Velocity group

Second corner reflection The first corner reflection appears as usual when the transducer is coupled to the probe at a certain distance from the V-butt weld. The second corner reflection appears if the transducer is positioned well above the V-hutt weld. If the weld is made of isotropic material the wavefront will miss (pass) the notch without causing any reflection or diffraction (see Fig. 3(a)) for this particular transducer position. In the anisotropic case, the direction of the phase velocity vector will differ from the 45° direction in the isotropic case. Moreover, the direction of the group velocity vector will no longer be the same as the direction of the phase velocity vector (see Fig. 3(b), 3(c)). This can be explained by comparing the corresponding slowness and group velocity diagrams. 

Figure 3 Snapshots of the shear wavefronts for different types of weld material to show the direction of the wave unit vector k (a phase velocity unit vector Cp ) and the group velocity unit vector...

Figure 4 Slowness and group velocity diagrams for isotropic weld material...

Travel time The larger group velocity for herringbone grain orientation (see Fig. 6(b)) explains the shorter travel time (see A-scans in Fig. 7 and 8). 

Using now the phase matching condition, it can be seen that besides the quasi shear wave (qSV) which is obtained as usual, a second quasi shear wave (qSV(2)) results from the upper quasi shear wave part. Since the direction of the group velocity vector points downwards this wave is able to propagate and can be seen in the snapshot (see Fig. 10) if a is properly adjusted, i.e. is pointing upwards as in Fig. 2. 

Fig 1. (a) Phase (b) group velocity dispersion curves for aluminium. Circles show minimum dispersion points. Diamonds show excitation positions for transducer designed for X/d = 2.4 where d is the plate thickness. 

Valdmanis J A and Fork R L 1986 Design considerations for a femtosecond pulse laser balancing self phase modulation, group velocity dispersion, saturable absorption, and saturable gain IEEE J. Quantum. Electron. 22 112-18... 

As described above for small a and NA, a fibre is single mode if U < 2.405. Here only one mode, with one group velocity, is possible. This lack of modal dispersion is why single-mode fibre dominates transport media in long-haul communication systems. 

Nonlinear dispersion becomes relevant at sufficient pulse powers. In some fibre stmctures tire interiDlay between tire nonlinear dispersion and tire group velocity dispersion can be used to produce non-dispersive waves called solitons. Solitons, altliough beyond tire scope of tliis treatment, may revolutionize tire communication systems of tire future. A full treatment of soliton tlieory can be found in [4, 261. 

A 0.2-mm thick hexadecane layer was placed on the oxazine solution. The vibrational coherence at the hexadecane/solution interface was pump-probed in a similar manner [27]. The light pulses traveled in the hexadecane layer and experienced group velocity dispersion before arriving at the interface. This undesired dispersion... 

The expression (1.9) for the group velocity of a composite of two plane waves is exact. 

Thus, the wave packet P(jc, 0 represents a plane wave of wave number ko and angular frequency mo with its amplitude modulated by the factor B(x, i). This modulating function B x, i) depends on x and t through the relationship [x — (dm/dA )o/]. This situation is analogous to the case of two plane waves as expressed in equations (1.7) and (1.8). The modulating function B(x, t) moves in the positive x-direction with group velocity given by... 

In contrast to the group velocity for the two-wave case, as expressed in equation (1.9), the group velocity in (1.16) for the wave packet is not uniquely defined. The point ko is chosen arbitrarily and, therefore, the value at ko of the derivative dco/dk varies according to that choice. However, the range of k is... 

Figure 1.5 shows the real part of the plane wave exp[i(A ox — coot)] with its amplitude modulated by B(x, t) of equation (1.20). The plane wave moves in the positive x-direction with phase velocity Uph equal to o)o/ko. The maximum amplitude occurs at x = v t and propagates in the positive x-direction with group velocity Ug equal to (dm/dA )o. 

As before, the wave packet is a plane wave of wave number ko and angular frequency mo with its amplitude modulated by a factor that moves in the positive x-direction with group velocity given by equation (1.16). Following... 

Thus, the wave packet (x, i) has the same value at point x and time t that it had at point x — ct at time t = 0. The wave packet has traveled with velocity c without a change in its contour, i.e., it has traveled without dispersion. Since the phase velocity is given by mo/ko = c and the group velocity Vg is given by (djco/dk)o = c, the two velocities are the same for an undispersed wave packet. 

Thus, the velocity v of the particle is associated with the group velocity Vg of the wave packet... 

The time A/ for a wave paeket to pass a given point equals the uncertainty in its position x divided by the group velocity Va... 

The phase veloeity for a partieular wave is Uph = A/X, where is a constant. What is the dispersion relation What is the group velocity ... 

This result is used to define the group velocity... 

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