Wave length, wave heigth, periode time and absorption width

by Povl-Otto Nissen

During the development of the two wave energy converters, WaveRocker (Da: Boelgevippe) and WaveSpinner (Da: BoelgeRok), it has been of interest to find the ideal diameter of the float in relation to the wave length. But the wave lengths do not appear in the tabels of the papers, published in connection with the Danish Wave Power Programme, (2), (3) og (4). Only the wave heights and the corresponding periodes of time occur. But it gives information enough to calculate the wave lengths with the help of some mathematic.
Important information can be found in the book ”Ocean Waves and Oscillating Systems” (Cambridge University Press 2002) by the Norwegian prof. Johannes Falnes (1). You need to focus on the so called dispersion relation, which is characteristic for oscillations in the water surface under the influence of gravity.
The dispersion relation (Falnes, p. 45) follows here:

where g is the acceleration caused by gravity: 9,82 m/s2

The symbol ω means the arc frequency and is 2π divided by the wave periode time T. It is the part per second of an arc of a circle, which a wave particle is running through during the movement of one wave. The time for that is the periode time T

The symbol k is the wave number, which is 2π divided by the wave length L.
Put together in the dispersion relation, you have

where L isolated gives:

This equation can be used to calculate the wave length L, when the wave periode time T is known.

The calculated wave lengths are shown in the following tabel. The calculations are made on the basis of the standard values of wave heights and periode times, which represent the five sea conditions, used in tests of wave energy converters within the Danish Wave Power Programme.

Sea conditions for standard test of wave energy converters
Hs [m] Tp [sek] Lp [m] Tz [sek] Lm [m] Pinf [kW/m]
1 5,6 49 4 25 2,3
2 7,0 76 5 39 11,5
3 8,4 110 6 56 31,2
4 9,8 150 7 76 64,6
5 11,2 195 8 100 115,4

Hs is the significant wave height from wave trougt to wave top.
Tp is the peak wave periode time.
Lp is the corresponding peak wave lenght.
Tz is the middle wave periode time.
Lm is the corresponding middle wave length.
Pinf is the power per metres wave front, facing the absorber of the wave converter.
The calculation formular is Pinf = 0,577*Hs2*Tz

A thumb rule says, that the middle wave lengths are approximatly 20 times the significant wave heigths.
To what extend, that is the case, can be found by compairing the numbers in the tabel.
As you see from the following curve, it is not quite linear:

This curve can be used to read the peak wave lengths, Lp, on the vertical axis as a function of the peak periode times, Tp, on the horizontal axis.
It can be used to read the middle wave lengths, Lm, as a function of the middle periode times, Tz, of the waves as well.

The float diameter compaired to the absorption width
When you are working with float based wave energy converters, a natural question is:
How big is it profitable to build the float?
Is there an ideal size of a float compaired to the wave length?

It would be reasonable to think, that there must be a maximum limit, be cause if it is too big, it will "shortcut" the wave form.

Falnes delivers in his book (p. 216) the proof, that the maximum absorped power only can be gained with floats, where the float diameter in the water surface is maximum the wave length, L, divided by 2π. That is D = L/2π, which also can be expressed D = 0,159 L, approximately a sixth part of the wave length. Falnes calls this the Absorption Width.

If he is right, it would not be useful and economical to build a float bigger than that. But it is not a fact, that it is ideal to go to this limit. Maybe it is even a good idea to make it smaller. Because the power absorption will also be influenced by the form, volume and weight of the float. Besides the float/machine systems ability to find a state of resonans with the waves, will be of outmost importance.

It would be reasonable to use the middle wave length as basis for calculating the dimensions of the float. Talking about diameter is only relevant, when the float is circular. The absorption width, D, is only relevant as a section of the wave length in the direction of the wave movements. It should give no limitations on dimension of the float in the directions parallel to the wave front.

Next tabel contains the absorption widths calculated on the basis of the middle wave lengths:

Wave lengths and absorption widths
Hs [m] Tp [sek] Lp [m] Tz [sek] Lm [m] D=Lm/2π [m]
1 5,6 49 4 25 4,0
2 7,0 76 5 39 6,2
3 8,4 110 6 56 8,9
4 9,8 150 7 76 12,1
5 11,2 195 8 100 15,9

As you can see, the size of the absorption widths variate strongly over the five standard sea conditions, in a way that makes it impossible to cover all the five sea conditions with one single float. The size of a float must therefore be chosen in accordance to the most common sea conditions at the actual position of the wave energy converter.
A good help to that is Energy Center Denmark's publikation Fakta om Bølgeenergi (2)and also the IEA Annex II report (6), which now is international recommended practises for testing and evaluating of ocean energy systems.

1. Falnes, Johannes, “Ocean Waves and Oscillating Systems”, Cambridge University Press 2002.
2. Energicenter Danmark, “Fakta om Bølgeenergi”, København 2002.
3. Bølgekraftudvalgets Sekretariat, ”Bølgekraft – forslag til forsøg og rapportering”, marts 1999.
4. Bølgekraftudvalgets Sekretariat, ”Bølgekraftprogram – Forslag til systematik …”, januar 2000.
5. Bølgekraftudvalgets Sekretariat, ”Bølgekraftprogram – Afsluttende rapport…”, august 2002.
6. IEA, Ocean Energy Systems, Annex II Report 2003.

The content of this paper is also inspired by reading many reports about wave energy converters, tested at the Institut of Water, Earth and Enviromental Technics at the University of Aalborg.
These reports are available via the Danish Wave Power Association.

Rev. 07-06-2007 and translated from Danish

Povl-Otto Nissen
Povlonis Innovation Project
Tangevej 47 A
6760 Ribe

web: www.povlonis.dk