Applications of Bessel ftns 例10.1 求四周固定的圆形薄膜的固有频率 10 1 02u ar (or) r2 a02 l=0有界 0
Cylindrical Functions Applications of Bessel ftns Eigenproblems Involving Bessel Eq Cooling of Cylindrical Body Plane Radial Oscillation of Circular Ring ~10.1 ¦o±½ / kªÇ ∂ 2u ∂t2 − c 2 1 r ∂ ∂r r ∂u ∂r + 1 r 2 ∂ 2u ∂φ2 = 0 u r=0k. u r=a = 0 u φ=0 = u φ=2π ∂u ∂φ φ=0 = ∂u ∂φ φ=2π C. S. Wu 1ù μê()
Applications of Bessel ftns 例10.1 求四周固定的圆形薄膜的固有频率 10 m)+ a2 ar r2 a02 l=0有界 0 q=2 现在要求的就是在上述边界条件的限制下,到底 许可哪些ω值,使得方程有非零解 u(r,, t=v(r, d)e
Cylindrical Functions Applications of Bessel ftns Eigenproblems Involving Bessel Eq Cooling of Cylindrical Body Plane Radial Oscillation of Circular Ring ~10.1 ¦o±½ / kªÇ ∂ 2u ∂t2 − c 2 1 r ∂ ∂r r ∂u ∂r + 1 r 2 ∂ 2u ∂φ2 = 0 u r=0k. u r=a = 0 u φ=0 = u φ=2π ∂u ∂φ φ=0 = ∂u ∂φ φ=2π y3¦Ò´3þã>.^e§. N= ω§¦§k") u(r, φ, t) = v(r, φ)eiωt C. S. Wu 1ù μê()
Applications of Bessel ftns 例10.1 求四周固定的圆形薄膜的固有频率 10 1 02u ar (or) r2 a02 l=0有界 0 将u(r,,t)=v(,)e代入上述方程及边界条 件,即得到v(T,)满足的本征值问题
Cylindrical Functions Applications of Bessel ftns Eigenproblems Involving Bessel Eq Cooling of Cylindrical Body Plane Radial Oscillation of Circular Ring ~10.1 ¦o±½ / kªÇ ∂ 2u ∂t2 − c 2 1 r ∂ ∂r r ∂u ∂r + 1 r 2 ∂ 2u ∂φ2 = 0 u r=0k. u r=a = 0 u φ=0 = u φ=2π ∂u ∂φ φ=0 = ∂u ∂φ φ=2π òu(r, φ, t) = v(r, φ)eiωt\þã§9>.^ §=v(r, φ)÷v ¯K C. S. Wu 1ù μê()
Applications of Bessel ftns 例10.1 本征值问题 10/0 1a2 rr(a-)r202 k2u=0 =0有界 0 了= 0 d=0 d=0 0o\=2 其中k=ω/c,待定 再今(0)=R()(),分离变量,就得到两个《类 常微分方程本征值问题
Cylindrical Functions Applications of Bessel ftns Eigenproblems Involving Bessel Eq Cooling of Cylindrical Body Plane Radial Oscillation of Circular Ring ~10.1 ¯K 1 r ∂ ∂r r ∂v ∂r + 1 r 2 ∂ 2 v ∂φ2 + k 2 v = 0 v r=0k. v r=a = 0 v φ=0 = v φ=2π ∂v ∂φ φ=0 = ∂v ∂φ φ=2π Ù¥k = ω/c§½ 2-v(r, φ) = R(r)Φ(φ)§©lCþ§Òü ~©§ ¯K C. S. Wu 1ù μê()
Applications of Bessel ftns 例10.1 本征值问题 10/0 1a2 rr(a-)r202 k2u=0 =0有界 0 了= 0 d=0 d=0 0o\=2 其中k=ω/c,待定 再令(,)=B(r)p(),分离变量,就得到两个 常微分方程本征值问题
Cylindrical Functions Applications of Bessel ftns Eigenproblems Involving Bessel Eq Cooling of Cylindrical Body Plane Radial Oscillation of Circular Ring ~10.1 ¯K 1 r ∂ ∂r r ∂v ∂r + 1 r 2 ∂ 2 v ∂φ2 + k 2 v = 0 v r=0k. v r=a = 0 v φ=0 = v φ=2π ∂v ∂φ φ=0 = ∂v ∂φ φ=2π Ù¥k = ω/c§½ 2-v(r, φ) = R(r)Φ(φ)§©lCþ§Òü ~©§ ¯K C. S. Wu 1ù μê()