# The waves and the wind. Wave height statistical forecasting

Predicting the height of the waves depending on the wind strength

Calculator for the forecast of wave height. It evaluates the height of significant waves, that is, giving a statistical forecast. For people interested, there are some arguments and formulas below the calculator.

#### The waves and the wind. Wave height statistical forecasting

Digits after the decimal point: 1
Height of significant waves

Limitations

In the article The waves and the wind. Calculation of wave characteristics, I reviewed how you can try to estimate its speed with the wave period. I also mentioned fully developed sea term - i.e., the wave that achieved its maximum properties with the current wind. Such a wave is in equilibrium with energy - as much energy is taken from the wind, as much is spent.

We have to remember that not every wave reaches such a state, as it's required for the wind to constantly blow over the entire surface, which the wave passes for some time. And the stronger the wind, the more time and distance are required to form such a wave. But certainly, when it has developed, its phase velocity will catch up with the wind speed.

It's time to tell you about forecasting wave height depending on the wind speed. This is needed, for example, when designing offshore and onshore facilities. Despite this, I could not find the information, on the calculation of the wave height depending on the wind strength, on the Internet for a long time. Finally, I have found some information in the "Shore Protection Manual" dating back to 1984. Later I have found a more recent book, "Coastal Engineering Manual," dated 2008. So the information and formulas below are from this book.

#### Statistical wave forecasting

The development of the wave theory began a long time ago, back in the late nineteenth century. But there was no direct study and model development to predict the waves' behavior before the Second World War. During and after the war, a base of observations that allowed us to start developing empirical models has been gathered.

The main tenet of the method of empirical forecasting is the assertion that the relationship between the dimensionless parameters of the waves obey universal laws (and all the models, in general, are trying to pick up the coefficients for the relations between the parameters so that they are close enough to conform to the parameters obtained as a result of actual observations).

One of the main laws is fetch-growth law. This law states that at constant wind speed and direction over a fixed distance (fetch), it can be expected that the waves will reach a stationary state, depending on the length of the acceleration (fetch-limited state of development). In such a situation, the wave height will be constant (in a statistical sense) over time but will vary along with the acceleration.

Acceleration - is a term used in Russian literature. The length of the body of water, where the wind affects the sea's surface in a constant direction, is understood as the acceleration of wind. I found this term here.

It would seem that with an increase of time and length of the wind acceleration, the wave can grow indefinitely, but that doesn't occur.

In the 1950s, researchers have found that the formation of waves is best described by the wave spectrum (distribution of wave energy depending on frequency) and the transfer of energy from the wind to the wave (there is some info about it in the first article). As mentioned above, the wave ceases to grow, reaching a steady-state energy balance, and it becomes fully developed sea.

The empirical relation for the height of the fully formed waves, which can serve as the upper limit of the wave height assessment for any wind speed, has been derived.

$H_f=\frac{\lambda_5 u^2}{g}$

where, $H_f$ - height of the fully formed wave
$\lambda_5$ - dimensionless coefficient approximately equal to 0.27
$u$ - wind speed
$g$ - acceleration of gravity.

Everything got more complicated. A large array of measurements, in particular in the research project on the North Atlantic JONSWAP (Joint North Sea Wave Project), was assembled. At the place of the first generation's wave prediction models came the second-generation model using the energy spectrum. In the early 1980s, there were wave models of the third-generation (3G). Actually, we hadn't reached the fourth-generation models yet. The most commonly used model is the third generation WAM model (Hasselmann, S., et al., WAMDI Group, The WAM model ‐ A third-generation ocean wave prediction model, J. Phys. Oceanogr., 18, 1775–1810, 1988.)

Of course, there are still shortcomings. For example, these models can not predict the waves in rapidly changing wind situations, but still, 3G models provide a good result.

In the pre-computer era, you could use a model built on the nomogram for wave heights forecasting in relatively simple situations, such as pre-assessment or small projects given, for example, in Shore Protection Manual.

There are 3 situations possible when the simplified prediction will give quite an exact estimation.

1. The wind is blowing in a constant direction over some distance and not limited by time (enough time) - then the growth of the wave is determined and limited by the length of acceleration (fetch-limited).
2. The wind rapidly increases within a short period of time and is not limited by distance (enough distance) - then the growth of the wave is determined and limited by elapsed time (duration-limited). This occurs very rarely in nature.
3. The wind is blowing in a constant direction at a sufficient distance and for a sufficient time so the wave will be fully formed (fully developed wave) under these conditions. Note that even in the open ocean, waves rarely reach the limit values at wind speeds greater than 50 knots.
Empirically, we obtained the following dependence when the length of the acceleration limits wave growth.

The time waves require under the wind influence at the velocity $u$ on the distance $X$ to achieve the maximum possible for a given distance height.
$t_{x,u}=77.23\frac{X^{0.67}}{u^{0.34}g^{0.33}}$

The relationship between the significant wave height $H_m_0$ and the distance $X$
$\frac{gH_{m_0}}{ u^2_f }=4.13*10^{-2}*(\frac{gX}{u^2_f})^{\frac{1}{2}}$

The relationship between the period of the wave $T_p$ and the distance $X$
$\frac{gT_p}{ u_f }=0.751*(\frac{gX}{u^2_f})^{\frac{1}{3}}$

The drag coefficient
$C_D=\frac{u^2_f}{U^2_{10}}$
$C_D=0.001(1.1+0.035U_{10})$

For a fully developed waves
$\frac{gH_{m_0}}{ u^2_f }=2.115*10^2$
$\frac{gT_p}{ u_f }=2.398*10^2$

Also the transition from the duration of the wind to the length of the acceleration (i.e. the wind impact for some time can be replaced by the wind at a distance)
$\frac{gX}{ u^2_f }=5.23*10^{-3}*(\frac{gt}{u_f})^{\frac{3}{2}}$

where
$U_{10}$ - wind speed at 10 meters height
$u_f$ - friction velocity.

Thus, if the duration of action and length of the wind's acceleration is known, it is necessary to select the most restrictive value. If the wave generation height is limited by time, it is necessary to replace it with an equivalent distance.

In case of shallow water equations remain valid except for the additional limitations under which the wave period can not exceed the following ratios
$T_p\approx9.78(\frac{d}{g})^\frac{1}{2}$,
where d - depth

Then the order of the wave height prediction for the shallow water is as follows:

1. Assess the wave period for a given distance and wind speed using the conventional formula.
2. In the case of shallow water, verify the conditions of the period and depth. If they are exceeded, take the boundary value.
3. In the case of the wave boundary value, find the distance corresponding to the generation of waves with such a period.
4. Calculate the height by the value of the distance.
5. If the wave height exceeds 0.6 of the depth values, limit this value.

Some more important notes

1. These empirical formulas are derived for relatively normal weather conditions and are not applicable for the assessment of the wave height in the event of, for example, a hurricane. Nomograms contained in the directory are built for the wind speed no higher than 37.5 m/s. For comparison, 33-42 m/s wind speed - the first category hurricane on the Saffir-Simpson scale.

2. These empirical formulas are used for statistical forecasting of wave heights, so the height of these formulas is nothing more than a significant wave height determined by the dispersion of the wave spectrum as follows: $H_{m_0}=4\sqrt{M_0}$.

This is a more modern definition of significant height of the waves, and the very first definition, which was given to Walter Munk during World War II, was: "The average height of the one-third of the highest waves." It was assumed that it mathematically expresses the estimate of wave heights, which a "trained observer usually gives." There is a few percent difference between these two definitions. The older definition commonly referred to $H_{1/3}$.

Thus, getting an assessment of significant wave height for the given conditions, it is necessary to realize that most waves (about 2/3) are below this height, BUT we can find waves that are larger than this height. It is believed that the Rayleigh distribution well approximates the statistical distribution of the wave height, so if we estimate 10-meter height, it can be expected that one of the 10 waves is greater than 10.7 meters, one of 100 waves is greater than 15.1 meters, one of 1000 waves is more than meters 18.6. More info.

In reality, due to the constantly changing conditions, a nearly two-fold excess is, of course, rare, but sometimes happens Rogue wave.

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