Nonlinear theory Lý thuyết sóng

HIGHER-ORDER WAVE THEORY Higher-Order Wave Theory Higher-Order Wave Theory • Stokes wave theory • Cnoidal wave theory • Solitary wave theory • Stream-function theory • Korteweg - de Vries equation • Boussinesq equation Wave Profiles from Different Theories Governing Equation and Boundary Conditions  2  2  0 x z  0 z  d z ( x, z , t )  ( x  L, z , t ) ( x, z , t )  ( x, z , t  T ) governing equation bottom BC periodicity DFSBC 2 p                    gz  CB (t ) z  ( x, t )    x   z   t        z t x z z  ( x, t ) KFSBC Characterize Nonlinearity H Wave steepness L d Relative water depth L H/L H Relative wave height  d/L d 2  L  H HL UR     d d  d Ursell number Regions of Validity for Wave Theories Stokes Finite-Amplitude Wave Theory Perturbation approach:   1      33  (velocity potential)   1  2  33  (surface elevation)   ka  2a H  L L (perturbation parameter) Stokes 2nd Order Wave Theory Wave Profile: H  2x t  cos    T   L H cosh(2d / L) x 4t     cosh(4d / L)  cos   L sinh (2d / L) T   L  Stokes 2nd Order Wave Theory Mass Transport in 2nd-Order Stokes Waves Mean drift velocity:  H  C cosh  ( z  d ) / L  U ( z)     L  sinh (2 d / L) Cnoidal Wave theory Periodic waves in shallow water (UR > 20) Solution expressed in elliptic integrals (K) and functions (cn):  x t ys  yt  Hcn  K ( k )   L T    , k    Cnoidal Wave Theory: Normalized Surface Profile Cnoidal Wave Theory In the limit: k  1, cn   solitary wave k= sinusoidal wave (Isobe, M 1985 ”Calculation and application of firstorder cnoidal wave theory,” Coastal Engineering, 9, 309-325) Solitary Wave Theory Wave form is entirely above the SWL  H x  Ct    sech    ho ho  u   h gho o (sech x = 1/sinh x) C  g ( H  ho ) Stream Function Theory Developed by Dean (1965, 1974) Describe wave through sine and cosine functions Determine coefficient values of each term so that the best fit in a leastsquare sense is obtained with respect to fulfilling the dynamic free surface boundary condition Boussinesq equation Shallow water  hydrostatic pressure   continuity   d  u   t t u u   3u u g  d momentum t x x x t Eliminate u   2  2   2  2   gd  gd   d t x x  d x2  Korteweg-deVries Equation u u u ho2  3u  gho  u  gho  t x x x ... Wave Theory: Normalized Surface Profile Cnoidal Wave Theory In the limit: k  1, cn   solitary wave k= sinusoidal wave (Isobe, M 1985 ”Calculation and application of firstorder cnoidal wave theory, ”... Stokes 2nd Order Wave Theory Mass Transport in 2nd-Order Stokes Waves Mean drift velocity:  H  C cosh  ( z  d ) / L  U ( z)     L  sinh (2 d / L) Cnoidal Wave theory Periodic waves... Wave Theory Perturbation approach:   1      33  (velocity potential)   1  2  33  (surface elevation)   ka  2a H  L L (perturbation parameter) Stokes 2nd Order Wave Theory