Seakeeping In Random Waves
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Assume known the ambient wave spectral density [math]S_{\zeta}(\omega_0)\,[/math] assumed unidirectional for simplicity
- [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;
Spectral analysis with forward-speed
- 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]?
Answer
- 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:
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]:
- 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