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1 60 0 ( ) Web Java Web maeno sci.u-ryukyu.ac.jp () () (3)

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3 0.. 3 () () (3) () () (3) () (3) () Hz 0000Hz Hz Hertz

4 4 0 A 4 440Hz 5 340m/s Maxwell Maxwell () () (roth = j + D) t (3) (rote = B) t Maxwell m/s m m

5 m 0 6 m 0 8 m m/s

6 Maxwell

7 7. d x(t) dt = ω x(t) (.) x(t) k m m d x(t) dt = kx(t) (.) k = mω (.) kx x = 0 x = 0 a x = a () () x = 0 (3) x = 0 0 x = 0

8 8 (4) x = 0 (5) (x < 0 ) (6) x = 0 0 (7) (8) kx(t) k(x(t)) 3 kx(t) kx(t) 0 kx(t) l θ = 0 ml d θ = mg sin θ (.3) dt θ mg sin θ T mg mg cos θ sin θ = θ 3! θ3 + 5! θ5 + }{{} (.4) 3 ml d θ = mgθ (.5) dt (harmonic oscillator) ( ) lθ l l d θ dt θ = π π 6 3 = 3 θ = π 6 (30 ) θ = π 6 = ! θ3 = ! θ5 = π 5 =

9 .. 9. x(t) = A sin(ωt + α) (.) x x e λt IV.. { x(t) = v0x t y(t) = v 0y t gt (.6) v 0x, v 0y x y g t = 0 x y () = ( ) ( ) (.7) x y x y m d x(t) dt = 0 m d y(t) dt = mg (v x0, v y0 ) (.6) g y g (.8) m d x(t) dt = m g (.9)

10 0 ( ) = ( ) ( ) (.0) r v v ω rad θ θ 4 r θ rθ r ω rω T n T = n π v = rω (v r ω) (.) θ r r r θ θ θ T = π ω, n = ω π π ω = = πn (.) T v = rω t ω t rω t (v = rω t = rω) t v = rω ω t ω t (v = rω ) x y r ω a = vω t t = rω ω t t = rω = vω = v r 4 90 v = rω rad (.3)

11 .. { x = r cos(ωt + α) y = r sin(ωt + α) (.4) v x = dx = rω sin(ωt + α) dt v y = dy dt r cos(ωt + α) (.5) x, y a x = d x dt a y = d y dt = rω cos(ωt + α) = ω x = rω sin(ωt + α) = ω y (.6) d x dt = ω x (.7) x, y a x, a y ω ω x d x dt = ω x d x dt = ω x ω ω ωt + α α t = 0

12 .. y = r sin(ωt + α) x = r cos(ωt + α) v y = rω cos(ωt + α) ωx = v y x ω mv y = p y p y mω y p y = mv y y + (p y) (mω) = r r..3 (.) dx = λx p7 dt λ dx dt = λx dt x dx x = λdt dx x = λdt (.8) phase space topological space

13 .. 3 dx dt = λx dx = λdt x dx dx dt dt x t t x t x dx dt x dx t dt dx dt 0 dx = a dx = adt dt dx dt 0 (dx) (dt) 0 0 dx x = λdt log x = λt + C C log x x (.9) x = ±e λt+c = ±C e λt (C = e C ) exp dx dt = λx x d dt = ω x dx dt dx d x dt dt = ω x dx dt (.0) ( ) dx dt ω x dx dt t U du dx m d x dt = du dx (.) «dx m = U + C (.) dt mv + U = ( ) dx dx = dx dt dt

14 4 ( ) dx = dt ω x + C (.3) C C < 0 0 = C = ω A 7 A x x = A sin θ 8 ( ) dx = dt ω x + ω A = ω (A x ) (.4) dx dt = ±ω A x (.5) dx = ±ωdt (.6) A x dx A x = A cos θdθ A sin θ = ±dθ (.7) x = A sin θ A > x x = A tan θ θ A < x (.7) dθ cos θ (.5) ± θ = ±ωt + α (.8) α x = A sin θ = A sin (±ωt + α) (.9) x = A sin ( ωt + α) = A sin ( (ωt α)) = A sin (ωt α + π) sin( θ) = sin(θ + π) (.30) α α + π (α ) 9 x = A sin(ωt + α) (.3) 7 C ω A A C = ω A 8 9 α α α + π

15 n n n n n A sin(ωt + α) A α A sin ωt + B cos ωt sin cos () () d n f A n dx n + A d n f n dx n + + A d f dx + A df dx + A 0f = 0 (.3) 0 f

16 6 A n x e λx d dx eλx = λe λx d dx eλx = λ e λx f = e λx ( An λ n + A n λ n + + A λ ) + A λ + A 0 e λx = 0 (.33) e λx 0 A n λ n + A n λ n + + A λ + A λ + A 0 = 0 (.34) n- n- λ, λ,, λ n, λ n n ( n ) f = a n e λ nx + a n e λ n x + + a e λ x + a e λ x n n n e λx (.35) d dx f + df f = 0 (.36) dx ( λ + λ ) f = 0 (.37) λ + λ = 0 λ + λ = 0 (λ + )(λ ) = 0 (.38) λ =, A, B f = Ae x + Be x (.39) λ + λ = (λ + )(λ ) (.36) ( ) d dx + f = 0 ( ) ( ) d d dx + dx f = 0 (.40) ( ) d dx f = 0 (.4) f = e x f = e x n- (.34) d df f k dx dx + k f = 0 (.4) λ kλ + k = 0 (λ k) = 0 (.43)

17 .. 7 λ = k e kx «d dx k f = 0 (.44) ««e kx d d dx k f = 0 dx k 0 «d dx k 0 = e kx «d dx k ( ) e kx f(x) = g(x)e kx «d dx k g(x)e kx = dg(x) dx ekx (.45) g(x) = x «d dx k «d dx k xe kx = d dx «d dx k 0 f(x) = xe kx (.46) xe kx kxe kx = dx dx ekx + x d dx ekx kxe kx = e kx (.47) λ e λx xe λx e λx, xe λx e λx, xe λx, x e λx x(t) = e λt d dt ( e λt ) = ω e λt (.48) λ = ω (.49) λ λ = ±iω 3 x(t) = Ce iωt + De iωt (.50) C, D x(t) x Ce iωt + De iωt (x(t)) = x(t) x(t) = x (t) + ix (t) (x(t)) = x (t) ix (t) x(t) (x(t)) = x(t) x (t) = 0 3 i i =

18 8 (x(t)) (x(t)) = C e iωt + D e iωt (.5) (x(t)) = x(t) e iωt e iωt D = C (C = D ) x(t) = Ce iωt + C e iωt (.5) C C = C R + ic I C R C I C C R C C I z = x + iy z = re iθ x, y r, θ x = r cos θ, y = r sin θ (.53) z = re iθ 4 e iθ = cos θ + i sin θ θ e iθ θ x = r cos θ, y = r sin θ r = x + y r = z r z 5 C = C e iα 6 x(t) = Ce iωt + C e iωt = C e iα e iωt + C e iα e iωt (.54) e iθ + e iθ = cos θ x(t) = C cos(ωt + α) (.55) 7 X 4 e iθ = n! (iθ)n n n cos sin n=0 (.68) (.67) 5 x + iy p x + y p x y q 6 C C C = C R + ic I C = (C R ) + (C I ) 7

19 t = 0 x = A 0 m d x dt = ω x ω A + (t = 0 0 ) x = A d x dt = ω x 0 = ω A x = A (.56) ω A ω A ω A x Aω t ω A x = A Aω t + Aω4 t x 4 Aω4 t 4 d x dt = ω x ( ω A ) 0 = ω A ( Aω4 t ) ω A = ω A + Aω4 t. x = A x Aω t x 4 Aω4 t 4 (.57) ( x = A ω t + 4 ω4 t 4 70 ω6 t 6 + ) 4030 ω8 t 8 + = A x = A cos ωt x(t) = n=0 ( ) n (n)! (ωt)n (.58) a n t n = a 0 + a t + a t + a 3 t 3 + (.59) n=0 a n t = 0 d x(t) dt = n(n )a n t n = a + 6a 3 t + a 4 t + 0a 5 t 3 + (.60) n=0 n = 0 n = n(n ) = 0 d x(t) dt = n(n )a n t n = a + 6a 3 t + a 4 t + 0a 5 t 3 + (.6) n= n

20 0 n(n )a n t n = ω n= n=0 a n t n (.6) t n (n ) t n ( n 0 ) a +3 a 3 t +4 3 a 4 t + +k(k )a k t k +(k + )ka k+ t k + = ω a 0 ω a t ω a t + ω a k t k ω a k t k + (.63) t t t k a = ω a 0, 6a 3 = ω a, a n t k k(k )a k = a k = ω a k ω k(k ) a k (.64) k k a k = ω (k )(k 3) a k 4 a k = = ω k(k ) ω (k )(k 3) a k 4 ω k(k ) ω (k )(k 3) ω (k 4)(k 5) a k 6 (.65) k ω a 0 ω k k ω 3 a ω k a k = ( ω ) k k! ( ω ) k k! a 0 a k k a n = ( ) n ωn (n)! a 0 a n+ = ( ) n ω n (n + )! a a 0, a sin θ = cos θ = (.66) ( ) n (n + )! θn+ (.67) ( ) n (n)! θn (.68) n=0 n=0 x(t) = a 0 cos ωt + a sin ωt (.69) ω.3

21 .3. y = A cos(ωt + α) y = B sin(ωt + β) sin cos α cos θ = sin(θ + π ) β = α + π, A = B cos Ae i(ωt+α) Ae i(ωt+α) A cos(ωt + α) A cos(ωt + α) e i(ωt+α) Ae i(ωt+α) = Ae iα }{{} =A e iωt = A e iωt A A A (A, α) }{{} }{{} A sin cos sin(a + B) = cos(a + B) = (.70) }{{} A sin(ωt + }{{} α ) (.7) }{{} A e iωt (.7) sin A cos B + cos A sin B cos A cos B sin A sin B cos(a + B), sin(a + B) e i(a+b) e i(a+b) = e ia e ib e i(a+b) = e ia e ib cos(a + B) + i sin(a + B) = (cos A + i sin A)(cos B + i sin B) cos(a + B) + i sin(a + B) = cos A cos B sin A sin B + i(sin A cos B + cos A sin B) (.73) (.74) cos θe iθ θ α α cos θ cos θ cos θ cos α sin θ sin α e iθ e iθ e i(θ+α) α e iα ( e iθ ) = e iθ (cos θ + i sin θ) = cos θ sin θ + i cos θ sin θ = cos θ + i sin θ cos θ + i sin θ (.75)

22 cos θ = cos θ sin θ sin θ = cos θ sin θ (.76) cos θ sin θ = cos(θ + π 4 ) cos(a + B) = cos A cos B sin A sin B ( cos θ + π ) = cos θ cos π 4 4 sin θ sin π 4 = cos θ sin θ A = θ, B = π 4 cos π 4 = sin π 4 = (.77) e iθ = cos θ + i sin θ ie iθ = i cos θ sin θ (.78) cos θ e iθ sin θ ie iθ cos θ = e iθ +) sin θ = ie iθ (.79) cos θ sin θ =( + i)e iθ ( + i)e iθ cos θ sin θ e iθ θ + i = e i π 4 e iθ + ie iθ = e iθ+i π 4 (.80) ( cos θ + π ) 4 + i = e i π 4 i + i 45 α e iα π 3 A cos(ωt) + A cos(ωt + π 3 ) (.8) i π 4 + i

23 .3. 3 Ae iωt + Ae iωt+i ( ) π 3 = A + e i π 3 e iωt (.8) + e i π 3 8 e i π 3 π 3 π e i π 3 = 3e i π 6 (.83) A sin(ωt + α) + B sin(ωt + β) = A + AB cos(α β) + B sin(ωt + γ) (.84) γ Ae iωt+iα + Be iωt+β A sin(ωt + α) + B sin(ωt + β) Ae iωt+iα + Be iωt+iβ = Ce iωt+iγ (.85) C γ t Ae iα + Be iβ = Ce iγ (.86) C Ae iα + Be iβ = Ce iγ ( Ae iα + Be iβ) ( Ae iα + Be iβ) = C A + ABe i(α β) + ABe i(α β) + B = C (.87) A + AB cos(α β) + B = C e iθ + e iθ = cos θ 9 C = A + AB cos(α β) + B 0 A sin(ωt + α) + B sin(ωt + β) = C sin(ωt + γ) (.88) γ C t A sin α + B sin β = C sin γ ωt = 0 (.89) A cos α + B cos β = C cos γ ωt = π (.90) cos γ + sin γ = γ 8 9 cos θ e iθ = cos θ + i sin θ e iθ = cos θ i sin θ 0 C ± C γ γ + π

24 4 e iγ e iγ = Ceiγ C = Ae iα + Be iβ A + AB cos(α β) + B (.9) β B γ cos γ = sin γ = A cos α + B cos β A + AB cos(α β) + B, A sin α + B sin β (.9) A + AB cos(α β) + B α A γ γ tan γ = A sin α + B sin β A cos α + B cos β (.93) Ae iωt e iθ θ i = e i π i π π i i π π i π 90 π 80 i -i - e iθ = cos θ + i sin θ cos θ 90 sin θ θ A.4 C

25 (.) p7 d dt x(t) = ω x(t) (.94) x(t) x(t) x(t) (linear) i t ψ = m x ψ (.95) ψ ( ) c t + x m φ = 0 (.96) φ (homegeneous) 0 (inhomogeneous non-homogeneous) 3 V ρ ( ) x + y + z V = ρ (.97) ε 0 V f (x), f (x),, f i (x), a f (x) + a f (x) + + a i f i (x) + = i a i f i (x) d dt x(t) = ω x(t) X(t) Y (t) X(t) + Y (t) X(t) Y (t) 3X(t) + 45Y (t) d ( ) 3X(t) + 45Y (t) dt = 3 d d X(t) + 45 dt dt Y (t) = 3ω ( X(t) 45ω Y (t) ) = ω 3X(t) + 45Y (t) (.98) 3X(t) + 45Y (t) d dt x(t) = ω (x(t)) X(t) Y (t) d ( ) 3X(t) + 45Y (t) dt = 3 d d X(t) + 45 dt dt Y (t) = 3ω (X(t)) 45ω (Y (t)) = ω ( 3 (X(t)) + 45 (Y (t)) ) ω ( 3X(t) + 45Y (t) ) (.99) 3

26 6 3X(t) + 45Y (t) 4 D DX = J (.00) X J 0 DY = 0 (.0) Y D(X + Y ) = J (.0) DX = J 0 DY = 0 D(X + Y ) = J DX = J, DY = J (.03) D(X + Y ) = J + J (.04) DX = J X J (source) ρ V = ρ ε 0 V 5 ρ ρ V = ρ ε 0 V ρ + ρ (V + V ) = ρ + ρ ε 0 (.05) ρ V ρ V ρ + ρ 6 n n f g, h, f f = 3g h f, g, h 7

27 .5. 7 n n f, f, f n a, a,, a n a f + a f + + a n f n a i n n n A sin(ωt + α) A sin(ωt + α) = A sin ωt + B cos ωt (.06) sin(α + β) = cos α sin β + sin α cos β A = A cos α, B = B sin α cos ωt sin ωt.4. (.) p7 d dt {z} x(t) {z} = ω {z} x(t) {z} (.07) d ( x ) d «dx dx Θ(x) = λθ(x) (.08) 8 d ( x ) d «Θ(x) λ dx dx «««a b x x c d {z y } {z } = λ {z} y {z } (.09).5 [ -] m d x dt = kx 8 Legendre Legendre

28 8 «dx m dt kx [ -] ml d θ dt = mg sin θ [ -3] k 0( ) m () x, y x = 0 () (3) y L o y x L [ -4] () () (3) d dx f(x) 3 d f(x) 0f(x) = 0 dx d dx f(x) 6 d f(x) + 9f(x) = 0 dx d 3 f(x) f(x) = 0 dx3 [ -5] d dx F (x) = kf (x) F (x) = Aekx A F (x) = X a n x n n=0 [ -6] d dx f(x) 6 d f(x) + 0f(x) = 0 dx [ -7] «dx = x dt t x = 0 dx dt 0 0 x = 0 x = 0 [ -8] 0 () i + e i 3π 4 () + e i π 3 + e i π 3 (3) + e i π 5 + e i 4π 5 + e i 6π 5 + e i 8π 5 [ -9] e iθ = cos θ + i sin θ () cos 3θ = cos 3 θ 3 cos θ sin θ () sin 3θ = 3 cos θ sin θ sin 3 θ (3) cos 4θ = cos 4 θ 6 cos θ sin θ + sin 4 θ (4) sin 4θ = 4 cos 3 θ sin θ 4 cos θ sin 3 θ

29 9. O x m d x dt dx = kx K dt (.) K dx dt x = Ae λt -kx -Kv v mλ Ae λt = kae λt KλAe λt (.) mλ + Kλ + k = 0 (.3) λ λ = K ± K 4mk m (.4) K 4mk [K 4mk > 0 ] λ λ = K + K 4mk λ = K K 4mk m m λ, λ t 0 x = C e K K 4mk m «t + C e K+ K 4mk m «t (.5)

30 30 K 4mk > 0 K m k! " # K 4mk < 0 K (dumped oscillation) (dump) e Kt λ = K + i 4mk K λ = K i 4mk K m m ««x = Ce K i 4mk K K+i 4mk K m t m t + De (.6) x D = C C = A eiα, D = A e iα ( x = A e K m t e i 4mk K m t+α «+ e i 4mk K m t+α «) ( ) = Ae K 4mk K m t cos t + α m (.7) Ae Kt 4mk K k K = 0 m m Kv K 4mk < 0 K m k K 4mk = 0 (.7) K = 0 x = A cos r! k m t + α

31 .. 3 (.4) x = Ce K m t p9 (.) p9 (.4) K 4mk = 0 λ = K p9 m ( d dt + K ) x = 0 (.8) m ( K = 4mk (.) ) p9 ( e K m t e K d m t dt + K ) ( d m dt + K ) m ( d dt + K ) x = 0 (.9) m (.9) (.8) (.8) te K m t te K m t d dt + K m d dt + K 0 m ( d dt + K ) te K m t = e K m t (.0) m (at + b)e K m t (.) (a, b ). m d x(t) dt = kx(t) + F cos ωt (.) O -kx x F cos ωt m d x(t) dt = kx(t) + F e iωt (.3).. (.3) x(t) 0 F e iωt (.3) (.3)

32 3 m d X(t) dt = kx(t) + F e iωt (.4) X(t) F e iωt ω X(t) X(t) = Ae iωt m d dt ( Ae iωt ) = kae iωt + F e iωt (.5) e iωt ω ma = ka + F (k mω )A = F ka (.6) A = X(t) = F k mω k mω F k mω eiωt (.7) x(t) = y(t) + X(t) m d dt (y(t) + X(t)) = k(y(t) + X(t)) + F eiωt (.8) y(t) m d y(t) dt = ky(t) (.9) (.8) (.4) (.9) (.9).. (.9) y(t) = Ce iω0t + C e iω0t (.0) k (ω 0 = m C, C ) y(t) = Ce iω 0t (.)

33 .. 33 e iωt e iωt e iωt e iωt (A, B )A cos ωt+b sin ωt k = m(ω 0 ) x(t) = Ce iω 0t + x(t) = Ce iω 0t + F k mω eiωt (.) F m ((ω 0 ) ω ) eiωt (.3) ω ω 0 ω ω 0 resonance C = 0 x(t) = F F m ((ω 0 ) ω ) eiωt = m ((ω 0 ) ω cos ωt () (.4) ) ω 0 > ω x F ω 0 < ω x=0 x=0 0 0 dx ωf dt t=0 = icω 0 + i m ((ω 0 ) ω ) C C = K K dx dt t=0 = 0 0 x(0) = K + F m ((ω 0 ) ω ) F 0 K = m ((ω 0 ) ω ) F x(t) = m ((ω 0 ) ω ) eiω0t + F m ((ω 0 ) ω ) eiωt = ω, ω 0 (.5) (.6) F ( e iωt m ((ω 0 ) ω e iω0t) (.7) )

34 34 ω=.ω 0 ω =.ω 0, ω =.ω 0, ω =.3ω 0 ω=.ω 0 (.7) ω=.3ω 0 ( ) ω = Ω + ω, ω 0 = Ω ω ω ω 0 Ω = ω + ω 0 ω e iωt e iω 0t = e iωt ( e i ωt e i ωt) = ie iωt sin ωt (.8) x(t) = if m ((ω 0 ) ω ) sin ωteiωt (.9) F Ω m ((ω 0 ) ω sin ωt ) F m ((ω 0 ) ω ) ω ω 0 ω = ω 0 (.7) (.7) k = mω ω 0 (ω 0 ) ω = (ω 0 ω)(ω 0 + ω) = ω Ω = 4 ωω (.30) F m ((ω 0 ) ω sin ωt = F ) m ωω F t sin ωt sin ωt = mω ωt (.3) sin x lim x 0 x F t = ω 0 mω x(t) = if t mω eiωt (.3) t 3 F = kx sin θ θ 3

35 .. 35 LC L di dt + Q C = 0 (.33) dq dt = I (.34) LC x Q dx I = dq dt dt d x di dt dt = d Q dt m L k C L di dt I= dq dt L C +Q -Q Q C m d x dt = kx L d Q dt = Q C r k m LC L d Q dt = Q C LC x Q I di dt m L k Q = 0 C kx Q C k C LC F cos ωt ω LC 4 4 R

36 I= dq dt [ -] L R C +Q -Q () () (3) t = 0 I = 0, Q = Q 0 t Q 0 [ -] k l m t = 0 A sin ωt k x = 0 x m x(t) [ -3] (.) x m d x dt dx = kx K dt [ -4].. m d dt x(t) = kx(t) K d x(t) + F cos ωt dt dt x(t) = kx(t) K d x(t) + F eiωt dt m d K < 4mk k mω [ -5] k = mω [ -6]

37 L k m y y y y y y m d y dt = ky + k(y y ) (3.) m d y dt = ky k(y y ) (3.) m d y dt = ky + ky (3.3) y y y y y -ky -ky y k(y y ) -k(y y ) m d y dt = ky ky (3.4) 3.. y = A e iωt, y = A e iωt mω A = ka + ka (3.5) mω A = ka ka (3.6) A, A A, A A, A A αa, A αa α A, A A A A, A ω

38 38 3 A k mω = A k A k = A k mω (3.7) A A A k mω = = ± (3.8) A k ω ±k = k mω mω = k k ω = ± ( ) k m (3.9) k 3k (3.8) ω = m ω = m ω = ±ω, ±ω y, y A = A y = y = C e iω t A = A y = y = C e iω t y = C e iωt + C e iωt y = C e iωt C e iω (3.0) t C, C 3 C, C 3.. C y = y k ω = m m m k m d y dt = ky k k ω = m = m m k x = x = ± 3 x y, y e iωt

39 m d y dt = ky C y = y 3k m m k k ( ) ! k m d y dt = 3k 3ky ω = m (3.0) C, C C, C

40 40 3 C = C = A y y = A (cos ω t + cos ω t) (3.) y = A (cos ω t cos ω t) (3.) t = 0 y = A, y = 0 0 cos ω t, cos ω t y, y y y cos cos ω t ω t cos ω t y C y y C y + y y y (3.3) + (3.4) (3.4) (3.3) (y + y ) = X, (y y ) = Y m d (y + y ) dt = ky ky (3.3) m d (y y ) dt = 3ky + 3ky (3.4) m d X dt = kx (3.5) m d Y dt = 3kY (3.6) k 3k X Y m m X, Y X, Y X, Y

41 y G = (y + y ) y R = y y X, Y y G, y R y G = X, y R = Y X, Y X, Y 3.. X Y 0 y (0) = y (0) = A ω y (0) = A, y (0) = A ω ( ) m dy + ( ) dt m dy (3.7) dt y = (X + Y ), y = (X Y ) m ( d(x + Y ) dt ) + m ( d(x Y ) dt ) = ( ) dx m + ( ) dy dt m (3.8) dt m y G y R ( ) m dyg + dt m ( ) dyr (3.9) dt ( ) dyg m y G dt m y R m 5 k(y ) + k(y y ) + k(y ) (3.0) k (X + Y ) + k Y + k (X Y ) = kx + 3 ky (3.) k X 3k Y X, Y 6 y, y X, Y 4 5 M m µ = Mm M + m 6 X Y X Y

42 m k y y y y 3 y y m d dt y = ky + k(y y ) = ky + ky (3.) m d dt y = k(y y ) + k(y 3 y ) = ky ky + ky 3 (3.3) m d dt y 3 = k(y 3 y ) ky 3 = ky ky 3 (3.4) m d dt (α y + α y + α 3 y 3 ) = K (α y + α y + α 3 y 3 ) (3.5) Y = α y + α y + α 3 y 3 m d Y dt = KY y, y, y 3 Y, Y, Y 3 y i (

43 (3.) (3.4) m d dt (y y 3 ) = k (y y 3 ) (3.6) Y = (y y 3 ) Y m k y y 3 y + αy + y 3 α (3.) + α(3.3) + (3.4) m d dt (y + αy + y 3 ) = ky + ky + α(ky ky + ky 3 ) + ky ky 3 = ( α)ky ( + ( α)ky ( α)ky 3 = ( α)k y α ) α y + y 3 y + αy + y 3 ( α)k y α α = α + α = α α (3.7) = α (3.8) α = ± ± Y Y 3 ( m d dt y + ) y + y 3 = ( ( )k ( ) ( m d dt y y 3 = k ( m d dt y ) y + y 3 = ( + ( )k y + y + y 3 ) y y 3 ) y y + y 3 ) Y = y + y + y 3, Y = y y 3, Y 3 = y y + y 3 (3.30) m d dt Y = ( )k Y m d dt Y = k Y m d dt Y 3 = ( + )k Y 3 (3.9) (3.3) 3.3. Y ( )k m

44 44 3 Y k m Y 3 ( + )k m y, y, y 3 y = Y + Y Y 3 (3.3) y = Y + Y 3 (3.33) y 3 = Y Y Y 3 (3.34)

45 Y = A sin ( )k t + α (3.35) m ( ) k Y = A sin m t + α (3.36) Y 3 = A 3 sin ( + )k t + α 3 (3.37) m y, y, y 3 A, A, A 3, α, α, α y, y, y 3 y sin (,, )

46 46 3 y, y, y 3 π 4 sin π 4 ( sin π 4, sin π 4, sin 3π 4 (,, ) ( ) sin π 4, sin 4π 4, sin 6π 4 ) (, 0, ) (,, ( sin 3π 4, sin 6π 4, sin 9π 4 ) ) ( sin pπ 4, sin pπ 4, sin 3pπ ) 4 y n = sin npπ 4 p =,, 3 (3.38) (3.39) p 4 p = 4 (0, 0, 0) p = 4 π p = 5 p = 3 5π 4 3π 4 ( ) (, = 3 sin π ) 3, 3 sin π 3 (, ) ( = 3 sin π ) 3, 3 sin 4π (3.40) 3 y n = npπ sin 3 3 (3.4)

47 m d dt m d y dt = ky + ky (3.4) m d y dt = ky ky (3.43) y y «««k k y = k k «8 y m d y dt «y ( ) y y y y «k k k k (3.44) «X = (y + y ) Y = (y y ) X ( ( ) m d dt a b ( ) k k k k y y = «( ) k k k k «y «(a + b) ( ) a b «= ( k + k k k) ( ) 9 k k k k y «) (3.45) «= k ( ) (3.46) «k 7 8 9

48 48 3 M M (eigenvector) λ (eigenvalue) N N N 0 0 M A v v. (v v ) M C B A = v v. A = λ C A (v v ) (3.47) M ( «) d y ( m = ) «k k dt y k k {z }» m d ( ) «y dt y {z } X m d X ( k k ) k k k( )» ( ) y = k y «{z } X y y «(3.48) dt = kx «X Y ( ) k k «k k m d Y = 3kY dt = 3k( ) (3.49)! m d y dt y m d dt X Y «= «= k 0 0 3k! ««k k y k k y ««(3.50) X Y

49 «(y, y ) (X, «Y ) k k k 0 k k 0 3k X Y y «=! y «! y = «X Y! (3.44) (3.5) m d X dt Y «=! y «(3.5) (3.5)! k k k k! «(X, Y )! X Y «(3.53)! k k k k «! = k 0 0 3k «(3.54) «k k M = T = k k! T = T k 0 MT 0 3k! «T MT 0 M T T MT M (diagonalize) M v v. M A v v. C B A = v v. C A M v = λ v (3.55) C A( ) λ( ) d dt 0 y y. C A = M 0 A y y. C A d y = M y (3.56) dt 0 m n n m

50 50 3 (3.55) v y d v = M v = λ v (3.57) dt d X = λx dt y v y v y λ v! k! 3k N N N N λ M v = λ v (M λi) v = 0 (3.58) I M λi (M λi) (M λi) v = 0 v = 0 (3.59) v = 0 M λi det(m λi) = 0 (3.60) N N λ N N λ N «k k M = k k k λ k diffet(m λi) = det k k λ «= ( k λ) k = 0 (3.6) ( k λ) = k k λ = ±k λ = k k k, 3k m d X dt = kx m d Y = 3kY dt λ ««k k k k λ = k (M λi) = λ = 3k (M λi) = (3.6) k k k k 0 λ = k «λ = 3k (normalization) p + = λ = 3k! λ = k λ = 3k! (3.64) «(3.63) «T! T =!! =! (3.65) (degenation)

51 T T N N N v, v,, v N T = v C B v C A v N CC AA (3.66) a b c d 0 ` ` T t = ` «a c b d v t v t. v t N C A t (transpose) ` «v t i 0 v i (3.67) C A M λ v λ v v v = 0 v M v t λ v v v M t v v t {z } M v v t M v {z λ } v v (3.68) =λ v =λ v t λ ( v v ) = λ ( v v ) λ λ v v = 0 j 0 v i v j = i j i = j (3.69) v i T t T = I T T t N N N V = a v + a v + a 3 v a N v N (3.70) N T T k λ k 0 k k λ k A = 0 0 k k λ ( k λ) 3 k ( k λ) = 0 ( k λ) `( k λ) k = 0 ( k λ) ( ( + )k λ)( ( )k λ) = 0 (3.7) λ = k, ( + )k, ( )k (orthogonal matrix) T

52 5 3 λ = k 0 0 k 0 k 0 k 0 k 0 a b c j A bk = 0 = 0 ak + ck = 0 b = 0, c = 0 A λ = ( ± )k k k 0 k k k 0 k k 0 a b c 0 0 (3.7) A A = 0 (3.73) b = a b = 0 0 A C k k 0 k k k 0 k k 0 0 (Y, Y, Y 3 ) m d dt s A Y Y Y 3 0 A C A = ( )ky ky ( + )ky 3 ( )k k ( + )k A (3.74) A (3.75) ( r s )k k, m m, ( + )k m 3.5 N N N N + n n + y n+ y n n n + k(y n+ y n ) k(y n+ y n ) k(y n y n ) m d y n dt = k(y n+ y n ) k(y n y n ) = ky n ky n + ky n+ (3.76)

53 3.5. N 53 n ( k y n y ) n + y n+ (3.77) y n y n + y n+ y n > y n + y n+ y n y n < y n + y n+ y n y n m d dt y y y 3. y N y N k k k k k k k k 0 0 = k k k k y y N y y y 3. y N y N (3.78) N =, N = 3 (3.4),(3.39) npπ y n = sin (3.79) N + N + e iωt y n (t) = sin npπ N + eiωt m d dt ( sin npπ ) N + eiωt mω sin npπ N + = (n )pπ (n + )pπ = k sin N + eiωt + k sin N + eiωt k sin npπ ( N + eiωt k (n )pπ (n + )pπ sin + sin sin npπ ) (3.80) N + N + N +

54 54 3 ( ) mω e i npπ N+ = k e i (n )pπ N+ + e i (n+)pπ N+ e i npπ N+ (3.8) e iθ i sin θ e i npπ N+ ) mω pπ i = k (e N+ + e i pπ N+ (3.8) ( ) e i pπ pπ (N+) e i (N+) e i pπ (N+) } {{ } b i sin e i = pπ pπ (N+) e i (N+) } {{ } ab pπ (N + ) 3 4 sin + e i pπ (N+) }{{} = a pπ (N + ) mω = 4k sin e i pπ (N+) pπ (N + ) k ω = m sin pπ (N + ) } {{ } a e i pπ (N+) } {{ } b ω n ( ) npπ k y n = sin N + N + sin m sin pπ (N + ) t + α p ( ) N npπ k y n = A p sin N + N + sin m sin pπ (N + ) t + α p p= A p, α p p mode y A sin (ω t + α ) y A y sin (ω t + α ) 3 = T A 3 sin (ω 3 t + α 3 ).. A N sin (ω N t + α N ) y N (3.83) (3.84) (3.85) (3.86) (3.87) (3.88) (3.89) T T = N + 3 sin θ = eiθ e iθ i π sin N + sin π N + sin 3π N +. sin Nπ N + k ω p = m sin pπ (N + ) sin π N + sin 4π N + sin 6π N +. sin Nπ N + sin 3π N + sin 6π N + sin 9π N +. sin 3Nπ N + sin Nπ N + sin Nπ N + sin 3Nπ N +. sin N π N + (3.90) (3.9)

55 3.5. N 55 T pq = N + sin pqπ N + T T t = T T = T (3.9) y i dy i dt Y = T t y( d Y dt = T t d y) dt Y A p, α p Y y = T Y y r N + T t T = I X T pq(t t ) qr = X T pqt rq = δ pr (3.93) q q p r 0 p = r NX T pqt pq = q= = = NX q= NX q= pqπ N + sin N + N + N + sin N+ X q= N sin pqπ NX N + + pqπ N + q= N + sin pqπ N + + p 0π N + sin N + + p (N + )π N + sin N + {z } =0 (3.94) 0 N q N N + sin exp 4(N + ) N+ X q= N e i pqπ N+ e i pqπ N+ = 4(N + ) N+ X q= N e i pqπ pqπ i N+ + e N+ (3.95)

56 56 3 q = N q = N + N + 4(N + ) ( ) N N N y n (t) = N p= (( ) ) npπ k A p sin N + N + sin m sin pπ t + α p (N + ) (3.96) N n x y(x) y n y(n x) L x = L N + m, k N m M N (M ) k K(N + ) K () n N + L x x n y(x, t) () m M k K(N + ) N 4 F = kx

57 sin npπ N + N + L x pπ N + = pπ L x (3.97) sin ( ) ( ) k m sin pπ K(N + ) pπ t + α p sin t + α (N + ) M p (3.98) (N + ) N N + N ( ) ( K(N + ) pπ K sin t + α M p (N + ) N pπ (N + ) = θ K pπ (N + ) sin t + α p = M }{{} (N + ) pπ θ pπ (N + ) sin M ) t + α p (3.99) (N + ) ( ) K pπ M θ sin θ t + α p (3.00) sin θ K N θ 0 lim = pπ θ 0 θ M t + α p y(x, t) = C p sin pπ ) K (pπ L x sin M t + α p (3.0) A p N + C p 5 p= C p sin pπ L x pπ K M p = p = 4 K p = L π M K p = L π M p = 3 L 3 K 3π M p = 4 L K 4π M r 5 C p = A p N Cp 0 N + N + A p

58 58 3 sin pπ L x x L p π L p (standing wave) 3.6. N N N n n+ x x n- n+ x n x n n M K(N + ) n N M d x n N dt = K(N + ) (x n x n + x n+ ) (3.0) (n + K(N + )(x n+ x n ) n K(N + )(x n x n ) ) x 0 x N+ 0 6 M d y(n x) N dt = K(N + ) (y((n ) x) y(n x) + y((n + ) x)) (3.03) d y(n x) KN(N + ) dt = (y((n + ) x) + y((n ) x) y(n x)) (3.04) M y(x, t) y x dy dx = lim y(x + x) y(x) x 0 x (3.05) x x + x y(x) x y(x + x) y(x) x x 6 0 N +

59 x x, x, x + x y y(x + x) y(x) y(x) y(x x) 7 x x d dy dy y dx = lim dx (x) dx (x x) x 0 x x = = lim x 0 = lim x 0 L N + N N N + (3.07) KL M (3.04) x = n x 8 KL y(x, t) = y(x, t) (3.08) t M x N y(x, t) ( y(x + x) y(x) x x (y(x + x) + y(x x) y(x)) x ) y(x) y(x x) x (3.06) d y dx = lim (N + ) x 0 L (y(x + x) + y(x x) y(x)) (3.07) t t x y 7 x + x x x x x x 0 8 N N + N (6 0 3 )

60 [ 3-] 3. K [ 3-] y, y, y 3 () y, y, y 3 () Y = 3 (y + y + y 3), Y = (y y ), Y 3 = (y 3 y ) (3) (4) (5) Y 4 = (y y 3) : [ 3-3] I n I n+ I n I n+ Q n I n I n+ = dq n dt (3.09) Q n, Q n, Q n+ I L n- L I n L I n+ +Q +Q n- n C C -Q -Q n- n Q n Q n + Q n+ = LC d dt Qn [ 3-4] x L C x l c L = l x, C = c x 9 9

61 N 4.. y(x, t) x x + y(x, t) x + dx x + dx + y(x + dx, t) x x + dx dx dx + y(x + dx, t) y(x, t) y(x, t) dx k dx k (y(x + dx, t) y(x, t)) k (y(x + dx, t) y(x, t)) k

62 6 4 k dx dx 0 dx K = kdx ( K ) dx 0 k (y(x + dx, t) y(x, t)) dx dx K (y(x + dx, t) y(x, t)) = K y x y K dx (4.) x f(x, ) dx = f(x + dx, ) f(x, ) (4.) x f(x, ) x f(x + x, ) f(x, ) = lim x 0 x (4.3) x 0 f(x, ) x = f(x + x, ) f(x, ) x + O( x) (4.4) O( x) x 0 x a x dx (??) b f(x, ) x = f(x + x, ) f(x, ) + O(( x) ) (4.5) x f(x, ) dx = f(x + dx, ) f(x, ) + O((dx) ) x }{{} (4.6) f(x + dx, ) = f(x, ) + f dx (4.7) x x dx f(x, ) f x dx O(dx ) x f a O( x) O(( x) ) x O( x) O( x) x 0 0 O( x) x 0 0 O(( x) ) O(( x) ) x x x 0 0 O(( x) ) ( x) 0 b dx dx 0 O((dx) )

63 y x > 0 K y x y K x x x+dx y y K (x, t) K x x (x+ dx, t) K y y (x + dx, t) K x x (x, t) = K y (x, t)dx (4.8) x ρ ρdx ρdx y t = K y x dx y t = K y ρ x ρdx (4.9) ρ M L K K L K = KL K ρ = KL M y(x, t) y x 4.. y(x, t) x, t x t y(x, t) = X(x)T (t) (4.0) y(x, t) = v t }{{} y(x, t) (4.) x = K ρ

64 64 4 K = v v ρ (??) X(x)T (t) = t X(x) d dt T (t) = v X(x)T (t) x v T (t) d dx X(x) v T (t) dt T (t) = X(x) dx X(x) d d t x (4.) t x x t 3 C d dt T (t) = Cv T (t) d dx d X(x) = C X(x) (4.3) dx T (t) = Cvt De + Ee Cvt, X(x) = F e Cx + Ge Cx (4.4) C y(x, t) = A e C(x+vt) + A e C(x vt) + A 3 e C( x vt) + A 4 e C( x+vt) (4.5) C C y(x = 0, t) = 0 y(x = L, t) = 0 x = 0 y(0, t) = A e Cvt + A e Cvt + A 3 e Cvt + A 4 e Cvt = 0 (4.6) t A 4 = A, A 3 = A (e ) ( ) y(x, t) = A C(x+vt) e C( x+vt) + A C(x vt) e e C( x vt) ( = A e Cvt Cx e ) ( e Cx + A e ) Cvt e Cx e Cx ( = A Cvt e ) ( + A e ) (4.7) Cvt e Cx e Cx x = L ( y(l, t) = A Cvt e ) ( + A e Cvt CL e ) e CL = 0 (4.8) e CL e CL = 0 e CL = CL = nπi 4 C = nπ L i (4.9) C 3 t x 4 e A = A = 0 e A = A = 0 A = πi, 4πi, A = nπi e A =

65 nπ L = k y(x, t) = ( A e ikvt + A e ikvt) ( e ikx e }{{ ikx ) } =i sin kx = ( ia e ikvt + ia e ikvt) sin kx (4.0) y(x, t) 5 A, A (ia ) = ia A = A y(x, t) = ( ia e ikvt ia e ikvt) sin kx (4.) A = A e iα k = nπ L = i A ( e ikvt+iα e ikvt iα) sin kx = 4 A sin kx sin(kvt + α) y(x, t) = n= C n sin nπx L ( ) nπvt sin L + α n n 4 A C n N (4.) (4.3) 4..3 e λx y(x, t) e λx+ωt Ω e λx+ωt = v λ e λx+ωt (4.4) Ω = ±vλ y(x, t) = Ae λ(x+vt) + Be λ( x+vt) (4.5) x = 0 y = 0 Ae λvt + Be λvt = 0 (4.6) B = A x = L y = 0 Ae λ(l+vt) + Be λ( L+vt) = 0 (4.7) Ae λ(l+vt) Ae λ( L+vt) = 0 e λl e λl (4.8) = 0 λl = nπi 5

66 d dx f(x) d f(x) 3f(x) = 0 dx d dx d ( ) ( ) d d dx 3 dx 3 dx + d d f(x) = 3f(x) f(x) = f(x) dx dx f(x) = Ae 3x + Be x ( ) v t x y(x, t) = 0 (4.9) ( t v ) y(x, t) = 0 x ( t v ) ( x t + v ) y(x, t) = 0 (4.30) x ( t + v ) y(x, t) = 0 (4.3) x ( t v ) x (4.3) x + vt 0 ( t v ) (x + vt) = x t (vt) v (x) = v v = 0 (4.33) x x + vt 0 (x + vt) e x+vt tan(x + vt) t v 0 F (x + vt) x ( t + v ) (4.34) x 0 x vt G (x vt) y(x, t) = F (x + vt) + G (x vt) (4.35) v f(x) x a f(x a) f(x + a) t = 0 G(x) t vt G(x vt) t = 0 F (x) t vt F (x + vt) G(x vt) v F (x + vt) v

67 C, D x + vt = C x vt = D (characteristic curve) 6 7 G(x vt) x v G(x) vt x + vt F(x+vt) left-moving t x - vt G(x-vt) F (x + vt) x v v F (x + vt) left-moving G(x vt) right-moving x = 0 x = L y(x) = 0 y(x = 0, t) = 0 F (vt)+g( vt) = 0 F G G(X) = F ( X) G(X) = F ( X) X x vt G(x vt) = F ( x + vt) y(x = L, t) = 0 F (L + vt) = F ( L + vt) (4.37) F (X) L F (X+ L) = F (X) x = 0 x = L F (X) L L L e i nπ L x n e i nπ L x right-moving y(x, t) = F (x + vt) F ( x + vt) (4.36) * +,"-/." 0 % ' F(x+vt) L t F(x+vt) G(x-vt) "! #$% & %' x x F(x+vt) G(x-vt) ( )!#$% & %' e iθ π e i(θ+π) = e iθ e i nπ L x L x L nπ n e i nπ L X F (X) = a n e i nπ L X (4.38) n= 6 7 x t

68 68 4 a n a 0 y(x, t) = = = n= a n e i nπ L (x+vt) a n }{{} n= = a n e iα n n= y(x, t) = n= n= e i nπ L vt ( e i nπ a n e i nπ L ( x+vt) L x nπ i e x) L }{{} =i sin nπ L x i a n e i nπ L vt+iαn sin nπ L x ( a n ) sin nπ L x sin ( nπ L vt + α n ) (4.39) (4.40) (transverse wave) (longitudinal wave) T ρ T x y(x, t) y (x, t) T T x x x + dx T T y x T = T T = T + + ( ) y x y x ( y x T ) (4.4)

69 y y ( ) y x + x x x + dx T y y (x, t) T (x + dx, t) x x T y y (x + dx, t) T x x (x, t) = T y (x, t)dx (4.4) x dx 0 T y (x + dx, t) x dx ρdx y (x, t) t ρ y t = T y x y t = T y ρ x (4.43) T v T ρ ρ x x + dx dx dx + y dx y dx x x y = 0 P 0 y x = 0 P 0 dx < 0 9 P P V = nrt P V dx + y ( x dx P 0 P 0 dx = P (x, t) dx + y ) x dx P (x, t) = P ( 0 + y P 0 y ) (4.44) x x y x + x x + x = x + x x 3 + x 4 P (x, t)s P (x + dx, t)s 8 9

70 70 4 ρsdx ρdx y (x, t) = t P y t (x, t) = P 0 ρ (x, t)dx x y (4.45) x (x, t) P 0 ρ P 0 = 0 5 N/m ρ =.kg/m 3 P 0 0 ρ = 5. =.9 0 m/s (4.46) 340m/s P V = P V γ = γ γ.4 ( P 0 (dx) γ = P (x, t) dx + y ) γ x dx P (x, t) = ( P 0 + y x ( ) γ P 0 γ y ) x (4.47) γ γp 0 ρ t y(x, t) = γ P 0 y(x, t) (4.48) ρ x γ PV = ( ) PV= ( ) m/s ρ 0 0 ( )=( ) ( )

71 x x x + dx ρ ρhdx h Y (x, t) Y (x, t) dx + Y (x + dx, t) H(x, t) ( ρh(x, t) (dx + Y (x + dx, t) Y (x, t)) = ρh(x, t) + Y ) (x, t) dx (4.49) x ρhdx () H(x, t) H(x, t) = h + η(x, t) η(x, t) ( ρh(x, t) (h + η(x, t)) ) (x, t) + Y ( x + Y (x, t) x dx ) = ρhdx = h h + h Y (x, t) + η(x, y) + η(x, t) Y (x, t) x x = h h Y (x, t) + η(x, y) = 0 x η(x, t) Y x (x, t) η Y 0 P 0 y H(x) y ρ g P 0 + ρg(h(x) y) H(x) 0 (4.50) (ρg(h(x) y)) dy = ρg ((H(x)) ) (H(x)) = ρg (H(x)) (4.5) P 0 3 x x + dx ρg (H(x + dx, t)) + ρg (H(x, t)) = ρgh(x, t) H dx (4.5) x = ( ) ( ) = = P 0 H(x + dx, t) P 0 H(x, t)

72 7 4 H(x, t) = h + η(x, t) η(x, t) ρgh η (x, t)dx (4.53) x ρhdx (??) η = h Y (x, t) x gh ρhdx η Y (x, t) = ρgh (x, t)dx (4.54) t x t Y (x, t) = gh η Y (x, t) (4.55) x ρ ρ ρ 4?? 4.3 x = 0 x = L 0 (fixed end ) x = L 0 y ( L, t ) = 0 K y x y x (free end) y (0, t) = (L, t) = 0 (4.56) x 4 gravity wavegravitational wave

73 x = 0, L 0 sin nπ L x x = 0, L 0 cos nπ L x x = L 0 cos nπ = 0 n n cos nπ = ± y(x, t) = n= C n cos ( ) (n + )π (n + )π x sin vt + α n L L (4.57) 4.4 (dimensional analysis) x = x 0 + v 0 t + at (4.58) x, x 0 [L] [m] t [T] [s] v 0 [LT ] [m/s] a [LT ] [m/s ] [L] [m] v 0 t [LT ][T]=[L] at [LT ][T ]=[L] m d x = kx k, m dt m k [T ] m, k [T ] k [MT ] [MLT ] [L] [MLT ] [M] [LT ] M L [M],[L] [T ] [T ] [M] K M [T ] r k [T] M [T ] π SI /60 m/s m/60 m/s m/ 60 = 360 x = x 0 + v 0t + at t /60 v 60 a 3600 x = vt (4.59) /60

74 74 4 [M][L][T] [Q] [I] s K ρ K [MLT ] ρ s [ML ] [LT ] K ρ P r [ML T ] [ML 3 ] P ρ γ 4.5 [ 4-] x θ θ G x N = πga4 ρπa x [ 4-] t y(x, t) = v y(x, t) x [ 4-3] (Hint:e ikx iωt ) t Y (x, t) = v x Y (x, t) M Y (x, t) [ 4-4] ɛ 0 µ 0 r Q Q F = Q Q 4πɛ 0 r r I I F = µ0iil ( l ) πr ɛ 0, µ 0

75 u(x, t) = v u(x, t) (5.) t x u(x, t) = F (x + vt) + G(x vt) (5.) u(x, t) = A sin (k(x vt) + α) (5.3) t = 0 u(x, 0) = A sin(kx + α) (5.4) v x (phase) k(x vt) + α x = 0, t = 0 α (initial phase) 3 π t = 0 kx x π k λ = π k k π k = π λ λ cos e iθ exp i 3

76 76 5 &% (' )' &*' +,-."!"#/0 "!#$ t = 0 x = 0! u(0, t) = A sin ( kvt + α) (5.5) kv ω π ω π π ω! u(x, t) = A sin (kx ( ( ωt + α) x = A sin π λ t ) ) (5.6) + α T! (??) k λ π ω T π! π λ T

77 v = λ T v = ω k (phase velocity) (groupe velocity) 5. 0 x 0 4 F (x + vt) G(x vt) x = 0 u(x, t) = F (x + vt) x = 0 u 0 u(0, t) = 0 F (vt) = 0 F 0 F ( vt) = 0 F F (vt) + G( vt) = 0 F (vt) + G ( vt) = 0 (5.7) F G G(x vt) = A sin (k(x vt) + α) F (vt) + A sin (k( vt) + α) = 0 (5.8) F (x + vt) = A sin (k( x vt) + α) = A sin (k( x vt) + α + π) (5.9) sin(θ + π) = sin θ F (vt) = ka cos (k( vt) + α) (5.0) F (x + vt) = ka cos (k( x vt) + α) (5.) x F (x + vt) = A sin (k( x vt) + α) (5.) 0 F (x + vt) x = 0 G( vt) = A sin ( kvt + α) F (vt) = A sin ( kvt + α + π) F (vt) = A sin ( kvt + α) π 4

78 78 5 π sin(θ + π) = sin θ 5.3 x k k ω x = x = 0 x { e u(x, t) = ikx iωt + Re ikx iωt x < 0 (5.3) P e ik x iωt x > 0 e ikx iωt Re ikx iωt x P e ik x iωt (R) 0 (P)

79 u(x = 0, t) x < 0 e iωt + Re iωt x > 0 P e ik x iωt x = 0 x = 0 u u x x=0 ik (??) (??) ik (??) + (??) + R = P (5.4) ik( R) = ik P (5.5) ik ( + R) ik( R) = 0 R = k k k + k (5.6) ik = i(k + k )P P = k k + k (5.7) P R k > k k < k ω ω k k > k π k > k k > k k k + ( ) k k + ( ) k P = kk k + k k k < k k > k k < k ( ) ( + ) k > k k > k

80 80 5 k > k π k π k k < k π k π k k < k k < k k > k k > k k < k π

81 (??) u t = v u x u(x, t) x ρ ρ ρ x ρ xv u(x, t) x x+ x ρv u x (x+ x, t) ρv u (x + x, t) x ρv u x u (x, t) ρv (x, t) x ρv u x (x + x, t) ρv u x ρ x u(x, t) = ρ xv u(x, t) (5.8) t x (x, t) = ρ xv u(x, t) + (5.9) x x x+dx 5 ρv u (x + x, t) x u t t ( ) ( ) = ρv u u (x, t) (x, t) t (5.0) x t ( x 0 ) u(x, t) = G(x vt) u x = G (x vt), u t = vg (x vt) ρv G (x vt) ( vg (x vt)) t = ρv 3 (G (x vt)) t (5.) u(x, t) = F (x + vt) x x + x t ρv 3 (G (x + x vt)) t ρv 3 (G (x vt)) t ρv 3 (G (x vt)) t ρv 3 (G (x + x vt)) t = ρv 3 G (x vt)g (x vt) x t (5.) U(x, t) x U(x, t) U t x = ρv3 G (x vt)g (x vt) x t (5.3) 5 F + ρv u u (x + x, t) F + ρv (x, t) F x x

82 8 5 ρv 3 G (x vt)g (x vt) x t U = ρv (G (x vt)) (5.4) U ( ) u ρ ( ) u t ρv x K kx v = ρ G(x vt) = A sin(k(x vt)) U = ρv A k cos (k(x vt)) (5.5) kv = ω U = ρω A cos (k(x vt)) v cos (k(x vt)) Ū = ρ ω A (5.6) u(x, t) ( ) u ρ ( ) u t ρv F (x + vt) x G(x vt) u(x, t) = F (x + vt) + G(x vt) ρ (vf (x + vt) vg (x + vt)) = ( ρv (F ) F G + (G ) ) ρv (F (x + vt) + G (x + vt)) = ( ρv (F ) + F G + (G ) ) ρv ( (F ) + (G ) ) (5.7)

83 ! "# %$ & F = m d x dt 5.6 u(x, t) = A sin (kx ωt + α) x = x = A 6

84 84 5 (wave packet) (??) v F (x + vt) + G(x vt) v p = ω sin(kx ωt) k A sin(kx ωt) A ( k ) k k ω ω k + k ω + ω k, ω k, ω e i((k k)x (ω ω)t) e i((k+ k)x (ω+ ω)t) e i((k k)x (ω ω)t) + e i((k+ k)x (ω+ ω)t) = e i(kx ωt) e i( kx ωt) + e i(kx ωt) e i( kx ωt) ( = e i(kx ωt) e i( kx ωt) + e i( kx ωt)) = e i(kx ωt) cos ( kx ωt) (5.8) e A+B = e A e B k, ω k, ω e i(kx ωt) e i(kx ωt) cos ( kx ωt) e i(kx ωt) k π π cos ( kx ωt) k k k cos ( kx ωt) k k e i(kx ωt) cos ( kx ωt) π ω k ( ( k cos ( kx ωt) = cos k x ω )) k t

85 !" v g v g (group velocity) ω ω k k ω + ω k + k J7;,K:L<GMON! #"$% & ' ( *)+, -.0/ :37;=<?>@7A=B0C D #"$% E& ' ( *)+, -.0/:3F476980:3G;=<H>@IGA=B:C

86 86 5 k dkf(k)e ikx iω(k)t (5.9) k f(k) k ω ω(k) k ω k k = k 0 f(k) 0 ω(k) = ω(k 0 ) + dω dk (k k 0) + (5.30) (k k 0 ) (??) e ik 0x iω(k 0 )t dkf(k)e i(k k 0)x i dω dk (k k 0)t x dω dk } {{ } x dω dk t dω t F (x t) dk e ik 0x iω(k 0 )t F (x dω t) (5.3) dk cos ( kx ωt) F (x dω t) dk F (x dω dk t)eik 0x iω(k 0 )t F (x) x dω(k) dk t v g = dω(k) (5.33) dk k v p = ω k v g = dω k dk ω(k) k (5.3) ω = vk v v k ω k ω = vk ω k ω = f(k) (dispersion relation) ω = vk ( ) 7 7

87 N (( N npπ k x n (t) = A p sin N + N + sin p= x n x k K v p = ω K = k K x sin, K m v g = dω k dk = x K x cos m k ω = m (5.36) sin K x p =,,, N K Nπ (N + ) x π x, k, m x K v p v g m sin ) ) pπ t + α p (N + ) (5.34) pπ k (N + ) x m sin pπ (N + ) (5.35) [ 5-] d λ λ λ > λ I II III A P I e ikx iωt + Re ikx iωt (x < 0) (R) B II Ae ik x iωt + Be ik x iωt (0 < x < d) 0 d III P e ikx iωt (d < x) R, A, B, P k π λ k π λ () x = 0, x = d () (R = 0 ) 8

88 88 5 [ 5-] () c ω k ω = ck n n k = nk ω = ck n n n () n k n(k) k n(k) (3) m ω k ω = ( k) m ω mv = p k p = mv k m (4) V ω = ( k ) + V k m [ 5-3] t u = v u v x t u = v x u M u k [ 5-4] " " #$ )"*+)",.-/ "! %! % & '( A A [ 5-5]?? [ 5-6] Z Z I = dt dx «u ρ «! u t ρv x I u(x, t) u(x, t) + δu(x, t) I δu(x, t) 0

89 u(x, t) = v u(x, t) (6.) t x x v u(x, y, t) = v t ( ) x + y u(x, y, t) (6.) ( ) u(x, y, z, t) = v t x + y + z u(x, y, z, t) (6.3) ) = (+ x + y z (Laplacian) (??) (6.4) ( ) v t u(x, y, z, t) = 0 (6.5) = v t (D Alenbertian) f(x + ) + f(x ) f(x) f(x) = lim x 0 ( ) (6.6) (Laplace) (d Alembert)

90 90 6 f(x + ) + f(x ) f(x + ) f(x ) f(x) 0 x y ( ) x + y f(x, y) = f(x +, y) + f(x, y) + f(x, y + ) + f(x, y ) 4f(x, y) lim (6.7) 0 ( ) f(x +, y) f(x, y) f(x, y + ) f(x, y ) f(x, y) f(x, y, z) = [ lim 0 ( ) f(x +, y, z) + f(x, y, z) + f(x, y +, z) + f(x, y, z) ] (6.8) + f(x, y, z + ) + f(x, y, z ) 6f(x, y, z) u(x, y, z, t) = U(x, y, z)e iωt ( ) t U(x, y, z)e iωt = v x + y + z U(x, y, z)e iωt ( ) ω U(x, y, z) = v x + y + z U(x, y, z) (6.9) ω U = U (6.0) v 6.. u(x, y, z, t) = e ipx+iqy+irz iωt exp ω v eipx+iqy+irz iωt = (p + q + r )e ipx+iqy+irz iωt (6.) ω = ±v p + q + r ω = 0 0 = U

91 , ( e ipx+iqy+irz iωt px + qy + rz p = (p, q, r) x = (x, y, z) p x p p a p ( x + a) = p x p a = 0 a p e ikx iωt k e i p x iωt " p #$ % '& ) +* -.! y θ x x x 3 λ x π λ x π λ x λ x π λ x λ x y z (λ x, λ y, λ z ) x y z (λ x, λ y, λ z ) (λ x ) + (λ y ) + (λ z ) = λ λ x λ π (k x, k y, k z ) p + q + r π λ λ 4 u(x, y, z, t) = dp dq dr ( f(p, q, r)e ipx+iqy+irz iωt + g(p, q, r)e ipx+iqy+irz+iωt) (6.) (p, q, r) f(p, q, r) ω = v p + q + r 3 x 4 (k, 0, 0) k k = π λ k = p p + q + r

92 9 6 u(x, y, t) u = u u (x, y, z, t) = dp dq dr ( f (p, q, r)e ipx iqy irz iωt + g (p, q, r)e ipx iqy irz+iωt) (6.3) p, q, r u (x, y, z, t) = dp dq dr ( f ( p, q, r)e ipx+iqy+irz+iωt + g ( p, q, r)e ipx+iqy+irz iωt) (6.4) u = u g(p, q, r) = f ( p, q, r) f(p, q, r) = g ( p, q, r) u(x, y, z, t) = = dp dp dq dq dr ( f(p, q, r)e ipx+iqy+irz iωt + f ( p, q, r)e ipx+iqy+irz+iωt) dr ( f(p, q, r)e ipx+iqy+irz iωt + f (p, q, r)e ipx iqy irz+iωt) (6.5) p p, q q, r r px + qy + rz ωt F (x + vt) x + vt (left-moving) G(x vt) x vt (right-moving) (p, q, r) F (px + qy + rz ωt) p, q, r f(p, q, r) = (F (p, q, r) ig(p, q, r)) F, G u(x, y, z, t) = = ( ) dp dq dr F (p, q, r)( e ipx+iqy+irz iωt + e ipx iqy irz+iωt + i G(p, q, r) ( e ipx+iqy+irz iωt e ipx iqy irz+iωt) dp dq dr (F (p, q, r) cos(px + qy + rz ωt) + G(p, q, r) sin(px + qy + rz ωt)) (6.6) ) 6.. (r, θ, φ) = r ( r ) + r r r sin θ ( sin θ ) + θ θ r sin θ φ (6.7) F = div(gradf ) x, y, z λ

93 r, θ, φ V = ε 0 ρ div A = r V = div(gradv ) = r grad,div, V gradv = e r r + e V θ r θ + e V φ r sin θ φ ( r ) A r + r r ( r V r r sin θ ) + θ (sin θa θ) + r sin θ θ r sin θ ( sin θ V θ ) + A φ φ (6.8) (6.9) V r sin θ φ (6.0) div r r r r r sin θ r λ π π k e ikr e ikr u = ω u r n e ikr ( r n e ikr) = ( r r ) (r n e ikr) r r = ( r ( r nr n e ikr + ikr n e ikr)) r = ( ( r nr n r e ikr + ikr n e ikr) + r ( n(n )r n e ikr + iknr n e ikr k r n e ikr)) = ( nr n e ikr + ikr n e ikr) + ( n(n )r n e ikr + iknr n e ikr k r n e ikr) = r n e ikr ( n(n + )r + ikr + iknr k ) (6.) n = ( ) e ikr = k eikr r r (6.) k = ω eikr v r e ikr k = ω r v

94 94 6 r = 0 r = 0 sin kr r = 0 5 r r r 4πr r 6 θ, φ θ, φ ( sin θ θ ( sin θ θ ) + sin θ ) φ Y (θ, φ) = λy (θ, φ) (6.3) 7 r R(r) cos θ, sin θe iφ ( ( r ) + r ) r r r θ V (r, θ) = ω V (r, θ) (6.4) V (r, θ) = R(r)e inθ ( r r ( r r ) n r ) R(r) = ω R(r) (6.5) R(r) r = ξ ω r = ω r ω R(r) = J(ξ) ( ( ξ ) ) + n ξ ξ ξ ξ J(ξ) = 0 (6.6) 6. Ae ikx iωt x = 0 ±Ae ikx iωt x = 0 x = 0 x = 0 y, t 5 sin θ lim = θ 0 θ 6 eikr r 7 Ym l (θ, φ)

95 Ae ipx+iqy iωt x = 0 y, t B ipx+iqy iωt 0 Ae ipx+iqy iωt x=0 + Be ipx+iqy iωt x=0 = 0 (6.7) B = A B = A x p p p q, ω q ω = v p + q p p p (p, q, r) ( p, q, r) 6.3 k k e ipx+iqy iωt Re ipx+iqy iωt P e ip x+iqy iωt y q y k = p + q, k = (p ) + q + R = P, p( R) = p P (6.8) P = p p + p, R = p p p + p (6.9)

96 96 6 θ θ q k, k q = k sin θ = k sin θ (6.30) 8 9 k π λ sin θ λ = sin θ λ (6.3) sin θ sin θ = λ λ (6.3) sin θ 0 sin θ = λ λ sin θ (6.33) θ λ > λ k > k 6.4 [ 6-] U = ω U U = R(r) cos θ R(r) 8 9 p gh

97 [ 6-]?? [ 6-3]?? (sin θ )

98

99 eikr x r A d x eik x x x x (7.) x =0 x x k x x x x z z = 0 e ikz z = 0 0 dx dr π 0 dy dφr (0, 0, z) (r, φ, 0) (0, 0, z) r + z (0, 0, z) A 0 dr π 0 dφr r + z eik r +z (7.) φ π r d ( e ik r +z ) = ik d ( ) r + z dr dr e ik r +z r = ik r + z eik r +z (7.3) d ( e ik r +z ) ik dr [ ] πa ik eik r +z = πa [ e i ] (7.4) 0 ik r = r = 0 πa ik eikz (7.5) A ik π (0, 0, z) e ikz ik π iλ z = 0 e ikz z (0, 0, z) (x, y, 0) e ikz e i

100 00 7 (0, 0, 0) (0, 0, z) (0, 0, 0) AB P AB ( ) P ( ) AB P ( ) R eikr R r ik πr eikr π 0 dθ π 0 eik R +r Rr cos θ dφ sin θr R + r Rr cos θ (7.6) R + r Rr cos θ = x dx = x = r R(r > R ) θ = π x = r + R Rr sin θdθ R + r Rr cos θ θ = 0 r R ike ikr dx r eikx = r+r r eikr [ e ikx] r R r+r = ( r eikr e ik(r+r) e ik(r R)) = (e ikr e ik(r+r)) r (7.7)

101 r eikr r eik(r+r) 7.3 ( ) 3 ( ) m m

102 0 7 y θ e iky y = 0 x x L L θ r r ik π 4 L L sin θ) eik(r+x dx r x r r >> L (7.8) r 000L 0. exp ik(r + x sin θ) r ikr e ik(r+x sin θ) = e ikr e ikx sin θ (7.9) r kx sin θ π e ikx sin θ 5 ik π eikr L L sin θ eikx dx r = ik π eikr = πr eikr kl = 0 sin (kl sin θ) kl = 00 θ sin θ π π θ = 0 θ = 0 kl θ = 0 θ = 0 k π λ λ kl = π L λ λ L 0 7 m L 0 3 kl [ e ik sin θ r [ e ikl sin θ e sin θ ] ikx sin θ L L ikl sin θ ] = i sin (kl sin θ) eikr πr sin θ (7.0) 4 5 x L L kl sin θ 6 L kl =

103 pt5.496pt 7.4 x y ( ) u(x, t) = v t x + y u(x, t) (7.) y x > 0 y = 0 y = L u(x, 0, t) = u(x, L, t) = 0y sin nπy L sin nπy L = ) (e inπy L e inπy L y π i L L y L L x k ω u(x, y, t) = Ce ikx iωt sin nπy (7.) L ( ω = v k + n π ) k = ± L ω v n π L (7.3) ω n k ( ω v + nx π ) > 0 L u(x, y, t) = Ce Kx iωt sin nπy L + De Kx iωt sin nπy L (7.4)

104 04 7 n π K = ± L ω v (7.5) (e Kx ) (e Kx ) ω v + ( nπ L ) πv > 0 ω < L πv (cut-off frequency) L 7.5?? A B A B A B A B A h + x B h + (L x) k, k ω ω k, ω k A B k ω h + x + k ω h + (L x) (7.6) ω k h + x + k h + (L x) (7.7)

105 A B x x 0 k x L x k h + x }{{ h + (L x) }}{{} = 0 =sin θ =sin θ sin θ k = sin θ k k sin θ = k sin θ (7.8) 0 B "!# $ %'&( )* +,"-'. 7 A sin k h + x + k h + (L x) 0 7 0

106 P ( ) P A Q A B ( ) P A A P A P ( ) P P P Q ( ) A A x x λ x sin φ sin φ x = λ sin φ 8 φ B A P x Q 8

107 (x, t ) (x, t ) (x, t ) e ik x x k x x 0 ( $! ) (x,t ) "#! (x,t ) ( $ (x x + ) cos 00x x = 0 ( 00x ) x = 0 x = 0 ( Euler-Lagrange ) )

108 08 7! (x -x+) " # $ cos 00x 9 comparable 0 ( ) 9 (classical) 0 comparable

109

110 0,, 4, 3, 3, 3, 9, 7, 49, 3, 7, 5, 40, 37, 4 N, 5