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Chapter 5
Fourier Series: Visualizing Joseph
Fourier’s Imaginative Discovery via FEA
and Time-Frequency Decomposition
P. J. Masson, P. M. Silveira, C. Duque, and P. F. Ribeiro
5.1 Introduction
Joseph Fourier was without a doubt one of the greatest minds whose work changed the face of
engineering. His contributions to physics and engineering are numerous; however, he is mostly
remembered for his work on heat transfer that leads to the widely used transform named after him.
Through a simple heat transfer experiment, Fourier noticed that the shape of temperature distribution in
a ring was varying with time to become eventually a sinusoidal distribution. He then had the idea of
representing the periodic temperature distribution by a sum of sinusoids giving birth to a novel signal
analysis method nowadays used in all disciplines of engineering.
Fourier analysis is one of the most utilized mathematical techniques across a wide range of
disciplines that covers from physics and engineering to biology, economics, oceanography, and other
areas. However, the geniality of Joseph Fourier in the process of his discovery is not well known and
recognized. This chapter attempts to reproduce some of the original experiments carried out that Joseph
Fourier conducted via the use of finite element analysis (FEA). The authors expect that the results will
bring more insight into the understanding of this revolutionary technique as well as illustrate the
fundamental need for an integration of physical intuition and mathematics for the progress of scientific
developments.
This chapter presents Fourier’s heat transfer experiment through the use of finite element
analysis. An iron ring is modeled and transient thermal analysis is performed to reproduce the data
Fourier obtained experimentally.
5.2 The historical context
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The life of Jean Joseph Fourier was an exciting and controversial one, to say the least. Among the
several facets of his life one can list: teacher, secret policeman, political prisoner, governor of Egypt,
mayor, friend of Napoleon and finally secretary of the Academy of Sciences in Paris.
Joseph Fourier was born in Auxerre, France, as the son of a tailor. He was recommended to the
Bishop of Auxerre, and was educated by the Benvenistes of the Convent of St. Mark. He accepted a
military lectureship on mathematics, took a prominent part in promoting the French Revolution, and was
rewarded by an appointment in 1795 in the École Normale Supérieure, and by a chair at the École
Polytechnique. A few years later, he traveled to North Africa with Napoleon Bonaparte and became
Governor of Lower Egypt. After he came back to France to become prefect of Isere, he resumed his work on
heat propagation. In 1822, he became permanent secretary of the French Academy of Sciences. In this year,
Fourier published his “Theorie Analytique de la Chaleur”; in this work he claimed that any function of a
variable can be decomposed in a sum of cosines of a multiple of that variable. Even though this statement was
not entirely correct, the breakthrough was there. Dirichlet later determined the restrictions to Fourier’s
statement. The bases left by Fourier were reworked by numerous mathematicians until 1829 when a final
demonstration was proposed by Sturm.
Fourier’s work was not limited to heat transfer and numerous discoveries have been attributed to
his genius such as greenhouse effect gases and our planet’s energy balance. [1-5]
5.3 The Experiment
Joseph Fourier was working on heat transfer / propagation when he started experimenting and
observing the heat transfer and temperature distribution along metallic materials. Having used a thin bar
first, he later on used a polished iron ring with a diameter of approximately 30 cm held in place by
wooden supports. Holes were drilled into the ring for thermometers. A time dependent experiment was
carried out where the ring was placed halfway into a furnace and then removed to an insulating bath of
sand. The initial distribution of temperature was of uniformly hot around one half and cold around the
other as shown in Fig. 5.2. Figure 5.1 was constructed using FEA and the characteristics of the ring used
by Joseph Fourier. The temperature then started to change as the heat flowed from hot to cold and
illustrated in Fig. 5.2 to Fig. 5.7. Fourier observed the nature of the distribution of the variation and
associated it with simple sinusoidal patterns. Despite the imprecision of the instruments and tools used,
Joseph Fourier’s genial mind was able to think of the initial distribution of temperature, and shape as
time went on, as a superposition of simple sinusoids that varied from peak to trough to peak an integer
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number of times along the circumference of the ring. As a result, Fourier could propose his creative and
revolutionary equation.

1
f (t )  a0  {a h cos( h 0 t )  bh sin( h 0 t )}
2
h 1
A summary of the idea that prompted the concept is depicted in Fig. 5.1a and Fig. 5.1b, where the
behavior of the temperature distribution along the spatial discontinuity versus time (5.1a) and as a
function of the number of Fourier series coefficients (5.1b) are respectively illustrated.
Figure 5.1 – (a) Temperature distribution around the discontinuity obtained by FEA for different
times.
Figure 5.1 – (b) Temperature distribution around the discontinuity as a function of the number of
Fourier coefficients .
5.4 The finite element analysis modeling simulation
As in Fourier’s experiment, an iron ring has been modeled in Finite Element Analysis software
COMSOL Multiphysics. The ring shown in figure 5.2 is composed of iron with a thermal conductivity and
heat capacity depending on temperature. The boundary is assumed adiabatic. Due to its large thermal
diffusion time, iron can accommodate a pretty steep temperature gradient which is set to maximum at
initial time. The initial temperature conditions are 600 K in half of the ring and 300 K for the second half.
At t = 0, heat transfer begins as represented in figures 5.2 to 5.7. Of course, as no external heat transfer
exists, the equilibrium temperature reached is the average 450 K. Fig. 5.8a shows the evolution of
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temperature along the mean radius of the ring; as time passes, the temperature profile shows less
harmonics content to end up being a pure sinusoid.
Fig. 5.2 – Geometry modeled in COMSOL Multiphysics.
Fig. 5.3 – Initial temperature distribution
Fig. 5.4 – Time step after initial distribution – variation seems to take a simple sinusoidal pattern
Fig.5.5 – Additional time step
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Fig. 5.6 – Additional time step
Fig. 5.7 – Additional time step
Fig. 5.8 – Final temperature distribution for the entire ring.
5.5 Time-frequency Analysis could have been discovered
Fourier reasoned that the higher frequency sinusoids would damp out rapidly. For example, a
sinusoid with twice the frequency would imply that the distance between hot peak and cold trough was
halved and on top of that the temperature gradient would be doubled. As a result, a sinusoidal
distribution with twice the frequency would dampen at four times the rate. Based on this observation,
the Fourier series was developed for steady-state signals which assume that all the sinusoidal
components would be present all the time. But in reality Joseph Fourier’s experiment was a timedependent one and time-varying components could have been included.
In order to exemplify to analyze the experiment with time-frequency resolution, the behavior of
the temperature distribution can be approximated by a quadratic waveform, whose initial discontinuity
changes and smoothes along the time as the temperature reaches stabilization, as illustrated in Fig. 5.9.
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0.8
0.6
0.3
0.4
0.2
0.2
0
0.1
-0.2
-0.4
1
0
-0.6
-0.1
0
0. 8
hot
0.02
0.04
0.06
0.08
0.1
-0.2
-0.3
0. 6
3.78
3.8
3.82
3.84
3.86
3.88
3.9
3.92
3.94
3.96
3.98
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x 10
Temperature
0. 4
0. 2
0
-0. 2
-0. 4
-0. 6
0.8
-0.
8
cold
0.6
0.4
-1
0
1
2
3
0.2
4
5
6
7
8
0
4
x
10
time
-0.2
-0.4
-0.6
-0.8
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
3200
Fig. 5.9 – Representation of temperature variation versus time
The time-varying components can be prompted and observed if Fourier had more advanced
mathematical tools (time-frequency analysis). Figure 5.10, for example, illustrates the solution of the
time-dependent experiment using a methodology developed [6] in which the behavior time-varying
harmonic components are observed as the temperature varies. This methodology based on Fourier’s
analysis can be accomplished by several approaches which uses a sliding window to overcome the
steady-state requirement of the Fourier series.
Fig. 5.10 – Decomposition of temperature variation by time-varying harmonics
Considering that during the first cycles of the temperature, the behavior is like a square
waveform, it is difficult to recompose the original signal using only sinusoidal components, even using a
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high order of components (N). In fact, since the square wave satisfies the Dirichlet conditions, the limit of
f(t) at the discontinuities, as N  , is the average value of the discontinuity. According to [7], when a
signal has a discontinuity of unity height, the partial sum of the components results in a maximum value
of 1.09, i.e., there will be an overshoot of 9% of the height of the discontinuity, independently of how
large N is. This behavior has been known as the Gibbs phenomenon, since it was the famous
mathematical physicist Josiah Gibbs, who investigated it and presented his explanation in 1899.
With his experience, Joseph Fourier had already inferred two important behaviors of the
harmonic components: (i) a fast decay of the higher frequency components along the time, as we can see
in Fig. 5.10 and, (ii) a fast decay of the higher order components at the discontinuities of the signal.
For this second behavior, although intuitive, it is possible to observe it easily by using another
methodology of decomposition, instead of the Fourier technique. In this case, for example, the
application of Wavelet Transform provides different components, each one localized inside a different
bandwidth. The higher frequency components, which are present at the discontinuities of the signal, can
be extracted and analyzed. Considering the signal shown in Fig. 5.9, one can perform its decomposition
using a wavelet based filter bank with six decomposition levels, whose results are shown in Figure 5.11. In
this case the Meyer wavelet has been used; taking into account that it is a very suitable wavelet to extract
signals with smoothed features. The Joseph Fourier’s observations are clearly depicted in this figure. First,
the higher the frequency component, the faster the decay along the time, which can be seen by
comparing the output signals (d1 to a6). Second, considering the zoomed details of the Fig. 5.11, one can
observe the existence of high frequency components in each discontinuity of the signal. These
characteristics can be observed at the different detailed levels shown in Figure 5.11 where the
decomposition of temperature variation using a wavelet based filter bank method has been used. Figure
5.11 also shows that the temperature discontinuity can be adequately represented by only six
components using a wavelet based filter bank decomposition, whereas the traditional harmonic
decomposition would require many times that number of components to achieve similar resolution since
they are all of steady-state nature. Direct physical interpretation of temperature variation from the
higher detail coefficients levels would, however, be meaningless.
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4
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Fig. 5.11 – Decomposition of temperature variation using a wavelet based filter bank
method
5.6 Conclusions
Joseph Fourier's many contributions to modern engineering science are so critically
important and so pervasive that he is rightly regarded as the father of modern engineering.
Great discoveries such as Fourier’s transform can be found through basic physics experiments
coupled with mathematics. Fourier’s physical intuition lead to one of the most used analysis
methods that changed the face of engineering. Finite Element Analysis allowed for a simple
reproduction of Fourier’s experiment of heat propagation / temperature variation through a
metallic ring. Simulated data gave a clear view of how Fourier first thought of representing
temperature distribution in a ring as a combination of sinusoidal functions and how this
experiment gave information about how harmonics content is modified in time. The use of
new signal processing methods, based on time-frequency decomposition, further illustrates
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Joseph Fourier’s physical intuition to visualize the time varying components long before the
mathematical foundation was developed.
5.7 References
[1] Joseph Fourier – Politician & Scientist, David A. Keston, Today in Science http://www.todayinsci.com/F/Fourier_JBJ/FourierPoliticianScientistBio.htm
[2] Grattam-Guiness, Ivor: Joseph Fourier (1768-1830): a survey of his life and work, The
MIT Press, 1972.
[3] Herivel, John: Joseph Fourier : The Man and the Physicist, Clarendon Press, 1975.
[4] Fourier, J.-B.-J. Mémoires de l'Académie Royale des Sciences de l'Institut de France
VII. 570-604 (1827) (greenhouse effect essay)
[5] The Project Gutenberg EBook of Biographies of Distinguished Scientific Men by
François Arago
[6] Duque, C., P.M. Silveira, T. Baldwin, P. F. Ribeiro, "Novel Method for Tracking TimeVarying Power Harmonic Distortions without Frequency Spillover," submitted to the PES
GM2008, Pittsburgh, 20 - 24 July, 2008.
[7] Oppenheim, A. V., Willsky A. S., Signals and Systems, Prentice-Hall, 1983.