1. Scattering by an Inhomogeneous Solid. II. The Correlation Function and Its Application
2. The volume average 〈f〉 of a functionfis defined as 1Vvf(r) dr, where the volumeVcovered by the integration is sufficiently large to extend over a representative sample of the porous material.
3. Absence of long-range order is assumed.
4. The variational principle which is being used here is not the familiar Helmholtz principle but rather its reciprocal: Eq. (5c), which would have been the Euler equation for the Helmholtz principle, appears here as a subsidiary condition. The Euler equation corresponding to (4) and (5) requires that σij be the symmetrized gradient of a divergence-free vector (i.e., the local fluid velocity); this requirement would have been a subsidiary condition in the case of the Helmholtz principle. The two principles are related in the same manner as the minimum potential energy and Castigliano principles in elasticity theory [see R. Courant and D. Hilbert,Methods of Mathematical Physics(Interscience Publishers, Inc., New York, 1953), Vol. I, Chap. IV, pp. 268–272].
5. The summation convention will be used throughout, i.e., it will be understood that repeated subscripts occurring in a single tensor or a product of tensors are to be summed over all coordinate directions; thus, σll represents σ11+σ22+σ33, or the trace of the stress deviation.