Energy flux of time-harmonic waves in anisotropic dissipative media

Vlastislav Cerveny


The energy flux of time-harmonic waves propagating in anisotropic dissipative media may be introduced in several ways. It is most common to consider the average energy flux, which is real-valued and time-independent. An extension of this definition is the complex Poynting vector, which is also time-independent, but complex-valued. The real part of it equals the average energy flux. Recently, also an instantaneous energy flux, which is complex-valued and time-harmonic, has found interesting applications. In this article, all these three energy flux quantities are investigated for general anisotropic dissipative media, mainly in relation to the reflection/transmission problem. Energy flux quantities related to the summary wavefield, composed of several waves, are derived. It is shown that, in a dissipative medium, also the interaction energy contributions must be considered in the summary energy flux in addition to the energy fluxes of the individual waves. The interaction energy fluxes in anisotropic dissipative media are derived and discussed in detail. In non-dissipative media, the interaction contributions vanish. It is proved that the interaction contributions, corresponding to the summary vanish even in dissipative media. They, however, play an important role in the summary average energy flux and in the summary complex Poynting vector. Using the results derived for energy fluxes of summary waves, the energy balance at a structural interface between two anisotropic dissipative media is studied. In this case the conservation laws for the complex Poynting vector and for the average energy flux contain interaction terms. In the R/T problem, at an interface between two anisotropic dissipative media, the number of interaction terms may reach eighteen. The conservation laws for the instantaneous energy fluxes, however, do not contain any interaction terms; they contain only the instantaneous energy fluxes of the individual waves (incident, reflected, transmitted). Using the conservation laws, the relevant energy-based R/T coefficients may be introduced. The most important role is played by the instantaneous energy flux R/T coefficients. They include the appropriate phase shifts, needed in the seismic ray theory. The complex Poyinting vector and average energy flux energy R/T coefficients are not as important, as they do not contain any phase information. The instantaneous energy flux R/T coefficients are closely connected with the energy normalized R/T coefficients, used broadly in the seismic ray method. All results derived here for anisotropic dissipative media are specified also for isotropic dissipative media. They remain valid even for non-dissipative media.

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In: Seismic Waves in Complex 3-D Structures, Report 11, pp. 301-314, Dep. Geophys., Charles Univ., Prague, 2001.
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