GUIDE WAVE ANALYSIS AND FORECASTING - WMO

this situation, the angular spreading factor is 30 per cent
(from Figure 3.3). This means that the height of swell
observed at the shore of Casablanca should be less than
√0.30 x 4.8 or 2.6 m.
The wave dispersion factor at any given forecast time
is the most difficult part of the forecast. A model spectrum
must be chosen, and the mathematical integration of
the frequency spectrum, S(f), should be carried out for
the range of frequencies at each forecast time. In this
example, Equation 4.1 is used in conjunction with the
JONSWAP spectrum (Figure 1.17) to determine the range
of important frequencies corresponding to f p = 1/T p =
0.099, i.e. between 0.7 f p and 2 f p (1/14.4 = 0.069 Hz and
1/5.04 = 0.199 Hz), and PNJ is used to determine the fraction
of energy in given ranges of frequencies (dispersion
factor) arriving at any given time at Casablanca.
To determine the energy values, use the distorted
co-cumulative spectra for wind speeds 10–44 kn as a
function of duration from PNJ (shown in Annex IV).
The wind speed is 15 m/s (30 kn). Determine where the
upper and lower frequencies intersect the 30 kn curve,
and find the E values that intersect with those points. For
f = 0.069 Hz, E = 51. For f = 0.199, E = 2.5. The difference
is 48.5. To calculate the dispersion factor at 36 h,
find the E values for the periods arriving at that time and
divide by the total energy, i.e. (51 – 31.5)/48.5 = 0.402.
Note that the PNJ curves are used only to find the
proportion of energy in the given frequency ranges; for
the significant wave height it is preferable to use the GD
curves (Figure 4.1).
The wave height is determined by multiplying the
square root of the dispersion factor (i.e. the dispersion
multiplier) times the square root of the angular spreading
factor (angular spreading multiplier) times H c. For 36
hours this is √0.402 x √0.3 x 4.8 = 1.7 m.
At t = 48 h a much larger part of the wave spectrum
is still present with periods shifted to lower values
(wave frequencies to higher values). The dispersion
factor is (51 – 12)/48.5 = 0.804, and the wave height
WAVE FORECASTING BY MANUAL METHODS 51
TABLE 4.9
Forecast of swell periods, wavelengths and heights at Casablanca for various times after the arrival of the longest period waves.
Hc and angular spreading and dispersion multipliers are also shown
(The bracketed values in the last two columns are based on the “uniform distribution” approximation.)
Arrival Periods Wave- Hc Angular Dispersion Arriving
time lengths spreading multiplier wave heights
(h) (s) (m) (m) multiplier (m)
30 14.4 – 13.2 323 – 272 4.8 0.55 0.35 (0.23) 0.9 (0.6)
36 14.4 – 11.0 323 – 189 4.8 0.55 0.63 (0.41) 1.7 (1.1)
42 14.4 – 9.4 323 – 138 4.8 0.55 0.75 (0.54) 2.0 (1.4)
48 14.4 – 8.2 323 – 105 4.8 0.55 0.90 (0.64) 2.4 (1.7)
54 13.2 – 7.3 272 – 83 4.8 0.55 0.84 (0.70) 2.2 (1.8)
60 11.0 – 6.6 189 – 68 4.8 0.55 0.73 (0.69) 1.9 (1.8)
66 9.4 – 6.0 138 – 56 4.8 0.55 0.62 (0.68) 1.6 (1.8)
72 8.2 – 5.5 105 – 47 4.8 0.55 0.40 (0.68) 1.1 (1.8)
is given by √0.804 x √0.3 x 4.8 = 2.4 m. The various
wave heights computed in this way are given in
Table 4.9.
A short cut: A rough approximation to the dispersion
factor may be obtained by simply assuming that the
fraction of the energy we are seeking is dependent only
on the proportion of the frequency range we are considering,
i.e. the ratio of the frequency range to the total
significant frequency range. For this example, the significant
frequency range is (0.199 – 0.069) = 0.13 Hz. For
the first row in Table 4.9 the frequency range is (0.076 –
0.069) = 0.007 Hz, which leads to a dispersion multiplier
of √(0.007/0.13) = 0.23. The appropriate estimates have
been entered in Table 4.9 in brackets.
It must be emphasised that this is a rough approximation.
It assumes that the energy distribution is
uniform across the frequencies, which we know to be
wrong (see Figure 1.17). As the worked example in
Table 4.9 shows, the errors may be acceptable if a rapid
calculation is required, although they are significant. The
peak of the storm will inevitably be underestimated and
the decay will also characteristically be too gradual.
Using these observations some subjective corrections
can be developed with experience.
This example has shown the common features of
swell development as a function of time: a general
increase in wave height at first and, over a long period
of time, a more or less constant height as the spectrum
reaches its greatest width. Since factors such as internal
friction and air resistance have not been taken into
account, it is likely that the swell would have died out
after about 60 h.
4.4.4 Further examples
Problem:
Assuming that swell originates from a small fetch area
of hurricane winds in a tropical cyclone, wind waves
have a characteristic height of 12 m over a width of