(*x *& x)2

&

1

2

2σx

(II-1-125)

σx 2π

where x is the mean of *x *and *σ*x is the standard deviation. The Gaussian cumulative probability or probability

distribution denoted by *P(x) *in Figure II-1-28, is the integral of p(x). A closed form of this integral is not

possible. Therefore, Gaussian distribution is often tabulated as the normal distribution with the mean x and

standard deviation *σ*x in handbooks (e.g., Abramowitz and Stegun (1965)), and is written as

(II-1-126)

σx

For zero mean (x = 0) and unit standard deviation (*σ*x = 1), the Gaussian probability density and distributions

reduce to

&

1

2

(II-1-127)

2π

m0

Φ(*x*) '

where the last integral is the *error function*.

(f) The probability of exceedence *Q(x) *may be expressed in terms of the probability of non-exceedence

P(x) as

(II-1-128)

σx

(g) This is the probability that *x *will exceed *x*1 over the time period *t*, and is shown as the shaded area

in the bottom lower end of Figure II-1-28. The probability of exceedence is an important design parameter

in risk-based design.

(h) In engineering practice, we are normally concerned with wave height rather than surface elevation.

However, to define wave height distribution, we only need to examine the statistics of the slowly varying

envelope of the surface elevation *η(t)*. With this approach, Longuet-Higgins (1952) found from statistical

theory that both wave amplitudes and heights follow the Rayleigh distribution shown in Figure II-1-29. Note

that this distribution can never be negative, it decays asymptotically to zero for large *x*, but never reaches

zero. The probability density function of the Rayleigh distribution and its cumulative probability are given

by

II-1-72

Water Wave Mechanics

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