FðixÞ ¼
Z þ 1
À1
f ðtÞe
Àixt dt
ð3:123Þ
f ðtÞ ¼
1
2p
Z þ 1
À1
FðixÞe
ixt dx
ð3:124Þ
FðixÞ is obtained from the Fourier series in complex form (Stearns and Hush
1990):
f ðtÞ ¼
X þ 1
n¼À1
c n e
inx 0 T
ð3:125Þ
c n ¼
w 0
2p
Z
Àp=w 0
Àp=w 0
f ðtÞe
Àinx 0 t dt
ð3:126Þ
To obtain the Fourier transform, the fundamental frequency of the function w 0
defining the interval between the frequencies of the harmonics, tends to zero, and
can then be represented as Dw. Thus, the integral lines of a discrete spectrum
converge tend toward a continuous spectrum. In other words, the period T = 2p/w 0
tends to infinity, which means that the time function is no longer repeated, and
becomes aperiodic.
Equation (3.125) can be written as
f p ðtÞ ¼
X þ 1
n¼À1
c n e
inw 0 T
ð3:127Þ
where p index indicates the hypothetical periodicity. From Eq. (3.126)
f p ðtÞ ¼
X þ 1
n¼À1
Dx
2p
Z
Àp=Dx
Àp=Dx
f ðsÞe
ÀinDxs ds e
inDxt
ð3:128Þ
¼
1
2p
X þ 1
n¼À1
Z
Àp=Dx
Àp=Dx
f ðsÞe
ÀinDxs ds
2
6
4
3
7
5 e
inDxt
Dx
ð3:129Þ
where s is a transitory dummy variable.
68
3 Characterization of Turbulent Flow in the Surface Boundary Layer