# Constructing Separable Non-$2\pi$-Periodic Solutions to the Navier-Lam\'{e} Equation in Cylindrical Coordinates Using the Buchwald Representation: Theory and Applications

## TL;DR

This paper extends previous work on cylindrical solutions to the Navier-Lame equation by constructing 18 new families of separable solutions with $2\pi$-aperiodic circumferential parts, useful for boundary conditions incompatible with $2\\pi$-periodicity.

## Contribution

It introduces a comprehensive set of separable solutions with elementary $2\\pi$-aperiodic functions, broadening the applicability of solutions in cylindrical elasticity problems.

## Key findings

- Constructed 18 families of $2\\pi$-aperiodic solutions.
- Demonstrated applications in solving boundary-value problems with non-periodic boundary conditions.
- Provided examples for open cylindrical shells and asymmetric problems.

## Abstract

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-Lam\'{e} equation; these solutions had the property of being $2\pi$-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 $2\pi$-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 $2\pi$-aperiodic functions. These solutions are useful for solving a wide variety of dynamical problems that involve cylindrical geometries and for which $2\pi$-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) $2\pi$-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 $2\pi$-periodic subsolutions that ensue when there is no explicit dependence on the circumferential coordinate.