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A wave mechanics approach is used to solve the Timoshenko beam equation, revealing that two waves exist. One is called the s a -wave and the other the s b -wave. These two waves are found to be the basic constituent components of the mode shapes of the beam. An experiment was carried out and the measured mode shapes of a free-free beam are shown to consist of one s a -wave and one s b -wave in superposition for each of the modes. The measured s a -wave and s b -wave exhibit the Rayleigh-Lamb first (with anomalous dispersion) and Rayleigh-Lamb second (with normal dispersion) flexural modes, respectively. The issue of the second spectrum is addressed and it is shown that, within the measurable frequency range, Rayleigh-Lamb second flexural modes are present in the free-free beam. The s b -wave is identified as the secondspectrum mode. The role of shear deformation is also investigated in explaining the basic difference in the behaviour of the s a -and s b -waves. This paper also contributes to a physical interpretation of the hyperbolic functions in the classical solution of beam vibration problems.

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