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1 ( ) / 32

2 : ( ) : ( ) :! ( ) / 32

3 ( ) / 32

4 ( ) ( ) ( : ) 1 2 (2014) 1 M2 2 ( ) / 32

5 ( ) ( ) ( : ) 1 2 (2014) YouTube, Chladni, 1 M2 2 ( ) / 32

6 ( ) ( ) ( : ) 1 2 (2014) YouTube, Chladni, 9/27( ) 1 M2 2 ( ) / 32

7 : (, ) ( ) ( ) / 32

8 (,, ) ( ) Entdeckungen über die Theorie des Klanges (, 1787) Die Akustik (, 1802) Neue Beyträge zur Akustik (, 1817) ( ) m n ( ) / 32

9 ( ) / 32

10 ( ) / 32

11 ( ) / 32

12 ( ) ( ) / 32

13 ( ) : ( ) ( ) / 32

14 1808 ( ) (1811, 1813, 1815) - (1850) - 2 ( u 4 ) t 2 = D u x u x 2 y u y 4 u = u(x, y, t) (x, y) t ( ) / 32

15 ( ) / 32

16 ( ) / 32

17 ( ) ( ) / 32

18 ( ) 1 ( ) 2 ( ) / 32

19 ( ) 1 ( ) 2 ( ) ( ) / 32

20 ( ) 1 ( ) 2 ( ) 20 ( 100 ) ( ) (1909 ) ( ) ( ) / 32

21 : 1 ( ) 2 ( ) 3 ( ) ( ) / 32

22 : 1 ( ) 2 ( ) 3 ( ) ( ) / 32

23 ( : ) ( ) (normal mode of vibration) ( ) / 32

24 : ( ) (, ) ( ) / 32

25 : ( ) (, ) ( ) T T = 2π l g (g = 9.8 [ms 2 ] ( ), l ) ( ) ( ) / 32

26 : ( ) ( ) / 32

27 : ( ) θ = A sin (ωt + ϕ), ω = 2π T ( ). A, ϕ T ( ω ) ( ) / 32

28 ( ) / 32

29 A 1 (x) = a 1 sin πx L, A 2(x) = a 2 sin 2πx L, A 3(x) = a 3 sin 3πx L, ( ) ( ) n A n (x) = a n sin nπx L. ( ) / 32

30 ( )? u n = A n (x)sin(ω n t + ϕ n ) = a n sin nπx L (ω n = nπc L ) ( nπc ) sin L t + ϕ n. (x A n (x) ) ( ) / 32

31 ( ) u = A 1 (x) sin (ω 1 t + ϕ 1 ) + A 2 (x) sin (ω 2 t + ϕ 2 ) + = n=1 A n (x) sin (ω n t + ϕ n ), A n (x) = a n sin nπx L. ( ) / 32

32 ( ) u = A 1 (x) sin (ω 1 t + ϕ 1 ) + A 2 (x) sin (ω 2 t + ϕ 2 ) + = A n (x) sin (ω n t + ϕ n ), n=1 A n (x) = a n sin nπx L. = A n (x) = a n sin nπx L ω n ( ) / 32

33 , ( ) / 32

34 : (the Tacoma Narrows Bridge) m ( ) / 32

35 ( [1]) ( ) / 32

36 ( ) ( ) / 32

37 ( ) / 32

38 ( [1]) ( WWW ) ( ) / 32

39 ( ) m = n m, n 1 m = n D 4 2 m = n D 4 3 m > n D 4 4 m > n D 4 5 m > n D 4 6 m > n D 4 7 m > n D 2 D n ( n, ). D 4 90 D ( ) / 32

40 ( ) m = n m, n 1 m = n D 4 2 m = n D 4 3 m > n D 4 4 m > n D 4 5 m > n D 4 6 m > n D 4 7 m > n D 2 D n ( n, ). D 4 90 D ( ) / 32

41 (Mary Désirée Waller, , ) Chladni Figures study in symmetry (1961) ( ) ( ) / 32

42 ( ) ( M2) ( [5], 2015/9/9) ( ) / 32

43 (oscillation, vibration) ( ) ( ;,, ) ( ) ( X ) ( ) / 32

44 ( ) 100 ( ) / 32

45 ( ) 100 ( ) / 32

46 Mary D. Waller, Vibrations of free square plates: part I. normal vibrating modes, Physical Society, Vol. 51 (1939), pp Mary D. Waller, Chladni figures a study in symmetry, G. Bell (1961). von Walter Ritz, Theorie der Transversalschwingungen einer quadratischen Platte mit freien Rändern, Annalen der Physik Volume 333, Issue 4, pp (1909).,,, ,,, Chladni, 2015, ,. ( ) / 32

47 : 1 ( ) 200 (2006) ( ) ( ) ( ) / 32

48 ( ) / 32

49 : ( 0) x kx (k, ) 2 mα(t) = kx(t). x (t) = α(t) = k m x(t) = ω2 x(t) (ω = x (t) = ω 2 x(t) k m ). ( ) / 32

50 : x(t) = C 1 sin ωt + C 2 cos ωt ( ) x(t) = A sin(ωt + ϕ) x (t) = ω 2 x(t) T = 2π ω = 2π m k. ( ) ( ) / 32

51 : x(t) = C 1 sin ωt + C 2 cos ωt ( ) x(t) = A sin(ωt + ϕ) x (t) = ω 2 x(t) T = 2π ω = 2π m k. ( ) ( ) ( ) / 32

85 4

85 4 85 4 86 Copright c 005 Kumanekosha 4.1 ( ) ( t ) t, t 4.1.1 t Step! (Step 1) (, 0) (Step ) ±V t (, t) I Check! P P V t π 54 t = 0 + V (, t) π θ : = θ : π ) θ = π ± sin ± cos t = 0 (, 0) = sin π V + t +V

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#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 = #A A A. F, F d F P + F P = d P F, F P F F A. α, 0, α, 0 α > 0, + α +, α + d + α + + α + = d d F, F 0 < α < d + α + = d α + + α + = d d α + + α + d α + = d 4 4d α + = d 4 8d + 6 http://mth.cs.kitmi-it.c.jp/

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m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m

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ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +

ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + 2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j

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