Abstract
A detailed solution to the transient interaction of plane acoustic waves with a spherical elastic shell was obtained more than a quarter of a century ago based on the classical separation of variables, series expansion, and Laplace transform techniques. An eight-term summation of the time history series was sufficient for the convergence of the shell deflection and strain, and to a lesser degree, the shell velocity. Since then, the results have been used routinely for validation of solution techniques and computer methods for the evaluation of underwater explosion response of submerged structures. By utilizing modern algorithms and exploiting recent advances of computer capacities and floating point mathematics, sufficient terms of the inverse Laplace transform series solution can now be accurately computed. Together with the application of the Cesaro summation using up to 70 terms of the series, two primary deficiencies of the previous solution are now remedied: meaningful time histories of higher time derivative data such as acceleration and pressure are now generated using a sufficient number of terms in the series; and uniform convergence around the discontinuous step wave front is now obtained, completely eradicating spurious oscillations due to the Gibbs' phenomenon. New results of time histories of response items of interest are presented.
Highlights
A detailed solution to the transient interaction of plane acoustic waves with a spherical elastic shell was obtained more than a quarter of a century ago based on the classical separation of variables, series expansion, and Laplace transform techniques (Huang, 1969)
A detailed solution to the transient interaction of plane acoustic waves with a spherical elastic shell was obtained more than a quarter of a century ago, two primary deficiencies existed, namely, meaningful solution histories of the higher time derivatives such as acceleration and pressure were not generated due to an insufficient number of terms in the series solution, and uniform convergence around the discontinuous step wave front was not obtained because of the Gibbs' phenomenon
These two deficiencies have been remedied with the advent and availability of more powerful computers and increased sophistication of computational algorithms
Summary
A detailed solution to the transient interaction of plane acoustic waves with a spherical elastic shell was obtained more than a quarter of a century ago based on the classical separation of variables, series expansion, and Laplace transform techniques (Huang, 1969). The asymptotic solution, exact within shell theory for all shell response items of the neighborhood of the first point ofimpact at very early time (roughly one twentieth of the time for the incident wave front to traverse the diameter of the shell), was previously found by Milenkovic and Raynor (1966) using the approach of geometric acoustics, and by Tang and Yen (1970) using Watson's transformation of the modal series together with the steepest descent method Because these results are for very early time, they are not suitable, and seldom used, for the validation of shell response results obtainable by other methods. Applying the Laplace transform with respect to T with s as the transform parameter, solving the wave equation for IT sra and the shell equation of motion for the deflections satisfying the initial and boundary conditions, the Laplace transformed solution of the modal radial deflection of the shell is obtained (Huang, 1969)
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