Test 1: Equations Summary
(These equations will be included with the Test)
• F ormI → F ormJ
1. C1 cos(wt) + C2 sin(wt) → A cos(wt + φ) where A =
arctan(−C2 /C1 ).
p
C12 + C22 and φ =
2. C1 cos(wt) + C2 sin(wt) → Eeiwt + Ēe−iwt where E = 12 (C1 − iC2 ) and Ē
is the complex conjugate of E.
3. You should be able to apply formula 2. to further find the complex form
of a Fourier Series and a Fourier Integral.
• Regarding the nonhomogenous ODE
ẍ + 2δ ẋ + ω 2 x = A cos(Ωt)
(1)
the forced solution is F orced=T ransient + Stationary where
√
√

δ 2 −ω 2 )t
(−δ− δ 2 −ω 2 )t
C1 e(−δ+
+
C
e

2
√
√
T ransient =
e−δt (C1 cos( ω 2 − δ 2 t) + C2 sin( ω 2 − δ 2 t)

C1 e−δt + C2 te−δt
Stationary = χ(δ, ω, A, Ω) cos(Ωt + φ)
when δ 2 > ω 2
when δ 2 < ω 2
when δ 2 = ω 2
where χ is called the Amplitude Response Curve given by
A
χ(δ, ω, A, Ω) = p
2
(ω − Ω) (ω + Ω)2 + 4δ 2 Ω2
and φ is a phase angle given by
φ = arctan
2δΩ
2
ω − Ω2
• The solution to eq. (1) when δ = 0 along with IC’s x(0) = x0 and v(0) = v0 is
x(t) = A cos(ωt) + B sin(ωt) = x0 cos(ωt) +
v0
sin(ωt)
ω
with the Total Energy (TE) of the wave being
T E = ω 2 x20 + v02 = ω 2 (A2 + B 2 )
• Parseval’s Theorem for Fourier Series and Fourier Integrals
1
L
∞
a20 X 2
|f (t)| dt =
+
an + b2n
2
−L
Z ∞
Z ∞ n=1
|f (t)|2 dt =
|fˆ(w)|2 dw
Z
L
2
−∞
−∞
1
• Fourier Series of f (t) on [−L, L]
f (t) = A0 +
∞
X
An cos(
n=1
nπ
nπ
t) + Bn sin( t)
L
L
where
1
A0 =
2L
Z
L
1
f (t)dt; An =
L
−L
Z
L
1
nπ
f (t) cos( t)dt; Bn =
L
L
−L
Z
L
f (t) sin(
−L
nπ
t)
L
• Fourier Integral of f (t)
Z
f (t) =
∞
A(w) cos(wt) + B(w) sin(wt)dw
(2)
0
where
1
A(w) =
π
Z
∞
1
f (t) cos(wt)dt; B(w) =
π
−∞
Z
∞
f (t) sin(wt)dt
−∞
• Fourier Transform of f (t)
1
f (t) = √
2π
Z
∞
fˆ(w)eiwt dw
(3)
−∞
where fˆ(w) is called the Fourier Transform of f (t) and is given by
Z ∞
1
ˆ
f (w) = √
f (t)e−iwt dt
2π −∞
• You should be familiar with how the complex Fourier integral (3) above arise
from the real Fourier integral (2)
√
• Relationships: A(w) and B(w) → E(w) = 21 (A(w)−iB(w)) → fˆ(w) = 2πE(w).
• Fourier Cosine Transform when f (t) is even
r Z ∞
2
fˆc (w) cos(wt)dw
f (t) =
π 0
where fˆc (w) is called the Fourier Cosine Transformation of f (t) and is given by
r Z ∞
2
ˆ
fc (w) =
f (t) cos(wt)dt
π 0
• You should learn the relationship of A(w) to fˆc (w)
• Fourier Sine Transform when f (t) is odd
r Z ∞
2
f (t) =
fˆs (w) sin(wt)dw
π 0
where fˆs (w) is called the Fourier Sine Transformation of f (t) and is given by
r Z ∞
2
fˆs (w) =
f (t) sin(wt)dt
π 0
• You should learn the relationship of B(w) to fˆs (w)
2