Wave number vector

Interpreting pjh as the wave number vector, we see that this is a wave packet function, and that

is the amplitude of the wave at the wave number p/h. Note that

and can also be regarded as the Fourier transforms of each other.8... 

The group velocity for multi-dimensional problem is a vector, decided by the variation of circular frequency with the wave number vector. It is only the real part of Eqn. (1.4.8) that is termed the group velocity - as discussed in Whitham (1978). Thus, this is the velocity at which the energy of a group of waves travel, centered about the middle of the wave number group. It is noted that the one-dimensional wave given by Eqn. (1.4.3), is non-dispersive with Vg = c. 

Thus, for a given shear at the interface given by, U — U2) and for a given oblique disturbance propagation direction at the interface indicated by the wave number vector k, instability would occur for all wave numbers k, given by... 

Note that the wave number vector makes an angle 7 with the x-axis, such that C0S7 = "Pr and the above condition can be conveniently written as,... 

Wave number, wave number vector and wave angular frequency are denoted k, k and co. The wave energy spectrum (divided by the water den-... 

Assuming that the wave number vector k = k/, Eq. (1.4) becomes the general formula of a plane wave. As defined, the wave number vector k indicates that the direction of the vector k is the propagation direction of the plane wave. 

When calculating the concentrational fluctuation spectral density, we must take into account that 1) fluctuations are isotropic so that spectral density can depend on the modulus but not on the direction of the wave-number vector 2) particles are solid spheres of finite volume and 3) decay of spontaneously originated fluctuations is governed by particle diffusion. As a result, we obtain the following formula ... 

It is easily understood that all spectral densities calculated in conformity with Equation 8.3 are proportional to the concentrational fluctuation spectral density. The proportionality coefficients depend on both the fluctuation frequency and the wave-number vector. 

Consider a monochromatic beam of electromagnetic waves with a given wave number vector ko (ko is a vector in the direction of propagation of the plane wave with absolute magnitude ko = 27t/A, where X is the wavelength), scattered from a system of N molecules. The scattered wave number vector is denoted by k. Two important assumptions are made regarding this process. (1) The incident and the scattered beam have the same wavelength, i.e.,... 

Since in many cases the velocity Vp is not sufficiently large to make a large shift, the interference of scattered lights from two laser beams at an angle reflected lights, which result in two different Doppler shifts, Aft>i and Aft>2. Then, subtracting Aft>2 from Aft) , we obtain the frequency of the interference, /d, called the Doppler frequency, which can be expressed as a function of the angle 0 by... 

The problem can be reduced to calculation of normal vibrational modes of a linear chain. The vibration frequency of such a chain is a periodic function of the wave number vector k ... 

The name recombination centres given to such states reflects the fact that they also accelerate the back reaction, that is electron-hole recombination. Their eflicacy depends on the fact that they make possible indirect transitions (i.e. transitions with change of the wave number vector, see Chapter 2) by taking up energy and momentum. In this way the establishing of the local equilibrium is accelerated [128, 129]. 

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