The Telegrapher's equations for a general nonuniform transmission line (NUTL) is analytically solved when the nonuniformities are of sub-wavelength scale. The proposed solution is based on an approximation of the chronological ordering operator for the Telegrapher's equations in an arbitrary NUTL. The proposed approximation is quite accurate when the transmission line has sub-wavelength nonuniformities. The scale of transmission line nonuniformity is assessed by using the concept of minimum resolvable length of nonuniformity (MRLN), which is based on the physically intuitive idea that electromagnetic waves are not affected by spectrally mild nonuniformities. The MRLN is inferred by using the spatial Fourier transform of the characteristics impedance of the NUTL. The proposed formulation is verified by using the measured scattering parameters of three different NUTLs.