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We derive an approximate analytic model for the eigenfunctions of underspread LTV channels which plays a fundamental role in the analysis/synthesis of the optimal precoding that maximizes the information rate in transmissions over linear time-varying (LTV) channels. We assume that channel status information is available to both receiver and transmitter and that the channel impulse response can be predicted exactly, within a finite duration interval, from its past evolution. More specifically, we use our analytic model to prove that the optimal precoding is equivalent to applying the water-filling principle in the time-frequency domain and to simplify the computation of the optimal precoder. We include numerical results to support the validity of our model.