of the potential problems caused by this form of artefact is that it can lead to misdiagnosis. This can arise due to loss of detail obscured by movement. This in turn makes it difficult to measure intervals and see waveforms, especially P waves and other smaller features such as the U wave. 

The approximation of Fresnel is scalar approximation. Let u(, r],0-0) be the scalar wave function of the laser beam falling onto the optical element, and u( X,y,Cl) will the be scalar wave function in the plane Z = Cl. Then [3,4]... 

Fig. 11 shows a composite model of the wave at U X =0.25. In the interfering wave on the upper and lower part of the insert metal, (a) is the same phase, and (b) is the opposite phase. A composite wave is attenuated by the weakened interference as the same phase, and is amplified by the strengthened interference as the opposite phase. 

In general, the phonon density of states g(a), da is a complicated fiinction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(co) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(ra) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g((o) = (oD-oig) and U = - 1), for each of the... 

The Debye model is more appropriate for the acoustic branches of tire elastic modes of a hanuonic solid. For molecular solids one has in addition optical branches in the elastic wave dispersion, and the Einstein model is more appropriate to describe the contribution to U and Cj from the optical branch. The above discussion for phonons is suitable for non-metallic solids. In metals, one has, in addition, the contribution from the electronic motion to Uand Cy. This is discussed later, in section (A2.2.5.6T... 

The fluctuations of the local interfacial position increase the effective area. This increase in area is associated with an increase of free energy Wwhich is proportional to the interfacial tension y. The free energy of a specific interface configuration u(r,) can be described by the capillary wave Hamiltonian ... 

U(qJ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical sbucture is discussed in detail in Section in.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, U(q ) is orthogonal and therefore has n n — l)/2 independent elements (or degrees of freedom). This transformation mabix U(qO can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schrodinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix U(qx) according to... 

The electronic wave functions of the different spin-paired systems are not necessarily linearly independent. Writing out the VB wave function shows that one of them may be expressed as a linear combination of the other two. Nevertheless, each of them is obviously a separate chemical entity, that can he clearly distinguished from the other two. [This is readily checked by considering a hypothetical system containing four isotopic H atoms (H, D, T, and U). The anchors will be HD - - TU, HT - - DU, and HU -I- DT],... 

As discussed in detail in [10], equivalent results are not obtained with these three unitary transformations. A principal difference between the U, V, and B results is the phase of the wave function after being h ansported around a closed loop C, centered on the z axis parallel to but not in the (x, y) plane. The pertm bative wave functions obtained from U(9, <])) or B(0, <()) are, as seen from Eq. (26a) or (26c), single-valued when transported around C that is ( 3 )(r Ro) 3< (r R )) = 1, where Ro = Rn denote the beginning and end of this loop. This is a necessary condition for Berry s geometric phase theorem [22] to hold. On the other hand, the perturbative wave functions obtained from V(0, <])) in Eq. (26b) are not single valued when transported around C. 

Vo + V2 and = Vo — 2 (actually, effective operators acting onto functions of p and < )), conesponding to the zeroth-order vibronic functions of the form cos(0 —4>) and sin(0 —(()), respectively. PL-H computed the vibronic spectrum of NH2 by carrying out some additional transformations (they found it to be convenient to take the unperturbed situation to be one in which the bending potential coincided with that of the upper electi onic state, which was supposed to be linear) and simplifications (the potential curve for the lower adiabatic electi onic state was assumed to be of quartic order in p, the vibronic wave functions for the upper electronic state were assumed to be represented by sums and differences of pairs of the basis functions with the same quantum number u and / = A) to keep the problem tiactable by means of simple perturbation... 

In the case of atoms (Section 7.1) a sufficient number of quantum numbers is available for us to be able to express electronic selection rules entirely in terms of these quantum numbers. For diatomic molecules (Section 7.2.3) we require, in addition to the quantum numbers available, one or, for homonuclear diatomics, two symmetry properties (-F, — and g, u) of the electronic wave function to obtain selection rules. 

Figure 2.3. A rigid piston drives a shock wave into compressible fluid in an imaginary flow tube with unit cross-sectional area. The shock wave moves at velocity U into fluid with initial state 0, which changes discontinuously to state 1 behind the shock wave. Particle velocity u is identical to the piston velocity.

The equation that expresses conservation of energy can also be determined by considering Fig. 2.3. Since the piston moves a distance u At, the work done by the piston on the fluid during this time interval is Pu At. The mass of material accelerated by the shock wave to a velocity u is PqU At. The kinetic energy acquired by this mass element is therefore (pqUu ) At/2. If the specific internal energies of the undisturbed and shocked material are denoted by Eq and E, respectively, the increase in internal energy is ( — o)Po V At per unit mass. The work performed on the system is equal to the sum of kinetic and... 

It is important to note that the state determined by this analysis refers only to the pressure (or normal stress) and particle velocity. The material on either side of the point at which the shock waves collide reach the same pressure-particle velocity state, but other variables may be different from one side to the other. The material on the left-hand side experienced a different loading history than that on the right-hand side. In this example the material on the left-hand side would have a lower final temperature, because the first shock wave was smaller. Such a discontinuity of a variable, other than P or u that arises from a shock wave interaction within a material, is called a contact discontinuity. Contact discontinuities are frequently encountered in the context of inelastic behavior, which will be discussed in Chapter 5. 

The propagation of a shock wave from a detonating explosive or the shock wave induced upon impact of a flyer plate accelerated, via explosives or with a gun, result in nearly steady waves in materials. For steady waves a shock velocity U with respect to the laboratory frame can be defined. Conservation of mass, momentum, and energy across a shock front can then be expressed as... 

If we accept the assumption that the elastic wave can be treated to good aproximation as a mathematical discontinuity, then the stress decay at the elastic wave front is given by (A. 15) and (A. 16) in terms of the material-dependent and amplitude-dependent wave speeds c, (the isentropic longitudinal elastic sound speed), U (the finite-amplitude elastic shock velocity), and Cfi [(A.9)]. In general, all three wave velocities are different. We know, for example, that... 

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