Spatial attenuation of bending waves in the infinite plates coupled by a viscous damping layer

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Plate vibrations can be reduced by a viscous damping layer between a primary excited plate and a secondary plate, of which frequency responses are considered by Ingard and Akay [J. Vib., Struct. Rel. Design 108, 178–184 (1987)]. For a free vibration, complex wave numbers are determined from a dispersion relation, where the imaginary part is related with the attenuation due to the damping layer. The steady state responses both in the plates and fluid layer are calculated when the primary plate is under a time harmonic line-driven force. Energy and power flow of a complex wave number are also considered. The complex wave number in a vibrating plate increases as the thickness of the damping layer decreases, such that the attenuation is greater and the period in space is shorter, and the waves in a damping layer become unstable. For a given thickness of a damping layer, as the exciting frequency is higher, the damping effects are weaker, since the boundary layer thickness becomes thinner. When a secondary plate of a same material is attached to attenuate the vibration of a primary plate, the secondary plate must be thicker than the primary plate to achieve an efficient vibration reduction. The bending waves in the primary plate are attenuated very fast, and become identical with the bending waves in the secondary plate at far field.