http://www.ike-dyn.ritsumei.ac.jp/ hyoo/wave.html 1 1, 5 3 1.1 1..................................... 3 1.2 5.1................................... 4 1.3.......................... 5 1.4 5.2, 5.3.................... 6 2 2 9 2.1 2.1-2.3......................... 9 2.2 [ ] 2.4......................... 11 2.3 [ ] 2.4......... 12 2.4 [ ].......................... 13 2.5 [ ]..................... 15 3 N 3 19 3.1 3.1.............................. 19 3.2 large N 3.2, 3.3................... 21 3.3.......................... 25 4 4 26 4.1 N 4.1, 4.4............ 26 4.2................................. 27 1

4.3 4.1....................... 29 4.4 8.3............................. 29 5 4 30 5.1 4.2....................... 30 5.2 4.2........................ 32 5.3 4.2..................... 32 5.4 6.4....................... 33 5.5 [ ] 4.2........... 33 6 5.5 35 6.1 [ ]............................. 35 6.2 [ ]............................... 39 6.3 [ ].......................... 42 7 7 44 7.1............................... 44 7.2................................. 45 7.3.............................. 46 7.4 2π........................... 46 7.5 [ ]................................... 47 7.6 [ ]........................ 48 8 50 8.1..................................... 50 8.2 A-D..................... 51 8.3............................ 53 8.4..................... 54 A 56 A.1........................................ 56 A.2................................. 57 A.3......................................... 58 A.4......................................... 61 A.5.................................... 62 A.6................................. 63 2

1 1, 5 1.1 1 1.1 k m mẍ(t) = kx(t) (1.1) k m ω2 0 (1.2) ẍ(t) + ω 2 0x(t) = 0 (1.3) x(t) = A cos( ω 0 t + φ) (1.4) 1.4 T = 2π ω 0 ν = 1 T 1 ω 0 3

E = 1 2 m{ẋ(t)}2 + 1 2 k{x(t)}2 = 1 2 m ω2 0A 2 (1.5) [Memo] x(t) = A sin( ωt + ϕ) (1.4) 1.2 5.1 5.1 mẍ(t) = kx(t) Γẋ(t) (1.6) k m ω2 0, Γ m γ (1.7) ẍ(t) + γẋ(t) + ω 2 0x(t) = 0 (1.8) γ/2 ω 0 ( ) x(t) = A e γt/2 cos ω0 2 (γ/2)2 t + φ (1.9) ω0 2 (γ/2)2 A e γt/2 5.2 [Memo] ω0 2 (γ/2)2 ω 0 4

1.3 5.5 mẍ(t) = kx(t) Γẋ(t) + F (t) (1.10) ω F (t) = F 0 cos ωt (1.11) mẍ(t) = kx(t) + F 0 cos ωt (1.12) k m ω2 0, 5 F 0 m f 0 (1.13)

ẍ(t) + ω 2 0x(t) = f 0 cos ωt (1.14) x(t) = A cos ωt (1.15) A ( ω 2 0 ω 2 )A = f 0 A = x(t) = ω 2 0 f 0 f 0 ω 2 0 ω2 (1.16) ω2 cos ωt (1.17) ω ω 0 [ ] (1.14) ẍ(t)+ ω 2 0x(t) = 0 (1.4) 1.4 5.2, 5.3 k m ω2 0, Γ m γ, F (t) f(t) (1.18) m (1.10) ẍ(t) + γẋ(t) + ω 2 0x(t) = f 0 cos ωt (1.19) x(t) = A cos ωt + B sin ωt (1.20) A, B ( ) ( ) ( ) ω0 2 ω 2 γ ω A f 0 = γ ω ω0 2 ω 2 B 0 (1.21) 6

A, B ( ) ( ) A f 0 ω0 2 ω 2 = B ( ω0 2 ω2 ) 2 + (γ ω) 2 γ ω (1.22) x(t) = A cos ωt + B sin ωt = A 2 + B 2 cos( ωt + φ 0 ) = f 0 ( ω 2 0 ω 2 ) 2 + (γ ω) 2 cos( ωt + φ 0) (1.23) tan φ 0 = B A = γ ω ω 2 0 ω2 (1.24) [Memo] (1.19) φ 0 f(t) = f 0 cos ωt 1 χ( ω) (1.25) ( ω 2 0 ω 2 ) 2 + (γ ω) 2 ω 5.7 ω ω 0 χ( ω) 7

ω ω 0 ω = ω 0 ω 0 χ( ω) χ( ω 0 ) 1 2 χ( ω) χ( ω 0 ) = 1 (1.26) 2 γ ω 0 ( ω 2 0 ω 2 ) 2 + (γ ω) = 1 2 2 ( ω 2 0 ω2 γ ω 0 ( ω 2 ω 2 0 γ ω 0 ω ω 0 ω 0 ) 2 ( ω + ω 0 ) 2 ( ω + ω 0 ) 2 = 2 ) 2 = 2 (1.27) ( ω 2 ω 2 0 ) 2 = 1 γ ω 0 ω 2 ω0 2 γ ω 0 ω 2 ω0 2 γ ω 0 (1.28) ω 0 ω ω 0 γ ω + ω 0 ω ω 0 γ 2 ω 0 ω = γ = Γ m (1.29) ω 0 ω ω 0 = γ ω 0 = Γ m [Memo] Q m k = Γ (1.30) mk Q Q ω 0 ω = mk Γ [ ] (1.19) ẍ(t) + γẋ(t) + ω 2 0x(t) = 0 (1.9) 8

2 2 2.1 2.1-2.3 2.1.1 2.5 m 1 ẍ 1 (t) = k 1 x 1 (t) + k 2 {x 2 (t) x 1 (t)} m 2 ẍ 2 (t) = k 2 {x 2 (t) x 1 (t)} + k 3 { x 2 (t)} (2.1) m 1 = m 2 M k 1 = k 2 = k 3 K (2.2) K M ω2 0 (2.3) ẍ 1 (t) = 2 ω0x 2 1 (t) + ω0x 2 2 (t) ẍ 2 (t) = ω0x 2 1 (t) 2 ω0x 2 2 (t) (2.4) 2.1.2 x 1(t) + x 2 (t) 2 x 2 (t) x 1 (t) q c (t) 1 2 {x 1 (t) + x 2 (t)} q r (t) 1 2 {x 2 (t) x 1 (t)} (2.5) 9

[Memo] q c (t) q r (t) x 1 (t), x 2 (t) (2.4) q c (t), q r (t) q c (t) = ω0q 2 c (t) (2.6) q r (t) = 3 ω0q 2 r (t) q c (t) = A c cos( ω 0 t + φ c ) q r (t) = A r cos( 3 ω 0 t + φ r ) (2.7) ω 0 3 ω0 2 2.6 2.1.3 E = 1 2 M{ẋ 1(t)} 2 + 1 2 M{ẋ 2(t)} 2 + 1 2 K{x 1(t)} 2 + 1 2 K{x 2(t) x 1 (t)} 2 + 1 2 K{x 2(t)} 2 (2.8) 10

q c (t) q r (t) E = 1 2 M{ q c(t)} 2 + 1 2 M{ q r(t)} 2 + 1 2 K{q c(t)} 2 + 1 2 3K{q r(t)} 2 (2.9) q c (t) = A c cos( ω 0 t + φ c ), q r (t) = 0 E c = 1 2 M ω2 0A 2 c (2.10) q c (t) = 0, q r (t) = A r cos( 3 ω 0 t + φ r ) E r = 1 2 M( 3 ω 0 ) 2 A 2 r (2.11) [Memo] E = 1 2 m ω2 A 2 (1.5) (2.10) (2.11) 2.2 [ ] 2.4 3 ( ) ( ) ( ) ẍ 1 (t) = ω0 2 2 1 x 1 (t) ẍ 2 (t) 1 2 x 2 (t) (2.12) ( ) x 1 (t) x(t) = x 2 (t) ( ) 2 1 H = 1 2 (2.13) ẍ(t) = ω 2 0Hx(t) (2.14) 3 11

H (i) H s s s ω 0 (ii) q(t) u x(t) H u x(t) 2.3 [ ] 2.4 H (2.13) ( ) 2 1 H = 1 2 (2.15) H s det (H s I) = 0 (2.16) I s = 1, 3 (2.17) s c = 1, s r = 3 (2.18) H (i) s c = 1 u c Hu c = s c u c (2.19) ( ) 1 1 u c = 0 (2.20) 1 1 u c ( ) u c = 1 1 2 1 1 2 u c u c = 1 s c = 1 (2.21) 12

1 ω 0 q c (t) u c x(t) = 1 2 (x 1 (t) + x 2 (t)) [Memo] (2.5) (ii) s r = 3 u r Hu r = s r u r (2.22) ( ) 1 1 u r = 0 (2.23) 1 1 u r ( ) u r = 1 1 2 1 (2.24) s c = 3 3 ω 0 q r (t) u r x(t) = 1 2 ( x 1 (t) + x 2 (t)) 2.4 [ ] ẍ(t) = ω 2 0Hx(t) (2.25) x(t) N (N 2) ω 0 M K Mẍ(t) = Kx(t) M 1 K = ω0h 2 ω0 2 H ẍ(t) = Hx(t) H ω0 2 H H H T H T = H 13

H Hu m = s m u m (2.26) u m 1 [ ] m = 1, 2, 3,... s 1 s 2 s 3... s N s m = s m+1 H ) U = (u 1 u 2 u N (2.27) U U U T U 1 U T = U 1 U 1 HU s 1 U 1 s 2 HU =... S (2.28) s N ẍ(t) = ω0hx(t) 2 (2.29) U 1 U 1 ẍ(t) = ω0u 2 1 Hx(t) (2.30) UU 1 = I I U 1 ẍ(t) = ω 2 0U 1 HUU 1 x(t) (2.31) 14

U 1 x(t) q(t) (2.32) q(t) = ω 2 0Sq(t) (2.33) S x(t) q(t) q(t) q(t) = U 1 x(t) m q m (t) m = 1, 2,..., N q m (t) = u m x(t) (2.34) U 1 = U T q m (t) q m (t) = s m ω 2 0q m (t) (2.35) ω m s m ω 0 (i) H s m s m s m ω 0 (ii) q m (t) u m x(t) (i)(ii) x(t) = m q m (t)u m = m A m cos( s m ω 0 t + φ m )u m (2.36) m x(t) = q m (t)u m = A m cos( s m ω 0 t + φ m )u m (2.37) u m A m cos( s m ω 0 t + φ m ) 2.5 [ ] 15

( 1) ( 7) ( 2) ( 8) [ ] 16

n 4 K n, n M n n x n (t) M 1 ẍ 1 (t) = K 1 x 1 (t) + K 2 {x 2 (t) x 1 (t)} (2.38) M 2 ẍ 2 (t) = K 2 {x 2 (t) x 1 (t)} K 3 M 1 = M 2 M K 1 = K 2 K (2.39) ω 0 K M ẍ 1 (t) = 2 ω0x 2 1 (t) + ω0x 2 2 (t) ẍ 2 (t) = ω0x 2 1 (t) ω0x 2 2 (t) (2.40) (2.41) ( ) x 1 (t) x(t) = x 2 (t) ( ) 2 1 H = 1 1 (2.42) ẍ(t) = ω 2 0Hx(t) (2.43) 1 (i) H s m s m ω m s m ω 0 (ii) u m 2 M = 70[t] = 70, 000[kg], K = 20, 000[kN/m] ω m (m = 1, 2) T m = 2π ω m (2.44) 4 17

3 F (t) = F 0 cos ωt (2.45) 5.11 ω ω m T = 2π ω T m = 2π ω m 2 m = 1 m = 2 4 18

3 N 3 3.1 3.1 3.1 3 ẍ 1 2 1 0 M = K 1 2 1 ẍ 2 ẍ 3 0 1 2 x 1 x 2 x 3 (3.1) ẍ(t) = ω 2 0Hx(t), ω 0 = K M (3.2) (i)(ii) (i) H s m s m ω m s m ω 0 (ii) u m x(t) = A m cos( s m ω 0 t + φ m )u m (2.37) H s det(h si) = 0 (3.3) s 1 = 2 2, s 2 = 2, s 3 = 2 + 2 (3.4) u Hu m = s m u m (m = 1, 2, 3) (3.5) u 1 = 1 1 2, u 2 = 1 1 0, u 3 = 1 1 2 (3.6) 2 2 2 1 1 1 19

3.2 x 1 (t) = A 1 cos( ω 1 t + φ 1 )u 1 (3.7) u 1 ω 1 x 2 (t) = A 2 cos( ω 2 t + φ 2 )u 2 (3.8) u 2 ω 2 x 3 (t) = A 3 cos( ω 3 t + φ 3 )u 3 (3.9) u 3 ω 3 20

3.2 large N 3.2, 3.3 3.5 N ẍ(t) = ω0hx(t) 2 (3.10) K ω 0 = H 2 1 0 M H x(t) q(t) (1) H Hu m = s m u m m (m = 1, 2,..., N) 3.4 (2) x(t) = A m cos( s m ω 0 t + φ m )u m (3.11) 21

(N ) 3.2.1 N = 3, 10, 30 1 ω m m 1: m ω m m vs ω m / ω 0 = s m N = 3, 10, 30. ω m m 3.2.2 N = 10 2 22

1 2: N = 10 m = 1, 2, 3, 4, 5. m = 6, 7, 8, 9, 10. 3.2.3 2 m m = 1 m m = 10 m m 23

m 3.3 E m = 1 2 M ω2 ma 2 m = s m 1 2 M ω2 0A 2 m (3.12) E m s m [ ] K A E = 1 2 M ω2 A 2 (1.5) N 3.2.4 N x(t) = A m cos( ω m t + φ m )u m (3.13) m=1 x(t) = A m cos( ω m t + φ m )u m (3.14) m: (3.14) 1 ω m m [ ] N N 3 [ ] 24

3: N = 10 3.3 N E = N i=1 1 2 M{ẋ i(t)} 2 + N i=0 1 2 K{x i+1(t) x i (t)} 2 (3.15) x 0 (t) = x N+1 (t) = 0 5 E = 1 2 M ẋ(t) ẋ(t) + 1 Kx(t) Hx(t) (3.16) 2 H 2 1 0 N x(t) = A m cos(ω m t + φ m )u m (3.17) m=1 6 1 m = m u m u m = (3.18) 0 m m E = = N m=1 N m=1 ( 1 2 Mω2 ma 2 m sin 2 (ω m t + φ m ) + 1 ) 2 Mω2 0s m A 2 m cos 2 (ω m t + φ m ) 1 2 Mω2 ma 2 m (3.19) Hu m = s m u m K = Mω 2 0 ω2 m = s m ω 2 0 5 M K E = 1 2 ẋ(t) Mẋ(t) + 1 x(t) Kx(t) 2 6 U T = U 1 U 1 U = U T U = I 25

4 4 N x(t) = A m cos( ω m t + φ m )u m large N m=1 ẍ(t) = ω 2 0Hx(t) N ẍ(t) = ω0hx(t) 2 N solve x(t) = N A m cos( ω m t + φ m )u m m=1 N solve 4.1 N 4.1, 4.4 ẍ(t) = ω0hx(t) 2 ẍ 1 (t) 2 1 x 1 (t) ẍ 2 (t) = ω0 2 1 2 1 x 2 (t)..... ẍ N (t) 1 2 x N (t) (4.1) N x 0 (t) = x N+1 (t) = 0 x 0 (t) ẍ 1 (t) 1 2 1 x 1 (t) ẍ 2 (t) = ω0 2 1 2 1 x 2 (t).... (4.2). ẍ N (t) 1 2 1 x N (t) x N+1 (t) n (n = 1, 2,..., N) ẍ n (t) = ω 2 0{x n+1 (t) 2x n (t) + x n 1 (t)} (4.3) 2 N ( T x(t) = x 1 (t) x 2 (t) x N (t)) (4.4) 26

nd d (4.3) ψ(nd, t) x n (t) = ψ(nd, t) (4.5) 2 ψ(nd, t) 2 ψ(nd + d, t) 2ψ(nd, t) + ψ(nd d, t) t 2 = ( ω 0 d) d 2 (4.6) N L = (N + 1)d d d 0 nd ξ 2 ψ(ξ, t) t 2 = ( ω 0 d) 2 2 ψ(ξ, t) ξ 2 (4.7) K M E ρ K = E S d ω 0 = 2 ψ(ξ, t) t 2 [Memo] M = ρsd (4.8) K M = E ρ d 2 (4.9) = E ρ E ρ 2 ψ(ξ, t) ξ 2 (4.10) v 4.2 N N N (2.43) ẍ 1 (t) 2 1 x 1 (t) ẍ 2 (t) = ω0 2 1 2 1 x 2 (t)..... ẍ N (t) 1 1 x N (t) 2 1 (4.11) 27

x 0 (t) = 0, x N+1 (t) = x N (t) (4.12) x 0 (t) ẍ 1 (t) 1 2 1 x 1 (t) ẍ 2 (t) = ω0 2 1 2 1 x 2 (t)..... ẍ N (t) 1 2 1 x N (t) x N+1 (t) (4.13) n (n = 1, 2,..., N) ẍ n (t) = ω 2 0{x n+1 (t) 2x n (t) + x n 1 (t)} (4.14) x n (t) = ψ(nd, t) d 2 ψ(nd, t) 2 ψ(nd + d, t) 2ψ(nd, t) + ψ(nd d, t) t 2 = ( ω 0 d) d 2 (4.15) N h = Nd d d 0 nd ξ 2 ψ(ξ, t) t 2 = ( ω 0 d) 2 2 ψ(ξ, t) ξ 2 (4.16) (4.12) ψ(0, t) = 0, ψ x (h, t) = 0 (4.17) ψ x x K M G 7 ρ K = G S d ω 0 = 2 ψ(ξ, t) t 2 M = ρsd (4.18) K M = G ρ d 2 (4.19) = G ρ 2 ψ(ξ, t) ξ 2 (4.20) 7 http://ja.wikipedia.org/wiki/ 28

4.3 4.1 v 8 E ρ v = T σ 1 κρ (4.21) [Memo] 4.1 4.2 4.4 8.3 E = (E x, E y, E z ) (x, y, z) t E i (x, y, z, t) 2 ( E i t 2 = c 2 2 ) E i x 2 + 2 E i y 2 + 2 E i z 2 c B B i (x, y, z, t) (4.22) 8 E ρ T σ κ 29

5 4 2 ψ t 2 = v2 2 ψ x 2 (5.1) ψ(0, t) = ψ(l, t) = 0 (5.2) N 5.1 4.2 [Memo] (step1) ψ(x, t) = f(x)g(t) (5.3) f(x), g(t) ω f(x), g(t) (step2) d 2 g dt 2 = ω2 g(t) d 2 f ( ω ) (5.4) 2 dx 2 = f(x) v f(x) (5.4) f(0) = f(l) = 0 ( ω ) ( ω ) f(x) = α sin v x + β cos v x 9 f(0) = 0 β = 0 f(l) = 0 m = 1, 2,... (5.5) ω L = mπ (5.6) v 9 ω ω f(x) = A cos v x + φ f(x) = B sin v x + ϕ OK 30

(step3) g(t) g(t) = A cos( ωt + φ) (5.7) ω = mπ L v g(t) = A cos ( mπ ) L vt + φ m (5.8) (step4) α = 1 ( mπ ) ( mπ ) ψ m (x, t) = A m cos L vt + φ m sin L x (5.9) x(t) = A m cos(ω m t + φ m )u m (3.11) (5.1) (step5) (5.1) ψ(x, t) = m=1 ( mπ ) ( mπ ) A m cos L vt + φ m sin L x (5.10) 31

5.2 4.2 T m λ m 10 ψ m (x, t + T m ) = ψ m (x, t) ψ m (x + λ m, t) = ψ m (x, t) (5.11) (5.9) T m λ m T m = 2L mv (5.12) λ m = 2L m λ m = vt m (5.13) v = 331[m/s] T m = 2L mv ν m 1 T m = m 2L v (5.14) (5.13) λ m ν m = v (5.15) m = 1 m = 2 m = 3 m = 4 2L v L v 2L 3v T m 2L λ m 2L L 3 ω m k m π L v π L 2π L v 2π L 3π L v 3π L L 2v L 2 4π L v 4π L 1: 5.3 4.2 T m λ m ω m 2π T m = mπ L v k m 2π = mπ λ m L 10 T m λ m (5.16) 32

(5.9) ψ m (x, t) = A m cos( ω m t + φ m ) sin(k m x) (5.17) ω m k m ω m = vk m (5.18) 5.4 6.4 (5.10) (5.10) ψ(x, t) = A m 2 m=1 ( mπ ) sin L (x vt) φ m = f(x vt) + g(x + vt) + A m 2 m=1 ( mπ ) sin L (x + vt) + φ m A m φ m f(s), g(s) s v v (5.19) 6.5 5.5 [ ] 4.2 2 ψ(x, t) t 2 = v 2 2 ψ(x, t) x 2 (5.20) 33

ψ(x + L, t) = ψ(x, t) (5.21) ψ(x, t) = m=0 ( ) { ( ) ( )} 2mπ 2mπ 2mπ cos L vt + φ m C m cos L x + D m sin L x (5.22) 34

6 5.5 2.5 5.13 N ν m 20 Hz - 20 khz 6.1 [ ] 6.1.1 11 11 7,8 35

[Memo] 8 A m m m L (5.14) ν 1 = 1 2L v (6.1) 6.1.2 ν : ν ν c 9 8 ν c 5 4 ν c 4 3 ν c 3 2 ν c 5 3 ν c 15 8 ν c 2ν c 2: ν c 2 2 12 36

[ ] 12 2 1/12 [Memo] 6.1.3 12 ν 1 L (5.14) ν 1 = 1 2L v = 1 T 2L σ (6.2) 12 37

8 L 9 L 4 5 L 3 4 L 2 3 L 3 5 L 8 15 L 1 2 L 9 ν c 8 ν 5 c 4 ν 4 c 3 ν 3 c 2 ν 5 c 3 ν 15 c 8 ν c 2ν c 3: 4: L {λ m = 2Lm : m = 1, 2,... } m = 1 m = 2, 3,... {ν m = mν 1 : m = 2, 3,...} L ν 1 = 1 { 2L v ν c ν c, 9 8 ν c, 5 4 ν c, 4 3 ν c, 3 2 ν c, 5 3 ν c, 15 } 8 ν c, 2ν c 13 13 38

ν 1 = v = v λ 1 2L [ ] (i) (ii) 3 3 6.2 [ ] http://otonanokagaku.net/feature/vol12/index.html http://homepage3.nifty.com/kuebiko/science/119th/sci 119.htm 6.2.1 RLC 5: C[F] L[H] 39

14 7 ν LC 1 ν r = 2π LC (6.3) ν [Memo] RLC - L M C K 1 ν = 1 K 2π M 6.2.2 700[kHz] 20[Hz] 20[kHz] ν r 700[kHz] 2 ν r 699.56[kHz] ν = 440[Hz] 6.2.3 2.5 2.9(a)(b) ν ν ν 2 ν + ν 2 ν 14 40

a(t) = A sin (2πνt π ν t) + A sin (2πνt + π ν t) (6.4) a(t) = 2A cos(π ν t) sin(2πνt) (6.5) a(t) 2.9(c) 2A cos(π ν t) sin(2πνt) 2 ν 2.9(c) T beat = 1 ν ν beat ν ν beat = ν (6.6) ν 2A cos(π ν t) [ ] (i) 41

(ii) (6.3) (6.6) 3 6.3 [ ] 6 6: 4.2 2 ψ(x, t) t 2 = v 2 2 ψ(x, t) x 2 (6.7) ψ(0, t) = 0, ψ x (h, t) = 0 (6.8) h x x λ 1 h λ 1 = 4h (6.9) λ 1 = vt 1 T 1 T 1 = 4L v = 4 ρ G h (6.10) 42

T 1 = 0.02h (6.11) E = 30[GPa] = 30 10 9 [N/m 2 ] ν = 0.2 ρ = 2.5[kg/m 3 ] G = E 2(1 + ν) = 1.25 1010 [N/m 2 ] (6.12) ρ G = 4.5 10 4 [s/m] (6.13) h[m] ρ T 1 = 4 h 0.002h[s] (6.14) G (6.11) 2.5 [ ] (i) (6.11) T 1 = 0.02h T 1 (ii) (i) 43

7 7 7.1 5.5 (5.20) ψ(x + L, t) = ψ(x, t) (7.1) ( ) { ( ) ( )} 2mπ 2mπ 2mπ ψ(x, t) = cos L vt + φ m C m cos L x + D m sin L x m=0 (7.2) t = 0 15 ψ(x, 0) = c 0 + m=1 ( ) 2mπ c m cos L x + m=1 ( ) 2mπ d m sin L x (7.3) ψ(x + L, 0) = ψ(x, 0) {f(x) : f(x) = f(x + L)} sin cos L f(x) = c 0 + m=1 ( c m cos 2π mx L ) + ( d m sin 2π mx ) L m=1 (7.4) {c m, d m } 15 C m cos φ m c m D m sin φ m d m 44

1 a(t) a(t + T ) = a(t) a(t) = c 0 + m=1 ( c m cos 2π mt ) T + m=1 ( d m sin 2π mt ) T (7.5) 7.2 ( exp 2π i mt ) = cos T ( 2π mt T ) ( + i sin 2π mt ) T (7.6) [ ] j i a(t) = c 0 + m=1 c m id m 2 ( exp 2π i mt ) T + m=1 c m + id m 2 c 0 m = 0 c m id m A m m = 1, 2,... 2 c m + id m m = 1, 2,... 2 a(t) = m= ( A m exp 2π i mt ) T ( exp 2π i mt ) T (7.7) (7.8) (7.9) {A m } 45

A m = A m (7.10) a(t) A m A m T m 7.3 a(t) m [ ] a(t) = m=0 A m = 1 T T 0 ( a(t) exp 2π i mt ) dt (7.11) T ( A m exp 2π i mt ) ( ) exp 2π i m t m Z T T t 0 T 1 T ( exp 2π i (m ) m )t dt = δ m,m (7.12) T 0 T [ ] a(t) 0 < x < T T 2 < x < T 2 A m = 1 T T/2 T/2 ( a(t) exp 2π i mt ) dt (7.13) T a(t) 7.4 2π T = 2π T = 2π a(t) a(t) = m= A m e imt (7.14) A m = 1 π a(t) e imt dt (7.15) 2π π [ ] 2π 46

7: m = 10 7.5 [ ] 7 a(t) = t π < t < π (7.16) a(t + 2π) = a(t) T = 2π (7.15) m = 0 A 0 = 0. m 0 π A m = 1 t e imt dt 2π π = 1 1 [ t e imt ] π ip 2π π + 1 1 π e imt dt ip 2π π = i 1 { π e imπ + π e imπ} + 1 1 [ e imt ] π m 2π m 2 2π π (7.17) = i m ( 1)m e imπ = e imπ = cos(mπ) = ( 1) m a(t) 47

a(t) = m 0 i m ( 1)m e imt (7.18) A m 1 m 7.6 [ ] [ 1] T = 2π a(t) (7.15) A m = 1 2π π π a(t) e imt dt (7.19) 1 π < t < 0 (1) a(t) = +1 0 < t < π 1 + t π π < t < 0 (2) a(t) = 1 t π 0 < t < π (7.20) π < t < π a(t + 2π) = a(t) [ 2] (7.20) {A m } (7.19) (7.19) π t < π 2N t 2N t = 2π t = n t = π n N N n < N (7.21) A m = 1 2N N 1 n= N a(πn/n) e imπn/n (7.22) fortrun, c++, Excel [ ] - (A-D) 1.5 8 DFT Discrete Fourier Transforamtion 48

A m a(πn/n) IDFT Inverse Discrete Fourier Transforamtion [ 3] M N M a M (t) A m e imt (7.23) m= M a(t) (7.20) plot M 2, 5, 10 [ ] m = M a M (t) a(t) (October, 2007) p23-28 49

8 CQ (October, 2007) p23-28, Blue Backs, Blue Backs 8.1 A A-D 1 T = 4.54[ms] ν = 220[Hz] [Memo] 50

A-D 3 T = 9.64[ms] ν = 104[Hz] 8.2 A-D A-D 7 T s ν s T s T T s T (ν s ν) (8.1) T s ν s D-A 1.9 51

ν max T min ν s ν max T s T min ν max 20[kHz] 0.05[ms] 20[kHz] 20[kHz] (*) ν max = 20[kHz] (T min = 0.05[ms]) (8.2) ν s ν s > 40[kHz] (T s < 0.025[ms]) (8.3) (*) A-D 20[kHz] 20[kHz] 1.5 RC CD DVD ν s = 44.1[kHz] T s = 0.0227[ms].wav ν max = 4[kHz] T min = 0.25[ms] 52

ν s = 8[kHz] T s = 0.125[ms] 8 8:.wav 2 16 = 65536 1.7 8.3 9: A 1 53

A 1 9 ν = 220[Hz] T = 4.54[ms] 5 [Memo] Octave.wav Octave http://www.gnu.org/software/octave/ 8.4 10: 3 3 10 ν = 104[Hz] T = 9.64[ms] 5 db 20 log 10 4 1 F 1 2 F 2 3 F 3 54

F 1 F 2 7-10 7-11 55

A H T = H - -, A.1 ( ) 2 1 H = 1 2 (A.1) H 11 11: H ( ) ( ) x, y i = 1 0, j = 0 1 H ( ) 2 a 1 H = ( a ) b ( ) 1, b 2 (A.2) (A.3) 56

H i, j a, b Hi = a Hj = b (A.4) ci + dj ca + db 12: H i, j ci + dj ca + db i, j Hi = a, Hj = b 1 det(h) = 3 [ ] H H 1 HH 1 = H 1 H = I (A.5) I H 11 H A.2 Hu u 11 Hu = su (A.6) s det(h si) = 0 (A.7) s 1 = 1, s 2 = 3 57

s 1 = 1, s 2 = 3 ( ) ( ) u 1 = 1 1, u 2 = 1 1 2 1 2 1 Hu 1 = u 1 Hu 2 = 3u 2 (A.8) (A.9) 13 i, j u 1, u 2 H H u 1 s 1 = 1 u 2 s 2 = 3 A.3 H i, j u 1, u 2 i, j u 1, u 2 ) U (u 1 u 2 (A.10) i, j u 1, u 2 Ui = u 1 (A.11) Uj = u 2 u 1, u 2 i, j U U 1 U ( ) U = 1 1 1 = 2 1 1 ( cos 45 o sin 45 o ) sin 45 o cos 45 o 45 o ( ) ( ) U 1 = 1 1 1 cos 45 o sin 45 o = 2 1 1 sin 45 o cos 45 o (A.12) (A.13) 45 o 58

13: H cu 1 + du 2 cs 1 u 1 + ds 2 u 2 u 1, u 2 Hu 1 = s 1 u 1, Hu 2 = s 2 u 2 H U 1, S, U 59

14: H U 1, S, U H (i) 45 o (ii) s 1 = 1 s 2 = 3 (iii)45 o U, U 1 H ( ) U 1 s 1 0 HU = S 0 s 2 (A.14) U 1 HU = S H = USU 1 (A.15) H 13 60

U 1 : 45 o i, j u 1, u 2 S: s 1 = 1 s 2 = 3 S U: 45 o i, j A.4 H H U 1, S, U U 1 U S s 1 = 1 s 2 = 3 s 1 s 2 S s 1 0 det(s) = 0 s 2 = s 1s 2 (A.16) [ ] 2 2 ( ) a 11 a 12 A = a 21 a 22 (A.17) det(a) = a 11 a 12 a 21 a 22 = a 11a 22 a 12 a 21 (A.18) H 2 1 det(h) = 1 2 = 3 (A.19) s 1 s 2 = 3 [ ] 2 2 det(a si) = 0 ( ) a 11 a 12 A = a 21 a 22 (A.20) (a 11 s)(a 22 s) a 12 a 21 = 0 (A.21) 61

s 1, s 2 (a 11 s)(a 22 s) a 12 a 21 = (s 1 s)(s 2 s) (A.22) s = 0 a 11 a 22 a 12 a 21 = s 1 s 2 (A.23) a 11 a 12 a 21 a 22 = s 1 0 0 s 2 (A.24) (Q.E.D.) H det(h) = det(s) H [ ] 3 3 3 3 A.5 H det(h) H 1 det(h 1 ) H det(h 1 ) = 1 det(h) = 1 3 det(h 1 ) det(h) = 1 (A.25) (A.26) [ 1] det(h) = 0 det(h 1 ) = det(h) = 0 H 1 [ 2] A, B det(a) det(b) = det(ab) (A.27) (A.26) H 1 H = I 62

A.6 H = ( a b ) b a (A.28) a b (1) H s 1, s 2 (2) H u 1, u 2 u i 1 (3) ) U = (u 1 u 2 (A.29) H H = USU 1? U 1 :? S U :?? :? (4) (a) U 1 (b) SU 1 (c) USU 1 H * 63