flexural deformations that are propagated in bars and plates. The length of flexural waves λ is always much greater than the thickness of the bar or plate. If the wavelength becomes comparable to the thickness of the plate, then motion in the wave becomes more complex and the wave is no longer said to be flexural. Examples of flexural waves are waves in a tuning fork, in the sounding boards of musical instruments, and in the exit cones of loudspeakers and those that arise during vibration of thin-wall mechanical structures (the frames of airplanes and motor vehicles, the roofs and walls of buildings, and other objects). Traveling flexural waves arise in very long bars and in large plates. On propagation of flexural waves each element of the bar is shifted perpendicular to the axis of the bar or to the plane of the plate. Dispersion is characteristic of flexural waves. The phase velocity of monochromatic flexural waves is proportional to the square root of the frequency. The group velocity of flexural waves is equal to twice the phase velocity. In bars and plates whose dimensions are limited in the direction of propagation of flexural waves, standing flexural waves arise as a result of reflections from the ends. Flexural waves are possible not only in flat but also in curved plates (shells).
I. A. VIKTOTOV