An acoustic acceleration wave is defined as a propagating singular surface (i.e., wavefront) across which the first derivatives of the velocity, pressure, or density exhibit jumps. In this talk, the temporal evolution of the amplitude and the propagation speed of such waves are investigated in the context of nonlinear acoustic propagation in rigid porous media. By considering the exact conservation/balance equations, it is shown that there exists a critical value, the constant α*(>0), of the initial jump amplitude such that the acceleration wave magnitude either goes to zero, as t→∞, or blows up, in finite time, depending on whether the initial jump amplitude is less than or greater than α*. In addition, stability is addressed; a connection to traveling wave phenomena is noted, for which an exact traveling wave solution is obtained; and a comparison with the linearized case, i.e., the well-known damped wave equation, is also presented. Finally, the numerical solution of an idealized, nonlinear initial-boundary value problem involving sinusoidal signaling in a fluid-saturated porous slab is used to illustrate the finite-time transition from acceleration to shock wave, which occurs when the initial jump amplitude exceeds α*. [Work supported by ONR/NRL funding.]