Many Uses for Diffusion Waves

Andreas Mandelis, in his interesting and informative article, remarks that only recently has significant progress in the subject occurred since the days of Anders Jonas Ångström. Not so! The neutron physics community has known and enjoyed “neutron waves” for the past half-century. (These are not the de Broglie waves associated with the particles but wavelike disturbances in a diffusing cloud of neutrons.) A description of the experiments with neutrons reached textbook-level in 1958. In their classic work, Alvin Weinberg and Eugene Wigner ¹ wrote, “This method of measuring the diffusion coefficient in a medium is analogous to Ångström’s cyclic method of measuring thermal conductivity.”

The elaborate experiments described by Mandelis are difficult, and the very simple model of transport used to analyze them appears acceptable, generally. But it will soon be necessary to go deeper. The proper description of these “waves” follows from a transport equation for the particles; the classical diffusion equation is but a crude approximation of the transport equation. For example, Mandelis notes the spurious instantaneous propagation of disturbances predicted by the diffusion equation. He and his colleagues know that a minor improvement gives the telegrapher’s equation, whose solutions propagate with finite speed.

The neutron physics community has spent decades walking the path from diffusion to transport, elucidating the effects of strong absorption, anisotropic scattering, and boundary effects in space and time. ² Many of us are saddened to find our colleagues in the field of diffusion waves, particularly those dealing with optical tomography, ³ reinventing the wheel by grappling with these issues. Visiting a neighbor’s garden can be very productive.