This article is concerned with longitudinal wave propagation in an isotropic thermoelastic material that is nearly incompressible at either uniform temperature or uniform entropy. A dimensionless effective bulk modulus X is defined such that X-+ oo corresponds to incompressibility at uniform temperature. For 0 < X < 1, both longitudinal waves are stable, but for X > 1, one is stable and the other unstable. If X = 1, there is only one propagating mode, and this is stable. Another dimensionless effective bulk modulus X is defined such that X -+ oo corresponds to incompressibility at uniform entropy. For 0 < X < oo, both longitudinal waves are stable. Making the identification X = x/(1-X), among others, enables the equivalence of the two types of near constraint to be demonstrated. In particular, X -* oo corresponds to-+ 1-, so that incompressibility at uniform entropy corresponds to near incompressibility at constant temperature. Many graphical results are presented to illustrate various points of theory.