Question 14.1: 1. At a site off a coast, measurements of wave heights have ......
1. At a site off a coast, measurements of wave heights have been made for 15 min durations at 3 h intervals. For a sample, the wave heights are as below. Check whether the data fit the Rayleigh distribution. The number of waves is given within an interval of wave height. The interval is the wave height below which the same is given minus the preceding height in the table:
Table 1
The significant wave height for the sample is 2.5 m. The total number of waves in the sample, N = 100.
2. The highest significant wave on each day has been recorded, to obtain 365 values for a year. Their distribution is given below:
Table 2
Estimate the significant wave height for a return period of 20 years.
Table 1
| H (m) | 0.25 | 0.5 | 0.75 | 1.0 | 1.25 | 1.50 | 1.75 | 2.0 | 2.25 | 2.5 | 2.75 | 3.0 | 3.25 | 3.5 |
| Number of waves | 3 | 7 | 7 | 7 | 12 | 11 | 10 | 8 | 9 | 13 | 7 | 2 | 3 | 1 |
Table 2
| H_{\mathrm{s}} (m) | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | 5.4 |
| Number of waves (>H_{\mathrm{s}}) | 340 | 128 | 84 | 73 | 48 | 37 | 18 | 15 | 6 | 2 | 1 |
1- P(H)=\frac{\text { number of waves exceeding } H}{N+1}, H_{ s }=2.5\, m \text {. }
Table 3
P(H) versus H/H_{\mathrm{s}} is plotted on probability paper for the Rayleigh distribution, which fits the data reasonably well (Fig. 14.25).
2. Table 4
P(H) versus H_{\mathrm{s}} is plotted on probability paper for the log-normal distribution (Fig. 14.26). The interval of measurement is 1 day = 1/365 yr. P( H_{\mathrm{s}} ) for a return period of 20 years is
P(H)=\frac{1}{20 \times 365}=1.37 \times 10^{-4}Extrapolating the straight line on the probability paper, H_{\mathrm{s}} corresponding to P(H) = 1.37 × 10^{-4} is the value H_{ s } \simeq 9 \,m .
| H/H_{s} | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 |
| Number of waves exceeding |
||||||||||||||
| H | 97 | 90 | 83 | 76 | 64 | 53 | 43 | 35 | 26 | 13 | 6 | 4 | 1 | 0 |
| P(H)(%) | 96 | 89 | 82 | 75 | 63 | 52 | 43 | 35 | 26 | 13 | 6 | 4 | 1 |
| H_{s} (m) | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | 5.4 |
| P(H_{s}) | 0.93 | 0.35 | 0.23 | 0.2 | 0.13 | 0.1 | 0.05 | 0.04 | 0.016 | 0.005 | 0.003 |
