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483F 3 6.........3... 6.4... 7 7.... 7.... 9.5 N (... 3.6 N (... 5.7... 5 3 6 3.... 6 3.... 7 3.3... 9 3.4... 3 4 7 4.... 7 4.... 9 4.3... 3 4.4... 34 4.4.... 34 4.4.... 35 4.5... 38 4.6... 39

5 4 5.... 4 5..... 4 5. (3... 4 5.3... 4 5.4... 4 5.5... 44.. x = r cos θ = r cos(ωt + δ y = r sin θ = r sin(ωt + δ =r cos θ ( ωt + δ π m d dt x = F c cos θ = mrω cos θ = mω x d dt x = ω x x = A cos(ω t + δ y = A cos(ω t + δ A = A,ω = ω,δ = δ A = A,ω = ω,δ δ = π A = A,ω = ω,δ δ = 4 π A = A, ω = ω,δ = δ A = A, ω = ω,δ δ = π 4 y = x d x dt = ω x d x dt + ω x = ( d dt + ω x =

y parametric curve cos(t, cos(t.8.6.4. -. -.4 -.6 -.8 - - -.8 -.6 -.4 -...4.6.8 x : A = A,ω = ω,δ = δ y parametric curve cos(t+pi/, cos(t.8.6.4. -. -.4 -.6 -.8 - - -.8 -.6 -.4 -...4.6.8 x : A = A,ω = ω,δ δ = π 3

y parametric curve cos(t+pi/4, cos(t.8.6.4. -. -.4 -.6 -.8 - - -.8 -.6 -.4 -...4.6.8 x 3: A = A,ω = ω,δ δ = 4 π y parametric curve cos(t, cos(*t.8.6.4. -. -.4 -.6 -.8 - - -.8 -.6 -.4 -...4.6.8 x 4: A = A, ω = ω,δ = δ 4

y parametric curve cos(t/+pi/4, cos(t.8.6.4. -. -.4 -.6 -.8 - - -.8 -.6 -.4 -...4.6.8 x 5: A = A, ω = ω,δ δ = π 4 d dt = d dt d dt d ( ( d d dt + ω = dt + iω dt iω ( ( d d dt + iω dt iω x = d dt eiωt = iωe iωt ( ( d dt iω x = ( d dt + iω x = x(t =e iωt x(t =e iωt x(t =c + e iωt + c e iωt x(t R x (t =c + eiωt + c i eiωt c + = c,c + = c c + = A eiδ,c = A e iδ x(t = A eiδ e iωt += A e iδ e iωt = A ei(ωt+δ + A e i(ωt+δ = A cos(ωt + δ 5

l θ 6:.3 dv (t :F (x = V (x dx m d dt x = F = dv dx m d dt x + dv dx = dx dt ( : kx m d dt xdx dt + dv dx dx dt = ( ( d dx dt m + V = dt = ( dx m + V = E ( dt ( dx m V dt V (x x x Taylor V (x =V (x +V (x (x x + V (x (x x x V (x = m d s = gm sin θ dt ml d θ = gm sin θ dt 6

k k k x,m 7: x,m sin θ θ l d θ dt = gθ d θ g dt = ω θ (ω = l h = l l cos θ V (x =gm(l l cos θ =gml( cos θ =lθ dv ds = dv dθ dθ = gml sin θ ds :.4. x,x m d dt x = kx + k(x x = kx + kx ( m d dt x = kx + k(x x = kx + kx ( 7

( + ( : m d dt x + m d dt x = m d dt (x + x : k(x + x +k(x + x = k(x + x X = (x + x ω I = m d dt X = kx k m X = A I cos(ω I t + δ I ( ( : m d dt x m d dt x = m d dt (x x : k(x x +k(x x = 3k(x x x = x x ( ω II = m d dt x = 3kx 3k m x =A II cos(ω II t + δ II A I,δ I,A II,δ II X = (x + x =A I cos(ω + δ I x = x x =A II cos(ω II + δ II x = X + x = A I cos(ω I + δ I +A II cos(ω II + δ II x = X + x = A I cos(ω I + δ I A II cos(ω II + δ II 8

x ( = d, x ( =,x ( =, x ( = x = ( ( d(cos(ω t + cos(ω II t = d cos (ω I + ω II t cos (ω I ω II t ( ( x = d sin (ω I + ω II t sin (ω II ω I t ( sin (ω II ω I t ( X x k k?. ( x x ( d dt ( = x x c c ω ( ω ( ω: ω ( ( c c ( = K K K K e iωt (c,c : const c c c c c c ( ( e iωt K K = K K ( ( K K c = K K c : = ( ( ω ω ( ω K K K ω K c c ( c c c c = e iωt (ω K c K c = (3 K c +(ω K c = (4 9

c (3 (ω K + (4 K (ω K (ω K K K c = (ω K (ω K K K = c = ω K c K ω K K K ω K = ( c,c ω = λ λ (K + K λ + K K K K = λ = ((K + K ± (K K +4K K ( ( = c c c µ ± = c c µ ± µ ± = λ ± K = ( (K + K ± (K K K K +4K K ( ( e I =, e II = µ I µ II ( { K K K ei = ωi e I K = = K K K eii = ωiie II ( x x = A I cos(ω I t + δ I e + A II cos(ω II t + δ II e + ω I λ ω II λ +

d dt x = [ ( ] e, e K P =(e, e λ K!! KP = (Ke,Ke = (λ e,λe ( λ = (e, e λ = PK K = ( λ λ :K KP = PK P (detp P KP = P PK = K PP KPP = K = PKP d x dt = Kx = PK P x P P d x dt = P PK P x d dt P x = K P x

x = P x d x dt = K x ex. K = ( k m { ( k m k m k m λ = k m = ω λ = 3k m = ω ( e = P = (, e =, P = P KP = = = k m 3 k k m k m 3 k m 3k m m k m k m k m k m x ( x = x x d x dt = K x : ( d dt x x ( = λ λ ( x x

d x dt = λ x d x i dt = λ ix i (λ i = ω i- ( m d x dt = kx + k(x x = x V (x,x m d x dt = k(x x kx = V (x,x x V (x,x = kx + kx + k(x x + const. V (x,x = kx + kx + k(x x + const. = k(x + x + k(x x + const. ( = (x + x ( x x ( x x + + k + const. = kx + 3 kx + const. (.5 N ( N+ x i ( N m d x dt = kx kx + k(x x = kx + kx m d x dt = k(x x +k(x 3 x =kx kx + kx 3. m d x n dt = k(x n x n +k(x n+ x n =kx n kx n + kx n+ m d x N dt = k(x N x N +kx N = kx N kx N + kx N+ x = x N+ = 3

( K k m k m k k m m k m. K = k k... m m........ k m... k k m m ω x n (t =v n cos(ωt + δ m d x n dt = mω v n cos(ωt + δ k(x n x n + x n+ =kcos(ωt + δ(v n v n + v n+ mω v n = k(v n + v n+ v n v n = A sin(nα + φ ( ( θ + θ θ θ sin θ + sin θ = sin v n + v n+ = A (sin ((n α + φ + sin ((n +α + φ = A sin(nα + φ cos α mω = k( cos α ω = k m ( cos α =k m sin α ω = 4k m sin α x = v cos(ωt + δ =, v = v = sin( α + φ =A sin φ = φ = x N+ = v N+ cos(ωt + δ =, v N+ = v N+ = A sin ((N +α = (N +α = iπ (i N 4

{ αi = iπ N+ (i =,,N k ω i = m sin α i ω i i N x n (t = A i sin(nα i cos(ω i t + δ i i= A i,δ i.6 N ( ( m d x dt = k(x x +k(x x =k(x x x = x x N+ = x N v = v sinφ = sin(α + φ =sin(α + φ sinφ = cos ( α + φ sin α α + φ = ±π v N = v N+ sin (Nα+ φ = sin ((N +α + φ = sin ((N +α + φ sin (Nα+ φ = cos (Nα+ α + φ sin α cos (Nα+ α + φ = sin Nα = α + φ = ±π Nα = iπ α i = iπ N i =,,N N ( 5

.7 N a =(N +a, n z = na u(t, z x n (t =u(t, na x n (t x n (t =u(t, na u(t, (n a n,a u(t, (n a =u(t, na+ u (t, na( a+ t x n (t x n (t =a u (t, na z x n+ (t x n (t =a u (t, (n +a z [x n+ (t x n (t] [x n (t x n (t] = a ( =m u t = ka u z u t = ka u z [ u z ka m = ka m = F a σ ka = F (, m a = σ( [ ] F = [f] σ [σ] = [MT [M =[ T ]=[( ] ] ] u (t, (n +a (t, na z v F σ u t = v u (wave equation z = a u (t, na z 3 3. x, t u(t, x 6

x 3 F x = T cos θ T cos θ T T = T = T y 4 F y = T sin θ T sin θ T θ Tθ = T (θ θ tan θ = u tan θ θ z F y = T ( u x = T x u x m = σ x u (t, x + x (t, x x m u t = F y = T x u x σ x u t = T u x x u t = T u σ x σ : y 3. u(t, x=asin(kx + φ cos(ωt + δ ( x n (t =A sin(nα + φ cos(ωt + δ u t = ω u(t, x, m u x = k u(t, x u t = v u x ω u = v k u k ω = vk ( ω = m sin α 3 4 7

. u(t, = u(t, = u(t, = A sin φ cos(ωt + δ = φ = u(t, =A sin(k cos(ωt + δ = k = iπ, i =,,, k i = iπ, ω i = vk i = v iπ. 5 u u (t, = (t, = x x x = x ( x N+ = x N u = ka cos(kx + φ cos(ωt + δ x u (t, = ka cos φ cos(ωt + δ = x φ = π u (k x (t, =ka cos + π cos(ωt + δ = k = iπ k i = iπ 3. u(t, =, u (t, = x φ = ( cos(k = k = i ( i π k i = π (i =,,, ( π u(t, x=asin x cos(ωt + δ, ω = vk = v π 5 ( 8

3.3 u(t, x= A i sin(k i x + φ cos(ω i t + δ i i= A i,δ i (i =,, (initial condition u(,x=f(x v(,x= u t (,x=g(x ( f( = f( = φ = u(,x= v(,x= A i sin(k i x cos δ i = f(x i= A i ω i sin(k i x sin δ i = g(x i= A i cos δ i = a i u(,x= a i sin(k i x=f(x i= A i ω i sin δ i = b i v(,i= b i sin(k i x=g(x i= a i < > f(x = a i sin(k i x a i i= sin(k j x x 9

: : = i= dxf(x sin(k j x a i sin(k i x sin(k j x a i dx sin(k i x sin(k j x i= dx sin(k i x sin(k j x I ij I ij = dx sin(k i x sin(k j x sin(k i x sin(k j x= (cos(k i k j x cos(k i + k j x i j k i + k j,k i k j I ij = [ ] sin((k i k j x sin((k i + k j x = k i k j k i + k j (( iπ sin((k i k j = sin jπ = sin((i jπ sin((k i + k j = sin((i + jπ= i = j I ij = dt dt cos((k i + k j x = I ij = δ ij i = j δ ij = i j ( = a i I ij = a j =( i=

a i = a j = dxf(x sin(k j x dxf(x sin(k i x < > a f(x = x <x< a ( x <a< a i a i = dxf(x sin(k i x [ = dx a x sin(k ix+ x = y ] a ( x sin(k ix k i = iπ = dx a ( x sin(k ix = dy a y sin(k iy dy a y sin(k iy (i : (i : dy a y sin(k i k i y= dy a y( sin(k iy iπ (i : a i = a dxx sin(k i x= 8a [ x cos(k i x+ ] k i ki sin(k i x (i : ( cos k i ( sin k i ( (n +π =(i =n + cos ( (n +π = sin = sin (( n + π = cos(nπ (( = cos n + π = sin(nπ =

a i = 8a ki cos(nπ = 6 8a (n+ π cos(nπ= 8a (n + π cos(nπ v(,x= A i ω i sin(k i x sin δ i = g(x= A i ω i sin δ i = δ i = < > 3 x =(a,a,a 3 a = a e + a e + a 3 e 3 e =(,, e =(,, e 3 =(,, n a =(a,a,,a n n a = a i e i i= a b a b = a b + a b + + a n b n e i e j = δ ij ( n a = a i e i i= a i = a e i f(x = a i = a i sin(k i x (sin(k i x e i i= (f, g= dxf(x sin(k i x dxf(xg(x 6? a e i

? < > u u (t, = (t, = x x u(t, x = v(t, x= A i cos(k i x cos(ω i t + δ i i= A i ω i cos(k i x sin(ω i t + δ i= cos u(,x=f(x,v(,x=g(x, f f ( = ( = x x f(x = a i cos(k i x g(x= a i = b i = i= b i cos(k i x, i= dx cos(k i xf(x dx cos(k i xg(x dx cos(k i x cos(k j x = k i = iπ dx [cos ((k i + k j x + cos ((k i k j x] = δ ij dx cos(k i x cos(k j x=δ ij cos(k i x 7 3.4 x ( f(x f(x = a + a i cos(k i x+ b i sin(k i x i= i= 7 sin cos 3

a i = b i = dx cos(k i xf(x dx sin(k i xf(x i =,,, i =,,, ( cos sin dx cos(k i x cos(k j x dx sin(k i x sin(k j x = = δ ij = = δ ij dx [cos((k i + k j x + cos((k i k j x] dx [ cos((k i + k j x + cos((k i k j x] dx cos(k i x sin(k j x (cos sin dx cos(k i x sin(k j x = = dx [sin((k i + k j x + sin((k j k i x] [ k i + k j cos((k i + k j x+ ] cos((k i k j x = k i k j 4

( a dxf(x= dx + a i cos(k i x+ b i sin(k i x = a = a a = dxf(x = i= i= dx cos(k xf(x, k = f(x f(x =f( x b i = b i = dx sin(k i xf(x = [ dx sin(k i xf(x+ dx sin(k i xf(x y = x = = dy sin(k i yf( y dy sin(k i yf(y ] dx sin(k i xf(x? cos a i = 8 x f(x sin(k i x cos(k i x x F (x f(x x F (x = f( x x< ( F (x = a + a i cos(k i x 8? i= 5

F (x cos F (x f(x x F (x = f( x x< ( F (x = b i sin(k i x i= F (x sin ex. a f(x = x x a ( x x a = dxf (x = dxf (x = dxf(x = a = a a i = dx cos(k i xf (x = dx cos(k i xf(x 8a = dx cos(k i xx i : i : i =u a u = 8a (cos(nπ = [ cos f(x = a 4a π 8a π ( π x n : 4a (m + π n : + 9 cos ( 6π x [ ( π sin x ( 3π 9 sin x + ( ] 5π 5 sin x + + ( ] π 5 cos x + + 6

(!! 9 < > cos(k n x= [e iknx + e iknx] sin(k n x= [e iknx e iknx] i f(x = a + a n cos(k n x+ b n sin(k n x n= = a + n= n= a n (eiknx + e iknx i = a + (a n ib n e ibnx + n= n= b n (eiknx e iknx n= (a n + ib n e iknx k n = nπ,k n = nπ (a n ib n n =,,, c n = a n = (a n + ib n n =,,, f(x = n= c n = = c n e iknx =(m n dxe iknx f(x e iknx e iknx i(k m k n c n = c n [e i(km knx] ==δ mn??? 9 x X f(x = a + a n cos(k nx+ n= X n= b n sin(k nx x > x< T = * k = nπ = n(π x x + x <x< 7

4 4. :u(t, x ( u t = v u x v t x u(t, x= ( t v ( t x + v x u(t, x = ( t + v ( x t v x u(t, x = ξ = x vt η = x + vt ξ f(ξ ( t + v ( ξ f(ξ = x t + v ξ df t dξ =( v + v df dξ = f(x vt g(x + vt u(,x=f(x+g(x=u (x u t (,x= vf (x+vg (x =v (x f (x g (x = v v (x x f(x g(x= v x dxv (x +f( g( f(x = u (x v x dx v (x + (f( g( f,g 8

g(x= u (x+ v x dx v (x (f( g( f,g u(t, x = [u (x vt+u (x + vt] + v x+vt x vt dx v (x ( ex. x >l u (x = a l ( x + l x l (v (t = y a -l O l x u(t, x= (u (x vt+u (x + vt y u(t,x -v a O u(t, x=acos(k(x vt+δ = A cos(kx ωt + δ v = ω k : v l x λ = π k k 9

4. f(ξ =f(x vt T = f(ξ = a + a n cos(k n ξ+ k n sin(k n ξ n= n= f(ξ = c n e iknξ c n = n= dxf(xe iknx k n = nπ = n k, k = π ( f(k n = c n π f(ξ = n= π = π = k c n e iknξ = π f(kn e iknξ = π f(kn π eiknξ = π n= k f(k n e iknξ, k f(ξ = dk f(ke ikξ ( π f(k = f(k = π ( lim = lim π π dxf(xe ikx dxf(xe ikx < : > a x vt f(x vt = x vt > 3

a ξ = x vt 8: f(k a π O k 9: f(k = π = a π dξf(ξe ikξ dξe ikξ = a π = a π k sin(k [e ik e ik ] ik [ e ik e ik ] = sin(k i 9,a = x x = dxδ(x = 3

ξ : δ(x =δ( x dxδ(x=f( δ(ax = a δ(x dxδ(x yf(y=f(x < > f(k = π δ(x = π dxδ(xe ikx = π dk π e ikx = π k dke ikx f(ξ = ikξ dk f(ke π = π = π ξ ξ = x dk π dξ f(ξ dξ f(ξ e ikξ e ikξ dke ik(ξ ξ = dξ f(ξ δ(ξ ξ =f(ξ 4.3 m = σ x v = T σ 3

y x : K = m ( u t = xσ( vf f = u x u = A sin(kx + φ cos(ωt + δ x,t = xσx (f = xσ T σ (f = xt (f y = u x = xf l = x + y x = ( x ( + f x = x +(f x x (+ (f x = x(f W = T l = xt (f U = W = T (f 33

: E = K + U = xt (f ɛ = E x = T (f < =energy flux> v = J = vɛ = vt(f = vt v ( u t T σ T v = T = T v σ T = Tσ = Z ( u t Z K = ( u xσ = t xσ( vf + vg = σv ( f + g U = ( u xt = x xt (f + g E = K + U = xt [ ( f + g +(f + g ] = xt [ (f +(g ] ɛ = E x = T [ (f +(g ] ( 34

4.4 4.4. u(t, = u x (t, = u i u r u(t, x =u i (t, x+u r (t, x = f(x vt+g(x + vt f( vt+g(vt= g(x = f( x u(t, x =f(x vt f( x vt u r (t, x= f( x vt f ( vt+g (vt = f ( x+g (x = x dx f ( x = x ( dx f ( x = f( x+f( x dx g ( x =g(x+g( g(x = f(x + const. g(x =f( x u r (t, x =g(x + vt =f( x vt u r (t, x=f( x vt 35

T,σ I x = T,σ v v 3: II 4.4. 3 (v = u I (t, x x< u(t, x= u II (t, x x> u I (t, x =u i (t, x+u r (t, x u II (t, x=u t (t, x u t (transmitted wave u i (t, x =f I (x v t u r (t, x =g I (x + v t u t (t, x=f II (x v t T T,v = σ σ u I (t, = u II (t, u I T x (t, = T u II (t, x ( ( f I ( v t+g I (v t = f II ( v t (5 T (f I( v t+g I(v t = T f II( v t (6 (6 t dt f I ( v t = v t d( v t f I v ( v t = (f I ( v t f I ( v (6 T ( f I ( v t+ g I (v t v v = T ( f II ( v t + const. v 36

(7 T v (f I ( v t g I (v t = T v f II ( v t (7 T = T = T σ = z : v T σ T v = z f I ( v t g I (v t= z z f II ( v t (8 (5+(8 ( f I ( v t= + z f II ( v t z f II ( v t= z f I ( v t z + z (5-(8 ( g I (v t= z f II ( v t= z z f II ( v t z z = (z z f I ( v t z + z g I (v, t= z z f I ( v t g I (t = z z f I (x z + z z + z x x + v t g I (x + v t= z z z + z f I ( x v t f II ( v t= z z + z f I ( v t v t = x v t t = t v x f II (x v t= z f I ( v (t xv z + z = z ( v f I (x v t z + z v 37

v x = v v >v 4: : v v : g I (x + v t= z z f I (x v t z + z : f II (x v t= z ( v f I (x v t z + z v = z ( v f I x v t z + z v z = z f I (x v t=acos(k (x v t = a cos(k x ω t g I (x + v t= z z a cos(k x + ω t z + z f II (x v t= z ( v a cos k ω t = z a cos(k x ω t z + z v z + z k = k v v (, ω = ω 4 38

x = v 5: 4.5 5 u(t, = h(t ( f( vt =h(t f(x =h t v u(t, x=f(x vt =h ( v (x vt = h ( v + t h(t =a cos(ωt ( f(x vt =a cos ω (t v x ( ω = a cos v x ω = a cos(kt ωt k = ω (ω = vk ( v 4.6 ( v t x + ω u(t, x = (ω = g l u(t, x=acos(kx ωt ω + v k ωu(t, x= ω = ω + v k,ω= ω + v k 39

ω ω k=k ω = vk ω k k 6: 6 v k = ω k = ω + v k k k?( f(x, t= dk f(ke ik(x v kt π ω k = ω + v k a k k < f(k = k k > a k ω k = ω k=k (= ω = ω(k + dω dk (k k + π = π dk f(ke i(kx ω kt = e i(k x ω t a π dk ae i(k x+k ω t dω dk k t e ik (x dω dk t = e i(k x ω t a π x v g t sin( (x v gt ( 4

v g = dω k dk = v k ω ω + v k ( ω = + v k 5 5. (3 ( (3 (3 ( ( (Maxwell (Schrödinger 5.. ρ(x, y, z, t ( ( ρ t v x y z ρ = (isotropic ( P v = = κ = ( P ρ Kρ ρ ρ S/N S N S/N 5. (3 P (x, y, z, t=f(x, t y,z f t v f x = n =(n x,n y,n z, n = n x + n y + n z = s = bn r = n x x + n y y + n Z z, r =(x, y, z ρ(x, y, z,t=f(s, t 4

f(s, t = s f s = n f s = n s ( f f(s, t = n = n ( s f s s = f n s = f s f v t s f = ρ(x, y, z, t=acos( k r ωt + φ n = k k,s= k r k 5.3 ρ(x, y, z,t=g(r, t r = x + y + z g(r, t=( r g r = r r ( r g g = r r g r =+ r r r g r r x = x x + y + z = x r ( ( ( r = r + r r r ( r = r ( r r + r 3 r = x x + y y + z z =3 = r r r r + 3 r = r + 3 r = r g = g r r + g r ( g t g v r r + g r g(r, t= f(r, t r g r = ( f + r r r g r = ( r r f + r = r f = r f + r f r f r = r 3 f f r f f r r + f r r g r r + g r = ( r r f + f + r r r 3 f f r r + f r r = r f r 4

v v x = p 7: t f v r r = ( f r t v f r = f f t v f r = f(r, t=h(r, vt ρ(x, y, z, t= a cos(kr ωt r ρ(x, y,z, t=g(r, t= f(x, t r 5.4 ρ(x, y, z,t=f(x, y, t=a cos( k r ωt k =(kx,k y, (ρ z ( ρ i e i(kxx+kyy ωt + ρ r e i(k xx+k yy+k z z ω t = ρ t e i(k xx+k y y+k z z ω t x = (y, z, t k y = k y = k y, =k y = k y, ω= ω = ω 43

y k k θ k θ x 8: r d r x x= 9: k y = k sin θ k y = k sin θ k sin θ = k sin θ ω = v k, ω = v k = v k sin θ sin θ = k k = ω ω snell 44

5.5 a cos(kr ωt+a cos(kr ωt r r a ( ( cos k(r + r k ωt cos (r r Typeset by AMS-ATEX 45

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