# Solving the Navier-Lame Equation in Cylindrical Coordinates Using the Buchwald Representation: Some Parametric Solutions with Applications

## TL;DR

This paper develops parametric solutions to the Navier-Lame equation in cylindrical coordinates using the Buchwald representation, enabling efficient solutions for elastic boundary value problems with physical parameters.

## Contribution

It introduces a method to decouple and solve the Buchwald potentials, providing new parametric solutions applicable to cylindrical elastic problems.

## Key findings

- Constructed three families of particular solutions with 2pi-periodic angular parts.
- Demonstrated application to a cylinder under sinusoidal surface pressure.
- Showed how solutions can be used for efficient problem-solving in elasticity.

## Abstract

Using a separable Buchwald representation in cylindrical coordinates, we show how under certain conditions the coupled equations of motion governing the Buchwald potentials can be decoupled and then solved using well-known techniques from the theory of PDEs. Under these conditions, we then construct three parametrized families of particular solutions to the Navier-Lame equation in cylindrical coordinates. In this paper, we specifically construct solutions having 2pi-periodic angular parts. These particular solutions can be directly applied to a fundamental set of linear elastic boundary value problems in cylindrical coordinates and are especially suited to problems involving one or more physical parameters. As an illustrative example, we consider the problem of determining the response of a solid elastic cylinder subjected to a time-harmonic surface pressure that varies sinusoidally along its axis, and we demonstrate how the obtained parametric solutions can be used to efficiently construct an exact solution to this problem. We also briefly consider applications to some related forced-relaxation type problems.