View source for Seakeeping In Random Waves

{{Ocean Wave Interaction with Ships and Offshore Structures
 | chapter title = Seakeeping In Random Waves
 | next chapter = [[Solution of Wave-Body Flows, Green's Theorem]]
 | previous chapter = [[Derivative Seakeeping Quantities]]
}}

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Assume known the ambient wave spectral density <math>S_{\zeta}(\omega_0)\,</math> assumed unidirectional for simplicity

<center><math> \frac{1}{2}A_i^2 = S(\omega_i)\Delta\omega \, </math></center>

* <math> \int_0^\infty S_\zeta(\omega) \mathrm{d}\omega = \sigma_\zeta^2 \equiv \,</math> Variance of the wave elevation of ambient random seastate, assumed Gaussian with zero mean

* Assuming that the <math>RAO(\omega)\,</math> of a seakeeping quantity <math> X(t) \,</math> has been determined from a frequency domain analysis;

<u>Spectral analysis with forward-speed</u>

<center><math> \omega = \left| \omega_0 - U \frac{\omega_0^2}{g} \cos \beta \right| </math></center>

* Ambient wave spectral density <math> S_\zeta(\omega_0)\,</math> is defined relative to the absolute wave frequency <math> \omega_0\,</math>.

* The <math> RAO_X(\omega) \,</math> is usually defined relative to the encounter frequency <math> \omega\,</math>.

* The relation of <math> \omega \leftrightarrow \omega_0 \, </math> is not single valued. The question thus arises of what is the <math>\sigma_X^2\,</math>?

<u>Answer</u>

* Given <math>\omega_0 \,</math>, a single value of <math>\omega\,</math> always follows.

* The opposite is not always true. Given <math>\omega\,</math> there may exist multiple <math>\omega_0\,</math>'s satisfying the encounter frequency relation.

* Therefore it is much simpler to parameterize with respect to <math>\omega_0\,</math>, even when the <math>RAO(\omega)\,</math> is evaluated as a function of <math>\omega\,</math>.

Proceed as follows:

Simply redefine the <math>RAO(\omega)\,</math> as follows:

<center><math> \left|RAO_3\right|(\omega) = \left|RAO_3\right| \left( \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \right) \equiv \left|RAO_3 \right| (\omega_0) \,</math></center>

New function of <math>\omega_0\,</math> by virtue of the <math> \omega \leftrightarrow\, \omega_0 </math> relation.

The standard deviation of heave follows by simple integration over <math> \omega_0\,</math>:

<center><math> \sigma_3^2 = \int_0^\infty \mathrm{d}\omega_0 S_\zeta \left(\omega_0\right) \left|RAO_3^*\left(\omega_0\right)\right|^2 </math></center>

* The opposite choice of parameterizing the above integral w.r.t. <math>\omega\,</math> ends up with a lot of unnecessary algebra.


[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]