Lossless propagation of one-dimensional, finite-amplitude sound waves exhibits a new mathematical technique for solving the finite-amplitude-wave problem. It involves expanding the wave equation's solution in a Taylor series about the boundary, followed by use of the differential equation to replace the Taylor-series derivatives by derivatives of the boundary function. This leads to a result that is formally identical with the Lagrange expansion formula and, thereby, to stability criteria. Two examples are included, one of which reproduces a classical result, while the other one illustrates the commencement of asymmetries in a propagating, finite-amplitude wave.

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The Ehrenfest model has been used to explain the ``irreversibility'' of thermodynamics and statistical mechanics. The modification described in this paper allows transitions to occur in both directions between the two ``boxes'' at each step of the model procedure. The equilibrium probability distribution is given in the form of a finite product, or in an iterated form particularly suitable for machine calculation. The analysis is illustrated by a simple model of an ionization-recombination process.

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