Frequency angular

The currents passing through the two coils are alternative with the angular frequency co one can write... 

In equation (A3.13.73), 8j is the average angular frequency distance between quantum states within level /... 

Here E(t) denotes the applied optical field, and-e andm represent, respectively, the electronic charge and mass. The (angular) frequency oIq defines the resonance of the hamionic component of the response, and y represents a phenomenological damping rate for the oscillator. The nonlinear restoring force has been written in a Taylor expansion the temis + ) correspond to tlie corrections to the hamionic... 

The previous subsection described single-experiment perturbations by J-jumps or P-jumps. By contrast, sound and ultrasound may be used to induce small periodic perturbations of an equilibrium system that are equivalent to periodic pressure and temperature changes. A temperature amplitude 0.002 K and a pressure amplitude 5 P ss 30 mbar are typical in experiments with high-frequency ultrasound. Fignre B2.5.4 illustrates the situation for different rates of chemical relaxation with the angular frequency of the sound wave... 

Figure B2.5.4. Periodic displacement from equilibrium through a sound wave. The frill curve represents the temporal behaviour of pressure, temperature, and concentrations in die case of a very fast relaxation. The other lines illustrate various situations, with 03Xj according to table B2.5.1. 03 is the angular frequency of the sound wave and x is the chemical relaxation time. Adapted from [110].

Figure B2.5.13. Schematic representation of the four different mechanisms of multiphoton excitation (i) direct, (ii) Goeppert-Mayer (iii) quasi-resonant stepwise and (iv) incoherent stepwise. Full lines (right) represent the coupling path between the energy levels and broken arrows the photon energies with angular frequency to (Aco is the frequency width of the excitation light in the case of incoherent excitation), see also [111].

Fig. 5. Langevin trajectories for a harmonic oscillator of angular frequency u = 1 and unit mass simulated by a Verlet-like method (extended to Langevin dynamics) at a timestep of 0.1 (about 1/60 the period) for various 7. Shown for each 7 are plots for position versus time and phase-space diagrams.

The cycle of oscillation is 0 to 2n, precisely the circumference of a circle. After one cycle of 2n radians is complete, another cycle begins, identical to the one before it. The angular frequency in radians co is related to the frequency expressed in units of complete cycles per second v as co = 2nv, whence... 

A 10,0-g mass connected by a spi itig to a statiotiaiy poitit executes exactly 4 complete cycles of harmonic oscillation in 1,00 s. What are the period of oscillation, the frequency, and the angular frequency What is the force constant of the spring ... 

An interesting variation of Raman spectroscopy is coherent anti-Stokes Raman spectroscopy (CARS) (99). If two laser beams, with angular frequencies CO and CO2 are combined in a material, and if cjj — is close to a Raman active frequency of the material, then radiation at a new frequency CJ3 = 2cJ2 — may be produced. Detection of this radiation can be used to characterize the material. Often one input frequency is fixed and the other frequency, from a tunable laser, varied until matches the Raman frequency. CARS has the capabiHty for measurements in flames, plasmas, and... 

Transverse electromagnetic waves propagate in plasmas if their frequency is greater than the plasma frequency. For a given angular frequency, CO, there is a critical density, above which waves do not penetrate a plasma. The propagation of electromagnetic waves in plasmas has many uses, especially as a probe of plasma conditions. 

A viscoelastic material also possesses a complex dynamic viscosity, rj = rj - - iv( and it can be shown that r = G jiuj-, rj = G juj and rj = G ju), where CO is the angular frequency. The parameter Tj is useful for many viscoelastic fluids in that a plot of its absolute value Tj vs angular frequency in radians/s is often numerically similar to a plot of shear viscosity Tj vs shear rate. This correspondence is known as the Cox-Merz empirical relationship. The parameter Tj is called the dynamic viscosity and is related to G the loss modulus the parameter Tj does not deal with viscosity, but is a measure of elasticity. 

The significance of G G tan 5, Tj, and Tj is that they can be determined experimentally and used to characterize real materials. These parameters depend on frequency and temperature, and this dependence can be used to define behavior. For example, viscoelastic fluids are often characterized by log—log plots of one or more of these quantities vs the angular frequency CO, as shown in Figure 21, which illustrates the behavior of a polymer melt (149). 

In a dilatational experiment, where the surface is periodically expanded and contracted, is a function of the angular frequency (co) of the dilatation as ia equation 3 where is the dilatational elasticity and Tj is the dilatational viscosity. 

The Nyquist critical frequency or critical angular frequency is... 

In Eq. (26), M is the hydrogen mass, X labels the mode, is the atomic eigenvector for hydrogen / in mode X, and co, is the mode angular frequency. is the number of quanta of energy Ao>, exchanged between the neutron and mode X. is a modified Bessel function. 

We cannot deal here with the details of the hydrodynamic Navier-Stokes equations and their consequences. For dimensional reasons one can derive the following expression [150] for the thickness of the boundary layer when the crystal rotates with angular frequency u>... 

Equation (4-53) describes the precession of the magnetization vector about the field vector with angular frequency yHo, in the absence of the rotating field W, (see Fig. 4-4A). 

There are many different solutions for X1 and X2 to this pair of coupled equations, but it proves possible to find two particularly simple ones called normal modes of vibration. These have the property that both particles execute simple harmonic motion at the same angular frequency. Not only that, every possible vibrational motion of the two particles can be described in terms of the normal modes, so they are obviously very important. 

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