Propagation in an Infinite Elastic Plate

At any single frequency an arbitrary force distribution in an infinite elastic plate must be expressible as a superposition of the plate's eigenvibrations. Since the force distribution may be varied in an infinite number of ways it follows that the plate must possess an infinity of eigenvibrations. Previous work has shown the existence of only a finite number, each being defined by a real propagation constant (unattenuated vibration: finite phase velocity) or an imaginary propagation constant (exponentially attenuated vibration: infinite phase velocity). This investigation yields the complementary infinity of eigenvibrations possessing complex propagation constants (exponentially attenuated vibrations: finite phase velocities). The probable structure of the complete eigenmode system in the low-frequency range has been hypothesized for a plate material with a Poisson's ratio of 0.31. The qualitative variation of this structure with changes in Poisson's ratio may be readily conjectured. The utility of the complex eigenvibrations is enhanced by transforming them into an equivalent set, each member of which is constant in phase throughout the plate. These reformulated eigenvibrations are unique in that they possess an exponential amplitude variation along the plate together with a superimposed sinusoidal variation.