Hamiltonian extension and eigenfunctions for a time dispersive dissipative string

TL;DR

This paper develops a Hamiltonian framework for a time dispersive dissipative string, explicitly constructing its extended system, deriving energy and momentum densities, and analyzing eigenmodes to better understand scattering and wave behavior.

Contribution

It introduces a Hamiltonian extension of the TDD string with hidden strings, providing explicit formulas and eigenmodes, advancing the theoretical understanding of dispersive dissipative systems.

Findings

Explicit Hamiltonian extension with hidden strings

Formulas for energy and momentum densities

Eigenmode expansion for the extended system

Abstract

We carry out a detailed analysis of a time dispersive dissipative (TDD) string, using our recently developed conservative and Hamiltonian extensions of TDD systems. This analysis of the TDD string includes, in particular: (i) an explicit construction of its conservative Hamiltonian extension, consisting of the physical string coupled to "hidden strings;" (ii) explicit formulas for energy and momentum densities in the extended system, providing a transparent physical picture accounting precisely for the dispersion and dissipation; (iii) the eigenmodes for the extended string system, which provide an eigenmode expansion for solutions to the TDD wave equation governing the TDD string. In particular, we find that in an eigenmode for the extended system the displacement of the physical string does not satisfy the formal eigenvalue problem, but rather an equation with a source term resulting from the excitation of the hidden strings. The obtained results provide a solid basis for the rigorous treatment of the long standing problem of scattering by a TDD scatterer, illustrated here by the computation of scattering states for a string with dissipation restricted to a half line.