16-05-2012, 01:37 PM
Fourier Series
fouriers.pdf (Size: 381.14 KB / Downloads: 48)
Just before 1800, the French mathematician/physicist/engineer Jean Baptiste Joseph
Fourier made an astonishing discovery. As a result of his investigations into the partial dif-
ferential equations modeling vibration and heat propagation in bodies, Fourier was led to
claim that “every” function could be represented by an infinite series of elementary trigono-
metric functions — sines and cosines. As an example, consider the sound produced by a
musical instrument, e.g., piano, violin, trumpet, oboe, or drum. Decomposing the signal
into its trigonometric constituents reveals the fundamental frequencies (tones, overtones,
etc.) that are combined to produce its distinctive timbre. The Fourier decomposition lies
at the heart of modern electronic music; a synthesizer combines pure sine and cosine tones
to reproduce the diverse sounds of instruments, both natural and artificial, according to
Fourier’s general prescription.
Fourier’s claim was so remarkable and unexpected that most of the leading mathe-
maticians of the time did not believe him. Nevertheless, it was not long before scientists
came to appreciate the power and far-ranging applicability of Fourier’s method, thereby
opening up vast new realms of physics, engineering, and elsewhere, to mathematical anal-
ysis. Indeed, Fourier’s discovery easily ranks in the “top ten” mathematical advances of
all time, a list that would include Newton’s invention of the calculus, and Gauss and Rie-
mann’s establishment of differential geometry that, 70 years later, became the foundation
of Einstein’s general relativity.
Dynamical Equations of Continuous Media.
The purpose of this section is to discover why Fourier series arise naturally when
we move from discrete systems of ordinary differential equations to the partial differential
equations that govern the dynamics of continuous mechanical systems. In our reconstrucd-
tion of Fourier’s thought processes, let us start by reviewing what we have learned.
In Chapter 6, we characterized the equilibrium equations of discrete mechanical and
electrical systems as a linear algebraic system
fouriers.pdf (Size: 381.14 KB / Downloads: 48)
Just before 1800, the French mathematician/physicist/engineer Jean Baptiste Joseph
Fourier made an astonishing discovery. As a result of his investigations into the partial dif-
ferential equations modeling vibration and heat propagation in bodies, Fourier was led to
claim that “every” function could be represented by an infinite series of elementary trigono-
metric functions — sines and cosines. As an example, consider the sound produced by a
musical instrument, e.g., piano, violin, trumpet, oboe, or drum. Decomposing the signal
into its trigonometric constituents reveals the fundamental frequencies (tones, overtones,
etc.) that are combined to produce its distinctive timbre. The Fourier decomposition lies
at the heart of modern electronic music; a synthesizer combines pure sine and cosine tones
to reproduce the diverse sounds of instruments, both natural and artificial, according to
Fourier’s general prescription.
Fourier’s claim was so remarkable and unexpected that most of the leading mathe-
maticians of the time did not believe him. Nevertheless, it was not long before scientists
came to appreciate the power and far-ranging applicability of Fourier’s method, thereby
opening up vast new realms of physics, engineering, and elsewhere, to mathematical anal-
ysis. Indeed, Fourier’s discovery easily ranks in the “top ten” mathematical advances of
all time, a list that would include Newton’s invention of the calculus, and Gauss and Rie-
mann’s establishment of differential geometry that, 70 years later, became the foundation
of Einstein’s general relativity.
Dynamical Equations of Continuous Media.
The purpose of this section is to discover why Fourier series arise naturally when
we move from discrete systems of ordinary differential equations to the partial differential
equations that govern the dynamics of continuous mechanical systems. In our reconstrucd-
tion of Fourier’s thought processes, let us start by reviewing what we have learned.
In Chapter 6, we characterized the equilibrium equations of discrete mechanical and
electrical systems as a linear algebraic system