29 21 3 1

1 2 1.1............................ 2 1.2................................... 2 2 3 2.1 Neumann...................... 3 2.2............................. 5 2.3................................. 5 3 2 11 3.1 Neumann...................... 11 3.2......................... 15 4 2 4.1............................... 2 4.2................................. 2 5 21 6 23 1

1 1.1 (1756 1827) ( U = λu, U =, U ) n 1 2 1.2 Mathematica λ YouTube (http://www.youtube.com/watch?v=s9gbf8yly) Neumann 2

2 2.1 Neumann Ω = (, 1) (, 1) 1 2 u c 2 t 2 u (x, y, t) = n = 2 u x + 2 u 2 y 2 u(x, y, ) = f(x, y), ((x, y) Ω, < t), ((x, y) Ω, < t), u (x, y, ) = g(x, y) t ((x, y) Ω). n Ω (x, y) Ω Fourier u(x, y, t) = 1 4 (a + b t) + cos mπx cos nπy + 1 2 + 1 2 m=1 n=1 [a mn cos(c m 2 + n 2 πt) + n=1 m=1 b mn c m 2 + n 2 π sin(c m 2 + n 2 πt) cos nπy[a n cos(cnπt) + b n cnπ sin(cnπt)] cos mπx[a m cos(cmπt) + b m cmπ sin(cmπt)], a mn = 4 b mn = 4 1 1 1 1 f(x, y) cos mπx cos nπydxdy, g(x, y) cos mπx cos nπydxdy. 3

2 U x + 2 U = λu ((x, y) Ω), 2 y2 U n = ((x, y) Ω) U U(x, y) = X(x)Y (y) X (x)y (y) + X(x)Y (y) = λx(x)y (y). X (x) X(x) = λ Y (y) Y (y). µ X (x) = µx(x), Y (y) = (λ µ)y (y). X () = X (1) = Y () = Y (1) =. µ = m 2 π 2, X(x) = A cos mπx (m =, 1, 2,, A ) λ µ = n 2 π 2, Y (y) = B cos nπy (n =, 1, 2,, B ) λ = (m 2 + n 2 )π 2, U(x, y) = C cos mπx cos nπy (C ) 4

2.2 λ = 2π 2 (m, n) = (1, 1) U(x, y) = C cos πx cos πy (C ) λ λ = 5π 2 (m, n) = (2, 1), (1, 2) U(x, y) = C 1 cos 2πx cos πy + C 2 cos πx cos 2πy (C 1, C 2 ) λ λ = 5π 2 (m, n) = (1, 7), (5, 5), (7, 1) U(x, y) = C 3 cos πx cos 7πy+C 4 cos 5πx cos 5πy+C 5 cos 7πx cos πy (C 3, C 4, C 5 ) λ cos mπx cos mπy cos mπx cos mπy = x = 2k 1 2m, y = 2l 1 (k, l = 1, 2, m) 2m a cos mπx cos nπy + b cos nπx cos mπy ( ) 2.3 ( ) a, b, m, n Mathematica 5

m = 6, n = 4, a = 1, b = 1 mm 2.1: cos 6πx cos 4πy + cos 4πx cos 6πy m = 3, n = 5, a = b = 1 mm 2.2: cos 3πx cos 5πy + cos 5πx cos 3πy 6

m = 14, n = 24, a = b = 1 mm 2.3: cos 14πx cos 24πy + cos 24πx cos 14πy m = 7, n = 11, a = b = 1 mm 2.4: cos 7πx cos 11πy + cos 11πx cos 7πy 7

m = 8, n = 4, a = b = 1 mm 2.5: cos 8πx cos 4πy + cos 4πx cos 8πy m = 8, n = 4, a = 2, b = 3 mm 2.6: 2 cos 8πx cos 4πy + 3 cos 4πx cos 8πy 8

Mathematica m = 6, n = 4, a = 1, b = 1 mm 2.7: cos 6πx cos 4π + cos 4πx cos 6πy 9

m = 8, n = 6, a = 4, b = 1 mm 2.8: 4 cos 8πx cos 6π + cos 6πx cos 8πy 2.7 2.8 2.5 2.6 m, n a, b a, b 1 a, b 1 2.8 a = b = 1 m, n 3 1

3 2 3.1 Neumann Ω = (x, y) R 2 ; x 2 + y 2 < 1 Neumann 2 u t 2 = c2 u ((x, y) Ω, < t), u n = ((x, y) Ω, < t), u(x, y, ) = f(x, y), U = 2 U r 2 u (x, y, ) = g(x, y) t + 1 r U r + 2 U θ 2. ((x, y) Ω). ( 2 u 2 t = u 2 c2 r + 1 u 2 r r + 1 ) 2 u ( < r < 1, θ 2π, < t < ). r 2 θ 2 u (1, θ, t) = ( θ 2π, < t). r u u = f(r, θ), = g(r, θ) ( < r < 1, θ 2π, t = ). t u(r, θ, t) = U(r, θ)t (t) λ s.t. U = λu,, T (t) = c 2 λt (t) 11

( 2 u (3.1) r + 1 2 r U n =. U (1, θ) = ( θ 2π) r u r + 1 ) 2 u = λu. ((x, y) Ω) r 2 θ 2 ((x, y) Ω) λ = U λ U(r, θ) = R(r)Θ(θ) R Θ + 1 r R Θ + 1 r 2 RΘ + λrθ =. RΘ r 2 R R + 1 R r R + 1 Θ 2 r 2 Θ =. r 2 R R + r R R + λr2 = Θ2 Θ. θ r µ r 2 R R + r R R + λr2 = µ, Θ2 Θ = µ. Θ 2π Θ = µθ Θ() = Θ(2π), Θ () = Θ (2π). Θ = A cos µθ + B sin µθ, (A, B ) µ = n (n =, 1, 2, ). 12

R µ = n 2. r 2 R R + r R R + λr2 = n 2. r 2 R + rr + (λr 2 n 2 )R =. Bessel r 2 R + rr + (λr 2 n 2 )R = Bessel R(r) = AJ n ( λr) + BY n ( λr) J n (x) Bessel Y n (x) Bessel ( 1) m ( x ) 2m+n J n (x) =, m!γ(m + n + 1) 2 m= Y n (x) = J n(x) cos(nπ) J n (x) sin(nπ) 1.8.6.4.2 -.2 -.4 2 4 6 8 1 3.1: Bessel.5.25 -.25 5 1 15 2 25 -.5 -.75-1 3.2: Bessel Y n (λr) r = R(r) r = B = R(r) = AJ n ( λr). 13

R (1) = R (1) = J n( λ) =. R (r) Ω λ J n(r) = λ l nm l nm Bessel J n(r) : J n(l nm ) =, < l n1 < l n2 < l n3 <. l 2 nm m N s.t. λ = l 2 nm. λ = lnm. U nm (r, θ) = J n (l nm r)[a sin(nθ) + b cos(nθ)] (3.1) = J n (l nm r)c cos(nθ + D) (A,B,C,D ) λ =, lnm 2 λ = U(r, θ) λ = lnm 2 U(r, θ) = CJ n (l nm r) cos(nθ + D) J n (l nm ) = l nm r = µ nk r = µ nk l nm (k = 1, 2, ) cos(nθ + D) = nθ = π 2, 3π 2, θ = π 2n, 3π 2n, U nm m=3,n=4 2 4 ( [2]) 14

3.2 Mathematica 1.5 -.5-1 -1 -.5.5 1.4.2 -.2 -.4-1 -.5 -.5.5 1.5 1-1 3.3: (n=1,m=3,a=1,b=1) 15

1.5 -.5-1 -1 -.5.5 1.4.2 -.2 -.4-1 -.5.5 1 -.5.5 16 1-1 3.4: (n=7,m=4,a=1,b=1)

1.5 -.5-1 -1 -.5.5 1.4.2 -.2 -.4-1 -.5 -.5.5 1.5 17 1-1 3.5: (n=1,m=7,a=1,b=1)

1.5 -.5-1 -1 -.5.5 1.4.2 -.2 -.4-1 -.5 -.5.5 1.5 18 1-1 3.6: (n=12,m=4,a=1,b=1)

3.7: Chladni[3] 3.1 U(r, θ) = CJ n (l nm r) cos(nθ + D) 3.3 3.6 Cladni Cladni 1,2 (m,n=1,1),(m,n=1,2) 4,6,8,11,12 3.3 3.6 19 4,6,8,11,12 3,5,9

4 4.1 a,b 1 m n 3 4.2 3.2 2

5 a cos(mπx) cos(nπy) + b cos(nπx) cos(mπy) (Neumann ) Needs["Graphics ImplicitPlot "] eigenn[m_, n_, a_, b_] := {ImplicitPlot[a*Cos[m Pi x]cos[n Pi y]+b*cos[n Pi x]cos[m Pi y] ==,{x,,1}, {y,,1}, PlotPoints -> 1, AspectRatio -> 1], Plot3D[a*Cos[m Pi x]cos[n Pi y] + b*cos[n Pi x]cos[m Pi y], {x,, 1}, {y,, 1}]} U(r, θ) = CJ n (l nm r) cos(nθ + D) (Neumann ) << NumericalMath BesselZeros For[n =, n < 2, n++, bpz[n] = BesselJPrimeZeros[n, 1, WorkingPrecision ->2]] Needs["Graphics ImplicitPlot "] eigennc1[n_, m_, a_, b_] := ImplicitPlot[BesselJ[n, bpz[n][[m]]*sqrt[x^2 + y^2]]* (a*sin[n*arg[x + I*y]]+b*Cos[n*Arg[x + I*y]]) ==, {x, -1, 1}, {y, -1, 1}, PlotPoints -> 1] eigennc2[n_, m_, a_, b_] := Plot3D[BesselJ[n, bpz[n][[m]]*sqrt[x^2 + y^2]]* (a*sin[n*arg[x + I*y]] + b*cos[n*arg[x + I*y]]), {x, -1, 1}, {y, -1, 1}, PlotPoints -> 1] 21

BesselJPrimeZeros[n,m] J n(x) m 22

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[1] : (1983 9 1 ) [2] (1998 ) [3] Entdeckungen uber die Theorie des Klanges : Chladni,Ernst Florens Friedrich 24