Question

Consider the wave elevation spectrum $S_{\eta\eta}(\omega)$ as shown in the figure. Then, the significant wave height is ________________ m.

• A. 2
• B. 4
• C. 6
• D. 8

Hint:

Significant wave height is always given by $H_s = 4\sqrt{m_0}$ where $m_0$ is the total area under the wave spectrum.

Answer 1

The Correct Option is D

The significant wave height $H_s$ is obtained using the spectral moment $m_0$:
\[ H_s = 4\sqrt{m_0} \]
where
\[ m_0 = \int_0^{\infty} S_{\eta\eta}(\omega)\, d\omega. \]
From the graph, the spectrum is piecewise linear and flat:
- From $\omega=0$ to $0.25$: triangular rise from 0 to 6.4.
- From $\omega=0.25$ to $0.50$: constant at 6.4.
- From $\omega=0.50$ to $1.0$: triangular fall back to 0.
Compute the area under the spectrum (i.e., $m_0$):
Triangle 1:
\[ A_1 = \frac{1}{2}(0.25)(6.4)=0.8 \]
Rectangle:
\[ A_2 = (0.50-0.25)(6.4)=1.6 \]
Triangle 2:
\[ A_3 = \frac{1}{2}(0.50)(6.4)=1.6 \]
Total:
\[ m_0 = A_1 + A_2 + A_3 = 0.8 + 1.6 + 1.6 = 4.0 \]
Thus:
\[ H_s = 4\sqrt{4} = 4 \times 2 = 8\;\text{m} \]
So the significant wave height is 8 m.