Free Vibration in a String Fixed at Both Ends 两端固定的弦(弦l)的自由振动 定解问题 02 a2 0 0<x<l.t>0 at2 ax2 0 t>0 u=0=(x) au at v(x)0≤x≤l t=0 尖
Free Vibration in a String Fixed at Both Ends Important Remarks Separation of Variables Orthogonality: Proof Summary Conclusions üà½u(ul)gdÄ ½)¯K ∂ 2u ∂t2 − a 2 ∂ 2u ∂x2 = 0 0 < x < l, t > 0 u x=0 = 0 u x=l = 0 t ≥ 0 u t=0 = φ(x) ∂u ∂t t=0 = ψ(x) 0 ≤ x ≤ l C. S. Wu 1ù ©lCþ{()
Free Vibration in a String Fixed at Both End 讲授要点 ③两端固定弦的自由振动 分离变量法 。本征函数正交性的证明 总结 若干重要评述 本征函数的正交性 。解的唯一性
Free Vibration in a String Fixed at Both Ends Important Remarks Separation of Variables Orthogonality: Proof Summary Conclusions ùÇ: 1 üà½ugdÄ ©lCþ{ ¼ê5y² o( 2 eZµã ¼ê5 )5 C. S. Wu 1ù ©lCþ{()
Free Vibration in a String Fixed at Both End 定解问题 考虑长为l、两端固定的弦的自由振动,方程及定 解条件为 a2u a2a 2 0x2 0<x<l,t>0 0 t>0 u4=0=o(x) v(x)0≤x≤l t=0 希望求得的特解具有分离变量的形式,即 u(a, t)=X(rr(t)
Free Vibration in a String Fixed at Both Ends Important Remarks Separation of Variables Orthogonality: Proof Summary Conclusions ½)¯K Äl!üà½ugdħ§9½ )^ ∂ 2u ∂t2 − a 2 ∂ 2u ∂x2 = 0 0 < x < l, t > 0 u x=0 = 0 u x=l = 0 t ≥ 0 u t=0 = φ(x) ∂u ∂t t=0 = ψ(x) 0 ≤ x ≤ l F"¦A)äk©lCþ/ª§= u(x, t) = X(x)T(t) C. S. Wu 1ù ©lCþ{()
Free Vibration in a String Fixed at Both End 寻找分离变量形式u(x,)=X(x)T()的解 u 2 0x2
Free Vibration in a String Fixed at Both Ends Important Remarks Separation of Variables Orthogonality: Proof Summary Conclusions Ïé©lCþ/ªu(x, t) = X(x)T(t)) ∂ 2u ∂t2 −a 2 ∂ 2u ∂x2 =0 C. S. Wu 1ù ©lCþ{()
Free Vibration in a String Fixed at Both End 寻找分离变量形式u(x,)=X(x)T()的解 u 2 0x2 0→X(a)r()=2x"(
Free Vibration in a String Fixed at Both Ends Important Remarks Separation of Variables Orthogonality: Proof Summary Conclusions Ïé©lCþ/ªu(x, t) = X(x)T(t)) ∂ 2u ∂t2 −a 2 ∂ 2u ∂x2 =0 =⇒ X(x)T 00(t)=a 2X 00(x)T(t) C. S. Wu 1ù ©lCþ{()