Flexural waves on narrow plates

Flexural wave speeds on beams or plates depend upon the bending stiffnesses which differ by the well-known factor $(1−ν2).$ A quantitative analysis of a plate of finite lateral width displays the plate-to-beam transition, and permits asymptotic analysis that shows the leading order dependence on the width. Orthotropic plates are analyzed using both the Kirchhoff and Kirchhoff–Rayleigh theories, and isotropic plates are considered for Mindlin’s theory with and without rotational inertia. A frequency-dependent Young’s modulus for beams or strips of finite width is suggested, although the form of the correction to the modulus is not unique and depends on the theory used. The sign of the correction for the Kirchhoff theory is opposite to that for the Mindlin theory. These results indicate that the different plate and beam theories can produce quite distinct behavior. This divergence in predictions is further illustrated by comparison of the speeds for antisymmetric flexural, or torsional, modes on narrow plates. The four classical theories predict limiting wave speeds as the plate width vanishes, but the values are different in each case. The deviations can be understood in terms of torsional waves and how each theory succeeds, or fails, in approximating the effect of torsion. Dispersion equations are also derived, some for the first time, for the flexural edge wave in each of the four “engineering” theories.