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3 Pages

### Midterm2_soln

Course: ME 631, Fall 2008
School: Kentucky
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Word Count: 724

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-it my 1a) = A cos cosh (k ( z + H ) )e L my my my -i t -it -it = Ak cos = Ak cos = Ak cos =0 w= sinh (k ( z + H ) )e sinh (k (- H + H ) )e sinh (0 )e z L L L b) 2 = 0 2 2 x 2 + 2 z 2 my my m -it -i t =0 + Ak 2 cos = - A cosh (k ( z + H ) )e cosh (k ( z + H ) )e cos L L L 2 m m 2 =k k = L L D , but to first order = w +u = = c) Kinematic BC : t x z t...

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-it my 1a) = A cos cosh (k ( z + H ) )e L my my my -i t -it -it = Ak cos = Ak cos = Ak cos =0 w= sinh (k ( z + H ) )e sinh (k (- H + H ) )e sinh (0 )e z L L L b) 2 = 0 2 2 x 2 + 2 z 2 my my m -it -i t =0 + Ak 2 cos = - A cosh (k ( z + H ) )e cosh (k ( z + H ) )e cos L L L 2 m m 2 =k k = L L D , but to first order = w +u = = c) Kinematic BC : t x z t z Dt Dynamic BC : 1 2 p + (u + w 2 ) + + gz = 0, t 2 2 t 2 - p = - 2 y 2 , but to first order 2 + g = 0 - t y 2 Take time derivative : 2 +g =0 t y 2 t - 2 +g =0 2 z z y 2 Substitute kinematic BC : 2 t 2 Ak m my my -it -i t A(-i ) 2 cos + cosh (kH )e cos sinh (kH )e L L L my -i t + gAk cos =0 sinh (kH )e L k m 2 k 3 Solve for 2 cosh (kH ) = + gk sinh( kH ) = + gk tanh(kH ) L d) c = k 2 k g k g + g (for shallow water), + (kH ) = H = + tanh(kH ) k k k -1 / 2 c 1 Hk 2 Set extrema to zero to find minimum : = + Hg k 2 Could determine this result by observation. e) For shallow water : k 3 1 k 3 = = + gk (kH ) + gk (kH ) . k 2 = 4Hk 3 + 2 gkH -1 / 2 2kH = 0 k 0, c c min = gH Hk 3 4 + 2 gkH (k 3 + gk (kH ) ) = 4Hk 2 + 2 gH (k 2 H + gH ) 2. Given : b= a gk a gk 1 + 2 e -kH + 1 - 2 e kH , 2 2 2 2 [1 sinh( kH ) + 2 cosh( kH )] - ( 2 - 1 ) sinh( kH ) = 0 - 1 gk gk a) Barotropic mode : 2 - 1 = 0 = gk b) Baroclinic mode : gk , c = k = g 1 , cg = = k k 2 g , dispersive k 2 [1 sinh( kH ) + 2 cosh( kH )] - ( 2 - 1 ) sinh( kH ) = 0 gk SW approximat ion : sinh( kH ) kH , cosh( kH ) 1 2 gk [1kH + 2 ] = ( 2 - 1 )kH gk ( 2 - 1 )kH = c= [ 2 ] = gH ( 2 - 1 ) g ( 2 - 1 ) H [ 2 ] k2 =k g ( 2 - 1 ) H [ 2 ] g ( 2 - 1 ) H [ 2 ] , cg = [ 2 non ] , - dispersive c) b = a gk a gk 1 + 2 e - kH + 1 - 2 e kH 2 2 a gk a gk a gk a Barotropic : b = 1 + 2 e -kH + 1 - 2 e kH = 1 + 2 e - kH = (1 + 1)e - kH = ae - kH 2 2 2 2 = ae - kH cos( kx - t ) + 1kH 2 gk cosh( kH ) + 1 sinh( kH ) Baroclinic : 2 = 2 , 2 ( 2 - 1 ) sinh( kH ) ( 2 - 1 )kH ( 2 - 1 )kH -kH a kH 2 2 a b = 1 + + 1 - ( - )kH e ( - )kH e 2 2 2 1 2 1 2 a 1 + 2 ( 2 - 1 )kH a1 ( 2 - 1 ) b=- 2 a 2 a 1 a (1 + kH ) = a - (1 - kH ) + 1 - =- 2 ( 2 - 1 )kH ( 2 - 1 ) ( 2 - 1 ) a1 =- cos( kx - t ) ( 2 - 1 ) 3. a) a cosh[k ( z + H )] sin(kx - t ), = gk tanh(kH ) k sinh(kH ) cosh[k ( z + H )] sinh[k ( z + H )] = u = a = w = a cos(kx - t ), sin(kx - t ) x z sinh(kH ) sinh(kH ) = For deep water, kH >> 1, sinh(kH ) cosh(kH ) e kH , sinh(k ( z + H )) cosh(k ( z + H )) e k ( z + H ) u = ae k ( z + H ) e -kH cos(kx - t ) = ae kz cos(kx - t ), w = ae kz sin(kx - t ) b) p 1 2 + (u + w 2 ) + + gz = 0, assume (u 2 + w 2 ) ~ small t 2 p + gz = - , Define p = p + gz t a 2 cosh[k ( z + H )] a 2 kz p = - e cos(kx - t ) for deep = cos(kx - ...

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