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Chapter 8

The Theory of Quasi-Determinism

Abstract

The theorem of the sufficient condition for a wave crest with a given very large elevation and the theorems of the sufficient and the necessary conditions for a wave with a given very large height are proven. These theorems imply that: if a wave crest of given exceptionally large elevation occurs at some point x_o, y_o, the random surface elevation and the relevant velocity potential will be close to well-defined deterministic functions η₁¯, ϕ₁¯ in a neighborhood of x_o, y_o; and likewise, if a wave of given exceptionally large height occurs at some point x_o, y_o, the random surface elevation and the relevant velocity potential will be close to well-defined deterministic functions η₂¯, ϕ₂¯ in a neighborhood of x_o, y_o. To follow the proofs given in this chapter it is necessary to know the basic concepts of stationary Gaussian processes given in Sections 4.1 and 4.2.

Keywords

Finite bandwidth; Gaussian; Generally nonhomogeneous in space; Large wave crests; Large waves; Random wave field; Stationary

8.1. The Necessary and Sufficient Condition for the Occurrence of a Wave Crest of Given Very Large Height

The condition

η(to)=b $η (t_{o}) = b$

(8.1)

where t_o is a given time instant and b is a given positive value, in the limit as b/σ → ∞, and is not only necessary but also sufficient in probability for the occurrence of a wave crest of given height b. (“A is sufficient in probability for the occurrence of B” means that “given A, the probability approaches 1 that B occurs.”)

Actually, the necessary and sufficient condition in probability is

η(to)=b,η˙(to)=0 $η (t_{o}) = b, \dot{η} (t_{o}) = 0$

(8.2)

However, the reasoning we are going to do is the same whether starting from condition (8.1) or from condition (8.2), and the conclusion is exactly the same. The advantage in dealing with condition (8.1) rather than with condition (8.2) is that one works with a 2 × 2 covariance matrix rather than 3 × 3, so the mathematical steps are lighter.

In order to prove that condition (8.1) is sufficient, let us consider the conditional probability of the surface elevation at any fixed time instant t_o + T given condition (8.1). We have

p[η(to+T)=u|η(to)=b]=p[η(to)=b,η(to+T)=u]p[η(to)=b] $p [η (t_{o} + T) = u | η (t_{o}) = b] = \frac{p [η (t_{o}) = b, η (t_{o} + T) = u]}{p [η (t_{o}) = b]}$

(8.3)

From Section 4.2 we know that

p[η(to)=b,η(to+T)=u]=12πM√exp[−12M(M11b2+2M12bu+M22u2)] $p [η (t_{o}) = b, η (t_{o} + T) = u] = \frac{1}{2 π \sqrt{M}} \exp [- \frac{1}{2 M} (M_{11} b^{2} + 2 M_{12} b u + M_{22} u^{2})]$

(8.4)

p[η(to)=b]=12πm0√exp(−b22m0) $p [η (t_{o}) = b] = \frac{1}{\sqrt{2 π m_{0}}} \exp (- \frac{b^{2}}{2 m_{0}})$

(8.5)

Hence, it follows that

p[η(to+T)=u|η(to)=b]=m02πM−−−−√exp[F(u)] $p [η (t_{o} + T) = u | η (t_{o}) = b] = \sqrt{\frac{m_{0}}{2 π M}} \exp [F (u)]$

(8.6)

where F(u) denotes the function

F(u)=−12M(M11b2+2M12bu+M22u2)+b22m0 $F (u) = - \frac{1}{2 M} (M_{11} b^{2} + 2 M_{12} b u + M_{22} u^{2}) + \frac{b^{2}}{2 m_{0}}$

(8.7)

which may be rewritten in the form

F(u)=−M222M(u−um)2+F(um) $F (u) = - \frac{M_{22}}{2 M} {(u - u_{m})}^{2} + F (u_{m})$

(8.8)

where u_m is the abscissa of the maximum:

um=−M12M22b $u_{m} = - \frac{M_{12}}{M_{22}} b$

(8.9)

(The determinant and the i,i cofactors of a covariance matrix are positive, and this is why we have concluded that the function F(u) has a maximum.) From Eqns (8.6) and (8.8) we get

p[η(to+T)=u|η(to)=b]=m02πM−−−−√exp[F(um)]exp[−M222M(u−um)2] $p [η (t_{o} + T) = u | η (t_{o}) = b] = \sqrt{\frac{m_{0}}{2 π M}} \exp [F (u_{m})] \exp [- \frac{M_{22}}{2 M} {(u - u_{m})}^{2}]$

(8.10)

Now let us obtain the cofactors M_ij and the determinant M. The random variables here are η(t_o) and η(t_o + T) and hence the covariance matrix is

CM=(m0ψ(T)ψ(T)m0) $C M = (\begin{matrix} m_{0} & ψ (T) \\ ψ (T) & m_{0} \end{matrix})$

(8.11)

so that

M11=m0,M22=m0,M12=−ψ(T),M=m20−ψ2(T) $M_{11} = m_{0}, M_{22} = m_{0}, M_{12} = - ψ (T), M = m_{0}^{2} - ψ^{2} (T)$

(8.12)

Here, it can be proven that F(u_m) = 0. However, this proof is not strictly necessary for our goal, so that we limit ourselves to note that the random variable η(t_o + T), given condition (8.1), has the following:

CONDITIONALAVERAGE=um $CONDITIONAL AVERAGE = u_{m}$

(8.13)

CONDITIONALSTANDARDDEVIATION=MM22−−−√ $CONDITIONAL STANDARD DEVIATION = \sqrt{\frac{M}{M_{22}}}$

(8.14)

which implies

CONDITIONALAVERAGE=ψ(T)ψ(0)b $CONDITIONAL AVERAGE = \frac{ψ (T)}{ψ (0)} b$

(8.15)

CONDITIONALSTANDARDDEVIATION<σ $CONDITIONAL STANDARD DEVIATION < σ$

(8.16)

The conditional average is a deterministic function of T; that is,

η¯(to+T)=ψ(T)ψ(0)b $\bar{η} (t_{o} + T) = \frac{ψ (T)}{ψ (0)} b$

(8.17)

Since the standard deviation of the random surface elevation with respect to this deterministic function is smaller than σ, the random function η(t_o + T) is asymptotically equal to the deterministic function

η¯(to+T) $\bar{η} (t_{o} + T)$

, if b/σ tends to infinity. Finally, given that

η¯(to+T) $\bar{η} (t_{o} + T)$

has its absolute maximum at t_o, and this maximum is b, we conclude that condition (8.1) is sufficient for the occurrence of a wave crest of given height b, if b/σ → ∞.

8.2. A Sufficient Condition for the Occurrence of a Wave of Given Very Large Height

The condition

η(to)=H2,η(to+T∗)=−H2 $η (t_{o}) = \frac{H}{2}, η (t_{o} + T^{*}) = - \frac{H}{2}$

(8.18)

where t_o is a given time instant and H is a given positive value, in the limit as H/σ → ∞ is sufficient and necessary in probability for the occurrence of a wave of given height H. Actually, this is not exactly the sufficient and necessary condition in probability. As to being sufficient: given condition (8.18) (with H/σ → ∞) the probability approaches 1 to have the occurrence of a wave height H plus a very small random difference of order (H/σ)⁻¹σ. As to being necessary: given a very large wave height H, the probability approaches 1 that the following occur:

η(to)=(12+δξ)H,η(to+T∗)=(−12+δξ)H $η (t_{o}) = (\frac{1}{2} + δ ξ) H, η (t_{o} + T^{*}) = (- \frac{1}{2} + δ ξ) H$

(8.19)

with

δξ $δ ξ$

being a very small random difference of order (H/σ)⁻¹. However, these very small random differences are negligible for the conclusions of this chapter. (They become nonnegligible in the problem of the probability of wave heights leading to Eqn (4.44). Sections 9.7–9.10 of a previous book by the author (2000) may serve to deepen this item.)

In order to prove that condition (8.18) is sufficient, let us consider the conditional probability of the surface elevation at any fixed time instant t_o + T given condition (8.18). We have

p[η(to+T)=u|η(to)=H2,η(to+T∗)=−H2]=p[η(to)=H2,η(to+T∗)=−H2,η(to+T)=u]p[η(to)=H2,η(to+T∗)=−H2] $\begin{matrix} p [η (t_{o} + T) = u | η (t_{o}) = \frac{H}{2}, η (t_{o} + T^{*}) = - \frac{H}{2}] \\ = \frac{p [η (t_{o}) = \frac{H}{2}, η (t_{o} + T^{*}) = - \frac{H}{2}, η (t_{o} + T) = u]}{p [η (t_{o}) = \frac{H}{2}, η (t_{o} + T^{*}) = - \frac{H}{2}]} \end{matrix}$

(8.20)

From Section 4.2 we know that

p[η(to)=12H,η(to+T∗)=−12H,η(to+T)=u]=1(2π)3/2M√exp{−12M[M33u2+2(M13−M23)12Hu+(M11+M22−2M12)14H2]}, $\begin{matrix} p [η (t_{o}) = \frac{1}{2} H, η (t_{o} + T^{*}) = - \frac{1}{2} H, η (t_{o} + T) = u] \\ = \frac{1}{{(2 π)}^{3 / 2} \sqrt{M}} \exp {- \frac{1}{2 M} [M_{33} u^{2} + 2 (M_{13} - M_{23}) \frac{1}{2} H u + (M_{11} + M_{22} - 2 M_{12}) \frac{1}{4} H^{2}]}, \end{matrix}$

(8.21)

p[η(to)=12H,η(to+T∗)=−12H]=12πM˜√exp[−12M˜(M˜11+M˜22−2M˜12)14H2], $p [η (t_{o}) = \frac{1}{2} H, η (t_{o} + T^{*}) = - \frac{1}{2} H] = \frac{1}{2 π \sqrt{\tilde{M}}} \exp [- \frac{1}{2 \tilde{M}} ({\tilde{M}}_{11} + {\tilde{M}}_{22} - 2 {\tilde{M}}_{12}) \frac{1}{4} H^{2}],$

(8.22)

where M_ij and M are the i,j cofactor and the determinant of the covariance matrix of η(t_o),

η(to+T∗) $η (t_{o} + T^{*})$

, η(t_o + T); that is,

CM=⎛⎝⎜ψ(0)ψ(T∗)ψ(T)ψ(T∗)ψ(0)ψ(T−T∗)ψ(T)ψ(T−T∗)ψ(0)⎞⎠⎟, $C M = (\begin{matrix} ψ (0) & ψ (T^{*}) & ψ (T) \\ ψ (T^{*}) & ψ (0) & ψ (T - T^{*}) \\ ψ (T) & ψ (T - T^{*}) & ψ (0) \end{matrix}),$

(8.23)

and

M˜ij ${\tilde{M}}_{i j}$

and

M˜ $\tilde{M}$

are the i,j cofactor and the determinant of the covariance matrix of η(t_o),

η(to+T∗) $η (t_{o} + T^{*})$

. (Note that

M˜ $\tilde{M}$

is equal to M₃₃.) Hence, it follows that

p[η(to+T)=u|η(to)=H2,η(to+T∗)=−H2]=M332πM−−−−√exp[F(u)] $p [η (t_{o} + T) = u | η (t_{o}) = \frac{H}{2}, η (t_{o} + T^{*}) = - \frac{H}{2}] = \sqrt{\frac{M_{33}}{2 π M}} \exp [F (u)]$

(8.24)

where F(u) here denotes the function

F(u)=−12M[M33u2+2(M13−M23)12Hu+(M11+M22−2M12)14H2−MM33(M˜11+M˜22−2M˜12)H24] $F (u) = - \frac{1}{2 M} [M_{33} u^{2} + 2 (M_{13} - M_{23}) \frac{1}{2} H u + (M_{11} + M_{22} - 2 M_{12}) \frac{1}{4} H^{2} - \frac{M}{M_{33}} ({\tilde{M}}_{11} + {\tilde{M}}_{22} - 2 {\tilde{M}}_{12}) \frac{H^{2}}{4}]$

(8.25)

which may be rewritten in the form

F(u)=−M332M(u−um)2+F(um) $F (u) = - \frac{M_{33}}{2 M} {(u - u_{m})}^{2} + F (u_{m})$

(8.26)

where u_m is the abscissa of the maximum:

um=M23−M13M33H2 $u_{m} = \frac{M_{23} - M_{13}}{M_{33}} \frac{H}{2}$

(8.27)

From Eqns (8.24) and (8.26) we get

p[η(to+T)=u|η(to)=H2,η(to+T∗)=−H2]=M332πM−−−−√exp[F(um)]⋅exp[−M332M(u−um)2] $p [η (t_{o} + T) = u | η (t_{o}) = \frac{H}{2}, η (t_{o} + T^{*}) = - \frac{H}{2}] = \sqrt{\frac{M_{33}}{2 π M}} \exp [F (u_{m})] \cdot \exp [- \frac{M_{33}}{2 M} {(u - u_{m})}^{2}]$

(8.28)

Now let us obtain the cofactors M_ij and the determinant M of the covariance matrix Eqn (8.23). When dealing with large covariance matrices it is convenient to resort to some compact symbols for the entries of these matrices. In this case let us define

a=ψ(0),b=ψ(T∗),c=ψ(T),d=ψ(T−T∗) $a = ψ (0), b = ψ (T^{*}), c = ψ (T), d = ψ (T - T^{*})$

(8.29)

so that the covariance matrix and the relevant cofactors are reduced to

CM=⎛⎝⎜abcbadcda⎞⎠⎟ $C M = (\begin{matrix} a & b & c \\ b & a & d \\ c & d & a \end{matrix})$

(8.30)

M11=a2−d2 $M_{11} = a^{2} - d^{2}$

(8.31)

M12=cd−ab $M_{12} = c d - a b$

(8.32)

M13=bd−ac $M_{13} = b d - a c$

(8.33)

M23=bc−ad $M_{23} = b c - a d$

(8.34)

M33=a2−b2 $M_{33} = a^{2} - b^{2}$

(8.35)

M=a(a2−d2)+b(cd−ab)+c(bd−ac) $M = a (a^{2} - d^{2}) + b (c d - a b) + c (b d - a c)$

(8.36)

CONDITIONALAVERAGE=um $CONDITIONAL AVERAGE = u_{m}$

(8.37)

CONDITIONALSTANDARDDEVIATION=MM33−−−√ $CONDITIONAL STANDARD DEVIATION = \sqrt{\frac{M}{M_{33}}}$

(8.38)

which implies

CONDITIONALAVERAGE=c−da−bH2 $CONDITIONAL AVERAGE = \frac{c - d}{a - b} \frac{H}{2}$

(8.39)

CONDITIONALSTANDARDDEVIATION<σ $CONDITIONAL STANDARD DEVIATION < σ$

(8.40)

Later we shall prove the inequality Eqn (8.40).

The conditional average is a deterministic function of T, that is,

η¯(to+T)=ψ(T)−ψ(T−T∗)ψ(0)−ψ(T∗)H2 $\bar{η} (t_{o} + T) = \frac{ψ (T) - ψ (T - T^{*})}{ψ (0) - ψ (T^{*})} \frac{H}{2}$

(8.41)

η¯(to+T) $\bar{η} (t_{o} + T)$

, if H/σ tends to infinity. Since

η¯(to+T) $\bar{η} (t_{o} + T)$

has its absolute maximum at t_o, and this maximum is H/2;

η¯(to+T) $\bar{η} (t_{o} + T)$

has its absolute minimum at

to+T∗ $t_{o} + T^{*}$

, and this minimum is −H/2.

we conclude that condition (8.18) is sufficient for the occurrence of a wave of given height H, if H/σ → ∞. This is provided that the wave crest at t_o and the wave trough at

to+T∗ $t_{o} + T^{*}$

in the deterministic wave

η¯(to+T) $\bar{η} (t_{o} + T)$

are the crest and trough of the same wave. Whether or not this condition is satisfied depends on the shape of the spectrum. This condition is satisfied with characteristic spectra of wind seas like JONSWAP or Pierson and Moskowitz.

It remains to prove the inequality Eqn (8.40). We must prove that

a(a2−d2)+b(cd−ab)+c(bd−ac)a2−b2<a $\frac{a (a^{2} - d^{2}) + b (c d - a b) + c (b d - a c)}{a^{2} - b^{2}} < a$

(8.42)

This may be rewritten in the form

a[1−c2+d2−2cd(ba)a2−b2]<a $a [1 - \frac{c^{2} + d^{2} - 2 c d (\frac{b}{a})}{a^{2} - b^{2}}] < a$

(8.43)

that is proven if we succeed in proving that

c2+d2−2cd(ba)>0 $c^{2} + d^{2} - 2 c d (\frac{b}{a}) > 0$

(8.44)

Here we note that

−1<ba<0 $- 1 < \frac{b}{a} < 0$

(8.45)

(b/a being the ratio between the absolute minimum and the absolute maximum of the autocovariance); as to the product cd it may be either positive or negative (according to the value of T). If cd > 0, the inequality is satisfied because the term −2cd(b/a) is greater than zero. If cd < 0 it follows that

c2+d2−2cd(ba)>c2+d2+2cd=(c+d)2>0 $c^{2} + d^{2} - 2 c d (\frac{b}{a}) > c^{2} + d^{2} + 2 c d = {(c + d)}^{2} > 0$

(8.46)

which completes the proof.

8.3. A Necessary Condition for the Occurrence of a Wave of Given Very Large Height

We shall prove that condition (8.18) is necessary in probability for the occurrence of a wave of given height H, as H/σ → ∞.

8.3.1. General Necessary Condition

A general necessary condition for the occurrence of a wave of given height H is that the surface elevation is

ξH $ξ H$

, with

ξ $ξ$

in (0,1), at a time instant t_o, and is

(ξ−1)H $(ξ - 1) H$

at a later time instant t_o + T (t_o being the instant of the wave crest and t_o + T the instant of the wave trough). The mathematical form of this general necessary condition is

η(to)=ξH,η(to+T)=(ξ−1)Hwith0<ξ<1 $η (t_{o}) = ξ H, η (t_{o} + T) = (ξ - 1) H with 0 < ξ < 1$

(8.47)

For focusing the general necessary condition, look at Fig. 4.4(a) (see Chapter 4), which shows two waves with a fixed height H and different values of

ξ $ξ$

and T(=τ).

8.3.2. The Probability P(H,T,ξ)

Let us consider the probability that the surface elevation at an instant t_o falls between

ξHandξH+dη1 $ξ H and ξ H + d η_{1}$

and, at a later time t_o + T, falls between

(ξ−1)Hand(ξ−1)H+dη2 $(ξ - 1) H and (ξ - 1) H + d η_{2}$

t_o, H, T, ξ being arbitrarily fixed, and dη₁, dη₂ being two fixed small intervals. The probability under examination is given by

P(H,T,ξ)=12πM√⋅exp{−12M[M11ξ2H2+M22(ξ−1)2H2+2M12ξ(ξ−1)H2]}dη1dη2 $P (H, T, ξ) = \frac{1}{2 π \sqrt{M}} \cdot \exp {- \frac{1}{2 M} [M_{11} ξ^{2} H^{2} + M_{22} {(ξ - 1)}^{2} H^{2} + 2 M_{12} ξ (ξ - 1) H^{2}]} d η_{1} d η_{2}$

(8.48)

Here M_ij and M are, respectively, the i,j cofactor and the determinant of the covariance matrix of η(t_o) and η(t_o + T):

CM=(ψ(0)ψ(T)ψ(T)ψ(0)) $C M = (\begin{matrix} ψ (0) & ψ (T) \\ ψ (T) & ψ (0) \end{matrix})$

(8.49)

It is convenient to resort to the compact symbols

a=ψ(0),c=ψ(T) $a = ψ (0), c = ψ (T)$

(8.50)

with which matrix and cofactors get the form

CM=(acca) $C M = (\begin{matrix} a & c \\ c & a \end{matrix})$

(8.51)

M11=a,M22=a,M12=−c,M=a2−c2 $M_{11} = a, M_{22} = a, M_{12} = - c, M = a^{2} - c^{2}$

(8.52)

Let us rewrite Eqn (8.48) in the compact form

P(H,T,ξ)=12πa2−c2√exp[−0.5f(T,ξ)(Hσ)2]dη1dη2 $P (H, T, ξ) = \frac{1}{2 π \sqrt{a^{2} - c^{2}}} \exp [- 0.5 f (T, ξ) {(\frac{H}{σ})}^{2}] d η_{1} d η_{2}$

(8.53)

where

f(T,ξ)=a[aξ2+a(ξ−1)2−2cξ(ξ−1)]a2−c2 $f (T, ξ) = \frac{a [a ξ^{2} + a {(ξ - 1)}^{2} - 2 c ξ (ξ - 1)]}{a^{2} - c^{2}}$

(8.54)

Let us develop the terms in the square parentheses on the right-hand side (RHS). We have

[.]=(2a−2c)ξ2−(2a−2c)ξ+a $[.] = (2 a - 2 c) ξ^{2} - (2 a - 2 c) ξ + a$

(8.55)

that may be rewritten in the form

[.]=2(a−c)(ξ−0.5)2+0.5(a+c) $[.] = 2 (a - c) {(ξ - 0.5)}^{2} + 0.5 (a + c)$

(8.56)

that together with Eqn (8.54) yields

f(T,ξ)=2a(a−c)(ξ−0.5)2+0.5a(a+c)(a+c)(a−c) $f (T, ξ) = \frac{2 a (a - c) {(ξ - 0.5)}^{2} + 0.5 a (a + c)}{(a + c) (a - c)}$

(8.57)

This may be rewritten in the form

f(T,ξ)=2aa+c(ξ−0.5)2+a2(a−c) $f (T, ξ) = \frac{2 a}{a + c} {(ξ - 0.5)}^{2} + \frac{a}{2 (a - c)}$

(8.58)

With the definitions (Eqn (8.50)) of the compact symbols a and c we arrive at

f(T,ξ)=2ψ(0)ψ(0)+ψ(T)(ξ−12)2+12ψ(0)ψ(0)−ψ(T) $f (T, ξ) = 2 \frac{ψ (0)}{ψ (0) + ψ (T)} {(ξ - \frac{1}{2})}^{2} + \frac{1}{2} \frac{ψ (0)}{ψ (0) - ψ (T)}$

(8.59)

8.3.3. Analysis of the Function f(T,ξ) $f (T, ξ)$

The second term on the RHS of Eqn (8.59), that is to say

12ψ(0)ψ(0)−ψ(T) $\frac{1}{2} \frac{ψ (0)}{ψ (0) - ψ (T)}$

is independent of

ξ $ξ$

, and its absolute minimum on the domain T > 0 occurs at

T=T∗ $T = T^{*}$

(given that

T∗ $T^{*}$

is the abscissa of the absolute minimum of ψ(T)). The first term on the RHS of Eqn (8.59), that is to say

2ψ(0)ψ(0)+ψ(T)(ξ−12)2 $2 \frac{ψ (0)}{ψ (0) + ψ (T)} {(ξ - \frac{1}{2})}^{2}$

is zero if

ξ=12 $ξ = \frac{1}{2}$

, and is greater than zero if

ξ≠12 $ξ \neq \frac{1}{2}$

. Therefore, the absolute minimum of the function

f(T,ξ) $f (T, ξ)$

for T in (0,∞) and

ξ $ξ$

in (−∞,∞) occurs at

T=T∗ $T = T^{*}$

and

ξ=12 $ξ = \frac{1}{2}$

8.3.4. Condition (8.18) Is Necessary

The fact that the absolute minimum of

f(T,ξ) $f (T, ξ)$

occurs at

T=T∗ $T = T^{*}$

ξ=12 $ξ = \frac{1}{2}$

implies

P(H,T,ξ)P(H,T∗,12)→0asH/σ→∞, $\frac{P (H, T, ξ)}{P (H, T^{*}, \frac{1}{2})} \to 0 as H / σ \to \infty,$

(8.60)

for every fixed pair

T,ξ $T, ξ$

with

T≠T∗ $T \neq T^{*}$

and/or

ξ≠12 $ξ \neq \frac{1}{2}$

. In other words: as H/σ → ∞, the probability that the surface elevation is

ξH $ξ H$

at an instant t_o and is

(ξ−1)H $(ξ - 1) H$

at an instant t_o + T, for any fixed pair

T,ξ $T, ξ$

with

T≠T∗ $T \neq T^{*}$

and/or

ξ≠12 $ξ \neq \frac{1}{2}$

, is negligible with respect to the probability that the surface elevation is

12H $\frac{1}{2} H$

at t_o and is

−12H $- \frac{1}{2} H$

to+T∗ $t_{o} + T^{*}$

. Hence, as H/σ → ∞, condition (8.18) becomes necessary in probability for the occurrence of a wave of given height H.

8.4. The First Deterministic Wave Function in Space and Time

“Given a very large wave crest of height b at a time instant t_o at a point x_o, y_o” is equivalent to given:

η(xo,yo,to)=b $η (x_{o}, y_{o}, t_{o}) = b$

(8.61)

This is what proceeds from Section 8.1. Here let us consider the conditional probability of the surface elevation at any fixed point x_o + X, y_o + Y at time instant t_o + T, given condition (8.61). We have

p[η(xo+X,yo+Y,to+T)=u|η(xo,yo,to)=b]=p[η(xo,yo,to)=b,η(xo+X,yo+Y,to+T)=u]p[η(xo,yo,to)=b] $p [η (x_{o} + X, y_{o} + Y, t_{o} + T) = u | η (x_{o}, y_{o}, t_{o}) = b] = \frac{p [η (x_{o}, y_{o}, t_{o}) = b, η (x_{o} + X, y_{o} + Y, t_{o} + T) = u]}{p [η (x_{o}, y_{o}, t_{o}) = b]}$

(8.62)

What is most important is that we make no restriction about whether the wave field is or is not homogeneous in space. For example, the wave field may be on the open sea, or before a long breakwater, or in the lee of a vertical breakwater.

Given that both η(x_o, y_o, t_o) and η(x_o + X, y_o + Y, t_o + T) represent stationary Gaussian processes of time, the steps to be done are the same leading from Eqns (8.1) to (8.10), and the result is

p[η(xo+X,yo+Y,to+T)=u|η(xo,yo,to)=b]=M222πM−−−−√exp[−(M222M)(u−um)2] $p [η (x_{o} + X, y_{o} + Y, t_{o} + T) = u | η (x_{o}, y_{o}, t_{o}) = b] = \sqrt{\frac{M_{22}}{2 π M}} \exp [- (\frac{M_{22}}{2 M}) {(u - u_{m})}^{2}]$

(8.63)

where

um=−M12M22b $u_{m} = - \frac{M_{12}}{M_{22}} b$

(8.64)

Now let us obtain the cofactors M_ij and the determinant M. The random variables here are η(x_o, y_o, t_o) and η(x_o + X, y_o + Y, t_o + T) and hence the covariance matrix is

CM=(σ2(xo,yo)Ψ(X,Y,T;xo,yo)Ψ(X,Y,T;xo,yo)σ2(xo+X,yo+Y)) $C M = (\begin{matrix} σ^{2} (x_{o}, y_{o}) & Ψ (X, Y, T; x_{o}, y_{o}) \\ Ψ (X, Y, T; x_{o}, y_{o}) & σ^{2} (x_{o} + X, y_{o} + Y) \end{matrix})$

(8.65)

We have

M11=σ2(xo+X,yo+Y) $M_{11} = σ^{2} (x_{o} + X, y_{o} + Y)$

(8.66)

M22=σ2(xo,yo) $M_{22} = σ^{2} (x_{o}, y_{o})$

(8.67)

M12=−Ψ(X,Y,T;xo,yo) $M_{12} = - Ψ (X, Y, T; x_{o}, y_{o})$

(8.68)

M=σ2(xo,yo)σ2(xo+X,yo+Y)−Ψ2(X,Y,T;xo,yo) $M = σ^{2} (x_{o}, y_{o}) σ^{2} (x_{o} + X, y_{o} + Y) - Ψ^{2} (X, Y, T; x_{o}, y_{o})$

(8.69)

The random surface elevation η(x_o + X, y_o + Y, t_o + T), given that η(x_o, y_o, t_o) = b, has

CONDITIONALAVERAGE=um $CONDITIONAL AVERAGE = u_{m}$

(8.70)

CONDITIONALSTANDARDDEVIATION=MM22−−−√ $CONDITIONAL STANDARD DEVIATION = \sqrt{\frac{M}{M_{22}}}$

(8.71)

which implies

CONDITIONALAVERAGE=Ψ(X,Y,T;xo,yo)σ2(xo,yo)b $CONDITIONAL AVERAGE = \frac{Ψ (X, Y, T; x_{o}, y_{o})}{σ^{2} (x_{o}, y_{o})} b$

(8.72)

CONDITIONALSTANDARDDEVIATION<σ(xo+X,yo+Y) $CONDITIONAL STANDARD DEVIATION < σ (x_{o} + X, y_{o} + Y)$

(8.73)

The conditional average is a deterministic wave function of X, Y, T that may be rewritten in the form

η¯(xo+X,yo+Y,to+T)=Ψ⌢(X,Y,T;xo,yo)[bσ(xo,yo)]σ(xo+X,yo+Y) $\bar{η} (x_{o} + X, y_{o} + Y, t_{o} + T) = \overset{⌢}{Ψ} (X, Y, T; x_{o}, y_{o}) [\frac{b}{σ (x_{o}, y_{o})}] σ (x_{o} + X, y_{o} + Y)$

(8.74)

where

Ψ⌢ $\overset{⌢}{Ψ}$

is the cross-correlation whose range is (−1,1) (cf. Section 7.3.3). From Eqn (8.73) we know that the standard deviation of the random surface elevation with respect to this deterministic function is smaller than σ(x_o + X, y_o + Y). Conclusion: the random surface elevation η(x_o + X, y_o + Y, t_o + T) is asymptotically equal to the deterministic wave function

η¯(xo+X,yo+Y,to+T) $\bar{η} (x_{o} + X, y_{o} + Y, t_{o} + T)$

, as b/σ(x_o, y_o) tends to infinity.

It must be pointed out that there is no restriction on the ratio σ(x_o + X, y_o + Y)/σ(x_o, y_o). In particular, x_o, y_o may be a point in the lee of a breakwater and x_o + X, y_o + Y may be a point on the wave-beaten wall of the breakwater, so that the ratio σ(x_o + X, y_o + Y)/σ(x_o, y_o) is very large. Nevertheless, if we know that at point x_o, y_o a wave crest occurs with a height b that is exceptionally large with respect to the root mean square surface elevation at this point, we may expect that the surface elevation at x_o + X, y_o + Y will be close to a well-defined deterministic wave function given by Eqn (8.74). Hence, even if the wave crest recorded in the lee is much smaller than the waves at the outer wall, the fact of having found a wave crest that is exceptionally large with respect to the average in the lee will enable us to predict how the waves are even at the outer wall.

8.5. The Velocity Potential Associated with the First Deterministic Wave Function in Space and Time

The form (Eqn (8.74)) of the deterministic wave function is effective for understanding the sense of the QD theory; however, for calculation, the form

η¯(xo+X,yo+Y,to+T)=Ψ(X,Y,T;xo,yo)σ2(xo,yo)b $\bar{η} (x_{o} + X, y_{o} + Y, t_{o} + T) = \frac{Ψ (X, Y, T; x_{o}, y_{o})}{σ^{2} (x_{o}, y_{o})} b$

(8.75)

is more straightforward. Associated with this deterministic wave function is a distribution of velocity potential in the water, which to the lowest order in a Stokes expansion is given by

ϕ¯(xo+X,yo+Y,z,to+T)=Φ(X,Y,T,z;xo,yo)σ2(xo,yo)b $\bar{ϕ} (x_{o} + X, y_{o} + Y, z, t_{o} + T) = \frac{Φ (X, Y, T, z; x_{o}, y_{o})}{σ^{2} (x_{o}, y_{o})} b$

(8.76)

The surface elevation (Eqn (8.75)) and the velocity potential (Eqn (8.76)) satisfy the linear flow equations. In particular, we shall prove that

η¯ $\bar{η}$

and

ϕ¯ $\bar{ϕ}$

satisfy the first linear flow equation under the hypothesis that η and

ϕ $ϕ$

satisfy this equation. That is to say, we shall prove that

η¯=−1g(∂ϕ¯∂T)z=0 $\bar{η} = - \frac{1}{g} {(\frac{\partial \bar{ϕ}}{\partial T})}_{z = 0}$

(8.77)

provided that

η=−1g(∂ϕ∂t)z=0 $η = - \frac{1}{g} {(\frac{\partial ϕ}{\partial t})}_{z = 0}$

(8.78)

With the formulas (8.75) and (8.76) of

η¯ $\bar{η}$

and

ϕ¯ $\bar{ϕ}$

, the equality (8.77) (to be proved) takes on the form

Ψ(X,Y,T;xo,yo)σ2(xo,yo)b=−1g[∂∂TΦ(X,Y,T,z;xo,yo)bσ2(xo,yo)]z=0 $\frac{Ψ (X, Y, T; x_{o}, y_{o})}{σ^{2} (x_{o}, y_{o})} b = - \frac{1}{g} {[\frac{\partial}{\partial T} Φ (X, Y, T, z; x_{o}, y_{o}) \frac{b}{σ^{2} (x_{o}, y_{o})}]}_{z = 0}$

(8.79)

where the term b/σ²(x_o, y_o) cancels. With the definitions (7.1) and (7.9) of

Ψ $Ψ$

and

Φ $Φ$

, the equality to be proved becomes

⟨η(xo,yo,to)η(xo+X,yo+Y,t+T)⟩=−1g[∂∂T⟨η(xo,yo,t)ϕ(xo+X,yo+Y,z,t+T)⟩]z=0 $〈 η (x_{o}, y_{o}, t_{o}) η (x_{o} + X, y_{o} + Y, t + T) 〉 = - \frac{1}{g} {[\frac{\partial}{\partial T} 〈 η (x_{o}, y_{o}, t) ϕ (x_{o} + X, y_{o} + Y, z, t + T) 〉]}_{z = 0}$

(8.80)

wherein the order “derivative with respect to T,” “average with respect to t” may be inverted, with the result that the equality to be proved becomes

⟨η(xo,yo,to)η(xo+X,yo+Y,t+T)⟩=[⟨η(xo,yo,t)(−1g)∂∂Tϕ(xo+X,yo+Y,z,t+T)⟩]z=0 $〈 η (x_{o}, y_{o}, t_{o}) η (x_{o} + X, y_{o} + Y, t + T) 〉 = {[〈 η (x_{o}, y_{o}, t) (- \frac{1}{g}) \frac{\partial}{\partial T} ϕ (x_{o} + X, y_{o} + Y, z, t + T) 〉]}_{z = 0}$

and this equality is proved since

η(xo+X,yo+Y,t+T)=(−1g)∂∂Tϕ(xo+X,yo+Y,z,t+T)z=0 $η (x_{o} + X, y_{o} + Y, t + T) = (- \frac{1}{g}) \frac{\partial}{\partial T} ϕ {(x_{o} + X, y_{o} + Y, z, t + T)}_{z = 0}$

(8.81)

as a consequence of Eqn (8.78). Equation (8.81) says that random surface elevation η and the relevant velocity potential

ϕ $ϕ$

satisfy the linear flow Eqn (8.78) at point x_o + X, y_o + Y, at time instant t + T.

8.6. The Second Deterministic Wave Function in Space and Time

“Given a very large wave of height H at a time instant t_o at a point x_o, y_o” is equivalent to

η(xo,yo,to)=H2,η(xo,yo,to+T∗)=−H2 $η (x_{o}, y_{o}, t_{o}) = \frac{H}{2}, η (x_{o}, y_{o}, t_{o} + T^{*}) = - \frac{H}{2}$

(8.82)

This is what proceeds from Sections 8.2 and 8.3.

Given condition (8.82) as H/σ(x_o, y_o) → ∞, the random surface elevation η(x_o + X, y_o + Y, t_o + T) is asymptotically equal to the deterministic function

η¯(xo+X,yo+Y,to+T)=Ψ(X,Y,T;xo,yo)−Ψ(X,Y,T−T∗;xo,yo)Ψ(0,0,0;xo,yo)−Ψ(0,0,T∗;xo,yo)H2 $\bar{η} (x_{o} + X, y_{o} + Y, t_{o} + T) = \frac{Ψ (X, Y, T; x_{o}, y_{o}) - Ψ (X, Y, T - T^{*}; x_{o}, y_{o})}{Ψ (0,0,0; x_{o}, y_{o}) - Ψ (0,0, T^{*}; x_{o}, y_{o})} \frac{H}{2}$

(8.83)

and the velocity potential associated with this deterministic wave function is

ϕ¯(xo+X,yo+Y,z,to+T)=Φ(X,Y,z,T;xo,yo)−Φ(X,Y,z,T−T∗;xo,yo)Ψ(0,0,0;xo,yo)−Ψ(0,0,T∗;xo,yo)H2 $\bar{ϕ} (x_{o} + X, y_{o} + Y, z, t_{o} + T) = \frac{Φ (X, Y, z, T; x_{o}, y_{o}) - Φ (X, Y, z, T - T^{*}; x_{o}, y_{o})}{Ψ (0,0,0; x_{o}, y_{o}) - Ψ (0,0, T^{*}; x_{o}, y_{o})} \frac{H}{2}$

(8.84)

The deterministic wave function (Eqn (8.83)) is obtained starting from condition (8.82) and reasoning as in Section 8.4. Then with the same reasoning done in Section 8.5, one can verify that deterministic wave function (Eqn (8.83)) and velocity potential (Eqn (8.84)) satisfy the linear flow equations.

8.7. Comment: A Deterministic Mechanics Is Born by the Theory of Probability

Given a wave with a known height H, if H/σ is very large, the conditional standard deviation of the random surface elevation is negligible with respect to the conditional average surface elevation. This conditional average represents a deterministic wave function of space and time. Hence, the actual (random) waves will be very close to this deterministic wave function. Associated with this deterministic wave function is a very precise distribution of velocity potential in the water.

The conclusion is that a deterministic mechanics consisting of the deterministic wave function (Eqn (8.83)) and the relevant velocity potential (Eqn (8.84)) is born by the theory of probability.

What the author finds exciting is that Eqns (8.83) and (8.84) hold for an arbitrary configuration of the solid boundary (of course, provided that the flow is frictionless). Only the relationship between functions Ψ and Φ and the directional spectrum S(ω,θ) of the incident waves changes with the solid boundary (several examples of this relationship are given in Chapter 7).

The sense of the QD theory is the following: if a wave with an exceptionally large height H occurs at some point x_o, y_o at a time instant t_o, it is most probable that the occurrence of this wave happens in a very precise (deterministic) way.

Note that the assumption that given wave height H is very large with respect to σ of the sea state may be consistent with the Stokes assumption that the wave height is small with respect to the bottom depth and the wavelength. Of course, what has been said in this section regarding the wave of given height H conceptually holds also for the wave crest of given height b.

8.8. Conclusion

In the spring of 1980 the author realized that a wave with a given very large crest height in a Gaussian sea state is close to the autocovariance (Eqn (8.17)) and published this result in papers (1982, 1983). After this result the author wondered whether determinism can be born within a chaotic process. In fact, the result suggested two opposite conclusions. Let us review these conclusions in simple words; to do this we shall resort to the evidence of numbers. (The reader should bear in mind that we are dealing with a Gaussian random process of unlimited duration and unlimited wave height). Let us assume that the ψ∗ of this stationary random Gaussian process is equal to 0.60. In this process, a wave with a crest height of b = 100σ will have a height H = 1.6 100σ = 160σ. Because of the statistical symmetry of the Gaussian process, a wave with an elevation of the trough of −100σ will have a profile opposite to that of the wave of crest height 100σ. Hence a consequence of Eqn (8.17) with our example is that

1. a wave of given crest height 100σ has a deterministic height of 160σ;

2. a wave of given height 160σ does not have a deterministic crest height: indeed, there is the same probability that this wave has a crest of 100σ or a trough of −100σ (which implies a crest of 60σ).

Item (1) suggests a positive conclusion: yes, determinism can be born within a chaotic process. Item (2) suggests the opposite conclusion. The author was able to solve this contradiction with the proof shown in Sections 8.2 and 8.3 of the present chapter: a wave of given very large height H is close to the deterministic wave function (8.41); what this implies, in our example, is that a wave of given height 160σ has a deterministic crest height of 80σ. This does not contradict what proceeds from Eqn (8.17), since the number of wave crests of 100σ is very small with respect to the number of wave heights of 160σ. In other words, the set of waves having the given height of 160σ consists of waves all with the crest height of 80σ, except for a very small fraction of anomalous elements of the set. This very small fraction includes the whole set of waves with the given crest height of 100σ. Hence, the general conclusion is: yes, determinism can be born within a chaotic process. This was called “quasi-determinism” because of the presence of the very small fractions of anomalous elements. The first time the term quasi-determinism appeared was in a paper by the author (1984), where Eqn (8.41) was also disclosed.

Some time later the author made substantial progress when Eqn (8.83) was applied to the space-time; this was in 1986, and the publication was in the paper (1989). Here, the author showed that a very large wave of given height H occurs at some fixed point x_o, y_o because this point is struck home by a well-defined wave group. This group has a deterministic (new) mechanics that we shall analyze in detail in the next chapter. Finally, in June 1987 the author applied Eqn (8.83) to Gaussian wave fields being nonhomogeneous in space (publication in the papers (1988) and (1997b)). It was an exciting experience! We shall see a few examples in Chapter 11. In particular, we shall see that a very large wave of given height occurs at a fixed point x_o, y_o far from a long breakwater, because of a collision of two wave groups. The logic is that of quasi-determinism: if a wave with a given very large height will occur at some fixed point, we can predict, with a probability approaching 1, the story of this occurrence. This result implies that “a deterministic mechanics can be born within a chaotic process,” which is an aspect of the possibility that determinism can be born within a chaotic process.

In the 1980s the author knew Rice’s work (1944, 1945, 1958) on the analysis of stationary random Gaussian processes, and the author did not know some subsequent works on the same subject. In particular, Lindgren (1972) had shown that the behavior of η_b(t) is well determined by the behavior of ψ(t) as b → ∞, where η_b is the random Gaussian function, given that at t = 0 there is a local maximum of ordinate b. This is essentially the same as Eqn (8.17). However, the approach was different from the author’s: in his paper Lindgren focused on mathematical random functions, whereas in the papers (1982, 1983) the author focused on waves. If the author had not reobtained independently Eqn (8.17) through the approach focused on waves, the author would not have arrived at the QD theory. Professor Leon Borgman played an important role with an open-minded review of the author’s early work (acknowledged in the paper (1983)).

The more difficult part of the basic proof of the QD theory is that concerning the necessary condition. The original forms the author gave in the 1980s were rather complicated, and only some years later (1997a) the author reached the simple form that is given in Section 8.3 of this chapter.

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Table of Contents for Chapter 8. The Theory of Quasi-Determinism

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Abstract

Keywords

8.1. The Necessary and Sufficient Condition for the Occurrence of a Wave Crest of Given Very Large Height

8.2. A Sufficient Condition for the Occurrence of a Wave of Given Very Large Height

8.3. A Necessary Condition for the Occurrence of a Wave of Given Very Large Height

8.3.1. General Necessary Condition

8.3.2. The Probability P(H,T,ξ)

8.3.3. Analysis of the Function f(T,ξ)<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:ce="http://www.elsevier.com/xml/common/dtd"><mrow><mi>f</mi><mrow><mo>(</mo><mi>T</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></mrow></mrow></math>

8.3.4. Condition (8.18) Is Necessary

8.4. The First Deterministic Wave Function in Space and Time

8.5. The Velocity Potential Associated with the First Deterministic Wave Function in Space and Time

8.6. The Second Deterministic Wave Function in Space and Time

8.7. Comment: A Deterministic Mechanics Is Born by the Theory of Probability

8.8. Conclusion

Table of Contents for
Chapter 8. The Theory of Quasi-Determinism

8.3.3. Analysis of the Function f(T,ξ) $f (T, ξ)$