Energy dissipation in flexural vibrations is considered by introducing into the Timoshenko beam equation viscous-damping terms proportional to the time rate of extensional strain. Two modes of wave transmission arise from the roots of a quartic equation. On the lower mode, a solution is indicated in terms of characteristic functions which represent the damped natural vibrations. For a specified range of values of damping, the amplitudes of natural vibrations are governed by characteristic damping factors which exhibit a maximum in the region of frequencies where wave velocities are dispersed. There can be no frequency cut-offs. In the higher mode of wave motion, the characteristic damping factors exhibit a monotone increase toward short wave-length limits analogous to those obtained in the Bernoulli-Euler-Sezawa equation.