Waterman and Truell [1] showed that the (complex) index of refraction of the effective medium is a simple function of the forward scattered amplitude of a single inclusion in the weak scattering regime. This amplitude can be determined in an exact manner for inclusions of simple form, so that one can compute, without difficulty, the real and imaginary parts of the effective index of refraction. These two functions do not generally obey, as they should, the Kramers-Kronig formulas. One can retain only the imaginary part (related to the attenuation) and determine the real part (related to the phase velocity) by means of one of the K-K formulas. This computation is carried out for a fibrous composite, submitted to both P and S polarized electromagnetic waves, over a wide range of frequencies. It is shown that the W-T formula provides predictions of the effective phase velocity that are consistent with the K-K relations only when the refractive index of the inclusions is larger than that of the host. Otherwise, consistency is only obtained at low frequencies and/or for very low inclusion concentrations.