29 2 1 2.1 2.1.1.,., 5.,. 2.1.1,. 2.2,., x z. y,.,,,. du dt + α p x = 0 dw dt + α p z + g = 0 α dp dt + pγ dα dt = 0 α V dα dt = 0 (2.2.1), γ = c p /c v., V = (u, w), = ( / x, / z).
30 2.1.1:
31., U p(z), ᾱ., ᾱ p z = g., ( ).,. u = U + u, w = w p = p(z) + p, α = ᾱ(z) + α (2.2.2) (2.2.1), (2.2),., 2. ( p ᾱ ( u u t + U u x + ᾱ p x = 0 ( ) w δ 1 t + U w + ᾱ p x z a ᾱ g = 0 ) ( t + U p α gw + pγ x t + U α x ) ( ) x + w α ᾱ δ 2 z t + U α x + w ᾱ z w ᾱ z = 0 ) = 0 (2.2.3), δ 1, δ 2,, 1 0.
32 2.3, g = 0, δ 1 = δ 2 = 1, ᾱ, p., x, y, t. u = S exp [i(kx + mz σt)], p = P exp [i(kx + mz σt)], w = W exp [i(kx + mz σt)] α = A exp [i(kx + mz σt)] (2.3.4), x, z k, m, σ.. (2.2.3),.,. (2.3.4) (2.2.3), S, W, P, A (ku σ) 0 ᾱk 0 S 0 (ku σ) ᾱm 0 W 0 0 ᾱ(ku σ) pγ (ku σ) P = 0 (2.3.5) ᾱk ᾱm 0 (ku σ) A. S, W, P, A,.. ᾱ(ku σ) 2 [ (ku σ) 2 + γ pᾱ(k 2 + m 2 ) ] = 0 (2.3.6) 4 4. 2, σ = ±ku,. 2, σ = ku ± (k 2 + m 2 ) 1/2 γr T.,., x U., (kx + mz = constant).. c = σλ/2π, λ. λ x = 2π/k, λ z = 2π/m, λ =
33 2π(k 2 + m 2 ) 1/2 *1. λ, c = σ(k 2 + m 2 ) 1/2 (2.3.7)., U = 0 (2.3). c = ± γr T (2.3.8). ( ). 2.4,., 0 = ᾱ p z g pᾱ = R T p ᾱ., p(z) = p(0)e z/h, ᾱ(z) = ᾱ(0)e z/h (2.4.9)., H = R T /g. U x U, U = 0. *1, k (k, m), k = (k 2 + m 2 ) 1/2.,. λ (λ x, λ z ), λ = 2π/(λ 2 x + λ 2 z)..
34 (2.2.3), u t + ᾱ p x = 0 w δ 1 t + ᾱ p z a ᾱ g = 0 ( ) ᾱ p α t ᾱ gw + pγ + w = 0 t z ( ) u x + w α ᾱ ᾱ δ 2 w z t z = 0.. (2.4.10) u = Sᾱ 1/2 e i(kx+mz σt), w = W ᾱ 1/2 e i(kx+mz σt) (2.4.11) p = Sᾱ 1/2 e i(kx+mz σt), α = W ᾱ 3/2 e i(kx+mz σt) (2.4.10), σ 0 k 0 0 δ 1 σ m + i ᾱ ig S ( ) 2α z W γr T ᾱ 0 g + i σ γr ᾱ z T σ P = 0 (2.4.12) k m + i ᾱ A 0 δ 2 σ 2α z.,. (2.4.9),. δ 1 δ 2 σ 4 {γr T (δ 1 k 2 + m 2 ) + g 4H [(2δ 2 1)γ + 2(1 δ 2 )] (2.4.13) + g(1 γ)(δ 2 1)im}σ 2 + k 2 g 2 (γ 1) = 0 2π/m 4πH, σ 2 g γr T m 2..,. δ 1 δ 2 σ 4 γr T (δ 1 k 2 + m 2 )σ 2 + γr T k 2 ḡ θ θ z = 0 (2.4.14) 4, 2 2. g = 0,., δ 2 = 0( ) δ 1 = 1. σ 2 = k 2 ḡ θ θ z (2.4.15) k 2 + m 2
35., c = σ k = ± k k 2 + m 2 ( ḡ θ θ ) 1/2 (2.4.16) z.,. ( θ/ z > 0),., ( θ/ z < 0),. (k 2 m 2 ), (2.4.15) ( ḡ σ ± θ ) 1/2 (2.4.17) θ z., - (Brunt- Vaisäla).,, (2.4.14) δ 1 = 0., σ 2 = k2 g θ (2.4.18) m 2 θ z., m 2 k 2 (2.4.15)., m m. e i(kx+mz σt) + e i(kx mz σt) = 2 cos mz e i(kx σt), σ m., c = σ k = ± 1 m ( ḡ θ ) 1/2 (2.4.19) θ z., k.,. (2.4.14), (δ 1 = 0) (δ 2 = 0),.,. (2.4.11) ᾱ 1/2,.,.
36 (2.2.3) (2.4.11).,., (2.2.3). u = Se (γ 1)z/γH e ik(x ct), w = 0, p = P e z/γh e ik(x ct), α = Ae (2γ 1)z/γH e ik(x ct). (2.4.20), c = ± γr T. (Lamb wave) *2., w = 0.,.,. 2.5.,., α = 0.,. (2.2.3). *2 w = 0, (2.4.20). w = 0 (2.2.3) ᾱ p t u t + ᾱ p x = 0, α + pγ t = 0, ᾱ p z a ᾱ g = 0 u x ᾱ α = 0 t 4, p 2 p z x 2 = g 2 p c 2 s x 2, 2 p t 2 + 2 p c2 s x 2 = 0 2., c 2 s = γr T., H(= R T /g) p = P e z/γh e ik(x c st) (2.4.21). u, α, p.
37,. u t + U u ( x w δ t + U w ) x. p z + ρ p x = 0 + ρ p z = 0 u x + w z = 0 (2.5.22a) (2.5.22b) (2.5.22c) = ρg (2.5.23) z = 0 H, z = 0 p 0 = g ρh., (2.5.22a), u = ψ(z)e ik(x ct) w = Φ(z)e ik(x ct) (2.5.24) p/ ρ = P (z)e ik(x ct) (U c)ψ(z) + P (z) = 0 dp (z) ikδ(u c)φ(z) + dz ikψ(z) + dφ(z) = 0 dz = 0 (2.5.25). 2 P (z), ψ(z), d 2 Φ(z) dz 2 k 2 δφ(z) = 0 (2.5.26)., δ = 1 δ = 0., Φ(z) = a 1 e kz + a 2 e kz δ = 1, Φ(z) = a 1z + a 2 δ = 0.., a 1, a 2, a 1, a 2.,. (δ = 1) a 1 = a 2 = a, Φ(z) = a(e kz e kz ) δ = 1 (2.5.27)
38,. Φ(z) = 2akz +, (δ = 0) a 2 = 0, a 1 = a, Φ(z) = az. (2.5.28) 2, ( )., d( p + p) dt = 0., z = H., p t + U p x + w p z = 0 at z = H (2.5.29) (2.5.27) (2.5.28), (2.5.25). (δ = 1). Φ(z) = 2a sinh(kz), ψ(z) = 2ia cosh(kz), P (z) = 2ia(U c) cosh(kz). (2.5.29),. c = U ± [ ] 1/2 gλ 2π tanh(2πh λ ) (2.5.30) (δ = 0). Φ(z) = az, ψ(z) = ia k. (2.5.29),., P (z) = ia(u c) k c = U ± gh (2.5.31)
39 (2.5.31), (shallow water wave). (2.5.30) H/λ, gλ c U ± 2π. (deep-water wave),. H/λ, (2.5.30) (2.5.31),,. gh = p 0 /ρ = RT, (2.5.31) c = U ± RT (2.5.32). (2.3.8). (2.5.31),.,.,., g ρ(h z) = p (2.5.33)., h. u = U + u, h = H + h (x, t), p = p(z) + p (x, z, t)., 1 p ρ x = g h x (2.5.34)., (2.5.22a) 1. u t + U u x + g h x = 0 (2.5.35) u h 2. (2.5.35) h z, u z u., z = 0 w = 0. (2.5.22a) 3 z = 0 h, w z=h = u x h.
40, 2 w z=h = dh dt = h t + u h x ( ρh) t., + ( ρuh) x = 0 (2.5.36) h t + U h t + H u x = 0 (2.5.37). h u z,. u = u 0 e ik(x ct), h = h 0 e ik(x ct), u 0 h 0. (2.5.35) (2.5.37),,. c = U ± gh, (external gravity wave).,, (internal gravity wave).. Haltiner and Martin(1957), 2 0., c = ρu + [ ρ U ] gλ(ρ ρ ) ± ρ + ρ 2π(ρ + ρ ) ρρ (U U ) 2 1/2 (ρ + ρ ) 2.,,.,. ρ = 0,., U = U, ρ = ρ.,. (Kelvin-Helmholtz wave).
41 2.6,.,.,.,. u t + u u x + v u h fv + g y x = 0 v t + u v x + v v h + fu + g y y = 0 (2.6.38a) (2.6.38b),.,.,.,. u x + v y + w z = 0 (2.6.39) (2.5.36), u h h t + (hu) x + (hv) y = 0 (2.6.40). (2.6.38a), (2.6.38b), (2.6.40) 3 (u, v, h) 3. (shallow-water equation). (2.6.38b) x, (2.6.38a) y,. (2.6.40),, ζ t + u ζ x + v ζ y ( u + βv = (f + ζ) x + v ) = (f + ζ) 1 dh y h dt (2.6.41)., f y, β = f/ y., ( ) d ζ + f = 0 (2.6.42) dt h. (ζ + f)/h., (potential vorticity)
42., Ertel(1942). (2.6.42),,.,.,. (ζ + f)/2, (2.6.42). (2.6.38a), (2.6.38b), (2.6.40), h = H + h, u = U + u, v = v. U, H U., U = g f H y (2.6.43)., H, U y. f y,. ( u δ t + U u x ) fv + g h x = 0 v t + U v x + fu = 0 h t + U h x + H u x + v H y = 0 (2.6.44),., δ dd/dt (D = u/ x + v/ y). x, y x y. v y, u., H,. u = u 0 e ik(x ct), v = v 0 e ik(x ct), h = h 0 e ik(x ct), (2.6.44). δ(u c)iku 0 fv 0 + gikh 0 = 0 (2.6.45) fu 0 + ik(u c)v 0 = 0 (2.6.46) ikhu 0 + H/ y v 0 + ik(u c)h 0 = 0 (2.6.47)
43 u 0, v 0, h 0,., 3. δ(u c) 3 (gh + f 2 /k 2 )(U c) fg k 2 H y = 0 (2.6.48)., U = 0, δ = 1 c = ± gh + f 2 /k 2 (2.6.49) c = 0. (inertial gravity wave), f = 0 (2.6.49). 1 2,. σ = kc = ±f (2.6.50), (1/k), gh.,, du dt = fv, dv dt = fu 2, d 2 u dt 2 + f 2 u = 0. u = u 0 e ±ift,. 2π/f = 12/ sin φ hour.,.,.,.
44 (2.6.48), δ = 0 *3. c = U + (f/h) H/ y k 2 + (f 2 /gh), q = f/h. q y, q y = f H H 2 y = f 2 U (2.6.51) gh 2, H(z) y., y. (2.6.51), c = U H q/ y k 2 + (f 2 /gh) (2.6.52). (Rossby wave) *4,., f., ( q y = β f H ) H /H = β + (f 2 U/gH) y H (2.6.53)., β = df/dy., k q. f, (2.6.52) q/ y (2.6.53)., y,. (2.6.44) δ = 0, (2.6.48). v., *3 δ = 0 fv = g h x. (2.6.54) fv + g h x = 0 v t + U v x + fu = 0 h t + U h x + H u x + v H y = 0. *4,. (2.6.52),.,,,.
45 u *5,. x y, (2.6.38a) (2.6.38b).,,.,. 2,., y,. D ζ = u/ x v/ x (2.6.55) (U = 0),. (2.6.45) e ik(x ct), v = if kc u (2.6.56) c (2.6.49), (2.6.55) D ζ = 1 + k 2 ghf 2 (2.6.57) I G. 1.,. (δ = 0), h. (2.6.54). v x = g k2 h. (2.6.58) f, (2.6.44) u h. 2 u x f 2 2 gh u = U 2 h (2.6.59) H x 2 *5 δ = 0, u = g 2 f 2 (hu Hu) x2., (hu Hu).
46. x, u x = ikug gh + (f 2 /k 2 ) h (2.6.60). (2.6.58) (2.6.60) (2.6.55),. D (Uk/f) ζ = (2.6.61) R (k 2 gh/f 2 ) + 1 (Rossby number), 0.1. 1,..
47 2.7,. 2., (H = 10km ) 300 m/s., 10 m/s.,.,. (δ = 1),., (Hinkelmann,1951; Phillips, 1960)., (2 ) (1 ) 3. c j (j = 1, 2, 3). c j, (u j, v j ) (h j )., (2.6.44), 3 u j, v j, h j 1 j. u j = U c j, δ = 1 (2.6.44) 2., u j = U c j. k, 3 u = a j u j e ik(x c jt) v j = if/k h j = g [ (2.7.62) 1 f 2 /k 2 (U c j ) 2] v = h = j=1 3 a j v j e ik(x c jt) j=1 3 a j h j e ik(x c jt) j=1., a j. u, v, h. u(x, 0) = u 0 e ikx, v(x, 0) = v 0 e ikx, h(x, 0) = h 0 e ikx (2.7.63), u 0, v 0, h 0. (2.7.63), u 0 = j=1 a j u j, v 0 = j=1 a j v j, h 0 = j=1 a j h j
48.,. c 1 = U U 1 + (ghk 2 /f 2 ) c 2 = U + gh + (f 2 /k 2 ) c 3 = U gh + (f 2 /k 2 ) (2.7.64) (2.7.62) (2.7.64), (2.7) a j. a 1 = ikf 1 ghv 0 gh 0 U 2 f 4 k 4 α 2 α a 2,3 = (u 0 /2α 1/2 ) ifk 1 (1 ± Uα 1/2 )v 0 + gh 0 2(Uf 2 k 2 α 1/2 ± α) (2.7.65), α = gh + f 2 k 2.. u = g f, h y = 0, v = g f h x u 0 = 0, v 0 = ikf 1 gh 0 (2.7.66). (2.7.65), j k 2 f 2 gh 0 a 1 = 1 U 2 f 4 k 4 α 3 Uα 3/2 gh 0 a 2,3 = 2(1 ± Uf 2 k 2 α 3/2 ) (2.7.67).,. α 1/2 > fk 1 U (2.7.68) v (v 1 a 1 = ifk 1 a 1 ) v 0, v 0 *6., *6 v 0 (a 1 v 1 ) v 0, v 1 a 1 v 0 = (1 U 2 f 4 k 4 α 3 ) 1 (2.7.68), U 2 f 4 k 4 α 3 1., v 1 a 1 v 0 1 + U 2 f 4 k 4 α 3, v 0 a 1 v 1.
49. (2.7.66)., a 2,3 a 1 = U gh f 2 k 2 gh. (2.7.69), (2.7.68).,. u 0,. (2.7.65) a 2,3 = 0, v 0, u 0. Ugh 0 u 0 = (2.7.70) gh + f 2 k 2 (2.7.68), α U 2 f 4 k 4 α 2. u = u 0 e ikx, (2.6.60). 2.8 2 1,.,, *7.,,,.,.,. *7 (2.6.44), x U u, y v. u, v,.,.,., (, ) 2.3.
50. u t fv + g h x = 0 v t + fu = 0 h t + H u x = 0 (2.8.71), (2.6.44) U = 0., 3., 2 (2.6.49) (2.6.52). H q/ y = 0,. Rossby(1938), Cahn(1945)., Schoenstadt(1977)., x u(x, t) x., ũ(k, t) = u(x, t) = u(x, t)e ikx dx (2.8.72) ũ(k, t)e ikx dk (2.8.73). v, h., (2.8.71) e ikx, x. x,., *8 dũ = fṽ ikg h dt dṽ dt = fũ d h dt = ikhũ (2.8.74), u ũ 0 = ū(k, 0) = u(x, 0)e ikx dx (2.8.75) *8, k., ũ(k, t)/ t dũ/dt.
51., v 0, h 0. ũ 0, ṽ 0, h 0 (2.8.74) *9, ( ) fṽ 0 ikg h 0 ũ(k, t) = ũ 0 cos νt + sin νt ν ( ) ṽ(k, t) = k2 ghṽ 0 + ikgf h 0 fũ 0 f 2 ṽ 0 ikgf h 0 sin νt + cos νt (2.8.76) ν 2 ν ν 2 h(k, t) = f 2 h0 ikhfṽ 0 ikhũ 0 sin νt + ν 2 ν., ( k 2 gh h 0 + ikhṽ 0 ν 2 ) cos νt ν = f 2 + k 2 gh (2.8.77)., ( ). U = 0, c 1 = 0, c 2,3 = ±ν/k., ( ) s ( ) T. ũ s (k) = 0 ( ) ṽ s (k) = ikg f 2 h0 ikfhṽ 0 f ν 2 ( ) f h 2 h0 ikfhṽ 0 s (k) = ν 2 (2.8.78) ũ T (k, t) = ũ 0 cos νt + f ν (ṽ 0 if 1 kg h 0 ) sin νt ṽ T (k, t) = f ν ũ0 + f 2 ν (ṽ 2 0 if 1 kg h 0 ) cos νt (2.8.79) h T (k, t) = ikh ν ũ0 sin νt + ikhf (ṽ ν 2 0 if 1 kg h 0 ). (2.8.78), ṽ s = ikgf 1 h s, ũ s = 0., (2.8.74) d/dt., (2.8.79), ((ṽ 0 ikgf 1 h 0 ) ũ 0 ). *9 : (2.8.74).
52,.,., u T (x, t) = 1 2π v T (x, t) = 1 2π h T (x, t) = i 2π, ũ 0 cos νt e ikx dk + 1 2π fũ 0 ν sin νt eikx dk + 1 2π khũ 0 ν sin νt e ikx dk + i 2π d(x, t) = v(x, t) g f f 2 h (x, t) x f ν d(k, 0) sin νt e ikx dk ν d(k, 0) cos νt e ikx dk 2 khf ν 2 d(k, 0) cos νt e ikx dk (2.8.80),. Schoenstadt(1977), (2.8.80). x, x L R ft = ght, *10. u T 1 2πα [ ũ(x/α, 0) cos(ft + φ) + d(x/α, 0) sin(ft + ψ)] v T 1 2πα [ ũ(x/α, 0) sin(ft + φ) + d(x/α, 0) cos(ft + ψ)] (2.8.81) h T 1 ( Hx) 2πα fα [ ũ(x/α, 0) sin(ft + φ) d(x/α, 0) cos(ft + ψ)], ψ φ., L R = gh/f, α = L 2 Rft. (2.8.81), u T, v T x t 1/2, h T t 3/2.,,., G. G = ν k = kgh f 2 + k 2 gh = gh f L 2 + L 2 R (2.8.82), 2πL = 2π/k., L R, c = gh,.,. (2.8.81), x *10 (2.8.80),.
53, x,., L 2 R /L2., (2.8.78). ( ) 1 L 2 ṽ s = R 1 + (L 2 R /L2 ) L 2 ṽ0 + ikf 1 g h 0 ( ) (2.8.83) 1 L 2 h s = R fṽ 0 1 + (L 2 R /L2 ) L 2 ikg + h 0,., ṽ 0 ikf 1 g h 0. L 2 L 2 R., (2.8.83) ṽ s = ṽ 0, hs = fṽ 0 ikg (2.8.84).,., L 2 L 2 R (2.8.83) ṽ s = ikf 1 g h 0, hs = h 0 (2.8.85).,.. L R,., L R,., 2., L R = c g /f(c g )., Okland(1970)., H 10 km, L R = 3000 km, 2πL R = 18000 km., 5000 km 2πL R., c g 100ms 1, L R 1000 km. 2πL R 6000 km,.,. f
54,.,.., h s (x) = h(x, 0) h T (x, 0), h T (x, 0) (2.8.79), h s (x) = h(x, 0) 1 2π Hfik ν 2 d0 e ikx dk (2.8.86)., d 0.,. 1 2π F (k) G(k) e ikx dk = F (x ) G(x x ) dx (2.8.87), F, G F, G., e x /L R e x /L R e ikx dx = 2L R = 2L Rf 2 1 + k 2 L 2 R ν 2., G(x) = e x /L R, F (x) = d(x, 0)/ x *11, (2.8.87), (2.8.86) h s (x) = h(x, 0) H e x x /L R d(x, 0) dx (2.8.88) 2L R f x., h s (x). h s (x) = h(x, 0) + H x x 2L 2 R f x x e x x /LR d(x, 0) dx (2.8.89), x L R., (2.8.89)., v s (x) = fg 1 h s / x, u s = 0 (2.8.90) 2,.,.,. *11 G(k) = 2L R f 2 /ν 2, F (k) = ik d(k).
55,,,. 2,,.,,.,,., L 2 /L 2 R., L R = c g /f, c g. L L R,. L L R,.,.
56 : (2.8.74) 3 s = (s 1, s 2, s 3 ) = (ũ, ṽ, h), (2.8.74) G ds = Gs (2.8.91) dt.,. 0 f ikg G = f 0 0 ikh 0 0 (2.8.92) G. G λ 1, λ 2, λ 3, p 1, p 2, p 3., G P = [p 1 p 2 p 3 ] *12. P 1 GP = diag(λ 1, λ 2, λ 3 ) (2.8.93) G, G 3., G. G, P 0 1 1 P = 1 if/ν if/ν (2.8.94) if/(kg) kh/ν kh/ν., G. P 1 GP = diag(0, iν, iν) (2.8.95), P s = P s. (2.8.91) P 1, ds dt = P 1 GP s (2.8.96) *12 P,.
57. (2.8.95), ds dt = diag(0, iν, iν) s (2.8.97). s = (s 1, s 2, s 3), s 1 = C 1, s 2 = C 2 e iν, s 3 = C 3 e iν (2.8.98), C 1, C 2, C 3. s P, (2.8.91). 0 1 1 s = C 1 1 + C 2 e iνt if/ν + C 3 e iνt if/ν (2.8.99) if/(kg) kh/ν kh/ν (2.8.76), (2.8.99) C 1, C 2, C 3 s (u 0, v 0, h 0 )., t = 0 (2.8.99) C 1, C 2, C 3.,. : (2.8.80) *13,. f(t) = g(k)e iθ(t,k) dk (2.8.100), θ(k, t) k, g(k) k. e iθ k, g(k) k,., f(t)., θ/ k = 0 *13, (http://www.ep.sci.hokudai.ac.jp/ yamasita/phys-math-approx.pdf).
58. k = k s, ε, f(t). f(t) ks +ε k s ε g(k)e iθ(t,k) dk (2.8.101) θ(t, k) k = k s, 2 θ(k) θ(k s ) + 1 2 θ 2 k 2 (k k s ) 2 k=ks (2.8.101)., g(k) g(k s ), k, ks +ε [ ] f(t) g(k s ) e iθ(t,k s) i 2 θ exp 2 k 2 (k k s ) 2 dk (2.8.102) k=ks k s ε., k = k s θ/ k = 0. g(k) = A(k)/2π, θ(k, t) = kx + νt, x t f(t)., (2.8.102), t f(t) 1 2π A(k s) e i(k sx+ν(k s )t) ks+ε k s ε., θ ν ν = f 1 + L 2 R k2 [ exp i t ] 2 ν 2 k 2 (k k s ) 2 dk (2.8.103) k=ks., L R = (gh) 1/2 /f. ν k 1 2, ν k = f fl 2 R k (1 + L 2 R k2 ), 2 ν 1/2 k = 2 fl 2 R (1 + L 2 R k2 ) 3/2, 0 = θ/ k = x + ( ν/ k)t k. t, x/l R k s = (L 2 R f (2.8.104) 2 t 2 x 2 ) 1/2., k = k s 2 ν k 2 = [f 2 L 2 R t2 x 2 ] 3/2 > 0 k=ks f 2 L R t 3
59. (2.8.103), { t p = (k k s ) 2 ν 2 k (k s)) 2 } 1/2. p, t. (2.8.103), f(t) 2 A(k s) e i(k sx+ν(k s )t) [ ( )] ( ) t 2 ν 1/2 exp ip 2 2 ν sgn k 2 k (k s) dp (2.8.105) 2., sgn(a) A 1, -1. Fresnel *14, f(t) 2π A(k [ s) e i(ksx+ν(ks)t) ( ) t 2 ν 1/2 exp i π ( )] 2 4 sgn ν k (k s) (2.8.106) 2 k 2. k s θ(k s ), k = k s 2 ν/ k 2,. f(t) ( L 2 R 2π f ) 2 t 2 1/2 [ (L 2 [L 2 R f A(k 2 t 2 x 2 ] 3/2 s ) exp i R f 2 t 2 x 2 ) 1/2 + π ] (2.8.107) 4 2 ν/ k 2, π/4 π/4. L 2 R (2.8.80) x L R ft(= ght) x, t (2.8.80). I(x, t) = 1 2π J(x, t) = 1 2π A(k) cos νt e ikx dk, A(k) sin νt e ikx dk. A(k)., 1. I(x, t) = 1 2π 1 { } A(k) e iθ1(k) dk + A(k) e iθ2(k) dk (2.8.108) 2 *14 Fresnel. 0 cos x 2 dx = 0 sin x 2 dx = 1 2 π 2
60., θ 1,2 (k) = kx ± νt., ν = f(1 + L R k 2 ) 1/2. (2.8.107), θ 1. θ 2., θ 2 / k = 0 k(= k 2s ), k 1s = k 2s, ν. θ 1 (k 1s ) = θ 2 (k 2s ), 2 ν k (k 1s) = 2 ν 2 k (k 2s) > 0 2 x L R ft, (2.8.104) k 1s = k 2s x/α, ν(k 1s ) = ν(k 2 s) f (2.8.109)., α = L 2 Rft., (2.8.107), θ 1 = kx + νt [ 2π f(t) α A(k 1s) exp i ft + π ( )] 2 4 sgn ν k (k s) 2 θ 2 = kx νt, [ 2π f(t) α A(k 2s) exp i ft π ( )] 2 4 sgn ν k (k s) 2., I(x, t) I(x, t) 1 2 2πα. (2.8.110) (2.8.111) { A(k1s )e i[ft+(π/4)] + A(k 2s )e i[ft+(π/4)]} (2.8.112), A(k), A( k) = A (k).,., (2.8.112), I(x, t) 1 { Re[A(k 2s )] cos(ft + π 2πα 4 ) + Im[A(k 2s)] sin(ft + π } 4 ) (2.8.113) *15., J(x, t) 1 2πα { Re[A(k 2s )] sin(ft + π 4 ) Im[A(k 2s)] cos(ft + π 4 ) } (2.8.114) *15 k 1s = k 2s,,. A(k 1s ) = A( k 2s ) = A (k 2s ) A e iθ + Ae iθ = 2(Re[A] cos θ + Im[A] sin θ)
61., I(x, t) 1 2πα A(k 2s ) cos(ft + φ k ), J(x, t) 1 2πα A(k 2s ) sin(ft + φ k )., { [ReA(k)] 2 + [ReA(k)] 2} 1/2 = A(k) (2.8.115)., φ k = π 4 arg(a(k))., (2.8.109) (2.8.115), (2.8.80) (2.8.81) *16. *16 3. ũ, (2.8.80) 3 ik ũ 0 (k)(b(k) ) B( k) = ik ũ 0 ( k) = ik ũ 0(k) = B (k), ik ũ 0 (k)., (2.8.115) A(k), ik ũ 0 (k). A(k s ),. A(k s ) = ik s ũ 0 = k s ũ 0 (k s )