Question 3.2: Complete Table Q1 by determining the values of the spectral ......
Complete Table Q1 by determining the values of the spectral energy density (S(f)) at frequencies f = 0.088 Hz and f = 0.113 Hz from the following data: wind speed U_{10} = 20 m/s, fetch length F = 220 km.
| f (Hz) | S(f) (m^{2}/s) |
| 0.050 | 0.0 |
| 0.063 | 0.2 |
| 0.076 | 5.6 |
| 0.088 | |
| 0.101 | 64.6 |
| 0.113 | |
| 0.126 | 13.8 |
| 0.138 | 9.8 |
| 0.151 | 7.0 |
| 0.164 | 5.0 |
| 0.176 | 3.6 |
| 0.189 | 2.7 |
| 0.201 | 2.0 |
| 0.214 | 1.5 |
| 0.227 | 1.1 |
| 0.239 | 0.9 |
Use the JONSWAP formula with U_{10} = 20 and F = 220000 to find
\begin{aligned}& \alpha=\frac{0.076}{\left(g F / U_{10}^2\right)^{0.22}}=0.0115 \\ & \left.f_p=\left(3.5 g / U_{10}\right)\left(g F / U_{10}^2\right)^{0.33}\right)=0.101 \end{aligned}Hence, for f = 0.088 Hz, σ = 0.07 giving S(f) = 20.1 m²/s, and for f = 0.113 Hz, σ = 0.09 with S(f) = 27.4 m²/s.
The moments of this spectrum cannot be calculated analytically but may be estimated by numerical integration. For \gamma =3.3, he following approximate relationships hold: T_{m01} = 0.8345Tp ; T_{z} = 0.7775T_{p}.
When \gamma = 1.0 the JONSWAP spectrum simplifies to the Pierson-Moskowitz spectrum. The JONSWAP spectrum has become one of the most widely used spectra, both in laboratory experiments and for design.
It should be borne in mind that the above three spectra have important limitations:
• They are not applicable to intermediate or shallow water conditions.
• The Pierson-Moskowitz spectrum is for fully developed seas only.
• The JONSWAP spectrum was developed under fetch-limited conditions.
• The Bretschneider spectrum accounts for duration and fetch limitation in an empirical manner.
• They are all single-peaked spectra. (Ochi and Hubble 1976 describe a method of modelling double-peaked spectra.)