# Seakeeping In Random Waves

Wave and Wave Body Interactions
Current Chapter Seakeeping In Random Waves
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Assume known the ambient wave spectral density $\displaystyle{ S_{\zeta}(\omega_0)\, }$ assumed unidirectional for simplicity

$\displaystyle{ \frac{1}{2}A_i^2 = S(\omega_i)\Delta\omega \, }$
• $\displaystyle{ \int_0^\infty S_\zeta(\omega) \mathrm{d}\omega = \sigma_\zeta^2 \equiv \, }$ Variance of the wave elevation of ambient random seastate, assumed Gaussian with zero mean
• Assuming that the $\displaystyle{ RAO(\omega)\, }$ of a seakeeping quantity $\displaystyle{ X(t) \, }$ has been determined from a frequency domain analysis;

Spectral analysis with forward-speed

$\displaystyle{ \omega = \left| \omega_0 - U \frac{\omega_0^2}{g} \cos \beta \right| }$
• Ambient wave spectral density $\displaystyle{ S_\zeta(\omega_0)\, }$ is defined relative to the absolute wave frequency $\displaystyle{ \omega_0\, }$.
• The $\displaystyle{ RAO_X(\omega) \, }$ is usually defined relative to the encounter frequency $\displaystyle{ \omega\, }$.
• The relation of $\displaystyle{ \omega \leftrightarrow \omega_0 \, }$ is not single valued. The question thus arises of what is the $\displaystyle{ \sigma_X^2\, }$?

• Given $\displaystyle{ \omega_0 \, }$, a single value of $\displaystyle{ \omega\, }$ always follows.
• The opposite is not always true. Given $\displaystyle{ \omega\, }$ there may exist multiple $\displaystyle{ \omega_0\, }$'s satisfying the encounter frequency relation.
• Therefore it is much simpler to parameterize with respect to $\displaystyle{ \omega_0\, }$, even when the $\displaystyle{ RAO(\omega)\, }$ is evaluated as a function of $\displaystyle{ \omega\, }$.

Proceed as follows:

Simply redefine the $\displaystyle{ RAO(\omega)\, }$ as follows:

$\displaystyle{ \left|RAO_3\right|(\omega) = \left|RAO_3\right| \left( \omega_0 - U \frac{\omega_0^2}{g} \cos\beta \right) \equiv \left|RAO_3 \right| (\omega_0) \, }$

New function of $\displaystyle{ \omega_0\, }$ by virtue of the $\displaystyle{ \omega \leftrightarrow\, \omega_0 }$ relation.

The standard deviation of heave follows by simple integration over $\displaystyle{ \omega_0\, }$:

$\displaystyle{ \sigma_3^2 = \int_0^\infty \mathrm{d}\omega_0 S_\zeta \left(\omega_0\right) \left|RAO_3^*\left(\omega_0\right)\right|^2 }$
• The opposite choice of parameterizing the above integral w.r.t. $\displaystyle{ \omega\, }$ ends up with a lot of unnecessary algebra.