The propagation of slow magnetohydrodynamic waves in vertical thin flux tubes embedded in a vertically stratified plasma in the presence of viscosity is shown here to be governed by the Klein-Gordon-Burgers (KGB) equation, which is solved in two limiting cases assuming an isothermal medium in hydrostatic equilibrium surrounded by a quiescent environment. The results presented here can be applied to, e.g., study the propagation of slow magnetohydrodynamic waves generated by the granular buffeting motion in thin magnetic photospheric tubes. When the variation in the reduced velocity occurs over typical lengths much larger than the gravitational scale height, the KGB equation can be reduced to a Klein-Gordon equation describing the propagation of an impulse followed by a wake oscillating with the frequency reduced by viscosity and the solution has no spatial or temporal decay. However, in the other limiting case, i.e., typical variations in the reduced velocity occur over characteristic lengths much smaller than the gravitational scale height, waves have a temporal and spatial decay.