The density perturbation is known as the acoustic approximation . A fluid has a pressure which is a function of density, temperature, and gravity forces. We shall assume that the gravity forces are relatively constant over the domain and do not exert any differential force on the fluid. The earth temperature field does vary as a function of position and in time, but the temperature changes are very small compared to the time it takes to make seismic measurements. Hence, we will assume that only the density is important and that the stress within the fluid is related to the strain as a function of density.
First, is defined as the Kronecker delta function,
The stress matrix is