Challenging previous understanding – physicists propose a wave-based theory of heat transport

Physicists have linked the Doppler effect to heat transport, suggesting wave-like properties in biological tissue, with implications for medical and cosmetic technologies.

When a train approaches or an ambulance approaches us with its siren blaring, we hear the sound at an increased frequency, which gradually decreases. As it passes, the frequency suddenly changes to a lower one, then decreases further. This commonly encountered phenomenon, known as the Doppler effect, can provide valuable insights into a seemingly unrelated field: heat transport.

Physics of heat transport

Although burns are painful for everyone, they cause a unique kind of pain in physicists. In addition to physical pain, they have yet to determine the exact mechanism that regulates heat transport in complex systems such as biological tissues. Is diffusion, related to the spread of initially aggregated molecules, or are acoustic-like wave phenomena responsible?

In a new study, published in International Journal of Heat and Mass Transfertheorists from the Institute of Nuclear Physics of the Polish Academy of Sciences in Krakow have explored this question using the telegraphic equation and the Doppler effect, familiar concepts from everyday life.

The telegraph equation and wave motion

Wave motion is described by an equation called the wave equation. However, as telegraph technology advanced in the late 19th century, it became clear that this equation required modification to accurately describe the transmission of Morse code messages. These modifications were needed to take into account the attenuation of the current passing through the medium in which it propagates, 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 telegraphic equation has found a new application: it has also begun to be used to describe phenomena related to the diffusion or transport of heat. This fact prompted us to pose an intriguing question,” says Dr. Katarzyna Gorska (IFJ PAN). “In solutions of the wave equation, ie without damping, the Doppler effect occurs. This is a typical wave phenomenon. But does it also occur in the solutions of telegraphic 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 tissue—heat flow could not be treated as a wave phenomenon. “

Doppler effect in physics

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. We understand local here 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 pick up a hot cup and it takes a second or two before we feel it burning. The phenomenon exhibits a certain delay; we say that it is not local, in other words painted in time. So do we see the Doppler effect in the generalized telegraphic equation that describes time-stained systems?”

Mathematical challenges and innovations

Addressing this question is challenging due to the mathematical complexity of the generalized telegraph equation, where derivatives and integrals occur simultaneously. However, the Krakow physicists demonstrated that solutions of the generalized telegraphic equation can be constructed from much simpler solutions of the local equation using a procedure known as dependence. Subordination replaces the complex physical time in the equations with a simpler internal time through a specific function that reflects temporal nonlocality. This simplification allows deriving the solutions of the equations.

“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 paper co-author Tobiasz Pietrzak, M.Sc, a student at the Interdisciplinary Doctoral School in Krakow, whose work was funded by a Preludium Bis grant. from the Polish National Science Center.

Results and Implications

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.

Reference: “Generalized Telegraph Equation with Moving Harmonic Source: Solution Using Integral Decomposition Technique and Wave Aspects” by T. Pietrzak, A. Horzela and K. Górska, 12 March 2024, International Journal of Heat and Mass Transfer.
DOI: 10.1016/j.ijheatmasstransfer.2024.125373