The chapter discusses the transverse vibrations of bars with uniform cross-sections, focusing on the corrections needed in the equation for transverse vibrations to account for "rotatory inertia" and shear deflection. The author, Prof. S. P. Timoshenko, presents an exact solution for a beam of rectangular section, where the breadth is either much larger or much smaller than the depth, leading to plane strain or plane stress conditions. The solutions are compared with previous work, confirming the importance of shear correction over rotatory inertia when the wavelength is large compared to the cross-sectional dimensions. Key points include: 1. The correction for shear is found to be four times more significant than the correction for rotatory inertia. 2. The frequency equation for the vibrations is derived, showing that the velocity of waves (V) is inversely proportional to the length (l) for long waves. 3. Approximate solutions for the frequency are provided, and the results are validated through comparisons with exact solutions for different cross-sections, such as circular sections. 4. The approximate formula derived in the earlier paper is shown to be accurate for various cross-sections, including circular sections. The chapter also touches on the application of these corrections to other shapes of cross-sections and the potential for extending the methods to longitudinal vibrations.The chapter discusses the transverse vibrations of bars with uniform cross-sections, focusing on the corrections needed in the equation for transverse vibrations to account for "rotatory inertia" and shear deflection. The author, Prof. S. P. Timoshenko, presents an exact solution for a beam of rectangular section, where the breadth is either much larger or much smaller than the depth, leading to plane strain or plane stress conditions. The solutions are compared with previous work, confirming the importance of shear correction over rotatory inertia when the wavelength is large compared to the cross-sectional dimensions. Key points include: 1. The correction for shear is found to be four times more significant than the correction for rotatory inertia. 2. The frequency equation for the vibrations is derived, showing that the velocity of waves (V) is inversely proportional to the length (l) for long waves. 3. Approximate solutions for the frequency are provided, and the results are validated through comparisons with exact solutions for different cross-sections, such as circular sections. 4. The approximate formula derived in the earlier paper is shown to be accurate for various cross-sections, including circular sections. The chapter also touches on the application of these corrections to other shapes of cross-sections and the potential for extending the methods to longitudinal vibrations.