As mentioned earlier, the effective seismic input method is derived by viewing soil-structure interaction as a scattering problem, wherein the presence of the dam causes the scattering of the free-field ground motion in the linear foundation domain. This is entirely analogous to acoustic scattering, wherein acoustic waves in a linear acoustic fluid are made to scatter by the presence of a solid body. In both cases, the waves in a linear background medium — the soil foundation or the acoustic fluid — are scattered by a solid structure placed in the free-field, and the solution is formulated by considering the scattered motion in the background medium, which replaces a distant excitation source with equivalent effective forces at the interface with the structure.
This shows that this scattered-motion approach to solving a scattering problem depends only on the linearity of the background medium, and not on the particular physical properties of the medium. Therefore, for dam analysis, we can consider the water domain and the foundation rock together as one free-field background domain, and the dam as the structure. However, we know the free-field ground motion in the foundation rock only, not in the water, so to determine that, we need an initial auxiliary analysis where the water domain plays the part of the structure. This results in a two-step analysis procedure, where we first analyze the auxiliary water-foundation rock to compute the free-field motion in the water, and then use this in the analysis of the whole dam-water-rock system.
Note that there is a physical significance to the auxiliary system, even though it is not a physical entity by itself. The earthquake ground motion affects the dam not only directly through its base, but also through pressure waves in the water, excited by the earthquake at the reservoir bottom. The auxiliary system brings in the effect of the far-field pressure waves in the final analysis of the whole system without needing to model a large length of the reservoir.