Chapter3.tables

Chapter3.tables - Page 61 Solutions to Laplace’s...

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Unformatted text preview: Page 61 Solutions to Laplace’s equation: (x,y,z), (r,S, jÞ, Ý_, j, zÞ; Helmholtz equation in (r,S, jÞ Equation 4 2 ®Ýx, y, zÞ = 0 separation. const. ! ! k=k 1 x +k 2 ŷ +k 3 z General solution: sum over all sep. constants # # conditions # # F k6k =0 ßcÝkÞe k 6 r + dÝkÞe k 6 r ÝÝax +bÞN k1,0 + Ýcy +dÞN k2,0 k 6 k = 0, k can be complex + Ýez + fÞN k3,0 Þ + Ýax +bÞÝcy +dÞÝez +fÞN |k|,0 à 4 2 ®Ýr, S, jÞ = 0 4 2 ®Ý_, j, zÞ = 0 |m|² § = 0, 1, 2... J, m= 0, ±1, ±2, ... F §,m ßa § r § + b § r ?§?1 àßc §,m Y §,m ÝS, jÞ + d §,m Q §,m ÝS, jÞà F J,m ßa Jm J m ÝJ_Þ + b Jm N m ÝJ_Þàe imj e ±Jz Q §,m ¸ K, cos S = ?1 N m ¸ K, _ ¸ 0; J can be complex Ý4 2 + k 2 Þ®Ýr, S, jÞ = 0 |m|² § = 0, 1, 2... F §,m ßa § j § ÝkrÞ + b § n § ÝkrÞàßc §,m Y §,m ÝS, jÞ + d §,m Q §,m ÝS, jÞà n § ¸ K, r ¸ 0; Function e ## k6r separation consts k 3 =± i|k 3 |; k 1,2 real e Ýk 1 x+k 2 yÞ e ± i |k 3 |z ===>sin, cos in one variable, exponential in other two ݧ ? mÞ! 1/2 |m| (-1) m ß 2§ + 1 à P § Ýcos SÞ e imj ; XX Y §,m D Y § v ,m v dI = N §,§ v N m,m v 4^ ݧ + mÞ! K J_ 2j+m Ý?1Þ j Fj ß à ; X a_ J m Ýa_ÞJ m Ýb_Þd_ = NÝa ? bÞ where J m ÝaÞ = J m ÝbÞ =0 0 j !Ý m + j Þ ! 2 ÝkrÞ § j § ÝkrÞ = ß ^ à 1/2 J §+1/2 ÝkrÞ; j 0 ÝxÞ = sin x ; j § ÝkrÞ k_¸K ¸ x 2kr Ý2§ + 1Þ!! ; k 6 k = 0 k 1,2 =± i|k 1,2 |; k 3 real e iÝ|k 1 |x+|k 2 |yÞ e ±k 3 z ===>sin, cos in two variables ,exponential in third Y §,m ÝS, jÞ J m ÝJ_Þ j § ÝkrÞ; n § ÝkrÞ |m|² § = 0, 1, 2... J, m= 0, ±1, ±2, ... § = 0, 1, 2... The boundary conditions:the boundary conditions on the solution restrict the separation constants to specific values. Uniqueness: solutions to4 2 ® = 0 in a finite volume, satisfying the same boundary conditions, are unique. ...
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