### ASA 126th Meeting Denver 1993 October 4-8

## 3aSA1. Reciprocity theorem for oscillations of inhomogeneous elastic
structures in a liquid---Modified reciprocity theorem and applications.

**Leonid M. Lyamshev
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*N. N. Andreev Acoustical Institute, Shvernik str. 4, 117036 Moscow,
Russia
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The oscillatory vibrations of inhomogeneous elastic structures are
considered. In particular, two problems involving plates and shells with
attached ribs excited by forces and moments are discussed. The formulation uses
the Green's function for the wave equation of sound propagation in a liquid,
and the equations of motion of plates and shells coupled with those of flexural
and torsional vibrations of ribs. An integral relationship connecting the
solutions of two self-conjugated boundary problems of inhomogeneous plates and
shells oscillations theory is obtained. This relationship is the reciprocity
theorem that is a precise mathematical formulation of the well-known Rayleigh
reciprocity principle for oscillating mechanical systems. If a liquid is moving
and there is a shear flow outside or inside of a shell (plate) then this
relationship is the modified reciprocity theorem for inhomogeneous elastic
structures oscillations in a flow that is the generalization of the Rayleigh
principle. Applications of the reciprocity theorems are discussed. Flexural
waves reflection, scattering, and radiation by elastic ribs in plates are
considered. Nonspecular reflections and resonance phenomena are observed. These
are similar to nonspecular reflection, sound scattering, and radiation by
elastic bodies in a liquid and are connected with interactions of flexural
waves in a plate with flexural and rotational waves in ribs.