The time behavior of the streaming potential is identical to that of differential fluid pressure when local laminar flow is maintained and electromagnetic relaxation is not important. The time evolution of the streaming potential waveform resulting from the application of a pressure pulse to one face of a cylindrical water-saturated porous structure is found to depend on the elastic properties of the fluid and rock frame and on the porosity and permeability of the core. The analysis of this quasi-static flow based on Biot’s soil consolidation work [Biot, J. Appl. Phys. 12, 155 (1941)] and advanced by J.R. Rice and M.P. Cleary [Rev. Geophys. Space Phys. 14, 227 (1976)] is completely equivalent to that obtained from Biot’s slow wave model [M.A. Biot, J. Acoust. Soc. Am. 28, 168 (1956)] in the limit of zero frequency. Both analyses result in a homogeneous diffusion equation in pore pressure (and streaming potential) where the diffusivity is expressed in terms of the bulk moduli of the fluid and solid constituents, bulk and shear moduli of the solid frame, fluid viscosity, porosity, and permeability. The predicted streaming potential transient response is in excellent agreement with experimental observations, permitting, for example, the extraction of sample permeability. Models assuming an incompressible frame are shown to be grossly inadequate in describing the temporal features of quasi-static flow.