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chpp12 - 12 u(x y z t u(x y z(x y z 12.1 Example 12.1 u T(x...

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Unformatted text preview: 12 . , , , ( ( u(x, y, z, t)), . u(x, y, z)). . ( (x, y, z)) ( , ) ( ). , , 12.1 Example 12.1 ( . ) , . . , u T (x + dx) u(x + dx) u(x) 2 1 T (x) O Solution t dx. 1 , 2 . , . x x + dx . : T2 cos 2 - T1 cos 1 = 0 : T2 sin 2 - T1 sin 1 = dm u t2 2 x x + dx x = 0, x x = l. u(x, t) T1 , T2 , x x , (1) (2) 0 cos 1 sin tan = (1) T1 = T2 = T (2) T dm = dx, ( u x - x+dx u x u x ). a= =T x 2u 2u dx = 2 dx x2 t T 1 T x . , ds = 2u 2u - a2 2 = 0 2 t x T t . du2 + dx2 - dx u x 2 (3) - 1 dx 1 2 u x 2 = 1+ , Hooke , u x dx ds 0 , T . ,T ,a . , f . u f dx T (x + dx) 2 1 T (x) u(x + dx) u(x) O x x + dx 2u 2u dx + f dx = 2 dx x2 t 2u 2u f - a2 2 = 2 t x x T , (4) f . ) , . Example 12.2 ( P (x, t)S P (x + dx, t)S u u + du x Solution u(x, t). ), x , (x, x + dx). x + dx x P (x, t) x t 2u t2 . , t, x ( P (x + dx, t)S - P (x, t)S = dm 2 dm = Sdx, . P 2u Sdx = 2 Sdx x t 2u P = 2 x t , Hooke , P ( P =E u x ) u x E Young . 2u 2u - a2 2 = 0 t2 x a = E , , ( 2u - a2 t2 2 , ) u=0 . (5) 2 2 2 2 + 2+ 2 2 x y z (6) Laplace . 12.2 Example 12.3 ( Solution x ) . , u(x, y, z, t) qx qx = -k u x x , , , (x, y, z) u x , . t . , qy = -k u y u qz = -k z k . Fourier . q = -k u (7) q u . 3 . . , dx, dy, dz, z dz dy dx y x dt x [(qx )x - (qx )x+dx ]dydzdt = = ,k [(qx )x - (qx )x+dx ]dydzdt = k , dt y k dt z k , k c . dm = dxdydz, k 2u 2u 2u + 2 + 2 x2 y z . 2 k x u x k u x - k x+dx u x dydzdt x dxdydzdt 2u dxdydzdt x2 2u dxdydzdt y 2 2u dxdydzdt z 2 dxdydzdt = dm c du udxdydzdt = cdxdydzdu k c = u - t . 4 2 u=0 (8) , . F (x, y, z, t), k 2 , , D, . , . . udxdydzdt + F (x, y, z, t)dxdydzdt = cdxdydzdu u - t 2 u= 1 F (x, y, z, t) f (x, y, z, t) c (9) 12.3 , . , . , . , , , u(x, y, z, t) = u(x, y, z) - 2 2 u=f f (10) u=- Poisson Laplace , . . , f =0 2 u=0 u(x, y, z) , (11) Poisson . 2 u=- 0 , 0 . = 0, 2 Laplace u=0 , 2u - a2 t2 u(x, y, z, t) u(x, y, z, t) = v(x, y, z)e-it . v(x, y, z) Helmholtz 2 2 u=0 v(x, y, z) + k 2 v(x, y, z) = 0 (12) k= a . 5 12.4 , Newton . , . , , u|t=0 = (x, y, z) , u t , u(x, y, z) u|t=0 = (x, y, z) , . t = (x, y, z) t=0 , -- , , -- . , . , , , ; , Example 12.4 Solution ( . , , : . , . . . . : , . ) , u|x=0 = 0 u|x=l = 0 Example 12.5 Solution x=0 , x=l (x ) , F (t) F (l - dx)S O x=0 u|x=0 = 0 x=l Newton F (t)S - P (l - dx, t)S = dm dx 0 F (t) = P (l, t) 6 . , F (l)S l - dx l (13) x=l , dx. 2u 2u = Sdx 2 t2 t Hooke P =E u x x=l , F (t) = 0, u x , F (t) = -k[u(l, t) - u0 ] u0 , u k + u x E = x=l u x 1 F (t) E (14) = x=l =0 x=l (15) k u0 E (16) , Example 12.6 Solution . ( u| = (, t) ) (17) . , , n (, t), , ( ) , , -qn | = (, t) , . qn - n -qn - , . d0 , , , qn |- - qn | = 0 , 0, qn |- - qn | = 0 7 , . . , - . , Fourier , q = -k u qn = q n = -kn ( u) = -k n n -k -k - , u n , = 0, u n . , u| u0 (, t) = -H(u| - u0 ) H . u + hu n h= H k = hu0 . u n u n - qn | = 0 - u n + (, t) = 0 - = 1 (, t) k (18) =0 (19) : Newton (20) : : 1 , . , u| = (, t) (21) u n = (, t) (22) u + hu n 1 = (, t) (23) , . 8 . . , ; . ( . ), , : -- , ; -- . , Problems 1. l ( 1) S t=0 F (x = 0) (x = l) F Figure 1: 2. u(r, t) u(r, t) 3. 4. M l a ( ( 2) 0) q1 q2 u(r, t) Figure 2: 9 ...
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