An approximation (the linear version of Burgers' equation with appropriate initial data) to a simple wave initial value problem for a set of two linear coupled dissipative partial differential equations is discussed. It has been shown that for the class of square integrable initial functions of which the spectra (Fourier-transforms) have bounded support 2 the approximation is valid for some finite interval of time [0, T()]. For some finite timeT 1 > T() the approximation may fail. However, fort, it is asymptotically valid again. For the class of initial conditions mentioned above expansions in series of the two solutions, which for every finite interval of time [0, ] are convergent, may be constructed.