第二十三讲柱函数(二 3231 Bessel方程的本征值问题 现在从一个具体问题入手,讨论 Bessel方程的本征值问题 [四鬧固定的圆形漶膜的固有频 注意,这个问题不同于过去讨论过的偏微分方程定解问题:现在并没有给出初始条件,所要 求的也不是描写圓形薄膜振动的位移如何随时间和空冋而变化·现在要求的是國有频率,即求出 给定偏微分方程和边界条件下的所有各种振动模弌的角频率·也正是因为现在的冋题中并没有给 出初始条件,所以也不能得出位移转动不变的结论 取平面极坐标系,坐标原点放置在圆形薄膜的中心.这样,偏微分方程和边界条件就是 1a2 0, (231a) r u=0有界 (23.1b) 现在要求的就是在边界条件(231b)和(231c)的限制下,到底许可哪些ω值,使得方程 23.1a)有非零解 u(r,o, t)=u(r, o)e (23.2) 将此解式代入方程(23.1)及边界条件(231b)和(23.lc),并令k=w/e,就可以得到 1 a/ au v-=0有界 0 le=0=叫 =2r 再令v(r,)=R(r)(刂),分离变量,就得到两个本征值问题 p"()+m2()=0 (0)=重(2丌),更(0)=更(2丌) 1d「dF(r) R(0)有界
Wu Chong-shi ✁✂✄☎ ✆ ✝ ✞ (✁ ) §23.1 Bessel ✟✠✡☛☞✌✍✎ ✏✑✒✓✔✕✖✗✘✙✚✛✜✢ Bessel ✣✤✥✦✧★✗✘✩ ✪✫✬ ✭✮✥ ✯✰✱✲✥ ✭✳✴✵ ✶✷✛✸✹ ✺✻✼ ✽✾✿❀❁❂✿❃❄❅❆❇❈❉❊ ✺✻❋●❍■❏❑▲ ▼◆❖P◗✛❘❙ ❚❃❯✼❱❲ ❳❨❩ ❬❭❪❫❃❴❵❛❜❝❞ ❡❢ ❣❡❤✐❥✩●❍❙❚❃❱ ❦❑❧♠✛♥❚ ▼ ▲❉❄❅❆❇❈❢♦♣P◗q❃❘❑rs❪❫t✉❃ ✈❧♠✩❯✇❱ ①②●❍❃ ✺✻ ③■❏❑▲ ▼◆❖P◗✛❘ ④❯✼⑤⑥ ▼❴❵⑦❫✼✐❃⑧❂✩ ⑨⑩❶❷❸❹❺✛❸❹❻❼❽❾✑ ✯✰✱✲✥ ❿➀✩➁➂✛➃➄➅✣✤➆➇➈➉➊➋➌ ∂ 2u ∂t2 − c 2 1 r ∂ ∂r r ∂u ∂r + 1 r 2 ∂ 2u ∂φ2 = 0, (23.1a) u r=0✳ ➈, u r=a = 0, (23.1b) u φ=0 = u φ=2π , ∂u ∂φ φ=0 = ∂u ∂φ φ=2π . (23.1c) ✏ ✑ ➍ ✪ ✥ ➋ ➌ ✑ ➇ ➈ ➉ ➊ (23.1b) ➆ (23.1c) ✥ ➎ ➏ ➐ ✛➑ ➒ ➓ ➔ → ➣ ω ★ ✛↔ ↕ ✣ ✤ (23.1a) ✳➙➛➜ u(r, φ, t) = v(r, φ)eiωt . (23.2) ➝➞➜➟➠✙ ✣✤ (23.1) ➡➇➈➉➊ (23.1b) ➆ (23.1c) ✛➢➤ k = ω/c ✛ ➋ ➔➥↕➑ 1 r ∂ ∂r r ∂v ∂r + 1 r 2 ∂ 2v ∂φ2 + k 2 v = 0, v r=0 ✳ ➈ v r=a = 0, v φ=0 = v φ=2π , ∂v ∂φ φ=0 = ∂v ∂φ φ=2π . ➦➤ v(r, φ) = R(r)Φ(φ) ✛➅➧➨➩✛➋ ↕➑➫✔✦✧★✗✘ Φ 00(φ) + m2Φ(φ) = 0, (23.3a) Φ(0) = Φ(2π), Φ 0 (0) = Φ 0 (2π) (23.3b) ➆ 1 r d dr r dR(r) dr + k 2 − m2 r 2 R(r) = 0, (23.4a) R(0)✳ ➈, R(a) = 0. (23.4b)
§231 Bessel方程的本征值问题 本征值问题(23.3)已经多次见到过,对应于它的本征值 m=0,1,2,3, 本征函数为 所以,在本征值问题(23.4)中,参数m2是已知的,而k2是本征值,待求 可以证明 PA((rdr=mR(G)r(T+dr ar, 所以,一定有本征值k2>0·通过作变换x=Mr,列(x)=B(r),就可以将微分方程(234)化为 Bessel方程,从而求得它的通解 R(r)=CJm(kr)+ DNm(kr) 考虑到边界条件(23.4b)的要求,R(O)有界,故D=0;又由于要求R(a)=0,就得到 将m阶Bsel函数Jn(x)的第i个正零点(由小到大排列记作r),i=1,2,3…,则本征值间 题(23.4)的解是 本征值 i=1,2,3, 237a) 本征函数Fmi(r)=Jm(kmr) 于是就求得了圆形薄膜的固有振动的角频率 其中m)是m阶Bes函数Jn(x)的第i个正零点 在上述求解过程中,实际上用到了有关J(x)零点的结论:当v>-1或为整数时,J(x)有 无穷多个零点,它们全部都是实数,对称地分布在实轴上
Wu Chong-shi §23.1 Bessel ➭➯➲➳➵➸➺➻ ➼ 2 ➽ ✦✧★✗✘ (23.3) ➾➚➪➶➹➑➘✛➴➷➬➮✥✦✧★ m2 , m = 0, 1, 2, 3, · · ·, ✦✧➱✃❐ Φm(φ) = cos mφ, sin mφ. ❒➥✛✑✦✧★✗✘ (23.4) ❿ ✛❮✃ m2 ➌ ➾❰✥✛Ï k 2 ➌✦✧★✛Ð✪✩ ➔➥Ñ Ò k 2 Z a 0 R(r)R ∗ (r)rdr = m2 Z a 0 R(r)R ∗ (r) dr r + Z a 0 dR(r) dr dR∗ (r) dr rdr, ❒➥✛✓✮✳✦✧★ k 2 > 0 ✩Ó➘Ô➨Õ x = kr ✛ y(x) = R(r) ✛ ➋ ➔➥➝➄➅✣✤ (23.4a) Ö❐ Bessel ✣✤✛✒Ï✪↕➮✥ Ó➜ R(r) = CJm(kr) + DNm(kr). (23.5) ×Ø➑ ➇➈➉➊ (23.4b) ✥ ➍✪✛ R(0) ✳ ➈ ✛Ù D = 0 ÚÛ Ü➬➍✪ R(a) = 0 ✛ ➋ ↕➑ Jm(ka) = 0. (23.6) ➝ m Ý Bessel ➱✃ Jm(x) ✥Þ i ✔ß➛❼ (Üà➑áâã) ä Ô µ (m) i ✛ i = 1, 2, 3, · · · ✛å ✦✧★✗ ✘ (23.4) ✥ ➜ ➌ ✦ ✧ ★ k 2 mi = µ (m) i a !2 , i = 1, 2, 3, · · · , (23.7a) ✦✧➱✃ Rmi(r) = Jm(kmir). (23.7b) ➬ ➌➋✪↕æ ✯✰✱✲✥ ✭✳çè✥é✴✵ ωmi = µ (m) i a c, (23.8) ê ❿ µ (m) i ➌ m Ý Bessel ➱✃ Jm(x) ✥Þ i ✔ß➛❼✩ ✑ëì✪➜➘ ✤ ❿ ✛íîëï➑æ✳ð Jν(x) ➛❼ ✥ñ✢❋ò ν > −1 ó❐ô✃õ✛ Jν(x) ✳ ö÷➪ ✔➛❼✛➮øùúû➌ í ✃ ✛➴üý➅þ✑íÿë✩
第二十三讲柱函数( 第3页 Zeros of the functions J,(a)& Nv(z) 1. Real zeros When v is real, the functions Jv(a)& Nv(z)each have an infinite number of zeros, all of which are simple with the possible exception of z=0. For non-negative v the sth positive zeros of these functions are denoted by ju,,s and nu, s respectively 3s2 3.831710.89358 7.015593.95768 312 38.6537310.173477.068058.59601 411.7915313.3236910.2223511.74915 514.9309216.4706313.3611014.89744 618.0710619.6158616.5009218.04340 721.2116422.7600819.6413121.18807 824.3524725.9036722.7820324.33194 927.4934829.0468325.9229627.47529 030.6346132.1896829.0640330.61829 2. McMahons expansions for large zeros -14(-1)(71-31)32(1-1)(8312-982+3779) 3(86) 64(-1)(89-1535415436272372-…,s>n,H=4D2 105(86 6= 3. Complex zeros of J,(a) When 12-l the zeros of J,(z) are all real. If v< -l and v is not an integer the number of complex zeros of J,(z) is twice the integer part of (-v; if the integer part of (-v)is odd two of these zeros lie on the imaginary axis 4. Complex zeros of Nv(a When v is real the pattern of the complex zeros of N,(z)depends on the non-integer part of Attention is confined here to the case v=n.a tive integer or zero
Wu Chong-shi ✁✂✄☎ ✆ ✝ ✞ (✁ ) ➼ 3 ➽ Zeros of the functions Jν(z) & Nν (z) 1. Real zeros When ν is real, the functions Jν(z) & Nν (z) each have an infinite number of zeros, all of which are simple with the possible exception of z = 0. For non-negative ν the sth positive zeros of these functions are denoted by jν,s and nν,s respectively. s j0,s j1,s n0,s n1,s 1 2.40483 3.83171 0.89358 2.19714 2 5.52008 7.01559 3.95768 5.42968 3 8.65373 10.17347 7.06805 8.59601 4 11.79153 13.32369 10.22235 11.74915 5 14.93092 16.47063 13.36110 14.89744 6 18.07106 19.61586 16.50092 18.04340 7 21.21164 22.76008 19.64131 21.18807 8 24.35247 25.90367 22.78203 24.33194 9 27.49348 29.04683 25.92296 27.47529 10 30.63461 32.18968 29.06403 30.61829 2. McMahon’s expansions for large zeros jν,s, nν,s ∼ β − µ − 1 8β − 4(µ − 1)(7µ − 31) 3(8β) 3 − 32(µ − 1)(83µ 2 − 982µ + 3779) 15(8β) 5 − 64(µ − 1)(6949µ 3 − 153855µ 2 + 1585743µ − 6277237) 105(8β) 7 − · · · · · · , s ν, µ = 4ν 2 , β = s + ν 2 − 1 4 π, for jν,s s + ν 2 − 3 4 π, for nν,s 3. Complex zeros of Jν(z) When ν ≥ −1 the zeros of Jν(z) are all real. If ν < −1 and ν is not an integer the number of complex zeros of Jν(z) is twice the integer part of (−ν); if the integer part of (−ν) is odd two of these zeros lie on the imaginary axis. 4. Complex zeros of Nν (z) When ν is real the pattern of the complex zeros of Nν(z) depends on the non-integer part of ν. Attention is confined here to the case ν = n, a positive integer or zero
231 Bessel方程的本征值问题 第4页 The figure 23. 1 shows the approximate distribution of the complex zeros of Nn(z) in the region I arg al<T. The figure is symmetrical about the real axis. The two curves on the left extend to infinit having the asymptotes Imz=±-ln3=±0.54931.. There are an infinite number of zeros near each of The two curves extending from z=-n to z=n and bounding an eye-shaped domain intersect the imaginary axis at the points ti(na+b), where Figure 23.1 Zeros of Nn(2) f-1=0.6274 and to=1.19968 is the positive root of cotht=t. There are n zeros near each of these curves. Complex zeros of No(z) Complex zeros of Ni(a) Real part Imaginary part Real part Imaginary part 2.403020.53988 0.78624 5.519880.54718 0.54841 7.01590
Wu Chong-shi §23.1 Bessel ➭➯➲➳➵➸➺➻ ➼ 4 ➽ Zeros of Nn(z) The figure 23.1 shows the approximate distribution of the complex zeros of Nn(z) in the region | arg z| ≤ π. The figure is symmetrical about the real axis. The two curves on the left extend to infinity, having the asymptotes Im z = ± 1 2 ln 3 = ±0.54931 . . .. . . There are an infinite number of zeros near each of these curves. The two curves extending from z = −n to z = n and bounding an eye-shaped domain intersect the imaginary axis at the points ±i(na + b), where Figure 23.1 Zeros of Nn(z) a = q t 2 0 − 1 = 0.66274 . . .. . . b = 1 2 q 1 − t −2 0 ln 2 = 0.19146 . . .. . . and t0 = 1.19968 . . .. . . is the positive root of coth t = t. There are n zeros near each of these curves. Complex zeros of N0(z) Complex zeros of N1(z) Real part Imaginary part Real part Imaginary part −2.40302 0.53988 −0.50274 0.78624 −5.51988 0.54718 −3.83353 0.56236 −8.65367 0.54841 −7.01590 0.55339
第二十三讲柱函数( 第5页 为了在分离变量法中的应用,自然要讨论上面得到的本征函数的正交归关系.下面,介绍 种略为不同的做法,可以同时得到本征函数的正交归一关系 首先,写出本征函数Rm1(r)=Jn(kmr)所满足的微分方程和边界条件, r2 Jm(kimi) (23.9a) Jn(0)有界,Jn(kma)=0. 同时,再写出函数R(r)=Jm(kr)所满足的微分方程和边界条件, 1d「dn(kr) dr Jm(O)有界 由于其中的k为任意实数,所以一般说来,不会有Jm(ka)=0 再用rJm(kr)和rJm(kmr)分别乘方程(239a)和(23.10a) d「dJn(kmin) m(kmir)Jm(er) Jm(mir)dram(kr2 72rJm(kmir)J m(kr)=0, 相减,并在区间0,]上积分,就得到 代入边界条件(23.9b)和(23.10b),可以将上面的结果化为 (kmi-k2)/Jm(kmir) Jm(kr)rdr=-kimiaJm(k a) 'm(kmia (2311) 我们对两个特殊情形感兴趣.第一种情形是k=km≠km这时就有Jm1(km (23.11)式的右端为0.但由于km≠km,所以 即对应于不同本征值的本征函数在区间0,a]上以权重r正交 另一种情形是k=km,这时(23.11)式的两端均为0.我们可以先将(23.11)式的两端同除以 k2,然后取极限k→km,这样就得到 Aim gJm(ka) J'm(kmia)=2 I'm(kimia 2313) 这正是本征函数Jm( kmir)的模方 如果将本征值问题(239)中r=a端的齐次边界条件(23.9b)改为第二类或第三类边界条件, 也可以类似地讨论 事实上,可以把这三种情形统一写成 1 d dR(n1+(k2- R(r)=0, (2314a)
Wu Chong-shi ✁✂✄☎ ✆ ✝ ✞ (✁ ) ➼ 5 ➽ ❐ æ✑➅➧➨➩✟ ❿✥➷ï✛ ✠✡➍✜✢ë❶↕➑✥✦✧➱✃✥ß☛☞✓ð❺✩ ➐ ❶✛✌✍ ✓✎✏❐✑✒✥✓ ✟✛➔➥✒õ ↕➑✦✧➱✃✥ß☛☞✓ð❺✩ ✔✕✛✖✗✦✧➱✃ Rmi(r) = Jm(kmir) ❒✘✙✥ ➄➅✣✤➆➇➈➉➊✛ 1 r d dr r dJm(kmir) dr + k 2 mi − m2 r 2 Jm(kmir) = 0, (23.9a) Jm(0)✳ ➈, Jm(kmia) = 0. (23.9b) ✒õ ✛➦✖✗➱✃ R(r) = Jm(kr) ❒✘✙✥ ➄➅✣✤➆➇➈➉➊✛ 1 r d dr r dJm(kr) dr + k 2 − m2 r 2 Jm(kr) = 0, (23.10a) Jm(0)✳ ➈. (23.10b) Ü ➬ê ❿✥ k ❐✚✛í ✃ ✛❒➥✓✜✢✣✛ ✑✤✳ Jm(ka) = 0 ✩ ➦ï rJm(kr) ➆ rJm(kmir) ➅✥✦✣✤ (23.9a) ➆ (23.10a) ✛ Jm(kr) d dr r dJm(kmir) dr + k 2 mi − m2 r 2 rJm(kmir)Jm(kr) = 0, Jm(kmir) d dr r dJm(kr) dr + k 2 − m2 r 2 rJm(kmir)Jm(kr) = 0, ✧★✛➢✑✩✪ [0, a] ë✫➅✛➋ ↕➑ k 2 mi − k 2 Z a 0 Jm(kmir)Jm(kr)rdr = r Jm(kmir) dJm(kr) dr − Jm(kr) dJm(kmir) dr r=a r=0 . ➠✙ ➇➈➉➊ (23.9b) ➆ (23.10b) ✛➔➥➝ë❶ ✥ñ✬Ö❐ k 2 mi − k 2 Z a 0 Jm(kmir)Jm(kr)rdr = −kmiaJm(ka)J0 m(kmia). (23.11) ✭ø➴➫✔✮✯✰✰✱✲✳✩ Þ ✓✎✰✰➌ k = kmj 6= kmi ✩➁õ➋✳ Jm(kmja) = 0 ✛✴➞ (23.11) ➟ ✥✵✶❐ 0 ✩✷ Ü ➬ kmj 6= kmi ✛❒➥ Z a 0 Jm(kmir)Jm(kmj r)rdr = 0, kmi 6= kmj , (23.12) ✸➴➷➬✑✒✦✧★✥✦✧➱✃✑✩✪ [0, a] ë➥✹✺ r ß☛✩ ✻✓✎✰✰➌ k = kmi ✛➁ õ (23.11) ➟ ✥ ➫ ✶✼❐ 0 ✩✭ø➔➥✕➝ (23.11) ➟ ✥ ➫ ✶✒✽➥ k 2 mi − k 2 ✛✡✾⑨❷➎ k → kmi ✛➁➂➋ ↕➑ Z a 0 J 2 m(kmir)rdr = − lim k→kmi kmia k 2 mi − k 2 Jm(ka)J0 m(kmia) = a 2 2 [J0 m (kmia)]2 . (23.13) ➁ß➌✦✧➱✃ Jm(kmir) ✥✿✣ ✩ ❀ ✬ ➝ ✦✧★✗✘ (23.9) ❿ r = a ✶✥❁➶➇➈➉➊ (23.9b) ❂❐Þ❃❄óÞ❅❄➇➈➉➊✛ ❆➔➥❄❇ý✜✢✩ ❈íë✛➔➥❉➁ ❅ ✎✰✰❊ ✓✖❋ 1 r d dr r dR(r) dr + k 2 − m2 r 2 R(r) = 0, (23.14a)