This paper presents a family of exact solutions for quasi-one-dimensional, transient acoustic wave propagation in ducts with mean temperature and area variations in the absence of mean flow. These solutions are obtained using a transformation of the spatial and acoustic variables in a manner suggested by the WKB approximation. Exact travelling wave-type solutions are obtained for a class of temperature and area profiles. These solutions differ from the classical travelling wave solution, however, in that the acoustic pressure and velocity are not algebraically related by the local value of the acoustic impedance, ρ̄(x)c̄(x). Although these solutions resemble the approximate, "high frequency", WKB form of solution of the wave equation, they have the interesting property that they are exact, regardless of the scale of the acoustic disturbance relative to that of the inhomogeneity. © 2001 Academic Press.