A class of irrotational, isentropic, and compressible flows is studied theoretically by formulating the density and the velocity potential in a Madelung transformation. The resulting nonlinear Schrödinger equation is solved in terms of similarity variables. One particular family of exact solutions, valid for any ratio of the specific heat capacities of the gas, permits explicit expressions of the fluid properties and velocities in terms of time and spatial coordinates. Analytically, the density is a Gaussian function of the similarity variable, while the temperature is a function of time only. This method is applicable in one (1D), two, and three dimensional geometries. As a simple example, a 1D gas column, with mass injection on one side and a steadily translating wall on the other, can be formulated exactly. The connection with the evolution of an unsteady velocity potential will also be examined.