From combustion to the wave nature of heat via the telegraphic equation

When a train approaches or an ambulance approaches us with its siren blaring, we hear the sound at an increased frequency, gradually decreasing. As it passes, the frequency suddenly changes to a lower one, then decreases further. This commonly encountered Doppler effect can be a valuable clue to the nature of a phenomenon seemingly completely unrelated to sound propagation: heat transport.

Burns are not pleasant for anyone, but they affect physicists doubly: not only do they suffer normally, in addition, they still do not know which mechanism is responsible for the transport of heat in systems as complex as biological tissues.

Is diffusion related to the spread of initially assembled molecules of matter? Or are wave phenomena similar to those known from acoustics responsible for heat transport?

A group of three theorists from the Institute of Nuclear Physics of the Polish Academy of Sciences (IFJ PAN) in Krakow decided to tackle the problem of heat transport using the telegraphic equation and the Doppler effect, known to us from everyday life (and primary school). . The results of the team’s work have just been published in International Journal of Heat and Mass Transfer.

In physics, wave motion is described by an equation called the wave equation. When telegraph technology was developing in the second half of the 19th century, it became clear that, in order to describe a message transmitted in Morse code, this equation had to be modified to take into account the attenuation of the current flowing through the medium in it. which it spreads, i.e. through the telegraph cable.

With telecommunications in mind, the telegraphic equation was then developed to describe how electric current propagates with attenuation along a spatial dimension.

“In recent years, the skillfully generalized telegraph equation has found a new application: it has also begun to be used to describe phenomena related to diffusion or heat transport. This fact prompted us to pose an intriguing question,” says Dr. Katarzyna Gorska. (IFJ PAN).

“In the solutions of the wave equation, i.e. without damping, the Doppler effect occurs. This is a typical wave phenomenon. But does it also occur in the solutions of the telegraph equations related to heat transport? If so, we would have an excellent indication that at least theoretically, there is no reason to believe that in damping systems—for example, in biological tissues—heat flow cannot be treated as a wave phenomenon.”

The classic Doppler effect is the apparent change in frequency of waves emitted by a source moving relative to an observer. When the distance between the source and the observer decreases, the maxima and minima of the emitted waves reach the receiver more often than when the distance between the source and the observer increases. In the case of sound waves, we can clearly hear that the sound of an approaching train or the siren of a speeding ambulance have significantly higher frequencies than when these vehicles are moving away from us.

Prof. Andrzej Horzela (IFJ PAN) points out, “The Doppler phenomenon occurs in the wave equations, which we say are local. Here we mean local in that there is no delay between action and reaction. The principles of mechanics, for example, are local – a change in the resultant force acting on a body immediately leads to a change in its acceleration.

“However, we all know that we can get a hot cup and before we feel it burning, it’s a second or two. The phenomenon exhibits a delay, we say that it is non-local, in other words, painted in time. So do we see the Doppler effect in the generalized telegraphic equation that describes time-stained systems?”

Easy to ask, harder to answer. The problem is in the math itself. If all we have in the equations are derivatives and constants, then there is usually little difficulty in finding solutions. This is the case in the wave equation. The issue becomes more complicated when the equation contains only integrals, but even then it can often be managed. Meanwhile, in the generalized telegraphic equation, derivatives and integrals occur simultaneously.

Therefore, at the heart of the work of the Cracow physicists was the proof that solutions of the generalized telegraphic equation can be constructed from much simpler to find solutions of the local equation. Here, a key role was played by the procedure known in stochastic process theory as subordination.

The following example helps us understand the concept of subordination. Imagine a man who has had too much to drink but bravely tries to walk a straight line. He takes a step and stands still, waiting for the world to stop spinning. He then takes another step, perhaps slightly longer or shorter than the previous one, and pauses again indefinitely.

The mathematical description of such motion, called random walk, should not be trivial at all. What really matters, however, is not how long our “wanderer” spends in a given location, but what distance he or she eventually covers.

If the time between the steps were equal, the description of the movement of sailors would be simpler and would correspond to the movement of a measured person – it would simply be the sum of a sequence of successive steps, followed without problems.

“In our approach, the dependence consists in replacing the uniformly passing physical time, in which the equations are complicated, with a definite internal time related to the physical time, which we do through a suitable function that contains information about the temporal non-locality of the process. This procedure simplifies the equations in a form that makes it possible to find their solutions,” says co-author Tobiasz Pietrzak, M.Sc, student at the Interdisciplinary Doctoral School in Krakow.

Solutions of the ordinary telegraphic equation show typical features of the Doppler effect. They indicate the presence of a clear, sharp frequency bend, corresponding to the moment when the source passes by the observer and there is a sudden, abrupt change in the pitch of the sound recorded by the observer.

Analogous behavior was observed by the Krakow physicists in the solutions of the generalized equation. Therefore, it appears that the Doppler effect is a fundamental feature of wave motion. However, that is not all. In the physical world, every wave has its own wave line, which, somewhat simplified, can be identified with its beginning and end. When looking at the front of the wave (and therefore its wavefront), the Doppler shift is easy to see.

It turns out that changes in wave frequency due to changes in the distance between the observer and the source occur even for waves that do not show the existence of a wave front, e.g. defined in an unlimited area.

Research into the wave aspects of heat propagation may seem like a very abstract consideration, but its translation into everyday practice seems quite real. Physicists from IPJ PAN emphasize that the knowledge they have obtained can be used especially in situations where heat transport is involved over short distances.

Examples include medical applications, where a better understanding of the mechanisms of heat transport may allow the development of safer techniques for working with laser surgical instruments or finding a method to remove excess heat from burned tissue. more efficient than before. Cosmetology, interested in minimizing the unwanted thermal effects that occur during cosmetic procedures, can also benefit.