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

Step 1: Treat the floating body as a linear system.
For a linear system, the RAO \( H(\omega) \) is the transfer function that relates the body's motion amplitude to the wave amplitude at each frequency \( \omega \).

Step 2: Apply the spectral input output relation.
For a linear system driven by a random input with spectral density \( S(\omega) \), the output spectral density equals the input spectrum scaled by the squared magnitude of the transfer function: \( S_R(\omega) = |H(\omega)|^2 S(\omega) \).

Step 3: Explain why it is squared magnitude, not the RAO itself.
A spectrum represents energy, which goes as amplitude squared, so turning a wave energy spectrum into a response energy spectrum needs \( |H(\omega)|^2 \), not just \( H(\omega) \) or \( |H(\omega)| \).

Final Answer:
The response spectrum is the wave spectrum multiplied by the squared RAO magnitude. \[ \boxed{S_R(\omega) = |H(\omega)|^2 S(\omega)} \]