Diffusive systems coupled to an oscillator: a Hamiltonian formulation

The aim of this paper is to study a conservative wave equation coupled to a diffusion equation : this coupled system naturally arises in musical acoustics when viscous and thermal effects at the wall of the duct of a wind instrument are taken into account. The resulting equation, known as Webster-Lokshin model, has variable coefficients in space, and a fractional derivative in time. The port-Hamiltonian formalism proves adequate to reformulate this coupled system, and could enable another well-posedness analysis, using classical results from port-Hamiltonian systems theory. First, an equivalent formulation of fractional derivatives is obtained thanks to so-called diffusive representations: this is the reason why we first concentrate on rewriting these diffusive representations into the port-Hamiltonian formalism; two cases must be studied separately, the fractional integral operator as a low-pass filter, and the fractional derivative operator as a high-pass filter. Second, a standard finite-dimensional mechanical oscillator coupled to both types of dampings, either low-pass or high-pass, is studied as a coupled pHs. The more general PDE system of a wave equation coupled with the diffusion equation is then found to have the same structure as before, but in an appropriate infinite-dimensional setting, which is fully detailed.