Seakeeping In Random Waves

Wave and Wave Body Interactions
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Seakeeping in random waves

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

$\frac{1}{2}A_i^2 = S(\omega_i)\Delta\omega \,$

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

Assuming that the $RAO(\omega)\,$ of a seakeeping quantity $X(t) \,$ has been determined from a frequency domain analysis;

Spectral analysis with forward-speed

$\omega = \left| \omega_0 - U \frac{\omega_0^2}{g} \cos \beta \right|$

Ambient wave spectral density $S_\zeta(\omega_0)\,$ is defined relative to the absolute wave frequency $\omega_0\,$ .

The $RAO_X(\omega) \,$ is usually defined relative to the encounter frequency $\omega\,$ .

The relation of $\omega \leftrightarrow \omega_0 \,$ is not single valued. The question thus arises of what is the $\sigma_X^2\,$ ?

Answer

The opposite is not always true. Given $\omega\,$ there may exist multiple $\omega_0\,$ 's satisfying the encounter frequency relation.

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

Proceed as follows:

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

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

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

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

$\sigma_3^2 = \int_0^\infty \mathrm{d}\omega_0 S_\zeta \left(\omega_0\right) \left|RAO_3^*\left(\omega_0\right)\right|^2$

The opposite choice of parameterizing the above integral w.r.t. $\omega\,$ ends up with a lot of unnecessary algebra.

Wave and Wave Body Interactions
Current Chapter	Seakeeping In Random Waves
Next Chapter	Solution of Wave-Body Flows, Green's Theorem
Previous Chapter	Derivative Seakeeping Quantities