The term a o is given by the mean value of f(t) in a period T:
a o ¼
1
T
Z T=2
ÀT=2
f ðtÞdt
ð3:114Þ
The Fourier series can also be expressed in terms of exponential functions, using
Euler’s formula:
e
iy
¼ cos y
ð Þ þ isin Y
ð Þ
ð3:115Þ
e
Àiy
¼ cos y
ð Þ À isin Y
ð Þ
ð3:116Þ
or:
cos y
ð Þ ¼ e
iy
þ e
Àiy
À
Á =2
or:
sin y
ð Þ ¼ e
iy
À e
Àiy
À
Á
=2i
Substituting in the above Fourier series:
f t
ð Þ ¼ a 0 þ
X 1
n¼1
a n cosðnx o tÞ þ
X 1
n¼1
b n sinðnx o tÞ
ð 3:117Þ
gives:
f t
ð Þ ¼ a 0 þ
X 1
n¼1
a n
e
inx 0 t
þ e
Àinx 0 t
2
þ
X 1
n¼1
b n
e
inx 0 t
À e
Àinx 0 t
2i
ð3:118Þ
given that 1=i ¼ Ài
f ðtÞ ¼ a 0 þ
X 1
n¼1
a n
e
inx 0 t
þ e
Àinx 0 t
2
À ib n
e
inw 0 t
À e
Àinw 0 t
2
ð3:119Þ
equivalent to:
f ðtÞ ¼ a 0 þ
X 1
n¼1
e
inxt a n À ib n
2
þ e
Àinxt a n þ ib n
2
ð3:120Þ
After some manipulation, the expression for the Fourier series as the summation
of exponential functions is obtained:
64
3 Characterization of Turbulent Flow in the Surface Boundary Layer