Question 3.1: a. Determine Hmax, Tmax, Hs, Ts, Hz and Tz. b. Plot a histog......
USING THE TIME SERIES DATA GIVEN IN TABLE 3.1
a. Determine H_{max}, T_{max}, H_{s}, T_{s}, H_{z} and T_{z}.
b. Plot a histogram of the wave heights using a class interval of 1 m.
c. Determine H_{max}, H_{s} and H_{rms} from H_{z}, assuming a Rayleigh distribution.
d. Calculate the value of f(h) at the centre of each class interval and superimpose the pdf on the histogram. (Note that we assume that the scale equivalence is f(h) ≡ n/NΔh.)
e. Suggest reasons for the anomalies between the results in (a) and (c).
| Table 3.1 Wave heights and periods | |||||
| Wave number | Wave Height H(m) | Wave Period T(s) | Wave number | Wave Height H(m) | Wave Period T(s) |
| 1 | 0.54 | 4.2 | 11 | 1.03 | 6.1 |
| 2 | 2.05 | 8.0 | 12 | 1.95 | 8.0 |
| 3 | 4.52 | 6.9 | 13 | 1.97 | 7.6 |
| 4 | 2.58 | 11.9 | 14 | 1.62 | 7.0 |
| 5 | 3.2 | 7.3 | 15 | 4.08 | 8.2 |
| 6 | 1.87 | 5.4 | 16 | 4.89 | 8.0 |
| 7 | 1.9 | 4.4 | 17 | 2.43 | 9.0 |
| 8 | 1.00 | 5.2 | 18 | 2.83 | 9.2 |
| 9 | 2.05 | 6.3 | 19 | 2.94 | 7.9 |
| 10 | 2.37 | 4.3 | 20 | 2.23 | 5.3 |
| 21 | 2.98 | 6.9 | |||
a. From Table 3.1
16th wave gives H_{max} = 4.89, T_{max} = 8.0
For H_{s} 16th, 3rd, 15th, 5th, 21st, 19th, 18th waves (21/3 = 7 waves) are the highest 1/3 of the waves
Average to obtain
H_{s} = 3.6 m T_{s} = 7.8 sFor H_{z}, T_{z} average all 21
H_{z} = 2.4 m T_{z} = 7.0 sThe histogram is shown in Figure 3.3.
c. From part (a)
\begin{aligned} H_z & =2.4 \mathrm{~m} \\ \therefore H_{\mathrm{rms}} & =1.13 \times 2.4=2.71 \mathrm{~m} \\ \therefore H_s & =1.414 H_{\mathrm{rms}}=3.83 \mathrm{~m} \\ \therefore H_{\max } & =H_s\left(\frac{1}{2} \ln N\right)^{1 / 2} \\ & =3.83\left(\frac{1}{2} \ln 21\right)^{1 / 2} \\ & =4.73 \mathrm{~m} \end{aligned}d. Using (3.2)
f(h)=\left(2h/H^{2}_{rms} \right) exp\left[-\left(h/H_{rms} \right) ^{2} \right] (3.2)
f(h)=\left\lgroup\frac{2h}{H^{2}_{rms} } \right\rgroup \exp -\left\lgroup\frac{h}{H_{rms} } \right\rgroup ²These results are also plotted in Figure 3.3.
e. Visually, the Rayleigh distribution is apparently not a good fit. However, this should be checked by undertaking a statistical goodness of fit test (see Chadwick 1989a for details).
| b. | |
| Class interval of wave height (m | No. of waves |
| 0–1 | 1 |
| 1–2 | 7 |
| 2–3 | 9 |
| 3–4 | 1 |
| 4–5 | 3 |
| (h) | f(h) | n ≡ f(h)N∆h |
| 0.5 | 0.13 | 2.8 |
| 1.5 | 0.3 | 6.3 |
| 2.5 | 0.29 | 6.1 |
| 3.5 | 0.18 | 3.8 |
| 4.5 | 0.078 | 1.6 |
| 5.5 | 0.0022 | 0.05 |
