Question

The RAO (Response Amplitude Operator) of a floating body with no forward speed is denoted by \(H(\omega)\).

If it encounters a wave spectrum \(S(\omega)\), then its response spectrum is ______.

• A. \(H(\omega)S(\omega)\)
• B. \(|H(\omega)|^2 S(\omega)\)
• C. \(|H(\omega)|^3 S(\omega)\)
• D. \(|H(\omega)|S(\omega)\)

Hint:

Remember that spectra represent energy, so amplitude ratios must be squared.

Answer 1

The Correct Option is B

This question tests the standard spectral relation used in seakeeping analysis to convert a wave spectrum into a ship response spectrum.

1. \(H(\omega)S(\omega)\): The RAO can be a complex number carrying phase information, so multiplying it directly with an energy spectrum does not give a physically meaningful energy quantity, so this option is wrong.
2. \(|H(\omega)|^2 S(\omega)\): A spectrum measures energy, which scales with amplitude squared, so passing a wave spectrum through a linear system scales it by the squared magnitude of that system's transfer function. This matches the standard seakeeping formula for response spectra, so it is correct.
3. \(|H(\omega)|^3 S(\omega)\): There is no physical basis for a cubic scaling in a linear spectral response relation, so this option is wrong.
4. \(|H(\omega)|S(\omega)\): This uses the magnitude to the first power, which would fit an amplitude spectrum, not an energy spectrum, so this option is wrong.

The correct choice is (B): the response spectrum equals the wave spectrum multiplied by the square of the RAO magnitude.