EM 1110-2-1100 (Part II)
30 Apr 02
(x & x)2
&
1
2
2σx
p(x) '
(II-1-125)
e
σx 2π
where x is the mean of x and σx 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 σx in handbooks (e.g., Abramowitz and Stegun (1965)), and is written as
x & x
p(x) ' N(x,σx)
(II-1-126)
P(x) ' Φ
σx
For zero mean (x = 0) and unit standard deviation (σx = 1), the Gaussian probability density and distributions
reduce to
x2
&
1
2
e
p(x) '
(II-1-127)
2π
m0
x
Φ(x) '
p(y) dy
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
x & x
Q x(t)>x1 ' 1 & P x(t) < x1 ' 1 & Φ
(II-1-128)
σx
(g) This is the probability that x will exceed x1 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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