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2015 9 30 ( ) 2015 9 30 1 / 32

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http://nalab.mind.meiji.ac.jp/koudai2015/ ( ) 2015 9 30 3 / 32

( ) ( ) ( : ) 1 2 (2014) 1 M2 2 ( ) 2015 9 30 4 / 32

( ) ( ) ( : ) 1 2 (2014) YouTube, Chladni, 1 M2 2 ( ) 2015 9 30 4 / 32

( ) ( ) ( : ) 1 2 (2014) YouTube, Chladni, 9/27( ) 1 M2 2 ( ) 2015 9 30 4 / 32

: (, 1756 1827) ( ) ( ) 2015 9 30 5 / 32

(,, ) ( ) Entdeckungen über die Theorie des Klanges (, 1787) Die Akustik (, 1802) Neue Beyträge zur Akustik (, 1817) ( ) m n ( ) 2015 9 30 6 / 32

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1808 (1776 1831) (1811, 1813, 1815) - (1850) - 2 ( u 4 ) t 2 = D u x 4 + 2 4 u x 2 y 2 + 4 u y 4 u = u(x, y, t) (x, y) t ( ) 2015 9 30 8 / 32

( ) 2015 9 30 9 / 32

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( ) ( ) 2015 9 30 9 / 32

( ) 1 ( ) 2 ( ) 2015 9 30 9 / 32

( ) 1 ( ) 2 ( ) ( ) 2015 9 30 9 / 32

( ) 1 ( ) 2 ( ) 20 ( 100 ) (1878 1909) (1909 ) ( ) ( ) 2015 9 30 9 / 32

: 1 ( ) 2 ( ) 3 ( ) ( ) 2015 9 30 10 / 32

: 1 ( ) 2 ( ) 3 ( ) ( ) 2015 9 30 10 / 32

( : ) ( ) (normal mode of vibration) ( ) 2015 9 30 11 / 32

: ( ) (, ) ( ) 2015 9 30 12 / 32

: ( ) (, ) ( ) T T = 2π l g (g = 9.8 [ms 2 ] ( ), l ) ( ) ( ) 2015 9 30 12 / 32

: ( ) 5 10 15 20 1.0 0.5 0.5 1.0 ( ) 2015 9 30 13 / 32

: ( ) 1.0 0.5 5 10 15 20 0.5 1.0 θ = A sin (ωt + ϕ), ω = 2π T ( ). A, ϕ T ( ω ) ( ) 2015 9 30 13 / 32

( ) 2015 9 30 14 / 32

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. ( ) 2015 9 30 14 / 32

( )? 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) ) ( ) 2015 9 30 15 / 32

( ) 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. ( ) 2015 9 30 16 / 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 ( ) 2015 9 30 16 / 32

, ( ) 2015 9 30 17 / 32

: (the Tacoma Narrows Bridge) 1940 11 7 19 m ( ) 2015 9 30 18 / 32

( [1]) ( ) 2015 9 30 19 / 32

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( [1]) ( WWW ) ( ) 2015 9 30 22 / 32

( ) 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 2 180 ( ) 2015 9 30 23 / 32

( ) 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 2 180 ( ) 2015 9 30 23 / 32

(Mary Désirée Waller, 1886 1959, ) Chladni Figures study in symmetry (1961) ( ) ( ) 2015 9 30 24 / 32

( ) ( M2) ( [5], 2015/9/9) ( ) 2015 9 30 25 / 32

(oscillation, vibration) ( ) ( ;,, ) ( ) ( X ) ( ) 2015 9 30 26 / 32

( ) 100 ( ) 2015 9 30 27 / 32

( ) 100 ( ) 2015 9 30 27 / 32

Mary D. Waller, Vibrations of free square plates: part I. normal vibrating modes, Physical Society, Vol. 51 (1939), pp. 831 844. 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. 737 786 (1909).,,, 2012 3.,,, Chladni, 2015, 2015 9 9,. ( ) 2015 9 30 28 / 32

: 1 ( ) 200 (2006) ( ) ( ) ( ) 2015 9 30 29 / 32

( ) 2015 9 30 30 / 32

: ( 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 ). ( ) 2015 9 30 31 / 32

: 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. ( ) ( ) 2015 9 30 32 / 32

: 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. ( ) 1.0 0.5 5 10 15 20 0.5 1.0 ( ) ( ) 2015 9 30 32 / 32