Seakeeping In Random Waves

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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



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

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

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

[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]

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]:

[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]
  • 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