Analytical representations for the transient admittance of a plate vibrating in a medium

Flexural vibrations of a plate contacting on one side with an ideal compressible liquid are considered. The plate is driven by a harmonic force uniformly distributed along a straight line. The transient admittance of the plate as a function of the distance from the line of the force application is shown to be representable as a sum of an integer function and an integer function multiplied by a logarithmic function. A procedure for determining the power series expansions of these functions is described, and the initial terms of the expansions are derived. The approximations formed by these initial terms and the asymptotic expansion at infinity are compared with the results of numerical calculations for several particular values of the parameters. Vibrations of a liquid with an impedance load at its surface are considered as an auxiliary problem, and, in the framework of this problem, the initial terms of the power series expansions of the integer functions, which appear in the expression for the transient admittance, are determined. The expansions obtained make it possible to raise the speed of the admittance calculations near the points of application of the driving force.