Wavenumber vector

The Fourier transform introduces the wavenumber vector , which has units of 1 /length. Note that, from its definition, the velocity spatial correlation function is related to the Reynolds stresses by... 

Figure 7.2 Coordinates for a plane wave. Having defined a reference point O (origin), an arbitrary spatial point of the wave travelling with wavenumber vector k into the K-direction is indicated by r. Cases (a) and (b) differ in the direction of the selected reference axis z (the quantization axis) which does or does not coincide with the direction of the wave, respectively. In case (c) the spin of an electron wave is also indicated it is shown as the double arrow pointing into the direction e against which the spin projection is assumed to...

The summations over Mf and ms are needed because no observation is made with respect to these final-state quantum numbers. ka and Kph are the wavenumber vectors of the Auger electron and the photoelectron, the minus sign indicates the correct asymptotic boundary condition for the wavefunctions, Vc is the Coulomb interaction between the electrons causing the Auger transition, and is the dipole operator causing the photoionization process. 

This type of gauge is called the Coulomb gauge. It reflects the condition of independence between the two components of the wave, since A oscillates in directions perpendicular to the wavenumber vector from Eq. (1.74). This corresponds to the introduction of the linearly polarized light as is shown in Table 1.4. 

The term ku represents a Doppler shift in the direction of the mean flow i/. In a frame of reference moving with the mean wind, the phase front propagates unifomily in the direction of the wavenumber vector k = lA + jl + kw with phase speed c = Cs = JyRt. This ideal sound wave is dynamically nondispersive, which means the phase speed is independent of the wavelength however, the wave energy certainly disperses geometrically flom a local source. 

By choosing the direction of r in parallel with the rf-th component of the wavenumber vector k, the correlation kernel is expressed as... 

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