Mandelis Replies: I was delighted to see the reactions to my article “Diffusion Waves and Their Uses” (Physics Today, August 2000, page 29). In addition to the three letters published here, I received several letters and comments privately. The spectrum of disciplines that encompass diffusion waves is much broader than my article indicates. The article acted as an awareness call to the scientific community regarding the strong interdisciplinary character and analytical–diagnostic potential of diffusion waves.
As Michael Trefry’s letter describes, the very recent use of diffusion-wave methods in the study of organic compound migration in stratified media can clarify the transport of cyclic diffusive transients subject to sinusoidal boundary conditions (his ref. 3). To my knowledge, the earliest applications of diffusion waves to mass transport with modulated input sources concern metals, electrolytes, dialysis membranes, and the study of harmonic atomic and molecular diffusion processes through polymers by means of pressure oscillations inside a vacuum chamber. 1 Oscillatory sorption measurements were reported even earlier. 2
Noel Corngold pointed out the very early work on neutron waves. These are harmonic fluxes of neutrons produced periodically in beryllium surrounded by heavy water (Corngold’s ref. 1, p. 212). Many similar harmonic physical phenomena can be described as diffusion waves.
Viscosity waves are still controversial, 3 but there is growing evidence that they are forced into existence by gravity-wave nonlinearities and reflections. Wavelengths can be 8–26 m. At the other extreme, thermal waves at high modulation frequencies can exhibit wavelengths of only a few microns.
The correspondence shows, however, that many scientists consider time-domain diffusive transport as “waves.” This is widespread in the literature, including the interesting cloud remote-sensing research based on radiative Green-function theory in the diffusion limit of the transport equation, reported by Anthony Davis (see his ref. 2). I object to this practice, since the “wave” label, mainly describing hyperbolic propagation (for example, D’Alembert traveling waves), is obviously inconsistent with the parabolic nature of diffusion. To be sure, diffusion waves, as the harmonic versions of the diffusion equation in the Fourier transform sense, qualify only as pseudowaves with many shortcomings of the time-domain equations. 4 The spectral decomposition of the time-domain hyperbolic equations with a diffusion term has been used in the study of wave propagation in poroelastic media, and many features of purely diffusion-wave behavior, such as strong spatial damping of the wave amplitude, have been noted, as in Trefry’s ref. 1.
The equivalent mathematical underpinnings between time-dependent diffusion and harmonic diffusion-wave equations tend to be glossed over by many researchers who assign to the latter periodic disturbances properties of propagating (hyperbolic) wave fields that properly belong to the former. Corngold’s argument that the telegrapher’s equation (otherwise known as “second sound,” 5 ) is an improvement over the infinitely fast diffusive propagation is correct; however, this has seen little experimental use within the (harmonic) diffusion-wave communities. I guess the main reasons are that the telegrapher’s equation is an ad hoc generalization of Fourier’s linear diffusion equation, and that there is no real problem in interpreting data by means of simpler diffusion-wave equations satisfying simultaneity rather than causality; and that the time-delayed expressions introduce relaxation times that are short compared to, say, conduction heat transfer times. This large time-scale difference between conductive transport and second-sound-type relaxation time tends to minimize any perceptible differences between instantaneous and time-delayed responses, offering mostly imperceptible exactitude at the expense of additional mathematical complexity.
I agree with Corngold that the introduction of the concept of neutron waves in the 1950s preceded the tremendous growth of diffusion-wave applications in, for example, the photoacoustic and photothermal communities in the last quarter century. Nevertheless, for a primarily experimental field, neutron waves do not seem to have been as important to neutron diffusion science as their more conventional time-resolved counterparts (Corngold’s ref. 1 and ref. 2 chap. 4.2). My bias against counting time-resolved diffusion as “diffusion waves” has led me to conclude that only recently has progress occurred in the diffusion-wave area, to which Corngold has objected. I am grateful, however, that he insightfully pointed out the need for more sophisticated analytical approaches to diffusion-wave applications as they spread across many disciplines, such as charge-carrier-wave dynamics and diffuse photon density waves. I agree with his exhortations for cross-fertilization between current diffusion-wave groups and workers in the broader transport physics areas. To date, the opportunity for cross-fertilization remains largely unexplored, exciting, and potentially fruitful territory, especially in the limiting case in which periodic diffusion lengths become commensurate with mean free paths of random microscopic and mesoscopic motion. 6,7
