SURFZONE WAVES I (PHASEAVERAGED)

SURF-ZONE WAVES I (PHASE-AVERAGED) Surf-Zone Wave Models I Phase-Averaged Models Existing Wave Decay Models • Battjes and Janssen (1978) • Thornton and Guza (1983) • Svendsen (1984) • Dally et al (1985); Dally (1990, 1992) • Larson (1995) Governing equation: d  F cos    P dx Energy dissipation from wave breaking: P gdH T (d  H ) P   F  Fs  d (bore theory) (Dally) Dally Breaker Decay Model (monochromatic waves) d   F cos     F  Fs  dx d F  gH 2C g Fs  gH s2Cg H s  d (linear wave theory)   0.15,   0.4 (Snell’s law yields change in wave angle) Random Waves Wave-by-wave approach: Transform each wave component representing the pdf individually across the profile – add together the components to obtain the statistical properties of the wave field at any location (no need to specify shape of the pdf except in the offshore) Rayleigh pdf is normally a good description in the offshore (Monte-Carlo simulation) Rayleigh PDF p( H )    H  2H exp      H rms  H rms        H 2  F ( H )   exp       H rms     Represent the Rayleigh pdf in the offshore by N waves: Hi, i = 1,N Rms wave height is given by: N  Hi  N N i 1 N  n( x )  q( x ) H rms  n( x) H i i1  N q(x) H i i 1 n=number of non-breaking waves q=number of breaking waves Define: H n2  n  Hi n i 1 H q2  q( x)  Hi q i 1 Let =q/N (ratio of breaking waves), then: H rms  (1  ) H n2  H q2 In the general case: non-breaking waves consists of unbroken and reformed waves H rms   H m2  H r2  H q2   m / N ,   r / N,   q / N      1 Wave Transformation d  Fi cos    dx non-breaking wave d   Fi cos     Fi  Fs  dx d breaking wave Sum over all waves: n d  dx  H i Cg cos    i Cg cos     i 1 q d  dx  H q i 1 i 1   Hi2Cg  2 d 2Cg  d Add equations: N d  dx  H i 1 q i Cg cos     i 1   Hi2Cg  2 d 2Cg  d Interchange derivation and summation, and develop: d  Cg cos  dx  N N H i1 i   1   d Cg  N   q H i   2d i1 q  1 N i 1  d  q q q C g cos    C g   H rms  Hi   d N  dx d  N q i 1  Substitute previous expressions: d  2 C g cos    H rms C g  (1  ) H n2Cg   d 2Cg   H rms dx d Or: d   Frms cos     Frms  Fstab  dx d Frms  gH rms Cg Fstab  g  (1   ) H n2   d  Cg Wave Decay Over Monotonic Profile Wave transformation neglecting breaking: H x2  C go cos o Ho Cg cos  Ratio of breaking waves:    d 2    exp    b     Hx     Rms Wave Height for Unbroken Waves b d H n2  H p( H ) dH bd  p ( H )dH Solve for a Rayleigh pdf: H n2  H x2    H x2   b2 d  1    Wave Decay Over Non-Monotonic Profile Breaking, unbroken, and reformed waves occur in the general case Fstab  g   H m2  H r2  1       d  Cg If a negative slope is encountered, wave reforming takes place H m2  1 k    b2 dk2C gk cos k  H x   k  H x   Cg cos        dk: smallest seaward depth Model closure requires a formula for the ratio of reformed waves:  F  Fstab      k   r  rms   Frms ,r  Fstab,r    = 0.5 (comparison with Monte-Carlo simulations) Validation of Random Wave Model SUPERTANK Laboratory Data Collection Project DELILAH Field Experiment SUPERTANK Data DELILAH Data Comparison Between Monte-Carlo Approach and Random Wave Model ... Rayleigh pdf in the offshore by N waves: Hi, i = 1,N Rms wave height is given by: N  Hi  N N i 1 N  n( x )  q( x ) H rms  n( x) H i i1  N q(x) H i i 1 n=number of non-breaking waves. .. breaking waves Define: H n2  n  Hi n i 1 H q2  q( x)  Hi q i 1 Let =q/N (ratio of breaking waves) , then: H rms  (1  ) H n2  H q2 In the general case: non-breaking waves consists of... all waves: n d  dx  H i Cg cos    i Cg cos     i 1 q d  dx  H q i 1 i 1   Hi2Cg  2 d 2Cg  d Add equations: N d  dx  H i 1 q i Cg cos     i 1   Hi2Cg  2 d 2Cg  d Interchange