On a check on the accuracy of Timoshenko's beam theory

In a recent article, Rneton (1) presented a comparison between the standing wave natural frequency predictions of Timoshenko beam theory (TBT) for a long beam of thin rectangular cross-section, when the flexural mode is sinusoidal in the axial co-ordinate x, and those of a plane stress elastodynamic solution; the latter may be regarded as the exact benchmark. Renton's main concern was the accuracy of the lower of the two frequency predictions of TBT (TBT1), although the results presented in Figure 1 of reference (1) also show the frequency prediction of the so-called second spectrum (TBT2), and re-presents evidence that this mode should be disregarded