課程名稱︰工程數學二
課程性質︰必修
課程教師︰謝傳璋
開課學院：工學院
開課系所︰工程科學及海洋工程學系
考試日期（年月日）︰98/04/09
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. Find the Fourier Series of the periodic function of period 2π:
f(x)= x^2/4 when |x|<π and verify the identities:
π^2/6 = 1+1/2^2+1/3^2+1/4^2+1/5^2+.......
π^2/12= 1-1/2^2+1/3^2-1/4^2+1/5^2-.......
2. Find the Fourier integral representation of the function:
┌
│ sinx |x|<π
f(x)= │
│ 0 |x|>π
└
3. Consier a real time signal f(t) for -∞ < t < ∞, and a truncated time
signal f (t) which is defined by : ┌
T │ f(t) -T/2 < t < T/2
f (t)≡│
T │ 0 otherwise
└
If the Fourier transform of f (t) is defined by:
T
∞ -iωt
F(ω)≡1/2π∫ f (t)*e dt
-∞ T
when this truncated time signal is expressed by its Fourier series,we have:
∞ inωot
f (t)= Σ Ｃn*e dt for -T/2 < t < T/2
n=-∞
T/2 -inωot
where ωo≡2π/T ; Ｃn = 1/T ∫ f (t)*e dt
-T/2 T
Find the relationship between F(ω) and Ｃn
4.(a) Solve the following wave equation of u(x,t)
d^2ｕ/dt^2 = c^2 d^2ｕ/dx^2 , 0 < x < L , 0 ≦ t ----(1)
┌
│ u(0,t)=0 0≦t<∞ -----(2) f(x)
B.C:│ u(L,t)=0 1.0│-------
└ │ ╱＼
│ ╱ ＼
I.C │／ ＼
┌ 0└──────── x
│ 2x/L ; 0≦x≦L/2 L
u(x,0)=│ ----(3)
│ 2(L-x)/L ; L/2≦x≦L
└
ｕ (x,0)=0
t
(b) Plot the shape of u(x,t) at t = L/2c and t = L/c
5. Use the method of Fourier transform,find the solution (you may express the
solution in double integral form) of the following heat equation:
du/dt=c^2 d^2ｕ/dx^2 -∞ < x < ∞ ; t > 0 ---(1)
I.C. u(x,0)=f(x) -∞ < x < ∞
6. Consider a vibrating rectangular membrane with fixed edge :
d^2ｕ/dt^2 = c^2*(d^2ｕ/dx^2 + d^2ｕ/dy^2) 0≦x≦a ; 0≦y≦b ; t>0 ---(1)
that satisfies the boundary condition u=0 on the boundary of the membrane
for all time t≧0. Find the "mode shape"and the corresponding "natural
frequency"of this vibrating membrane.
y
│ u=0
b ├─────┐
│ │u=0
u=0 │ │
└─────┴─x
0 u=0 a
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