A transport-theoretical model which generalizes the classical partial differential equations for wave propagation is presented. The 3- dimensional space is assumed filled up homogeneously and isotropically by bundles of linear ideal flexible strings, with linear density \(\rho\), along which transversal and longitudinal waves propagate with the same speed, v. Let \(\vec r\) be a point of \({\mathbb{R}}^ 3\) and \(\vec r+\sigma \vec s\), \(\sigma\in (-\infty,\infty)\), a string which, at equilibrium, is passing through \(\vec r\) in direction \(\vec s\) (\(\vec s\in \Omega\), the unit sphere), at time t. Let \(\vec A(\vec r,\vec s,t)\) denote the displacement of the string. It satisfies the wave equation (along the string) (\(\vec s.\text{grad})^ 2\vec A-v^{-2} \partial^ 2\vec A/\partial t^ 2=0.\)
A related quantity is the momentum density \(\vec i(\vec r,\vec s,t)=(v\rho /2)[v^{-1}\partial /\partial t-\vec s.\text{grad}]\vec A(\vec r,\vec s, t)\), which is easily shown to satisfy the very simple transport equation \(v^{-1}{\vec \partial}i/\partial t=-\vec s.\text{grad} \vec i\) or, possibly, a similar one with the additional source-term \(\vec F(\vec r,\vec s,t)/2v\), if an external force \(\vec F\) acts on (the unit length of) the string. This force could be, at least partly, the effect of the displacements (or the momenta) of all the strings belonging to the same bundle (i.e. passing through the same point \(\vec r)\), so that we are led to the following transport equation \[ v^{-1} \partial \vec i/\partial t=-\vec s.\text{grad} \vec i+\int^{\infty}_{-\infty}dt'\int_{\Omega}O\quad \underset \tilde{} {\;}(\vec s,\vec s',t,t')\vec i(\vec r,\vec s',t')d\Omega '+\vec j(\vec r,\vec s,t), \] where \(\underset \tilde{} O(...)\) is a \(3\times 3\) matrix kernel and \(\vec j\) the “true” source term. A large part of the paper is devoted to the restrictions which should be imposed on the kernel in order to satisfy isotropy, space-inversion, time-invariance, causality, time-reversal, conservation of mechanical work in periodic phenomena and generalizations of Newton’s third law (forces, torques and tensions exerted by strings interacting on each othermust be locally balanced). The consequences of other assumptions, such as \(\vec A(\vec r,\vec s,t)=\vec A(\vec r,-\vec s,t)\) (the two strings passing through \(\vec r\) in the opposite directions are identified) are also investigated. The most important particular case is that of the “strong coupling limit”. A spherical harmonics expansion with respect to the directional variable \(\vec s\) shows, in fact, that in the limit of an extremely strong local interaction an asymptotic distribution of momenta, locally at equilibrium, exists and can be represented by a low-order expansion of \(\vec i(\vec r,\vec s,t)\), whose coefficients satisfy partial differential equations of the Maxwell type.
Thus the classical description of the wave phenomena is recovered as a limit case of the proposed transport-theoretical model, more precisely when, by virtue of the low-order asymptotic distribution existing in the case of an extremely strong coupling, the directional variable \(\vec s\) can be essentially disregarded (much in the same way as in the derivation of the diffusion equation from the classical transport equation). To foretell the fortune, as an effective tool of theoretical physics, of this otherwise new, interesting and well-constructed model is, of course, a difficult affair. But the reading of the paper sould be anyway profitable to any transport theorist.