Constitutive equations and reciprocity

The differential equations that must be satisfied for most fields (physical variables) in geophysics are determined primarily by conservation equation s which relate the divergence of the flux of the field, the field's time ra te of change, and its sources and sinks. These conservation (or equilibrium ) equations do not provide sufficient constraints to determine the fluxes a nd fields even when boundary conditions (both in space and time) are specif ied. To constrain the fields completely, it is necessary to introduce the p roperties of the media; that is, the constitutive equations. Because the co nservation equations can be determined without considering the properties o f the media, these equations are valid for the most general media; that is, heterogeneous, anisotropic, time-varying, nonlinear, etc. media in which m any held variables can interact or be coupled. When the fields can be descr ibed by 'self-adjoint' differential equations in space-time, these media ex hibit reciprocity; that is, upon interchange of 'sources' and 'detectors', the same result is obtained. We show that viscoelastic, elastodynamic probl ems relating to generalized Kelvin-Voigt and generalized Maxwell media sati sfy the conditions for reciprocity. In addition, we show that the introduct ion of tensor 'densities' (which relate the inertial-force density to the p article-acceleration, particle-velocity and particle-displacement vectors i n the inertial force's constitutive equation) do not invalidate the recipro city conditions. The two constitutive equations (the stress/strain and the inertial-force ones) lead to dispersion and attenuation in the propagation of the fields even though none of the material constants in the constitutiv e or conservation equations is complex (i.e. with real and imaginary parts) . Complex material properties cannot exist in nature for actual materials o r media; nor can the material constants or properties be functions of frequ ency. However, 'apparent' or 'equivalent' properties may be complex and fun ctions of frequency if time-harmonic fields are assumed to exist.