chap03.dvi

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1 99 3 (Coriolis) cm m (free surface wave) 3.1 Φ 2.5 (2.25) Φ

2 100 3 r =(x, y, z) x y z F (x, y, z, t) =0 ( DF ) Dt = t + Φ F =0 onf =0. (3.1) n = F/ F (3.1) F n Φ = Φ n = 1 F F t Vn on F = 0 (3.2) Φ (3.1) (kinematic condition) z = ζ(x, y, t) (3.1) F F = z ζ (x, y, t) = 0 (3.3) (3.1) (3.2) Φ ζ (x, y, t) (dynamic condition) 2.4 (2.18) Φ t + 1 Φ Φ + gζ =0 onz = ζ (x, y, t). (3.4) 2 (3.1) (3.4) ζ Φ (3.4)

3 (ζ) (3.4) Φ ζ Φ ζ (3.4) (3.1) ζ 2 Φ t 2 ζ = 1 Φ g t + O(Φ2 ), (3.5) ζ t + Φ + O(ζΦ)=0. z (3.6) + g Φ z + O(Φ2 )=0 onz = ζ (3.7) z = ζ ζ z =0 ( ) Φ Φ(x, y, z, t) =Φ(x, y, 0, t)+ζ +. (3.8) z z=0 (3.7) z =0 O(Φ 2 ) (linearized free-surface condition) 2 Φ t 2 + g Φ z =0 onz =0. (3.9) Φ z = ζ (3.5) ) ( )[ D p Φ = Dt( ρ t + Φ t + 1 ] 2 Φ Φ + gz =0 onz = ζ. (3.10)

4 Φ t 2 + g Φ ( Φ ) z +2 Φ + 1 t 2 Φ ( Φ Φ ) =0 onz = ζ. (3.11) O(Φ 2 ) (3.9) (3.8) ζ (3.5) z =0 (3.11) O(Φ 2 ) 2 Φ t 2 + g Φ z = 2 Φ Φ t + 1 Φ g t ( 2 Φ z t 2 + g Φ ) z on z =0. (3.12) 3.1 (3.2) F (x, y, z) = 0 n = F/ F x = a cos θ, y = b sin θ cos ϕ, z = b sin θ sin ϕ n 1 = ɛ cos θ/, n 2 = sin θ cos ϕ/, n 3 = sin θ sin ϕ/. = sin 2 θ + ɛ 2 cos 2 θ, ɛ= b/a. 3.2 (3.1) (3.4) ζ (3.8) (3.12) 3.2 ( z = h ) x 3.1 a ω z = ζ(x, t) =a cos(ωt kx) (3.13)

5 z k (wavenumber) SF O c λ k = x 2π/λ S-oo Soo (3.13) ωt kx SB -h x 3.1 t + δt x x + δx ωt kx = ω(t + δt) k(x + δx), δx δt = ω c>0 (3.14) k (3.14) c (phase velocity) x ωt + kx f(ωt kx) f t + c f x = ( ω ck ) f = 0 (3.15) Φ Φ 3.1 S F S B x ± S ± Φ [L] [F ] [B] 2 Φ x + 2 Φ = 0 2 z2 for z 0, (3.16) 2 Φ t + g Φ =0 2 z onz =0, (3.17) Φ =0 z onz = h. (3.18)

6 104 3 S ± x f(ωt kx) Φ(x, z, t) =Z(z) sin(ωt kx). (3.19) (3.16) Z(z) d 2 Z dz 2 k2 Z = 0 (3.20) Z(z) =D 1 e kz + D 2 e kz (3.21) D 1 D 2 [ F ] [ B ] } D 1(ω 2 gk)+d 2(ω 2 + gk) =0, (3.22) D 1 e kh D 2 e kh =0. D 1 = D 2 =0 ω 2 gk ω 2 + gk e kh e kh =0, (3.23) k tanh kh = ω2 g K (3.24) k ω, g D 1 e kh = D 2 e kh 1 2 D (3.25) Φ(x, z, t) =D cosh k(z + h) sin(ωt kx) (3.26)

7 [ F ], [ B ] D (3.13) (3.5) ζ = 1 ( ) Φ = D ω cosh(kh) cos(ωt kx) g t z=0 g = a cos(ωt kx). (3.27) ga D = (3.28) ω cosh kh Φ = ga cosh k(z + h) sin(ωt kx). (3.29) ω cosh kh Φ(x, z, t) =Re [ φ(x, z) e iωt ], (3.30) φ(x, z) = iga cosh k(z + h) e ikx (3.31) ω cosh kh e iωt (3.30) φ(x, z) f(z) (3.24) (3.14) (3.24) c c = ω g k = k tanh kh = gλ 2πh tanh 2π λ (3.32) λ (3.32)

8 106 3 (3.32) (3.24) (dispersion relation) (3.24) k y = tanh kh (3.24) K k tanh kh k k = k 0 (h ) tanh kh 1 k 0 = K k O K k0 K<k k 0 K (3.32) (h ) (h 0) gλ c = ( h ), (3.33) 2π c = gh ( h 0 ) (3.34) h/λ 0 (3.33) tanh kh 1 kh =2πh/λ 2.65 λ 2.4h 1.0% 3.3 r 0(t) =(x 0(t),z 0(t)) d r 0 dt = u( r 0,t) (3.35)

9 r 0 r =(x, z) O(a) r =(x, z) d r 0 dt = u(r, t)+(r 0 r ) u + O(a 3 ) (3.36) u(r,t)= Φ O(a) (3.29) Φ cosh k(z + h) (x 0 x) = dt = a sin(ωt kx), x sinh kh (3.37) Φ sinh k(z + h) (z 0 z) = dt = a cos(ωt kx) z sinh kh (3.37) (3.36) (3.24) (3.29) dx 0 dt dz 0 dt cosh k(z + h) = aω cos(ωt kx) sinh kh + 1 cosh 2k(z + h) cos 2(ωt kx) 2 ωka2 + O(a 3 ), sinh 2 kh (3.38) sinh k(z + h) = aω sin(ωt kx)+o(a 3 ) sinh kh (3.39) O(a 2 ) (Stokes drift) (3.38) T E 1 T T 0 Edt (3.40) M = ρ 0 h dx 0 dt dz = 1 ρωa 2 2 tanh kh = ρga2 c (3.41)

10 108 3 c (3.32) O(a 2 ) O(a 2 ) ([ Note ] ) O(a 2 ) M = ζ ρ Φ x a Φ dz ρ (ζ + a) x z=0 (3.41) [ Note ] = 1 2 ρga2 k ω (3.42) O(a 2 ) Φ (2) Φ (2) (3.12) (3.12) (3.30) (3.31) 2 Φ (2) t 2 + g Φ(2) = Q on z =0, (3.43) z [ Q 2 Φ Φ t + 1 ( Φ 2 Φ g t z t 2 + g Φ )] z z=0 [ =Re i 3 g 2 a 2 2 ω k2( ) ] 1 tanh 2 kh e i 2(ωt kx) (3.44) z = h 2(ωt kx) [ ] Φ (2) =Re φ (2) (x, z) e i 2ωt, φ (2) (x, z) =D cosh 2k(z + h) cosh 2kh i 2kx e D (3.43) D = i 3 g 2 a 2 4 ω k(1 tanh 2 kh ) ( 2 tanh kh tanh 2kh ) (3.45) (3.46) (3.45) φ (2) (x, z) =i 3 8 ωa2 cosh 2k(z + h) sinh 4 kh e i 2kx. (3.47)

11 (h ) (3.44) Q =0 (3.47) h Φ (2) 3.3 O(a 2 ) (3.8) (3.4) ( ζ = 1+ζ + ){ z 1 ( Φ g t )}z=0 Φ Φ = 1 ( Φ g t Φ Φ 1 ) Φ 2 Φ + O(Φ 3 ) g t z t z= ω k (3.32) ω, k δω = ω 2 ω 1, δk = k 2 k 1 (3.48) [ ] ζ =Re A 1 e i (ω 1t k 1 x) + A 2 e i (ω 2t k 2 x) [ { } ] =Re A 1 1+ A2 e i (δω t δk x) e i (ω 1t k 1 x) A 1 (3.49) (3.49) { } δk δω (group velocity) (3.14) c g = δω (3.50) δk

12 π δk cg 2π k c= ω k 3.3 δω/δk δω 0 δk 0 δω t δk x t x (3.49) c g c g = dω dk = d(kc) = c + k dc dk dk = c λ dc dλ. (3.51) (3.24) c g c c g = 1 2 c [ 1+ 2kh sinh 2kh (3.52) (h ) (h 0) ]. (3.52) c g = 1 c 2 ( h ), (3.53) c g = c ( h 0 ) (3.54) (ω 1,k 1) (ω 2,k 2) k

13 [ ] α ζ =Re A e α(k k) 2 e i {ω(k )t k x} dk π (3.55) α α α lim α π e αk2 = δ(k) (3.56) α 1 k = k ω(k )=ω(k)+(k k) dω dk (3.57) (3.55) α(k k) 2 + i {ω(k )t k x} = i (ωt kx) 1 ( x dω ) 2 4α dk t α { (k k)+ i ( x dω )} 2 2α dk t (3.58) k [ { ζ Re A exp 1 ( x dω ) 2 } ] 4α dk t e i (ωt kx). (3.59) oo α cg=d ω/dk 2π k c= k ω 3.4 (3.59) x =(dω/dk)t dω/dk

14 112 3 x =(dω/dk)t dω/dk (3.51) 3.4 c (3.32) (3.52) c = tanh kh, (3.60) c c g = c [ kh ] (3.61) c c 2 sinh 2kh kh 0 kh 3 (3.60) kh (3.61) kh (3.61) kh 3.5 V ] [ ] 1 1 E = ρ[ 2 q2 + gz dv = 2 Φ Φ + gz dv. (3.62) V (t) V ρ V (t) (3.62) V S U n de dt = ρ d [ 1 ] dt 2 q2 + gz dv V (t) [ ] [ ] 1 1 = ρ t 2 q2 + gz dv + ρ 2 q2 + gz U n ds. (3.63) S

15 V(t) S n ( 1 p 2 q2 + gz = ρ + Φ ). (3.64) t [ 1 ] ( Φ ) t 2 q2 = t Φ (3.65) 3.5 [ ( de Φ dt S = ρ Φ p t n ρ + Φ ] )U n ds. (3.66) t S S F S H S S on S C U n =0, Φ on S H = Un = Vn, n (3.67) Φ on S F = Un, p =0 n V n (3.2) (3.67) (3.66) de dt = Φ Φ pv n ds + ρ ds (3.68) S H S t n (3.68) (3.68) W D de/dt 0 0 Φ Φ W D pv n ds = ρ ds. (3.69) S H S t n

16 114 3 (V n =0) (3.69) 0 (3.69) y x (3.62) ζ [ 1 ] E = ρ 2 Φ Φ + gz dz. (3.70) h ζ = 1 Φ (3.71) g t h o z SF SB a z=0 δx z=ζ(x,t) n S+ n x 3.6 Φ (3.70) E = 1 0 [ ( Φ ) 2 ( Φ ) 2 ] 2 ρ + dz + 1 x z 2 ρg ζ2 + O(Φ 3 ) (3.72) h

17 (3.30) Re [ Ae iωt] Re [ Be iωt] = 1 2 Re[ AB ]. (3.73) B B (3.31) z E = 1 ( ) gak ρ cosh 2k(z + h) dz + 1 ω cosh 2 kh 4 ρga2 h = 1 4 ρga ρga2 = 1 2 ρga2 (3.74) ρga 2 /4 (3.74) (3.41) E = Mc (3.75) M c (3.68) (3.68) y x δx S(x) S(x + δx) ( E ) Φ t δx = ρ Φ t x dz S(x+δx) S(x) = ρ x 0 h Φ Φ t x dz δx + O(Φ3 ) (3.76)

18 116 3 Φ (3.30) (3.31) (3.73) E t = 1 2 ρ [ 0 ( ) φ ] Re iωφ x x dz = x h [ 1 4 ρga2 ω k (3.52) (3.74) { 1+ 2kh sinh 2kh }] (3.77) E t + ( ) cge = 0 (3.78) x (3.15) (3.78) E c g 3.5 (3.73) (3.77) 3.7 x x =0 z O Reflected Wave Incident Wave x x =0 Φ =0 atx = 0 (3.79) x -h x 3.7 (3.29) Φ s = ga cosh k(z + h) { } sin(ωt + kx) + sin(ωt kx) ω cosh kh = 2ga cosh k(z + h) cos kx sin ωt (3.80) ω cosh kh

19 (3.79) ζ s = 1 ( ) Φs =2acos kx cos ωt (3.81) g t z=0 (3.80) (3.81) (standing wave) x = mπ/k (m =0, ±1, ±2, ) x =(m + 1 )π/k (x =0) 2 (loop) (node) (3.79) x = mπ/k Φ/ x =0 (3.80) x = mπ/k x l λ/2 l λ =2l/m (m =1, 2, ) ω m (3.24) ω m = g mπ ( ) mπh tanh (3.82) l l T m T m = 2π = 2 l 1, ( m =1, 2, ) (3.83) ω m m gh m =1 (seiche) (x, y)

20 118 3 h λ h z 0 z ( ) ( Φ 2 ) Φ = h z x + 2 Φ (3.84) 2 y 2 z=0 z = h (3.84) c = gh 2 Φ t 2 ( 2 ) = Φ c2 x + 2 Φ, c = gh (3.85) 2 y 2 Φ z Φ = Φ(x, y, t) (3.85) (x, y) C B Φ (3.85) φ Φ =0 oncb (3.86) n Φ(x, y, t) =φ(x, y) sin ωt (3.87) 2 φ x + 2 φ 2 y + 2 k2 φ =0, k = ω c (3.88) (Helmholtz s equation) (3.88) C B x =0,a y =0,b (3.86) φ x =0, φ y =0, at x =0,a at y =0,b (3.89)

21 (3.88) φ(x, y) =X(x)Y (y) (3.88) X + p 2 X =0, Y + q 2 Y = 0 (3.90) k 2 = p 2 + q 2 (3.89) X = C cos(px), p = mπ a Y = D cos(qy), q = nπ (3.91) b C D φ(x, y) =X(x)Y (y) φ(x, y) =A mn cos mπ a x cos nπ b y (3.92) {( ) m 2 ( ) n 2 } k 2 = π 2 + (3.93) a b y x=0 O y=b y=0 x=a 3.8 x m, n A mn (3.93) k y (3.83) (3.92) 3.6 (3.80) (3.79) x =0 x =0 (p =0)

22 x =0 α x =0 α 0 α ((3.80) [ 3.6]) 3.8 x =0 (3.69) 0 Φ Φ W D = ρ t x dz = ρ 0 2 Re iωφ φ dz x h x=0 h x=0 W D

Gmech08.dvi

Gmech08.dvi 51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r