# The “Wave Bridge” for bypassing oceanic wave momentum

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## Abstract

Here, we introduce and investigate the concept of the Wave Bridge that can bypass the momentum of oceanic waves about ocean objects. The Wave Bridge is composed of a wave energy absorber on the upstream side of an ocean object, and a wave maker on its downstream side. The wave absorber and the wave maker are mechanically connected in such a way that the wave energy absorbed on the upstream side is simultaneously used by the wave maker downstream of the ocean object to generate waves. The Wave Bridge therefore protects the ocean object from waves by transferring incident wave energy from the upstream to the downstream. Furthermore, since the wave absorbed upstream is the same as the one generated downstream, the corresponding horizontal forces are equal in magnitude and opposite in sign and hence cancel each other, resulting in a zero net horizontal force on the Wave Bridge and its supporting structure. Our experimental results show a wave protection efficiency of up to 97 % and a horizontal force protection efficiency of up to 80 %. We also investigate the effect of the finite height of the Wave Bridge and the resulting wave energy leakage underneath the plungers on the overall protection efficiency. The Wave Bridge and its variants may reduce the costs of offshore structures by reducing the wave loads, provide calm water in the midst of an energetic ocean for future offshore cities, and conserve energy of dynamic position systems by reducing the wave-induced disturbances of vessels.

## Keywords

Experimental hydrodynamics Wave load protection Offshore structures High-efficiency wave energy absorption Wave maker## 1 Introduction

Specifically in our Wave Bridge design, we use wedge-shaped plunger-type wave energy harvester and wave maker that, if properly made for a specific ocean environment, can theoretically have an efficiency of unity (i.e., absorb the entire energy of incoming waves). In this case, the entire wave energy and wave momentum can be bypassed about the ocean structure. Thus, the protected ocean structure does not experience any waves or any wave-induced horizontal forces. A plunger-type wedge-shaped wave maker (and an equal wave absorber) is an asymmetric two-dimensional wedge-shaped float with one vertical side and one sloped side. As this plunger moves in the vertical direction, it generates waves on the sloped side, while not much waves will be excited on the flat side. These devices are typically surface piercing and their vertical extent may or may not reach the bottom. In this sense, they are almost the opposite of surge-type converters that are hinged at the bottom and, depending on their designs, may or may not reach the free surface.

Plunger-type wedge-shaped wave makers and wave energy converters have been subjects of extensive studies since decades ago. A finite depth wedge in an infinite depth water was studied analytically and experimentally by Wang (1974). For the case of a finite depth water and for a wedge that has a flat vertical side on one side and a sloped surface on the other, Wu (1988) investigated the radiated waves to the right numerically using a boundary collocation method (BCM), where the no-flow boundary condition on the flat side of the plunger is assumed to extend to the seabed. For the case of a finite height plunger Wu (1991) used a boundary element method to determine radiated waves on both sides of the plunger. These numerical results agree well with our investigations (Ellix and Arumugam 1984; Patel and Ionnaou 1980; Henderson et al. 2006). Plunger-type wedge-shaped floaters have also been proposed for energy extraction (e.g. Hager et al. 2012; Count 1980; Evans 1976; Madhi et al. 2014) and as a break water (Hales 1981; He et al. 2012; Dong et al. 2008).

In this paper, we experimentally investigated both the wave protection efficiency and the horizontal momentum bypassing the efficiency of the Wave Bridge in a two-dimensional case with monochromatic, linear waves. We show via wave tank tests that a wave protection efficiency of up to 97 % and a horizontal force bypassing efficiency of up to 80 % can be obtained. Both maximum efficiencies are obtained when the frequency of the incident wave is close to the natural frequency of the Wave Bridge, but the bandwidth of acceptable perforation is relatively broad. If the Wave Bridge is deployed in deep water, the plungers will not extend to the seabed. Hence, wave energy will partly leak underneath the plungers in the protected zone resulting in waves in the protection zone. Numerical investigation, based on the boundary collocation method, are performed to address the leakage issue and therefore help to determine the theoretical efficiency of the Wave Bridge to create a protected zone.

Incident wave force, particularly its horizontal component, clearly increases the structural costs of offshore structures. The presented Wave Bridge provides a simple design to significantly reduce such forces on offshore structures. This is more highlighted in areas with poor foundation and in deep waters. Wave Bridge may also help conserving energy of dynamic positioning system by reducing the wave-induced motion of a vessel. The efficiency and economic advantages of the Wave Bridge idea is more pronounced if a single Wave Bridge is deployed about a group of offshore structures, say, about an offshore wind farm. Wave Bridge is a broadband wave protection mechanism that can theoretically be used both in shallow and deep waters.

## 2 Experimental details

### 2.1 Wave tank and wave maker

### 2.2 Wave Bridge

The main frame of the Wave Bridge supports the forces from the bearings and the hinge and keeps the structure in position. Standard T-slotted aluminum profiles were used to build the main frame. The main frame occupies a space of 1.22 m \(\times \) 1.25 m \(\times \) 0.45 m, which fits into the wave tank. Four linear, closed bearings (e) with a dynamic load capacity of 1 kN are used to guarantee low friction. They are guided on two 1.27 cm rods (f) and align each plunger. The distance between the vertical sides of the plungers is set to 1.36 m.

*x*-,

*y*- and

*z*-direction).

### 2.3 Wave measurement

*E*per unit length is

*g*is the gravitational acceleration in m/s\(^2\), and

*H*is the wave height in

*m*. The wave energy does not depend on the wave frequency or the wavelength. Thus, the efficiency for wave transmission \(\eta _\mathrm{t}\), reflection \(\eta _\mathrm{r}\), and protection \(\eta _\mathrm{p}\), are defined as

The wave surface height was obtained using an optical measurement system. Images of the waves are recorded at 30 frames per second at three locations along the wave tank. Camera 1, which is placed 5 m upstream from the center of the Wave Bridge, records the incident and reflected waves, while camera 3 records the transmitted wave 5 m downstream, as shown in Fig. 2. The wave parameters between the absorbing and emitting plungers are recorded by camera 2, mounted at the center. All cameras were mounted outside the wave tank, recording the waves through the transparent wave tank panels from a horizontal distance of 40 and 20 cm below mean water level. Snell’s law shows that the wave heights recorded by the cameras are underestimated by about 6.5 % for all of the waves measured in the current study. Yet, by evaluating the Wave Bridge’s performance using wave height ratios, the error due to refraction in the wave tank glass cancels out in the current results. The cameras have a resolution of 1080 pixels \(\times \) 1920 pixels. Each camera covers a view of approximately 70 cm by 40 cm. Additional light sources, shining from below into the wave tank, were used to increase the contrast at the water surface. To transform the wave height from the pixel domain to the physical domain in the laboratory coordinate system, a calibration grid of four black dots with known vertical and horizontal distance is attached to the transparent side panels of the wave tank. The vertical and horizontal numbers of pixels between the dots are related to the physical distances. Each pixel correlates to an area of 0.41 mm \(\times \) 0.41 mm. Each frame of the videos is processed in MATLAB. The Canny edge detection technique (MathWorks, Inc., Boston, MA, USA) was used to extract the history of the free surface profiles. A method, developed by Goda and Suzuki (1976), based on trigonometric considerations, is then used to obtain the incident and reflected wave heights \(H_\mathrm{i}\) and \(H_\mathrm{r}\) from cameras 1 and 3. The wave within the Wave Bridge (camera 2) is not fully formed, since the distance between the plungers is in the range of the wavelength and the method by Goda and Suzuki leads to significant errors. Therefore, conservative approximation is used and the wave height \(H_\mathrm{p}\) is determined as the maximum wave height measured within the Wave Bridge.

### 2.4 Force measurement

Horizontal forces due to absorption are counteracted by the forces of the wave-generating plunger and hence minimal horizontal net forces are expected to act on the system if a high wave transmission is achieved. However, reflection of the waves causes a horizontal force on the plunger and thus the supporting structure must provide the necessary counterforce. To minimize the horizontal net forces on the structure, the wave transmission must be maximized. To measure the resulting horizontal forces, four strain half-bridges, built from two active, 120 \(\Omega \) resistance strain gauges with a gauge factor of \(k=2.03\) were used. For each side of the Wave Bridge, one of these half-bridges was glued onto the top side (*k*) and one on the bottom side of the hollow aluminum profile (*j*) connecting the plungers to the linear bearings. Figure 5b shows the basic pattern of one half-bridge used on one side of the aluminum profile making the half-bridge sensitive to both axial and bending strain. While one of the strain gauges of a half-bridge is mounted in the direction of axial/horizontal strain, the second gauge acts as a Poisson gauge and is mounted perpendicular to the principal axis of the strain.

*Catman Easy*software was used to read the data at a frequency of 200 Hz. Using the gauge factor

*k*and the Poisson’s ratio of \(\nu _\mathrm{Alu}=0.34\) for aluminum, the strain \(\epsilon _\mathrm{i}\) of each half-bridge \(i\in (\mathrm{Em,top;~Em,bot; ~Ab,top;~Ab,bot})\) was calculated by

*H*,

*B*,

*h*and

*b*as the outer and inner dimensions of the hollow aluminum profile.

## 3 Results and discussion

Two types of tests were conducted to experimentally assess the performance of the Wave Bridge. One test included the assessment of the Wave Bridge for a variable frequency ratio of the incident wave frequency to the natural frequency of the Wave Bridge (\(\omega /\omega _0\)), while the wave steepness *ka* was kept in the range of \(0.09<{ka}<0.13\). For the other test series, the frequency ratio was kept constant, while ka of incident waves was varied. For all experiments, a plunger-type wave maker was placed about 17.1 m from the center of the Wave Bridge inside the wave tank with a mean water depth of 0.7 m. Cameras 1 and 3 recorded the surface elevation of the incident, reflected and transmitted waves, while camera 2 recorded the waves between the plungers, which were 48 cm submerged. Although a wave-absorbing beach was used at the end of the wave tank, the limited length of the tank led to wave reflection. Thus, the measurements were stopped when the waves reflected from the back wall returned to the wave-emitting side of the Wave Bridge.

*h*. This leads to a frequency ratio range of 0.66–1.38 for all of the cases tested throughout the current study. To assess the influence of the incident wave frequency on the Wave Bridge performance, the steepness of the incident wave was kept in the range of \(0.09<{ka}<0.13\), ensuring linear waves. Each of the averaged measurement points in Fig. 6 includes three actual measurement results.

*a*/

*h*values.

*ka*is directly proportional to the increase of the incoming wave height, since for these tests the frequency and thus

*k*were kept constant. Therefore, the absolute force on a fixed plunger increases with higher ka values. Yet, the horizontal net force induced on the system in an operating mode stays nearly on an equal level. Waves in the protected zone are partly caused by leakage of incident wave energy underneath and through the side gaps between the plunger and the wave tank. Yet, by designing customized plungers, the side gaps were reduced to a minimum of 0.5 cm each. In the current investigation, the incident wave steepness, ka, is below 0.3 and thus the linear theory is applicable.

## 4 Wave leakage effects

In a real-life implementation, the wave absorber and wave emitter of the Wave Bridge will not reach the sea floor. Waves will partially leak underneath the plungers and decrease the protection efficiency of the Wave Bridge. Here, we present an estimation of percentage of energy leakage for a single plunger as a function of the ratio of the plunger height to the water depth i.e., *d* / *h* (Fig. 9). We would like to emphasize that the goal of this short section is not to present the solution to the wave maker or wave absorber problem, as such solutions are already provided by several prior works, e.g., Wang (1974), Wu (1988, 1991) for a plunger as a wave maker, and Hager et al. (2012), Count (1980), Evans (1976), Madhi et al. (2014) for a plunger as a wave absorber.

*d*in a water of depth

*h*as shown in Fig. 9. Water is assumed to be incompressible and inviscid, and the velocity field is assumed to be irrotational, such that potential flow theory applies. The governing equation and linearized boundary conditions, respectively, at the air–water interface, seabed and on the plunger read

*S*the boundary of the plunger,

*n*normal to the surface

*S*,

*g*the gravitational acceleration, and subscripts of \(\varphi \) indicate partial derivatives. We denote the velocity potential on the right side of the plunger as \(\varphi _\mathrm{r}\), and on the left side as \(\varphi _\mathrm{l}\), i.e.,

*n*) and the propagating wave (index 0):

To numerically solve this problem, we use a variation of the boundary collocation method (see e.g., Wu 1988). Let the entire water column be evenly divided into \(M-1\) sections (\(\delta h=h/(M-1)\)). Thus, there will be *M* nodes for which boundary conditions have to be satisfied, each at the depth of \(z_j=-j ~\delta h\), \(j=0,\ldots ,M-1\). Assume out of these *M* nodes, *q* are along the side of the plunger. In other words, *q* nodes are above \(z=-d\) or \((q-1)\delta h<d\). Obviously, the void area below the plunger has \(p=M-q\) nodes. Each of the Eqs. (4.9) and (4.10) now lead to *q* equations, and each of Eqs. (4.11) and (4.12) lead to *p* equations, making the total number of equations \(2(q+p)\). If *n* terms are chosen in the velocity potential expansion, the total number of unknowns in the series expansion (4.6) and (4.7) is \(2(n+1)\). Therefore, \(n+1\le M\), where equality gives a full rank equation and greater gives an overdetermined equation. To reduce errors in matrix operations, an overdetermined system is solved using least squares method.

The boundary collocation method is now implemented on an oscillating plunger to numerically estimate the amount of wave energy that leaks beneath the plungers of the Wave Bridge leading to the appearance of waves in the protected zone. The diffraction problem can be solved via BCM in a similar manner. In this case, the plunger is fixed and a linear wave impinges from the right on the plunger front. In numerical implementation, the stroke amplitude is set to zero (\(S_0=0\)) and an incident wave \(\varphi _{\mathrm{r},\mathrm{in}}=C_0\cos h k_0(z+h)\mathrm{e}^{-ik_0 x} \mathrm{e}^{-i\omega t}\) is added to the potential of the wave on the right of the plunger \(\varphi _\mathrm{r}\). In both cases, the number of nodes and evanescent wave modes are chosen as \(M=200\) and \(n=8\), yielding convergence.

*d*/

*h*(i.e., the plunger depth normalized by the water depth) and \(\lambda /h\) (i.e., the wavelength scaled by the water depth). The vertical axis, \((a_\mathrm{l}/a_\mathrm{r})^2\) is the ratio of square of amplitude of radiated waves behind (i.e., leaked) and amplitude of waves in front of the plunger, hence providing a measure of the energy leakage. We call waves on the back side of the plunger

*leaked*waves because an ideal wave maker is expected to send all the input energy to the right-going (i.e., desired) waves. Therefore any energy propagating in the opposite direction (leftward in Fig. 9) is leaked underneath the plunger. Figure 10a shows that radiated wave leakage is higher for longer waves and shorter plunger drafts.

We also present the leakage due to incident wave (the diffraction problem) in Fig. 10b. Here incident wave (amplitude \(a_\mathrm{i}\)) is assumed to arrive from \(x = +\infty \) and therefore \(a_\mathrm{l}\) (amplitude on the left side of the plunger as shown in Fig. 9) is the amplitude of the leaked wave. Therefore, the quantity \((a_\mathrm{l}/a_\mathrm{i})^2\) shows the ratio of the leaked energy underneath the plunger to the energy of the incident wave. A trend similar to the radiation problem is observed. Specifically, longer waves leak more, and shorter plungers lead to higher leakages, as expected.

The numerical results for the wedge-shaped plunger imply that care must be taken in selecting the dimensions of the Wave Bridge plungers. The dimensions are dependent on the water depth, the dominant wavelength, and the target magnitude of the protection for the offshore structure. Higher protection requires a deeper draft of the plungers relative to the water depth. This is because the ratio of the left-going wave amplitude to the right-going wave amplitude approaches unity for most wave conditions when the draft of the body is small.

## 5 Conclusion

A new concept, called the Wave Bridge, for bypassing oceanic waves and horizontal momentum about ocean objects was experimentally and numerically investigated. The Wave Bridge is composed of a wave energy absorber and a wave maker that are connected via a linkage mechanism. The energy absorbed upstream of an ocean object by the wave energy absorber is mechanically transferred to the wave maker that is installed on the downstream side of the object. In other words, ocean wave energy is absorbed on the upstream side of the ocean object and spent to generate similar waves downstream of that object. Hence the structure is nearly *cloaked* against the waves, since waves are bypassed around the ocean object using the Wave Bridge. Furthermore, the horizontal forces on the wave energy absorber and wave maker are the same, but opposite. As a result, the net horizontal force on the structure protected by the Wave Bridge is relatively low. We have shown via laboratory experiments that a wave protection efficiency of up to 97 % and a horizontal force protection efficiency of up to 80 % is achievable. The numerical results show that the Wave Bridge protection depends on the plunger draft, the water depth, and the wave length. For a deployment of the Wave Bridge in a real sea environment, further studies are needed to investigate the performance of the device under nonlinear and broadband waves conditions as well as the effects of scaleup.

The Wave Bridge idea can be extended for shielding against omnidirectional waves by, say, attaching a number of finite-width bridges at different angles about the structure, or, by a cylinder-like Wave Bridge that fully surrounds the offshore structure.

## Notes

### Acknowledgments

We would like to thank Marcus Lehman and Ryan Elandt for their help. Support from the American Buearu of Shipping is gratefully acknowledged.

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