Seakeeping In Random Waves

Wave and Wave Body Interactions
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Assume known the ambient wave spectral density [math]\displaystyle{ S_{\zeta}(\omega_0)\, }[/math] assumed unidirectional for simplicity

[math]\displaystyle{ \frac{1}{2}A_i^2 = S(\omega_i)\Delta\omega \, }[/math]

• [math]\displaystyle{ \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]\displaystyle{ RAO(\omega)\, }[/math] of a seakeeping quantity [math]\displaystyle{ X(t) \, }[/math] has been determined from a frequency domain analysis;

Spectral analysis with forward-speed

[math]\displaystyle{ \omega = \left| \omega_0 - U \frac{\omega_0^2}{g} \cos \beta \right| }[/math]

• Ambient wave spectral density [math]\displaystyle{ S_\zeta(\omega_0)\, }[/math] is defined relative to the absolute wave frequency [math]\displaystyle{ \omega_0\, }[/math].

• The [math]\displaystyle{ RAO_X(\omega) \, }[/math] is usually defined relative to the encounter frequency [math]\displaystyle{ \omega\, }[/math].

• The relation of [math]\displaystyle{ \omega \leftrightarrow \omega_0 \, }[/math] is not single valued. The question thus arises of what is the [math]\displaystyle{ \sigma_X^2\, }[/math]?

Answer

• Given [math]\displaystyle{ \omega_0 \, }[/math], a single value of [math]\displaystyle{ \omega\, }[/math] always follows.

• The opposite is not always true. Given [math]\displaystyle{ \omega\, }[/math] there may exist multiple [math]\displaystyle{ \omega_0\, }[/math]'s satisfying the encounter frequency relation.

• Therefore it is much simpler to parameterize with respect to [math]\displaystyle{ \omega_0\, }[/math], even when the [math]\displaystyle{ RAO(\omega)\, }[/math] is evaluated as a function of [math]\displaystyle{ \omega\, }[/math].

Proceed as follows:

Simply redefine the [math]\displaystyle{ RAO(\omega)\, }[/math] as follows:

[math]\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) \, }[/math]

New function of [math]\displaystyle{ \omega_0\, }[/math] by virtue of the [math]\displaystyle{ \omega \leftrightarrow\, \omega_0 }[/math] relation.

The standard deviation of heave follows by simple integration over [math]\displaystyle{ \omega_0\, }[/math]:

[math]\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 }[/math]

• The opposite choice of parameterizing the above integral w.r.t. [math]\displaystyle{ \omega\, }[/math] ends up with a lot of unnecessary algebra.

Ocean Wave Interaction with Ships and Offshore Energy Systems