Lý thuyết sóng tuyến tính linear wave theory

LINEAR WAVE THEORY Linear Wave Fundamentals C a = Wave Am plitude To describe a wave: H, L, T, C Wave steepness: H/L Wave speed: C = L/T Linear Wave Theory – Principle of Superposition Wave Classification: Spectrum capillary waves, gravity waves, infra-gravity waves, tidal waves Wave Classification: Generation * impulse (free wave) - tsunami * constant forcing (forced) - tide, wind Mechanism: Wind Generating Waves Plane Water Surface Wind High p Low p Water Surface /w waves at time t1 Wind Direction friction friction Water Surface /w waves at time t2 Wind Direction Reduced Static Pressure Stream Lines Wave Classification: Water Depth DEEP WATER Wave Direction Wave Base Water Particle Movem ent SHALLOWER WATER Motion of water particles Wave Classification: Motion Progressive waves Standing waves Linear Wave Theory (Airy theory) Assumptions: • incompressible, homogeneous fluid • neglect surface tension and Coriolis • uniform and constant pressure at the surface • ideal fluid (no viscosity) • horizontal, impermeable bottom • small wave height and invariant form • long-crested waves (2D approach) George Airy, 1801-1892 Basic Relationships Wave profile:  2x 2t  H   a cos     cos  kx  t  T   L k 2 2 ,  L T Wave phase speed: C L   T k Dispersion relationship Wave phase speed: C C gL  2d    2  L  gT  2d    2  L  Wave length: L gT  2d     CT 2  L  Dispersion Mechanism t1 t2 Waves (short, irreg.) Swell (long, reg.) Asymptotic Solutions 10 2 d/ L Deep water: (d/L >1/2)  2d    1  L  gT  2d  gT   1.56T  2  L  2 Shallow water: (d/L < 1/25) L) h( 2 d/ ta n ( nh ta C 2 ) d/L 0.1 2d  2d    L  L  gT  d  L    gdT 2  L  = tanh(2d/L) Co  = gT  d  gT Lo    1.56T  2  L  2 Shallow 0.01 0.01 Transitional 0.1 Deep 10 2 d/L gT  2d     gd 2  L  Practical Solution of the Dispersion Relationship  2d  L  Lo    L  (iterative solution required) d d  Lo L L Lo  2d     d d  L  (from Table) Hyperbolic Functions y  sinh x  0.5( e x  e x ) dy / dx  cosh x y  cosh x  0.5(e x  e  x ) dy / dx  sinh x y  x  ex  e x ex  e x Variation of Wave Parameters with d/Lo Wave Period vs Depth n1  t T1 n2  t T2 n1  n2  T1  T2 For a simple harmonic wave train, the wave period is independent of depth! Water Particle Velocity Water Particle Velocity u H gT cosh  2( z  d ) / L   2x 2t  cos    L cosh(2d / L ) T   L w H gT sinh  ( z  d ) / L   x t  sin    L cosh(2 d / L ) T   L Particle Acceleration ax  g H cosh  ( z  d ) / L   x t  sin    L cosh(2 d / L) T   L az   g H sinh  2( z  d ) / L   2x 2t  cos    L cosh(2d / L ) T   L Water Particle Motion: Equations   H cosh  2( z  d ) / L   2x 2t  sin    sinh(2d / L) T   L H sinh  2( z  d ) / L   2x 2t  cos    sinh(2d / L) T   L A H cosh  2( z  d ) / L  sinh(2d / L) B H sinh  2( z  d ) / L  sinh(2d / L ) 2   1 A2 B Water Particle Motion: Trajectory Shallow water: H L H zd A , B 2 d d Deep water: AB H z / L e Pressure Under Waves Pressure Under Waves p '  g cosh  2( z  d ) / L  H cosh(2d / L)  2x 2t  cos     gz  pa T   L Atmospheric pressure Dynamic component due to acceleration p  p ' pa Water surface profile Static component of pressure (relative pressure) Pressure Under Waves Kz  cosh  2( z  d ) / L  cosh  d / L  Pressure response factor p '  g (K z  z ) At bottom (z = -d): Kz  K  Water level from pressure:  cosh  2d / L  N ( p  gz ) gK z (N = for linear waves) Wave Energy Potential energy: x L EPT   x d ( EPT )  gzdm 1 d ( EPT )  gd L  gH L 16 EP  EPT  Enowav  d  dm  (d  )dx z gH L 16 w dz u dx Kinetic Energy: x L  EK    d (E x d K ) gH L 16 d ( EK )   u  w2 dxdz w dz u dx Total Energy: E  EK  EP  gH L (per wave and unit width) E  gH (per wave and unit surface area) 10 Wave Energy Flux Definition (rate of energy transfer): P L x L   p D udzdx x d pD  p  gz 1 2kd  L P  gH     EnC  sin 2kd  T Group Velocity t1 t2 t3 10 t4 t5 t6 C > Cg Distance from Wavemaker Cg Group Velocity 1 4d / L  L C g  1   nC  sinh(4d / L )  T Cg  Lo  Co 2T (deep water) Cg  L  C  gd T (shallow water) 11 Superposition of Waves Two waves:   1  2   x 2t  H  2x t  H cos    cos   T1  T2   L1  L2 Envelope:  L L T T  env   H cos   x   t  TT  L1 L2  Envelope of Superimposed Waves Cnode = Cg Energy Transport 12 Summary of Linear Wave Theory 13 .. .Wave Classification: Spectrum capillary waves, gravity waves, infra-gravity waves, tidal waves Wave Classification: Generation * impulse (free wave) - tsunami * constant... Stream Lines Wave Classification: Water Depth DEEP WATER Wave Direction Wave Base Water Particle Movem ent SHALLOWER WATER Motion of water particles Wave Classification: Motion Progressive waves Standing... Motion of water particles Wave Classification: Motion Progressive waves Standing waves Linear Wave Theory (Airy theory) Assumptions: • incompressible, homogeneous fluid • neglect surface tension