Kert Tamm

Waves in microstructured solids and negative group velocity

Waves with negative group velocity (NGV) were discovered in optics by Sommerfeld and Brillouin, and experimentally verified in many cases, for example in left-handed media. For waves in solids, such an effect is described mostly in layered media. In this paper, it is demonstrated that in microstructured solids, waves with NGV may also exist leading to backwards pulse propagation. Two physical cases are analysed: a Mindlin-type hierarchical (a scale within a scale) material and a felt-type (made of fibres) material. For both cases, the dispersion analysis of one-dimensional waves shows that there exists certain ranges of physical parameters which lead to NGV. The results can be used in dispersion engineering for designing materials with certain properties.

Dispersion Analysis of Wave Motion in Microstructured Solids

The Mindlin-type model is used for describing longitudinal waves in microstructured solids. This model involves explicitly the internal parameters and therefore tends to be rather complicated. An hierarchical approximation is derived, which is able to grasp the main effects of dispersion with wide variety of parameters. Attention is paid to the internal degrees of freedom of the microstructure and their influence on the dispersion effects. It is shown how the internal degrees of freedom can change the effects of dispersion.

On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes

Boussinesq-type wave equations involve nonlinearities and dispersion. In this paper a Boussinesq-type equation with displacement-dependent nonlinearities is presented. Such a model was proposed by Heimburg and Jackson for describing longitudinal waves in biomembranes and later improved by Engelbrecht, Tamm and Peets taking into account the microinertia of a biomembrane. The steady solution in the form of a solitary wave is derived and the influence of nonlinear and dispersive terms over a large range of possible sets of coefficients demonstrated. The solutions emerging from arbitrary initial inputs are found using the numerical simulation. The properties of emerging trains of solitary waves are analysed. Finally, the interaction of solitary waves which satisfy the governing equation is studied. The interaction process is not fully elastic and after several interactions radiation effects may be significant. This means that for the present case the solitary waves are not solitons in the strict mathematical sense. However, like in other cases known in solid mechanics, such solutions may be conditionally called solitons.

Original article: On the propagation of 1D solitary waves in Mindlin-type microstructured solids

The Mindlin model and hierarchical approach by Engelbrecht and Pastrone are used for modelling 1D wave propagation in microstructured solids. After introducing the free energy function, one gets from Euler-Lagrange equations a system of equations of motion. Making use of the slaving principle, a nonlinear hierarchical wave equation can be derived. Equations are solved numerically under localized initial conditions. For numerical integration, the pseudospectral method based on the Fourier transform is used. The influence of free energy parameters on the character of dispersion and wave propagation is studied. Numerical results of hierarchical approximation and the full equation system will be compared and the quality of the approximation will be discussed.

Electromechanical coupling of waves in nerve fibres

The propagation of an action potential (AP) in a nerve fibre is accompanied by mechanical and thermal effects. In this paper, an attempt is made to build up a mathematical model which couples the AP with a possible pressure wave (PW) in the axoplasm and waves in the nerve fibre wall (longitudinal—LW and transverse—TW) made of a lipid bilayer (biomembrane). A system of differential equations includes the governing equations of single waves with coupling forces between them. The single equations are kept as simple as possible in order to carry out the proof of concept. An assumption based on earlier studies is made that the coupling forces depend on changes (the gradient, time derivative) of the voltage. In addition, it is assumed that the transverse displacement of the biomembrane can be calculated from the gradient of the LW in the biomembrane. The computational simulation is focused to determining the influence of possible coupling forces on the emergence of mechanical waves from the AP. As a result, an ensemble of waves (AP, PW, LW, TW) emerges. The further experiments should verify assumptions about coupling forces.

On Nonlinear Waves in Media with Complex Properties

In nonlinear theories the axiom of equipresence requires all the effects of the same order to be taken account. In this paper the mathematical modelling of deformation waves in media is analysed involving nonlinear and dispersive effects together with accompanying phenomena caused by thermal or electrical fields. The modelling is based on principles of generalized continuum mechanics developed by G.A. Maugin. The analysis demonstrates the richness of models in describing the physical effects in media with complex properties.

Modeling of complex signals in nerve fibers

Experiments have demonstrated that signals in nerve fibers are composed by electrical and mechanical components. In this paper a coupled mathematical model is described which unites the governing equations for the action potential, the pressure wave in the axoplasm and the longitudinal and the transverse waves in the surrounding biomembrane into one system of equations. As a solution of this system, an ensemble of waves is generated. The main hypotheses of such a model are related to the nature of coupling forces between the single waves in the ensemble. These coupling forces are assumed to have bi-polar shapes leading to energetically stable solutions. The in silico modeling demonstrates the qualitative resemblance of computed wave profiles to experimental ones. The ideas of possible experimental validation of the model are briefly described.

Soliton trains in dispersive media

In this paper two Boussinesq-type mathematical models are described which lead to solitonic solutions. One case corresponds to microstructured solids, another case to biomembranes. The emergence of soliton trains in both cases is demonstrated by using numerical simulation. The pseudospectral method guarantees the high accuracy in computing. The significance of the nonlinearities—either deformation-type or displacement-type, is demonstrated.In this paper two Boussinesq-type mathematical models are described which lead to solitonic solutions. One case corresponds to microstructured solids, another case to biomembranes. The emergence of soliton trains in both cases is demonstrated by using numerical simulation. The pseudospectral method guarantees the high accuracy in computing. The significance of the nonlinearities—either deformation-type or displacement-type, is demonstrated.