672
H The Fourier Decomposition
Using the Euler identity exp(iθ ) = cos θ +i sin θ , the Fourier series has the more
succinct form
ξ(x, t) =
∞
j =−∞
c j (x) exp (−2π ij t/T ) .
(H.4)
with the coefficients c j (x) found from
c j (x) =
1
T
T
0
dt ξ(x, t) exp (2π ij t/T ) .
(H.5)
The complex coefficients c j are related to the real coefficients a j and b j through
c j = (1/2)
a j + ib j
.
(H.6)
Note that the frequencies of the Fourier component waves, f j = j/T , are integer
multiples of a lowest frequency, f 1 = 1/T , which is called the ‘fundamental
frequency’. Each of sine or cosine waves with frequencies which are an integer
multiple of a fundamental are called ‘harmonics’. The fundamental frequency is
also called the ‘first harmonic’, corresponding to j = 1. The ‘second harmonic’,
also called the ‘first overtone’, has a frequency f = 2f 1 , etc.
Th terminology ‘harmonic analysis’ comes from the nomenclature of music. But
the properties of harmonic analysis can now be found in pure mathematics, with
no reference to sound or music, even though many terms, such as harmonic, have
survived the abstraction.
H.2 The Fourier Transform
For waves which do not repeat in time, the period of the lowest frequency wave in
the Fourier decomposition may be taken as infinity. Analogously, waves which do
not repeat in space, or which are not maintained between physical boundaries, will
have Fourier components which have wavelengths approaching infinity. In these
cases, the Fourier series becomes a sum over a continuous set of frequencies or
wavelengths. The series is then referred to as a Fourier transform. 1
To get an expression for the Fourier transform, first consider the Fourier series
for a function with a finite period T . Combining (H.4) and (H.5), this series can
expressed as
1 An approximate transform, called the Fast-Fourier-Transform, or FFT, is used in computers for
rapid analysis.
H The Fourier Decomposition
Using the Euler identity exp(iθ ) = cos θ +i sin θ , the Fourier series has the more
succinct form
ξ(x, t) =
∞
j =−∞
c j (x) exp (−2π ij t/T ) .
(H.4)
with the coefficients c j (x) found from
c j (x) =
1
T
T
0
dt ξ(x, t) exp (2π ij t/T ) .
(H.5)
The complex coefficients c j are related to the real coefficients a j and b j through
c j = (1/2)
a j + ib j
.
(H.6)
Note that the frequencies of the Fourier component waves, f j = j/T , are integer
multiples of a lowest frequency, f 1 = 1/T , which is called the ‘fundamental
frequency’. Each of sine or cosine waves with frequencies which are an integer
multiple of a fundamental are called ‘harmonics’. The fundamental frequency is
also called the ‘first harmonic’, corresponding to j = 1. The ‘second harmonic’,
also called the ‘first overtone’, has a frequency f = 2f 1 , etc.
Th terminology ‘harmonic analysis’ comes from the nomenclature of music. But
the properties of harmonic analysis can now be found in pure mathematics, with
no reference to sound or music, even though many terms, such as harmonic, have
survived the abstraction.
H.2 The Fourier Transform
For waves which do not repeat in time, the period of the lowest frequency wave in
the Fourier decomposition may be taken as infinity. Analogously, waves which do
not repeat in space, or which are not maintained between physical boundaries, will
have Fourier components which have wavelengths approaching infinity. In these
cases, the Fourier series becomes a sum over a continuous set of frequencies or
wavelengths. The series is then referred to as a Fourier transform. 1
To get an expression for the Fourier transform, first consider the Fourier series
for a function with a finite period T . Combining (H.4) and (H.5), this series can
expressed as
1 An approximate transform, called the Fast-Fourier-Transform, or FFT, is used in computers for
rapid analysis.