Constructing separable non-2π-periodic solutions to the Navier-Lamé equation in cylindrical coordinates using the Buchwald representation: Theory and applications
Advances in Applied Mathematics and Mechanics
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In a previous paper (Adv. Appl. Math. Mech., 10 (2018), pp. 1025-1056), we used the Buchwald representation to construct several families of separable cylindrical solutions to the Navier-Lamé equation; these solutions had the property of being 2π-periodic in the circumferential coordinate. In this paper, we extend the analysis and obtain the complementary set of separable solutions whose circumferential parts are elementary 2π-aperiodic functions. Collectively, we construct eighteen distinct families of separable solutions; in each case, the circumferential part of the solution is one of three elementary 2π-aperiodic functions. These solutions are useful for solving a wide variety of dynamical problems that involve cylindrical geometries and for which 2π-periodicity in the angular coordinate is incompatible with the given boundary conditions. As illustrative examples, we show how the obtained solutions can be used to solve certain forced-vibration problems involving open cylindrical shells and open solid cylinders where (by virtue of the boundary conditions) 2π-periodicity in the angular coordinate is inappropriate. As an addendum to our prior work, we also include an illustrative example of a certain type of asymmetric problem that can be solved using the particular 2π-periodic subsolutions that ensue when there is no explicit dependence on the circumferential coordinate.