Design of Wave Event/Return Periods

Question 1

Stat basics

Answer 1

P(x) is the probability that X<=x
-X is a random variable, continuous in the case of hydrodynamics
-CDF

p(x) = dP/dx = PDF
-used to get moments and expected values

mj = integral from -ve to +ve infinity of x^j.p(x) dx
*m0 = 1
*m1 = x bar
*m2 = mean square of X

mu_j = integral from -ve to +ve infinity of (x - x bar)^j .p(x) dx
*mu0 = 1
*mu1 = 0
*mu2 = variance

Question 2

skewness

Answer 2

-defines the symmetry around x bar
-lambda

Question 3

kurtosis

Answer 3

-defines the peaked-ness
- beta

Question 4

statistically stationary

Answer 4

X(t) but p(x:t) = p(x)
-means all the statistics from the a problem can be taken from one sample
*short term oceans states are statistically steady (20mins to 3hrs) but long term not due to storm development and decay

Question 5

temporal description

Answer 5

LRWT
-m1 = eta bar = 0
-m2 - sum to infinity of a_m^2/2 –> distribution of the wave components E summed over the f range

Question 6

probabilistic description

Answer 6

-for a measured eta(t),the time that eta approxiates eta_j is T(eta_j)
- pdf p(eta_j) is lim as T -> 0 of T(eta_j)/T, T being the total time of the measured sample
*approxiamtely gaussian or normal but taken as gaussian and independent of eta bar since it is zero and X=eta in this case

Question 7

spectral description

Answer 7

mean of eta(t) square = sum of a_m^2/2, the sum of the variance of its wave components = sigma_eta^2
-total E is the sum of the E for each wave
-since variance is proportional to the avergae E per unit ____

S_etaeta(omega) = mean of eta(t) square /delta omega = omni directional spectral density function

E(omega) = rho.g.S_etaeta(omega) = wave E spectra

sigma_eta^2 = S_etaeta(omega) . delta omega = a_m^2/2 for small delta omega
- rearrange for eta(x,t)

Question 8

spectral bandwidth

Answer 8

-epsilon
-gives the relative width of the spectrum
-most sea states are approximated as narrow banded
-defines the number of f involved

Question 9

linear wave statistics

Answer 9

for random waves the periods are taken at zero up crossings and the wave heights as the difference between the max and the min surface elevation
-alternative is taking period based on local crests

as spectral bandwidth tends to 1, crests occur above and below the SWL (gaussian)

as spectral bandwidth tends to 0, crests occur above and troughs below the SWL (rayleigh)
-assumes linear waves
- rayleigh vs gaussian can be plotted on a Pp(eta/sigma) to eta/sigma graph

Question 10

significant wave height

Answer 10

Hs = H_1/3
-the expected value for the highest 1/3 of the waves
-related to what is observed as the highest wave

Question 11

second order wave stats

Answer 11

-no effect on H since the crests get higher while troughs get shallower
-rayleigh cant capture it because specrtal bandwidth doesnt tend to 0

Empirical fits
*forristall 1978:
- for H
- 2 parameter weibull fit
- based on measure hurricane data of the gulf of mexico, only 116 hrs

*forristall 2000:
-captures second order crest height change
-simulated a wide range of seas using second order theory
-2 parameter weibull as well
-captures directionality which can be crucial

Question 12

spectral representation of directionality

Answer 12

treated as independent from frequency

S_etaeta(omega,t) = D(theta).S_etaeta(omega)
-same directional distribution applied to all f but incorrect due to variation in swell and wind waves
-hard to separate swell and wind waves

Question 13

notes on spectral directionality solutions

Answer 13

Normal
-A is a normalising coefficient so integral from zero to 2pi of D(omega,t) = 1
-sigma_theta approximately 10 to 20 degrees for a north sea storm

Mitsuyasm:
- s=150 for small spread and 4 for large spread but typically 7 to 15 in the northern sea storms
-cos raised to the s instead of the 2s is for the distribution of variance/energy

General
-theta_m is the mean direction
-all the ones in the notes give similar solutions

Question 14

effect of directionality

Answer 14

-max elevation @ a point is unchanged linearly but nonlinearly the wave slope changes
-important for structures with large or multiple legs
-unidirectional waves are long crested with constant heights
-multidirectional waves are short crested with varying length, lengths decrease with increased directionality

Question 15

linear directionality solutions

Answer 15

eta(x,y,t) = sum over f, sum over dir. of am.cos(omega_m.t - (km in x).x - (km in y).y + alpha_m)
-wave number split into x/y components
-wave particle kinematics are effected

for u
- sum the linear solution over f/dir. and split km again
-in the cosh/sinh use km^2 = sum of the two km components, both squared
-still suffers from high f contamination
* velocity reduction factor avoids this, added to the unidirectional velocities to account for directionality, usually 0.8-0.9

-design assuming unidirectional is conservative for fixed structures but nit for floating ones since they are under rolling motion

-directional second order random wave theories exist as do full nonlinear directional wave models

Question 16

numerical solutions

Answer 16

Dold and Pereque:
-gives unsteady solutions, unidrectional, based on time marching the FSBC**
- needs eta(x) and phi(x,z=eta) @ a certain time

____, swan and taylor
-gives exact, directional, fully nonlinear solution ***
-3D equivalent of Dold and Pereque
-no breaking or overturning waves
-horizontal bed
-beach _______

Sobey
-nonlinear and based on local fit to measured data
-computationally efficient
-accurate in deep water but concerns in shallow
-can include directionality

Baldock and Swan
-nonlinear based on global fit to measured eta(t)
-accurate at any water depth
-computationally intensive
-limited to unidirectional
-small errors in steep waves