Appendix 4
Calculation of the Variances of νφ, νη. νθ
A4.1. Variance of the νφ and νη, variables
From relationship [3.50] in section 3.3.2 of Chapter 3 we know that the azimuth angle φi of the waves with random trials is linked to the νφ variable by the following expression:
The computation of the pdf established in [3.52] enables us to formulate the sought after variance, i.e.:
is the equation in which D is the variation domain of νø, i.e. the interval [-1 +1]. Integral [A4.2] can then take the detailed form below:
[A4.3]
By using expression [A4.1], we carry out the change of variable leading to integral [A4.4]:
Analytical solution of this integral allocate the value ½ to the variance:
[A4.5]
Regarding υη, a similar calculation leads to the determination of the integral [A4.6]:
A4.2. Variance of the υθ variable
We recall that the definition of the υθ variable ...
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