UNIDAD RESPONSABLE: DIR.GRAL ADMINISTRACION DE INFRACCIONES

Regular waves or monochromatic waves may be written mathematically in terms of a single amplitude A and frequency 𝜔 as

𝜂(𝑥, 𝑧, 𝑡) =𝐻

2 𝑠𝑖𝑛(𝜔𝑡 − 𝑘𝑥 + 𝜙) = 𝐴 𝑠𝑖𝑛(𝜔𝑡 − 𝑘𝑥 + 𝜙) (2.30)

Needless to say, regular waves bear little resemblance to those in the sea which are comprised of multiple waves interacting from different directions. The figure overleaf gives

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an example of how it possible to construct a more realistic wave by the simple superposition of component regular waves.

Figure 2:6: Example wave surface elevation, which is constructed by adding 4 regular waves of different height and period

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Regular waves, however, offer a useful starting point for physical modelling as they allow for the observational response to very specific input conditions. This is crucial when one chooses to regard certain ‘real’ waves as a superposition of regular waves at different frequencies because, by linearity, the response of the real wave is then the sum of the individual responses of its constituent regular waves.

2.2.1 Regular Wave Reflection

If a monochromatic incident wave meets a plane vertical barrier then perfect reflection is said to occur. The interaction of the incident and reflected wave in this scenario sets up a standing wave with an antinode at the barrier and at half wavelength intervals thereafter, with nodes in between as depicted in Figure 2:7.

Wave record analys

Figure 2:7: Standing wave (clapotis) system, perfect reflection from a vertical barrier (Coastal engineering research centre, 1984)

The following is the theory for partial standing waves as given by Ippen (1966).

Assuming a linear incident wave

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and a linear reflected wave

𝜂𝑟 = 𝑎𝑟sin (𝑘𝑥 + 𝜔𝑡 + φ𝑟) (2.32)

Boccotti (2000), writes that one cannot exclude some phase angle between the reflected and the incident waves and this is why the phase angle φ𝑟 is an unknown.

One obtains by superposition of the incident and reflected wave trains

𝑛𝑡𝑜𝑡𝑎𝑙= 𝜂𝑖+ 𝜂𝑟 = 𝜂𝑖 = 𝑎𝑖sin(𝑘𝑥 − 𝜔𝑡) + 𝑎𝑟sin (𝑘𝑥 + 𝜔𝑡 + φ𝑟) (2.33)

Assuming that the barrier is at x = 0 and that φ𝑟 = phase shift =  radians, then becomes

𝜂𝑡𝑜𝑡𝑎𝑙 = 𝑎𝑖sin(𝑘𝑥 − 𝜔𝑡) − 𝑎𝑟sin (𝑘𝑥 + 𝜔𝑡) (2.34)

Adding 𝑎𝑟sin (𝑘𝑥 + 𝜔𝑡) to both components of (2.34) in opposite directions we get

𝜂𝑡𝑜𝑡𝑎𝑙= (𝑎𝑖+𝑎𝑟)sin(𝑘𝑥 − 𝜔𝑡) − 𝑎𝑟[sin(𝑘𝑥 − 𝜔𝑡) + sin(𝑘𝑥 + 𝜔𝑡)] (2.35)

The first component of the above equation describes a progressive wave of amplitude (𝑎𝑖+ 𝑎𝑟) while the second component is a standing wave of amplitude 𝑎𝑟. The above can

be re-written as 𝜂𝑡𝑜𝑡𝑎𝑙 = (𝑎𝑖+𝑎𝑟)sin(𝑘𝑥)cos (𝜔𝑡) − (𝑎𝑖+𝑎𝑟)cos(𝑘𝑥)sin (𝜔𝑡) (2.36) Evaluating 𝜕𝜂𝑡𝑜𝑡𝑎𝑙 𝜕𝑡 |𝑥 = 0 (2.37)

for any x,, i.e. at the nodes visible in Figure 2:7; gives

[𝜔𝑡]_𝜂 𝑡𝑜𝑡𝑎𝑙𝑚𝑎𝑥 = 𝑡𝑎𝑛 −1_[−𝑎𝑖+ 𝑎𝑟 𝑎𝑖+ 𝑎𝑟 cot (𝑘𝑥)] (2.38) With [𝑘𝑥]𝜂_{𝑡𝑜𝑡𝑎𝑙𝑚𝑎𝑥} = 𝑛𝜋 𝑎𝑛𝑑 (2𝑛 + 1) 𝜋 2⁄ for 𝑛 = 0,1,2, … (2.39)

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Substituting these latter two expressions into (2.39) gives the following maximum and minimum amplitudes of the partial standing wave:

𝑎𝑚𝑎𝑥= 𝑎𝑖+ 𝑎𝑟 (2.40) 𝑎𝑚𝑖𝑛 = 𝑎𝑖− 𝑎𝑟 (2.41) Therefore 𝐻𝑚𝑎𝑥= 2(𝑎𝑖+ 𝑎𝑟) (2.42) 𝐻𝑚𝑖𝑛= 2(𝑎𝑖− 𝑎𝑟) (2.43) With 𝐻𝑚𝑎𝑥+ 𝐻𝑚𝑖𝑛 2 = 𝐻𝑖 (2.44) 𝐻𝑚𝑎𝑥− 𝐻𝑚𝑖𝑛 2 = 𝐻𝑟 (2.45) Then 𝐻𝑚𝑎𝑥− 𝐻𝑚𝑖𝑛 𝐻𝑚𝑎𝑥+ 𝐻𝑚𝑖𝑛 =𝐻𝑟 𝐻𝑖 = 𝑘𝑟 (2.46)

Which is known as Healy’s formula.

The pure standing wave profile occurs when the incoming wave train hits a breakwater orthogonally; however, this setup is rarely the case in basin facilities, except for specific coastal engineering studies. In practice wave reflections in a flume or basin are undesirable and measures and put in place to reduce them. Even though reflection in a basin can be reduced, they are still an ever present feature. This results in a partial standing wave profile which is discussed in more detail in section 3.7.

Ouellet & Datta (1986) state that after the wave generator, wave absorber is the most important mechanism in a wave flume or basin. Wave absorbers can be broadly classified into two main categories: active and passive wave absorbers. In their survey of wave generating facilities, the authors mention for passive absorbers, the beach of constant slope seems to be the most popular arrangement. Additionally other materials such as transversal bars, horsehair and wire screen are used in conjunction. The authors suggest that; reflections

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of up to 10% are to be expected even for well-designed beaches and that the % reflection tends to increase with reduced wave height. It does not appear possible to attain reflection coefficients below 10% for absorbers shorter than 0.5 to 0.75 of a wavelength. It is also reported that parabola shaped beaches generally exhibit lower reflection coefficients.

Goda & Ippen (1963) theoretically analysed and tested wave absorbers composed of vertical mesh screens aligned normal to the direction of wave propagation. Four different screen absorbers where tested under deepwater wave conditions, and results compared favourably with theory. Reflection was shown to be dependent upon screen spacing, but not so dependent on the number of screens "provided the number is fairly large." They also stated that the screen absorber must be at least as long as the wavelength of the incident wave. Hughes (1993) summarises an extensive experimental program to develop an efficient wave absorber made of wire mesh screens carried out by Jamieson and Mansard (1987);

• The frontal area of the supporting framework should be minimal.

• High porosity mesh screens work best for absorbing energy from high

steepness waves.

• Low porosity screens work best for absorbing energy from low steepness

waves.

• Mesh screen porosity should decrease toward the rear of the wave absorber.

• Screen locations should be selected to correspond approximately to node

locations of the partial standing waves.

• Wider screen spacing is required for steeper waves and longer wave periods,

with sheet spacing progressively decreasing toward the rear of the wave

absorber.

Active wave absorbers can be position or force feedback based. Milgram (1970) developed a position feedback controller which uses the signal from upstream wave gauges to estimate the reflected wave travelling towards to wavemaker. The driving signal to the wavemaker can then be adjusted based on the information attainable from the wave probes. The author reported that there was a significant level of error between the actual wave field generated compared to the desired result. Force feedback allows for direct measurement of the wave field at the wavemaker’s surface. Salter (1981) noted that basin reflection made experimental measurements difficult (with amplitude variations of up to 30%) To address

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this he developed a force feedback absorption system in the wide tank. This allowed the incident wave field to be determined and proved an effective way of cancelling out any reflected waves from the wavemaker surface. Active feedback systems like this eliminate the need for the wave board transfer function, and they also help to minimize second-order effects that arise when driving the wave board with a first-order signal. HMRCs basin at Pouladuff utilised such an active absorber developed by Edinburgh Designs (Bolton King & Rogers, 1997).

Several publications exist that explain different experimental techniques for measuring wave reflection. A good summary of these publications is presented by (Isaacson, 1991). The author compares three different reflection analysis methods. A two probe method, using one phase angle developed by Goda & Suzuki (1976), a three probe method using two phase angles by Funke & Mansard (1980) and three probe method developed by Isaacson himself. The author owns method was found to be the least accurate as it does not utilise the phase angles between the probes. The three probe method by Funke and Mansard was shown to give the best results, hence this method is utilised in this work.

In document JURISDICCION MINISTERIO DE JUSTICIA Y SEGURIDAD (página 82-88)