# Stationary viscoelastic wave fields generated by scalar wave functions

## TL;DR

This paper introduces a refined Helmholtz decomposition for vector fields in viscoelastic media, showing that divergence-free parts depend on only two scalar functions, which simplifies solving boundary value problems in physics.

## Contribution

It presents a new decomposition of divergence-free vector fields into polarized components and demonstrates the completeness of this representation for stationary wave fields in viscoelastic media.

## Key findings

- Each scalar function satisfies a Helmholtz equation.
- The decomposition simplifies boundary value problem solutions.
- Application to torsional deformation of cylindrical viscoelastic media.

## Abstract

The usual Helmholtz decomposition gives a decomposition of any vector valued function into a sum of gradient of a scalar function and rotation of a vector valued function under some mild condition. In this paper we show that the vector valued function of the second term i.e. the divergence free part of this decomposition can be further decomposed into a sum of a vector valued function polarized in one component and the rotation of a vector valued function also polarized in the same component. Hence the divergence free part only depends on two scalar functions. Further we show the so called completeness of representation associated to this decomposition for the stationary wave field of a homogeneous, isotropic viscoelastic medium. That is by applying this decomposition to this wave field, we can show that each of these three scalar functions satisfies a Helmholtz equation. Our completeness of representation is useful for solving boundary value problem in a cylindrical domain for several partial differential equations of systems in mathematical physics such as stationary isotropic homogeneous elastic/viscoelastic equations of system and stationary isotropic homogeneous Maxwell equations of system. As an example, by using this completeness of representation, we give the solution formula for torsional deformation of a pendulum of cylindrical shaped homogeneous isotropic viscoelastic medium.