1 capillary-gravity wave 1) 2) 3, 4, 5) (RTI) RMI t 6, 7, 8, 9) RMI RTI RMI RTI, RMI 10, 11) 12, 13, 14, 15) RMI 11) RTI RTI y = η(x, t) η t
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1 1 capillary-gravity wave 1) ) 3, 4, 5) (RTI) RMI t 6, 7, 8, 9) RMI RTI RMI RTI, RMI 10, 11) 1, 13, 14, 15) RMI 11) RTI RTI y = η(x, t) η t + ϕ 1 η x x = ϕ 1 y, η t + ϕ η x x = ϕ y. (1) 1
2 ϕ i i (i = 1, ) u i u i = ϕ i i = 1 (i = ) - { [ ϕ 1 (1 A) t + 1 ( ϕ1 ) ( ) ]} ϕ1 + x y { [ ϕ (1 + A) t + 1 ( ϕ ) ( ) ]} ϕ + Agη = σκ, () x y A = (ρ ρ 1 )/(ρ 1 + ρ ) ρ i i (i = 1, ) g ( ) [ ( ] 3/ σ = σ 0 /(ρ 1 + ρ ) σ 0 κ κ = η x 1 + η x) i (i = 1, ) ϕ i = 0 (1) () η = Re{A 1 e iωt }cos(kx), A 1 C. k ω = Agk + k 3 σ/ (A = 1) Concus 3) Concus Vanden-Broeck 4) Schultz 5) A = 1 i = Concus 1 A 1 A > 0 RTI RTI RMI 11) η ϕ i (i = 1, ) ω ϵ ( ϵ 1) η = ϵη (1) + ϵ η () +, ϕ i = ϵϕ (1) i + ϵ ϕ () i +, (1) () ϵ ω = ω 0 + ϵω 1 + ϵ ω + (3) ϵ η () = η (1) = Re{A 1 e iω0t }coskx, ω 0 = Agk + k3 σ, ϕ (1) 1 = ω 0 k Re{iA 1e iω 0t }e ky coskx (y < 0), ϕ (1) = ω 0 k Re{iA 1e iω 0t }e ky coskx (y > 0) (4) [ Re{A () eiω 0t } + Re{A () 0 } ] coskx, ϕ () 1 = Re{B () 1 eiω0t }e ky coskx + Re{B () 01 eiω0t } (y < 0) ϕ () = Re{B () eiω 0t }e ky coskx + Re{B () 0 eiω 0t } (y > 0)
3 A () = ( Ak(A 1) ), A () 0 = ( k A 1 ), 4 1 3k3 σ k3 σ ω0 ω0 ) ] iω 0 [(1 3k3 σ + A (A B () ω0 1 = 1 ) ( ), 4 1 3k3 σ ω0 ) ] iω 0 [(1 3k3 σ A (A B () ω0 = 1 ) ( ), 4 1 3k3 σ ω0 ω 1 = 0, (5) B () 01 B() 0 η (3) ϕ (3) i (1 A)B () 01 (1 + A)B() 0 = iaω 0(A 1 ) (i = 1, ) [ ] η (3) = Re{A (3) 13 e3iω 0t } + Re{A (3) 11 eiω 0t } coskx [ ] + Re{A (3) 33 e3iω0t } + Re{A (3) 31 eiω0t } cos3kx, ϕ (3) 1 = Re{ 13,1 e3iω 0t }e ky coskx [ ] + Re{ 33,1 e3iω0t } + Re{ 31,1 eiω0t } e 3ky cos3kx (y < 0), ϕ (3) = Re{ 13, e3iω 0t }e ky coskx [ ] + Re{ 33, e3iω 0t } + Re{ 31, eiω 0t } e 3ky cos3kx (y > 0), A (3) 13 = k 3A( A) (7 A) + ( ) 3k3 σ k3 σ 4ω (A 1 ) 3, ω0 0 13,1 = ikω 0 3A( A) (15 + A) + ( ) 3k3 σ k3 σ 4ω (A 1 ) 3, ω0 0 13, = ikω 0 3A( A) (7 + A) + ( ) 3k3 σ k3 σ 4ω (A 1 ) 3, ω0 0 A (3) 11 = k 16 A( + A) 7 + 3A ( ) + ( 1 3k3 σ ω0 A 1 + 3k3 σ ω 0 ) + 9k3 σ 4ω0 A 1 A 1, A (3) 33 = k A ( ) k3 σ 3 3A ( ) 1 3k3 σ ω0 ω0 1 ( ) 11 ( ) 3A ( ) 3 + A ( ) k3 σ k3 σ 4 1 3k3 σ 16 3k3 σ 3ω ω0 ω0 ω0 0 (A 1) 3, 33,1 = iω ( 0 k A(3) k 48 (A 1) ) A 1A (), 33, = iω ( 0 k A(3) k 48 (A 1) 3 3 ) A 1A (), 3
4 A (3) 31 = k A(1 A) 8A + 15 ( ) ( ) k3 σ 4 1 3k3 σ 48 ω0 ω0 ]} + 55k3 σ 3ω0 A 1 A 1, 31,1 = iω ( 0 3k A(3) 31 1 A 1A () + A 1A () 0 + k ) 8 A 1 A 1, ) 3,1 = iω 0 ( 3k A(3) A 1A () A 1A () 0 + 5k 4 A 1 A 1 (6) A 1 A 1 ω = k ω A A( + A) ( ) + 9k3 σ A + ( 8 1 3k3 σ ω 0 4ω k3 σ ω 0 ) A 1 (7) (7) 16) (5) A () (6) A(3) 33 ω 0 = 3/k 3 σ ω0 = k 3 σ n (1) η (n) t 1 ( ϕ (n) 1 y + ϕ(n) y ) = R 1, ϕ(n) 1 y + ϕ(n) y = R, η (n) A (n) nn ϕ (n) i (i = 1, ) B (n) nn,i A (n) nn = ik ω 0 (B (n) nn,1 B(n) nn, ) + R 3 (8) R i (i = 1,, 3) (1) - () n n (1 A)ω0B (n) nn,1 + n (1 + A)ω0B (n) nn, ( ω + nk 0 k + (n 1)k ) σ ( ) B (n) nn,1 B(n) nn, R 4 () (8) (9) A (n) nn = ( R 1 (n+1)k3 σ ω 0 R O(ϵ), O(ϵ ), O(ϵ n 1 ) ω 0 = R 4 (9) ) (10) ω0 = (n + 1)k3 σ, (11) A (n) nn B (n) nn,i (i = 1, ) n Ag < 0 (11) Christodoulides Dias 17) (n = ) nk nω 0 (4) k ω 0 n n = , 11) 4
5 Fig. 1: A = 0., g = 1.0,σ = 0.5 (a) (b) (c) (d) Hou 13, 14, 15) RTI 11, 18) 11) RMI(g = 0) x(β, 0) = β, y(β, 0) = 0; γ(β, 0) = sinβ, (1) RTI x(β, 0) = β, y(β, 0) = 0.1cosβ; γ(β, 0) = 0, (13) β γ = Γ/ β, (Γ = ϕ 1 ϕ, ) 11) (1) RMI 19, 0) RMI RTI (4) A 1 A 1 = i (RMI), A 1 = 1 (RTI). (14) k = 1 ω 0 RMI ω0 = σ/ RTI ω0 = Ag + σ/ Fig. 1 A = 0., g = 1.0,σ = 0.5 ϵ ϵ = 0.1 A < 0 t ( ) N t = N = 51 RTI capillary-gravity standing wave solutions (13) t = 5.0 t =
6 Fig. : A = 0.5, g = 1.0, σ = 1.5 (a) (b) (c) (d) 6
7 Fig. 3: RMI (a) A = 0., σ = 1.0 (c) (a) (b) A = 1.0, σ = 1.0 (d) (b) (a) t = (b) β = ±π β = 0 (c) (d) (3) T = π/ω T = T = O(ϵ ) Fig. RTI A = 0.5, g = 1.0, σ = 1.5, ϵ = 0.1 t = N = 51 (A > 0) ω0 > 0 ω 0 (a) (b) Fig. 1 (c) (d) T = T = O(ϵ) Fig. 1 RMI Fig. 3 A (a) [(c)] A = 0. (b) [(d)] A = 1.0 σ = 1.0 t = N = 51 (1) RMI (1) γ(β, 0) ϵ ϵ = O(1) (a) (c) (c) (d) (a) [(c)] t 50 (b) [(d)] t = 6 ω0 > 0 ω0 < 0 ω 0 = 0 ω0 = σ/ σ 0 RMI 7
8 Fig. 4: A = 0., g = 5.0, σ =(a).01 (b).0 ( ) (c) 1.99 Fig. 5: A = 0. g = 5.0 t = (a) 84.0, (b) 34.3, (c) 8.0 (a) (c) Fig. 4 (a) (c) 8
9 Fig. 6: A = 0., g = 5.0 σ = (a).01, (b).0 & 1.99 (c) (a) A > 0 ω 0 = 0 ω 0 Fig. 4 A = 0. g = 5.0 ω 0 σ =.0 N N = 51 [(b) 0 t 8.0 (c) 0 t 0.0] N = 104 [(b) 8.0 t 3.0 (c) 0.0 t 5.5] N = 048 [(b) 3.0 t 34.3 (c) 5.5 t 7.6] t N = t = N = 048 t = (a) N N = 51 (b) t = 34.3 (c) t = 7.6 (a) (b) (c) (b) (ω 0 = 0) (c) ω0 < 0 (c) Fig. 5 (b) (c) Fig. 4 (b) (c) ω0 > 0 (a) ( t = 84.0) γ t = 83.0 ω0 A > 0 Fig. t 100 ω0 < 0 Fig. 4 (b), (c) ω0 < 0 10, 11) Fig. 6 Fig. 4 Fig. 5 ω 0 (σ =.0) σ =.01 σ = 1.99 ω 0 (c) (a) σ = σ =.0 t = 5 σ = 1.99 t = 0 (b) Fig. 4 (b) (c) O(1) (a) O [ (ω 0 ) 1] ω 0 = O(10 1 ) (c) sinusoidal σ A g ω 0 0 Fig. 7 (11) Γ γ = γ/s β s β s β (11) n n = 3 n ) Ascher, Ruuth, Wetton ARW 1) t 0 t 1.5 t = t 1.36 t = N 0 t 1.5 N = t 1.36 N = 048 9
10 Fig. 7: (a) t = (b) 1.5 (c) 1.31 Γ γ A = 1.0, g = 1.0,σ = /3 (b) (c) γ Γ (13) t = 1.36 t = 1.36 γ Fig. 7 (c) wave packet t = 1.36 A < 0 4 ω0 > 0 ω 0 = 0 (11) e i(kx ω0t) + c.c. n ω/k 3 multiple scale (C) REFERENCES 1) Whitham G B in Linear and Nonlinear Waves (John Wiley & Sons, 1999) ) Okamoto H and Shoji M in The mathematical theory of permanent progressive water-waves, (World Scientific, 001) 3) Concus P 196 J. Fluid Mech ) Vanden-Broeck J-M 1984 J. Fluid Mech ) Schultz W W, Vanden-Broeck J-M, Jiang L and Perlin M 1998 J. Fluid Mech [references therein] 10
11 6) Richtmyer R D 1960 Comm. Pure. Appl. Math , Meshkov E E 1969 Fluid Dyn ) Matsuoka C, Nishihara K and Fukuda Y 003 Phys. Rev. E , (E) 8) Matsuoka C and Nishihara K 006 Phys. Rev. E (references therein), (E) 9) Matsuoka C and Nishihara K 006 Phys. Rev. E (R), ) Matsuoka C 008 Phys. Scr. E T ) Matsuoka C 009 Phys. Fluids ) Leppinen D and Lister J R 003 Phys. Fluids ) Hou T Y, Lowengrub J S and Shelley M J 1993 J. Comput. Phys ) Hou T Y, Lowengrub J S and Shelley M J 1997 Phys. Fluids ) Hou T Y, Lowengrub j S and Shelley M J 001 J. Comput. Phys ) Hakim V 1998 in Hydrodynamics and Nonlinear Instabilities, ed. by Godréche and Manneville (Cambridge) 95 17) Christodoulides P and Dias F 1994 J. Fluid Mech ) Ceniceros H D and Hou T Y 1998 Math. Comput ) Wouchuk J G and Nishihara K J 1996 Phys. Plasmas , ) Nishihara K, Wouchuk J G, Matsuoka C, Ishizaki R and Zhakhovsky V V to be published in Philos. Trans ) Ascher U M, Ruuth S J and Wetton B 1995 SIAM J. Num. Anal
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More information) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
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A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
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