Until the beginning of the 19th century, heat was a mystery. What the hell is it?Is it a liquid like water?It does seem to be flowing, but you can't hold it in your hand or see it. Although you can measure a hot object indirectly by tracking the temperature change as it cools, no one knows what's going on inside the object.
The secret of heat is revealed by a person who often feels cold. Orphaned at the age of 10, Fourier was frail as a teenager suffering from indigestion and asthma. As an adult, he believes that calories are essential for health. Even in the summer, he would stay in an overheated room and wrap himself in a thick coat. In all aspects of his scientific career, Fourier was focused and obsessed with heat. He invented the concept of global warming and was the first to explain how the greenhouse effect would regulate the average temperature of the Earth.
In 1807, Fourier used calculus to solve the mystery of heat flow. He proposed a partial differential equation that can be used for how the temperature of an object (such as a red-hot iron rod) changes during cooling. Fourier was astonished to find that no matter how uneven the temperature of the rod was at the beginning of the cooling process, this partial differential equation could be easily solved.
Imagine a long, thin cylindrical iron rod that is heated unevenly in the blacksmith's forge, with hot and cold spots scattered around it. For simplicity's sake, let's assume that the iron rod has a completely insulated sleeve on the outside so that the heat is not lost. In this case, the only way for heat to flow is to diffuse from hot to cold spots along the length of the iron rod. Fourier postulated (and experimentally confirmed) that the rate of temperature change at a point on an iron rod is proportional to the mismatch between the temperature at that point and the average temperature of its adjacent points on either side. By adjacent points I mean two points that are infinitely close to each other on either side of the point we are interested in.
Under these idealized conditions, the physical process of heat flow becomes simpler. If a point is colder than its neighbors, it heats up;If a point is hotter than its neighbors, it cools down. The greater the mismatch, the faster the temperature will equilibrate. If the temperature of a point is exactly equal to the average temperature of its neighbors, everything is in equilibrium, the heat no longer flows, and the temperature of this point will remain the same for the next instant.
By comparing the instantaneous temperature of one point with the instantaneous temperature of its neighbors, Fourier established a partial differential equation, what we now call the heat conduction equation. It contains derivatives of two independent variables: one is the infinitesimal change in time (t) and the other is the infinitesimal change in position (x) on the iron rod.
The difficulty with Fourier's self-imposed problem is that the initial arrangement of hot and cold spots can be haphazard. To address this general problem, Fourier proposed a solution that seemed overly optimistic, even reckless. He claimed that an equivalent sum of simple sine waves could be used instead of any of the initial temperature distribution modes.
Sine waves are his building block, and he chose them because they make the problem much simpler. He knew that if the temperature distribution started out in a sine wave pattern, it would remain that pattern as the rod cooled down.
The point is, the sine waves don't move around, they stay there. Indeed, when their hot spots cool down and cold spots heat up, the sine waves weaken. But this attenuation is easy to deal with, it simply means that the temperature change tends to flatten over time. As shown in the figure below, the initial temperature distribution pattern (dashed sine wave) gradually weakens and looks like a solid sine wave.
Importantly, when the sine waves weaken, they are stationary. That is, they are standing waves.
If Fourier could find a way to decompose the original temperature pattern into sine waves, he would be able to solve the heat flow problem for each sine wave separately. He already knows the answer to the question: every sine wave decays exponentially, and its decay rate depends on how many peaks and troughs it has. Sine waves with more crests decay faster because their hot and cold spots are more tightly packed together, which makes the heat exchange between them more rapid, and thus the thermal equilibrium is reached faster. After understanding the attenuation of each sine wave, all Fourier had to do was put them back together to solve the original problem.
The difficulty with all this is that Fourier inadvertently invokes the infinite series of the sine wave. Once again, he summoned the "Golem" of Infinity to calculus, and Fourier did so more desperately than his predecessors. Instead of using the infinite series sum of triangle fragments or triangle numbers, he casually adopted the infinite series sum of waves. This is reminiscent of Newton's treatment of infinite series sums of power functions, except that Newton never claimed that he could describe arbitrarily complex curves with such terrible features as discontinuous jumps or sharp turns.
Fourier, on the other hand, claimed that curves with turns and jumps did not intimidate him. In addition, Fourier's sine waves arise naturally from the differential equations themselves, and in the sense that they are the natural vibrational modes or inherent standing wave modes of the differential equations, which are made for the heat flux body. Newton did not consider power functions as building blocks, while Fourier saw sine waves as building blocks, and thought they were organically matched to the heat flow problem.
Although Fourier's bold use of sine waves as building blocks has sparked controversy and created a thorny rigor problem (it took mathematicians a century to solve it), in our time, Fourier's great ideas have played an important role in technologies such as computer speech synthesizers and MRI scans for medical imaging.