Уравнения математической физики. Мицик М.Ф. - 9 стр.

              u = u (x; y; t ) (              n        OU −               )
                      − u ′x                           − u ′y
       cos α =                         , cos β =                        ,
                1 + (u ′x ) + (u ′y )            1 + (u ′x ) + (u ′y )
                           2         2                      2         2


                                  1
       cos γ =
                     1 + (u ′x ) + (u ′y )
                                  2            2    .

                                                                                                    (
                                                                                                   o ∆ x2 + ∆ y2       )
            :
                 cos α = −u ′x , cos β = −u ′y , cos γ = 1 ,
                   (28)                                 :
                                               − T ∫ u ′y d x1 − u ′x dy1 .                                                (29)
                                                            Γ1

                                               dx1 = dx                          dy1 = dy ,                 dx, dy −
                                                                 (S )                              σ.                             ,
                              (29)                                                                  σ,
                                                                                                              ,             ,
a x = u ′y , a y = −u ′x .
       − T ∫ u ′y d x1 − u ′x dy1 = −T ∫ u ′y d x − u ′x dy = T ∫ (u ′x′ + u ′y′)dS .
            Γ1                                      σ                                      σ
                                                                                                        (          ,         ),
                           OU                                                              ,
                                                                      ρ g (x; y; t ) .

                          − ∫ ρ g (x; y; t )dS .
                              S
                                           ,                                                            ρ−
                                                                                   (S ′)
                          ρ ∫ utt′′(x; y; t ) dS ,
                              S
                t ≥0

                          ∫ [ρ u ′′ − T (u ′′ +u′′ ) + ρ g ]dS = 0 .
                          S
                                      tt           xx            yy                                                        (30)

          (S ) −                                                                               ,
                   (30)                                                      .
                 ρ utt′′ − T (u ′xx′ +u ′yy′ ) + ρ g (x; y; t ) = 0 ,