Wave energy (ocean waves)

Wave Dragon

Wave power is the power of waves work to do - destructive work in the sinking or damaging of ships or devastation on the coast, but also useful work in wave power plants .

In order to estimate this wave energy for certain regions of the sea (e.g. the planned location of a wave power plant), a wave theory is required, which summarizes the shape and the play of forces of the waves in mathematical formulas. Among various approaches, the linear theory (Airy theory) is predominantly used today, which is also the basis for the following considerations.

Requirements and assumptions for estimating the wave energy

The linear wave theory is not presented here as such, but only the conditions under which it depicts the wave activity well and can then also be used to estimate the wave energy. The most important requirements are:

• The wave height is much smaller than the wave length (distance between two successive wave crests).
• The wave height is much smaller than the water depth.
• The wave height is independent of the wave length and period .

The empirical data (measurements on the seas with measuring buoys) confirm the existence of the first requirement: by far the most common are waves with a height of 0.5 to 2 m. Most wavelengths are over 10 m, and wavelengths of 100 m and more are not uncommon. The second condition is also met in most cases, because the water depth is known to be mostly far greater than the wave heights mentioned above.

However, the following statements do not apply to shallow water zones (water depths of a few meters). The third condition is only approximately fulfilled, because the empirical data show at least a stochastic relationship ( correlation ) between wave height and wave length, i.e. H. high waves have a certain probability that they have a longer wavelength than low waves.

Calculation of the wave energy

For practical purposes, it is less the wave energy as such that is relevant, but the amount of energy that can be converted per period of time . This is also known as the energy flow or power . According to the linear wave theory, the flow of energy is in a wave

${\displaystyle P={\frac {\rho g^{2}}{32\pi }}\cdot T\cdot H^{2}\qquad (1)}$

where P = power in kW / m (kilowatts per meter of wave roll or crest), ρ is the density of the sea ​​water , g is the acceleration due to gravity , π is the number of circles = 3.14159…, T = period of the waves in Seconds and H = wave height in meters. The period T ("time") is defined as the length of time from the arrival of a wave crest at a certain point until the arrival of the next wave crest. “Wave roller” is understood to mean the entire body of the wave with the height H and a lateral extent that is viewed across the direction of movement of the wave. Since the waves near the coast are mostly parallel to it, the term “length of the coastline” (towards which the wave rolls) is used instead of “length of the wave cylinder”. Also with the term “length of the crest of the wave” meant the same thing. So much for the theory. For practical use - e.g. B. to estimate the energy flow of the waves in a certain sea or coastal area - you have to insert empirically determined mean values ​​into the above formula. These originate from B. from measuring buoys, which are installed all over the seas and continuously register the wave height and the period. Therefore one can also base the calculations on long-term mean values, such as B. made available on the Internet by the Royal Meteorological Institute of the Netherlands (see web links). The wave height is output as a " significant wave height " , which is the mean value of a third of the highest waves in a measurement period. If one wants to insert this value into the formula (1), the relation applies: ${\displaystyle (\mathrm {g} /\mathrm {cm} ^{3})}$ ${\displaystyle (\mathrm {m} /\mathrm {s} ^{2})}$ ${\displaystyle H_{s}}$

${\displaystyle H=H_{s}/{\sqrt {2}}\qquad (2)}$

inserted into the above formula (1) results

${\displaystyle P={\frac {\rho g^{2}}{64\pi }}\cdot T\cdot H_{s}^{2}\qquad (3)}$

This formula is therefore suitable for practical application wherever measurement results for the significant wave height and period are available; in addition one needs the values ​​g and ρ. These last two values ​​hardly change for a specific geographic location and can therefore be regarded as constants. If their numerical value for the respective location is not known, average values ​​of or are used . The only variables left in (1) or (3) are T and H or ; all other quantities can be combined to form a constant . You then get ${\displaystyle 9{,}81\,\mathrm {m} /\mathrm {s} ^{2}}$ ${\displaystyle 1025\,\mathrm {kg} /\mathrm {m} ^{3}}$ ${\displaystyle H_{s}}$

${\displaystyle P\approx 0{,}490605\cdot T\cdot H_{s}^{2}\approx 0{,}5\cdot T\cdot H_{s}^{2}\qquad (4)}$

When a wave power plant z. If, for example, a section of the wave rollers of 3 m is recorded, the significant wave height is 2 m and the waves follow one another at a distance of 10 seconds, the result is an energy supply of 0.5 by 10 seconds by 2 to the power of 2 m by 3 m Width of the power plant = 60 kW. If the efficiency of the wave power plant is 40%, it produces 60 kW times 0.4 = 24 kW, i.e. 24 kWh in one hour and 576 kWh in one day (24 hours).

Speed ​​of sea waves

The above formula for the wave energy appears paradoxical in that it states that a wave series with a long period delivers more energy than a wave series of the same wave height but short period. But that means nothing else than that the wave series, whose waves arrive at a wave power plant faster one after the other, delivers less energy than a wave series with the same wave height, but less frequently arriving waves. The puzzle solves considering the fact that ocean waves do not have a uniform speed; they reproduce at different speeds, which is commonly referred to as dispersion . It follows that waves with a long period are faster than those with a short period. This can be seen in the following transformation of formula (3).

In linear wave theory, the following applies in general ${\displaystyle c={\frac {L}{T}}\qquad (7)}$

with the speed c (m / s) and the wavelength L (m).

Furthermore applies to deep water waves ${\displaystyle L={\frac {g}{2\pi }}T^{2}\qquad (8)}$

With this one can express the energy flow as a function of c and H _s :

${\displaystyle P={\frac {\rho g}{32}}\cdot c\cdot H_{s}^{2}\qquad (9)}$

This means that if the period is doubled, the wavelength becomes four times as large, but because of its double speed of propagation, half as many waves per hour arrive at a wave power plant, but these waves transport twice the amount of energy - per wave quadruple energy.

Formula (9) breaks down the wave energy into the following components: With some components of the constants, it represents the area of ​​a cross section of the wave roller. Multiplied by the length of the shaft roller (and other components of the constants) this becomes the mass of the shaft. If you consider half the wave height as the height of the fall, the potential energy of the wave results . Finally, the quantity c together with the mass represents the kinetic energy of the wave. It should be noted that with c the propagation speed of the wave is meant and no transport of water masses, such as the z. B. would be the case after a dam burst when a tidal wave rolls through a valley. The water movements in the wave are not linear-horizontal, but circular in vertical planes (see water wave ). ${\displaystyle H_{s}^{2}}$

Web links

• Royal Dutch Meteorological Institute

Individual evidence

1. ^ Graw, Kai-Uwe: Wave energy - a hydromechanical analysis. Wuppertal 1995, pp. 5–8 on the Internet (PDF; 37.9 MB)
2. ^ Parsons, Jeffrey: Linear (airy) Wave Theory. Washington 2004 On the Internet ( Memento from June 29, 2010 on the Internet Archive )
3. ↑ Graw, pp. 5-8
4. ^ Graw, Formula 5.15
5. ^ Graw, Formula No. 4.14
6. ^ Graw, Table 4.2