## Abstract

Future designs of systems for power extraction from ocean waves will likely involve a periodic array of absorbing units. We report an asymptotic theory of scattering and radiation by a linear array of heaving buoys in a channel and attached to power-takeoff devices. The spacing between buoys is assumed to be comparable to the incident wavelength and sea depth but much greater than the buoy size. The effects of extraction rate on the buoy motion, transmission and reflection coefficients for a range of frequencies in and outside the band gap are studied. It is found that strong reflection for frequencies inside the band gap of Bragg resonance reduces the extraction efficiency significantly. For comparison an alternate theory for the efficiency away from the band gap is derived using Froude–Krylov approximation. The predictions confirms and complements the asymptotic theory.

## 1. Introduction

Extensive theoretical studies have been devoted to the potential of power extraction from sea waves by an isolated unit such as a buoy, a raft or an oscillating water column (see reviews by Newman 1979; McCormick 1980; Falnes 2002; Mei *et al.* 2005; Cruz 2008). To achieve power output comparable to a conventional power plant or a wind-turbine farm, a large array of absorbing units is necessary. The possible effects of hydrodynamic interactions among units in any geometrical deployment are therefore of design interest.

Several methods have been developed to compute the scattering and radiation by a finite number of stationary or floating bodies. In particular, Falnes (1980), Falnes & Budal (1982) and Falnes (1984) have examined the case of large separation where hydrodynamic interactions between bodies are weak. Kagemoto & Yue (1986) have used eigenfunction expansions and addition theorems of Bessel functions to derive a numerical method to treat scattering by a few fixed vertical cylinders of circular cross sections. Infinite and semi-infinite lines of vertical cylinders have been treated semi-numerically by Linton & Evans (1992), Linton & Mclver (1996) using multi-pole expansions. Numerical studies of a finite number of cylinders have been reported by Linton & Mclver (1996), Chamberlain (2007), Peter *et al.* (2006), Peter & Meylan (2007), McIver (2002), Siddorn & Eatock Taylor (2008) and Mavrakos & McIver (1997) who also considered wave power extraction without giving numerical results. These methods can, in principle, deal with several cylinders of different sizes, but become computationally intensive for a large number of cylinders.

It is well known that, if resonated, a single buoy with one degree of freedom (e.g. heave) can absorb all the energy within the length of 1/*k* of the incoming wave front. If roll is also allowed and optimized then the length can be doubled. Thus for high efficiency the ideal spacing *d* between adjacent buoys is 1/*k*, i.e. *k**d* = 1. For environmental and navigational considerations, future wave-power farms will likely involve two-dimensional arrays with much larger spacing. Now it is known in general physics that when *k**d* = *n**π*, where *n* is an integer, waves are strongly scattered owing to Bragg resonance. It is therefore of interest to predict the effect of Bragg resonance on energy absorption and buoy response.

In this article, we shall first extend the multiple-scale analysis of Li & Mei (2007*a*) for a periodic array of fixed slender piles to small movable buoys which scatter, radiate and absorb energy by its heave motion only. To simulate their potential for power absorption we assume that each buoy is attached to a linear device that converts the mechanical energy of the buoy to electricity. No phase control of the power-takeoff system is assumed. Analytical results for the scattering coefficients, energy absorption rate and buoy motion will be derived and discussed.

## 2. Scales and normalization

Consider a linear array of small buoys in a long channel of constant width *d* and mean depth *h*, as shown in figure 1. Simple harmonic waves arrive from one end of the channel . In the framework of potential theory for inviscid and incompressible fluid, the mathematical problem is equivalent to an infinitely long strip of buoys in a rectangular lattice, attacked by a plane wave with crests parallel to the edge of the strip.

The geometry has three sharply different length scales: the small radius *a* of the buoy, the water depth *h* and the large horizontal extent of the array. These scales will be, respectively, referred to as the micro-scale for the near field, the meso-scale for the intermediate field, and the macro-scale for the far field. In all three fields the vertical displacements of the free surface (*η*) and of the buoys (ζ) are characterized by the incident wave amplitude *A*. Let us first introduce the following dimensionless variables, distinguished by primes, for the intermediate field,
2.1
2.2
We shall assume that the wavelength, mean depth and buoy separation are of the same order of magnitude, so that
2.3
are both of order unity but the buoy radius and drafts are small:
2.4
In the near field of a buoy the physics is controlled by the much smaller radius *a*, hence it is proper to employ the micro-scale coordinates, distinguished by bars and defined by:
2.5
It is known that the scattered wave from a small cylinder is smaller than the incident wave by a factor of order (*k**a*)^{2}, as in the case of sound (e.g. Mei *et al.* 2005). It will also be shown in the next section that a small heaving buoy extracts a fraction (*k**a*)^{2} of the incoming energy. Therefore, significant scattering and radiation effects can be expected to accumulate in an array with *O*(1/*μ*^{2}) buoys, or after a distance of *h*/*μ*^{2}. We therefore define the following macro-scale coordinates for the far-field to describe the evolution of wave envelopes:
2.6

## 3. Away from Bragg resonance

As a reference for later comparison, we first study the performance of an array of buoys well separated from each other without Bragg resonance. In this case, interaction between buoys can be neglected as a first approximation. Consider one small buoy in a plane incident wave of frequency *ω* and amplitude *A*. The potential of the incoming wave is:
3.1
where *ω* and *k* are related by the dispersion relation
3.2
Because of the small size of buoys, the scattered and radiated waves are negligible. Froude–Krylov approximation can be applied so that the hydrodynamic pressure on each buoy is dominated by the undisturbed incoming wave (see Newman 1979). The vertical exciting force on the buoy is therefore
3.3
Let us assume that an energy extraction device is attached to each buoy and exerts a load force i*ω*λ_{g}ζ, where λ_{g} denotes the extraction rate. Since the added buoyancy force due to heave is −*π**a*^{2}*ρ**g*ζ, Newton’s law gives
3.4
where *M* = *ρπa*^{2}*H* by Archimedes principle. It follows that
3.5
Use has been made of the fact that *ω*^{2}*H*/*g* ∼ *k**H* = *O*(*μ*) ≪ 1. Thus the inertia of a small buoy is relatively unimportant. As a consequence, the draft of the buoy *H* is much less relevant than its lateral dimension *a*. The time-averaged rate of energy extraction by a single buoy is given by
3.6
The fraction of power extracted by one buoy from the incident energy flux across a channel width *d* is then
3.7
where *E*_{inc} denotes the rate of energy influx over a width *d* and
3.8
is the group velocity of the incident wave.

Let the normalized frequency and extraction rate be denoted by then in dimensionless form and 3.9

Consider now a large number of buoys along the centreline of a channel and spaced at the same distance *d*. On the macro-scale of the total array, the mathematical effect of many point absorbers can be replaced by a continuous distribution. The fraction of the incoming energy absorbed within a unit distance of the macro-scale coordinates must be
3.10
In terms of the macro coordinate *X* = *μ*^{2}*x*′, the spatial rate of change of is
3.11
Thus the fraction of energy remaining at the end of the array *X* = *L* (i.e. *x*′ = *L*/*μ*^{2}) is and the extraction efficiency is
3.12
Clearly the efficiency depends on the number of buoys *N* = *L*/(*μ*^{2}*d*′), the frequency of the incoming wave through *ω*′ and *C*_{g}′, and the extraction rate λ_{g}′. For sufficiently large *L*, approaches unity. The dependence of efficiency on *k*′ and the extraction rate is plotted in figure 2*a* for a fixed *L* = 1. For a given *L*, the optimal extraction rate for maximum efficiency is found from
which gives the optimal extraction rate
3.13
Thus the optimum extraction rate should be higher for longer waves. Figure 2*b* shows the efficiency for a few *L*/*d*′, when the extraction rate is optimized for every *k*′. One sees that at high frequencies, a long enough array can extract all the incoming energy, and a larger array is better for low-frequency waves.

We now turn to the physics of Bragg resonance.

## 4. Scattering by an array of fixed buoys

### (a) Linearized dimensionless equations

As it is standard in linearized theories, the problem of wave–body interaction is equivalent to the sum of two hydrodynamic problems: scattering by stationary bodies and radiation by body motion. The two problems are coupled by the dynamics of the floating bodies. Let us first study the diffraction of a wave-train by fixed buoys. The following symbols for different parts of the physical domains are employed: *Ω*_{F} for the fluid domain, *S*_{F} for the free surface, *S*_{W} for the lateral surface of the buoys and *S*_{B} for the bottom surface of the buoys. The seabed is the entire horizontal plane *z*′ = −1. In terms of the meso-scale coordinates, the dimensionless governing equations for the scattering potential *Φ*′ is
4.1a
4.1b
4.1c
4.1d
4.1e
4.1f

We shall now find the law for the slow evolution of the envelope from the fast variations between buoys, by the method of multiple scales (homogenization).

### (b) Envelope equations by multiple-scale analysis

Substituting into equation (4.1) the following expansion
4.2
with *ϕ*_{n}, *n* = 0, 1, 2, … being dimensionless functions of (*x*′,*y*′,*z*′;*X*;*T*), we obtain the following equations from (4.1*a*) and (4.1*b*)
4.3a
4.3b
From successive orders of these and the remaining equations in (4.1), we obtain boundary-value problems for the wave-scale variations in a unit cell as shown in figure 3. Because each cell is one period in a very large array, we invoke Bloch’s theorem (Ashcroft & Mermin 1976) which states that the solution *ϕ*_{n} should be of the form e^{±ikx′}*f*(** x**′), where

*f*(

**′) is periodic in**

*x**x*′ with the period

*d*′. Since we shall focus on the state of Bragg resonance with

*k*

*d*′ =

*π*, Bloch’s theorem implies 4.4

#### (i) Leading order

At the leading order *O*(*μ*^{0}), the governing equations are homogeneous:
4.5a
4.5b
4.5c
4.5d
As reasoned in Li & Mei (2007*a*,*b*), since *μ* = *a*/*h* ≪ 1, *H*′ ≪ 1 and *d*′ = *O*(1), the areas of the buoy surfaces *S*_{B} and *S*_{W} are of the order *O*(*μ*^{2}). Hence, the buoys have negligible effects on the waves at the leading order, i.e. the boundary conditions on *S*_{B} and *S*_{W} are ineffective until at higher orders. The formal solution is the sum of left- and right-going plane waves in free space
4.6
where
4.7
The no-flux boundary conditions at *y* = ±*d*/2 and Bloch conditions (4.4) are trivially satisfied. The envelope functions α^{±}(*X*,*T*) are yet to be found.

#### (ii) First order

From equation (4.3), the first-order *O*(*μ*) meso-scale problem in the unit cell is inhomogeneous, and governed by
4.8a
4.8b
4.8c
4.8d
4.8e
4.8f
together with the Bloch conditions (4.4). Despite the large factor *μ*^{−2}, the integrated effects of the boundary value in (4.8*c*) and (4.8*d*) are of order 1 since the area of the buoy surface is of order *μ*^{2}. An important estimate of *ϕ*_{1} near and on the buoy may be deduced. In terms of the meso-scale coordinates, we have, in the neighbourhood of the buoy,
Now equations (4.8*c*) and (4.8*d*) imply that
4.9
hence
4.10
in the neighbourhood of the buoy.

We next apply Green’s formula to *ϕ*_{1} and the homogeneous solutions
4.11
over the volume of cell *Ω*_{F}:
4.12
which can be rewritten after using the boundary conditions in equation (4.8) as:
4.13
This is the solvability condition for the inhomogeneous problem of *ϕ*_{1} and gives a constraint on *ϕ*_{0}. To the leading order, the left-hand side of equation (4.13) can be simplified to
The integral over the free surface on the right-hand side gives
In view of equation (4.10), the second term in the surface integral over the buoy can be neglected with an error of *O*(*μ*). The remaining surface integral on the buoy is
Let us now use the fact that the buoy draft is small *H*′ ≡ *H*/*h* = *O*(*μ*), so that
4.14
from equation (4.7). In terms of the micro-scale variables and so that , we have
Hence, equation (4.13) becomes
which can be simplified to the coupled-mode equations on the macro-scale
4.15
where *Ω*_{0} is the dimensionless coupling constant
Equation (4.15) can be rewritten in physical coordinates as
4.17

These equations govern the envelopes of the forward and backward waves in an array of fixed small buoys, and are similar to those for wave propagation in an array of slender piles or over a periodic seabed (see Naciri & Mei 1988; Li & Mei 2007*a*,*b*). The difference with Li & Mei (2007*a*,*b*) is because of the fact that in the present problem the scattering effect comes from the bottom of the buoy and not from the lateral wall.

### (c) Macro-scale dispersion relation and band gap

As the first application of equation (4.15) let us consider that periodically modulated wave envelopes in an infinite array,
4.18
where *Ω* corresponds to a frequency detuning of *μ*^{2}*Ω*. The homogeneous equations (4.15) have non-trivial solutions only if
4.19
which is plotted in figure 4. For real *Ω* three regions can be distinguished. In either *Ω* ≤ 0 or *Ω* ≥ 2*Ω*_{0}, *K*_{S} is real, hence waves propagate. However, inside the band gap defined by 0≤*Ω* ≤ 2*Ω*_{0}, *K*_{S} is imaginary, hence propagation is forbidden. The wave train can only decay in distance. The spatial rate of amplitude decay is proportional to Im(*K*_{S}), which is maximum at *Ω* = *Ω*_{0}.

The presence of the band gap and its limits have been confirmed by numerical computation of the band structure but without the small-buoy assumption, using a standard numerical approach of solid-state physics (Garnaud 2009).

### (d) Scattering by an array of finite width

Let there be a finite array of fixed buoys in 0 < *X* < *L*, where *L* corresponds to the physical width (*L*/*μ*^{2})*h*. An incoming wave slightly detuned from Bragg resonance
4.20
arrives from with *K*_{0}≡*C*_{g}′*Ω*. Let the scattering potential be
4.21
on the incidence side and
4.22
on the transmission side. *C*_{R} and *C*_{T} are, respectively, the complex reflection and transmission coefficients. The potential inside the array is
4.23
where are defined by
4.24
By differentiating equation (4.15), we find
4.25
with
4.26

Requiring continuity of the leading-order pressure and horizontal velocity at the edges of the array, we must have
4.27a
4.27b
where *L*^{+} means slightly greater than *L* and *L*^{−} slightly less than *L*. Using equations (4.20) and (4.22) in (4.26), we obtain
4.28a
4.28b
It follows that
4.29a

The general solutions of equation (4.24) are of the form The six coefficients and can be found from the four conditions (4.27a), (4.27b) and the coupled equations (4.15). Finally, the amplitudes inside the array are given by 4.30a and 4.30b In the open sea, the transmission and reflection coefficients are 4.31 and 4.31b

The numerical results are qualitatively similar to those of vertical piles. As shown in figure 5, the scattering coefficients depend strongly on the detuning frequency *Ω*. Outside the band gap, *Ω*/*Ω*_{0}<0 or *Ω*/*Ω*_{0}>2, the scattering effects of the array are weak, as *C*_{R} is small and *C*_{T} is close to unity. Both are oscillatory in *Ω*/*Ω*_{0}. Inside the band gap, 0 ≤ *Ω*/*Ω*_{0} ≤ 2, propagation is inhibited. The reflection coefficient is close to 1 and the transmission coefficient close to 0. Inside the array the free surface profile is oscillatory in space if *Ω* is outside the gap and exponentially attenuating if inside, as shown in figure 6.

Next let us examine the effects of buoy motion on waves.

## 5. Envelope of radiated waves

In this section, we allow the small buoys to heave either freely or partially constrained by energy-absorbing devices. Since *k*′*d*′ = *π*, if one buoy goes up then its immediate neighbours must go down, and vice versa. Across the wide array of many buoys, the amplitude of waves and buoy displacements will be a slowly varying function of space. The normalized heave amplitude of the *m*th buoy can therefore be expressed as:
5.1
The buoy displacement ζ_{0}(*X*,*T*) is yet unknown and will be found later by additional account of the buoy dynamics. Let the centre of the *m*th buoy be located at *x*′_{m} = *m**d*′. For mathematical convenience, we express the term (−1)^{m} as:
5.2
in the kinematic boundary conditions on the buoys. Denoting the radiation potential by *ϕ*_{R} and expanding
5.3
the governing equations become
5.4a
5.4b
5.4c
5.4d
5.4e
where and are, respectively, the bottom and lateral boundaries of the *m*th buoy. Referring to figure 3, we require the radiation potential to be anti-periodic in *x*′ with the period *d*′,
5.5
for *i* = 1,2, ….

From equation (5.4), it is evident that the zeroth order radiation potential is of the form
5.6
where the long-scale functions β^{±} represent the unknown amplitudes of the propagating waves. The first-order problem is governed by
5.7a
5.7b
5.7c
5.7d
5.7e
Note that as the buoy radii and drafts are of order *μ*, we have *x*′ = *x*_{m}′+*O*(*μ*) for . The problem (5.7) is similar to the scattering problem (4.8) except for the additional term proportional to the buoy displacement, which is also anti-periodic with a period *d*′.

As in the scattering problem, we derive the solvability condition for *ϕ*_{1} by applying Green’s formula to *ϕ*_{1} and over a cell *Ω*_{F}.

It is easily checked that the only change is in the surface integral over the buoys
5.8
where . Using equation (4.14), we get
5.9
This finally gives us the envelope equations
5.10
where *Ω*_{0} was defined in equation (4.16). In physical terms, equation (5.10) reads
5.11
The pair of equations in (5.10) expresses the coupling between the amplitude of the right- and left-going waves and the unknown buoy motion.

We must now examine the buoy displacement induced by the waves, in order to relate ζ_{0} to α^{±} and β^{±} in equation (5.10).

## 6. Buoy dynamics

The forcing term ζ′ in the long-scale equation for the radiation problem is related to the scattering and radiation potentials given, respectively, by equations (4.6) and (5.6). By Froude–Krylov approximation, the leading order vertical forces on a buoy by the scattering and radiation potentials are given, respectively, by 6.1 6.2 where the forces are normalized according to 6.3 Use is made of the fact that the neighbouring buoys move in opposite phases.

Let us assume that the energy extraction device exerts a force
6.4
on the *m*th buoy. Applying Newton’s law to the *m*th buoy, we get
6.5
which gives us in dimensionless form
6.6
Again the mass of the buoy can be ignored so that
6.7
where for brevity we denote
6.8
which is the same as equation (3.9). Using equation (6.5) we can rewrite equation (5.10) as:
6.9
which couple the radiation and scattering components. The scattering amplitudes α^{±} are already found in the previous section and serve as forcing terms here.

We can now study the envelopes owing to waves interacting with a finite array of energy-extracting buoys.

## 7. Radiation by a finite array of energy-extracting buoys

### (a) Solution for radiation amplitudes

Consider again an array of finite but large width (*L*/*μ*^{2})*h* forced to oscillate by the incident and scattered waves and subjected to the reactive forces from the extractors and from the buoy motion. The potential outside the array must satisfy the radiation condition, hence
7.1a
7.1b
where the complex coefficients *C*^{+} and *C*^{−} are yet unknown. As in the scattering problem, we introduce by defining and assume the solution inside the array to be of the form
7.2
Continuity of pressure and horizontal velocity are required at the edges of the array
7.3a
7.3b
Using equations (7.1) and (7.2), we get
7.4a
7.4b
It follows that
7.5a
which implies that and . To complete the analytical solution, let us write the forcing term in equation (6.7) as:
7.6
Using equation (4.30), we find
7.7a
7.7b
To find , it is simpler to use linearity and just to solve first for one of the complex exponentials on the right-hand side of equation (6.7). Let the response to the forcing (e^{iKSX},e^{−iKSX}) be denoted by , respectively, then
7.8a
7.8b
where *K* ≡ *Ω*/*C*_{g}′ and *K*_{0} ≡ *Ω*_{0}/*C*_{g}′. By cross-differentiating these two equations can be decoupled to give
7.9a
7.9b
with
7.10
being the natural wave number of the radiated wave. As is complex for any non-zero λ_{g}′, *K*_{R} will be likewise, implying spatial attenuation. Since *K*_{R}≠*K*_{S}, the spatial period of the forcing term differs from the natural period in equation (6.7) and no resonance is expected. Variations of the real and imaginary parts are shown in figure 7 as functions of the detuning frequency *Ω* and the extraction rate λ_{g}′. One sees that there is no band gap and that all radiated waves are spatially attenuated. The special case of free buoys without extractors will be treated later.

The general solutions of equation (7.9) are: 7.11a 7.11b

For brevity we let
Because of the coupling by equation (7.8), only two of the four coefficients are independent. Two relations among them can be found by invoking equation (7.8) at any *X*∈[0,*L*], say *X* = 0,
7.12
7.13
which give
7.14a
7.14b
with . From the two boundary conditions (7.5), we get
7.14c
7.14d
We can now solve for for all *i* = 1,2,3,4:
7.15a
7.15b
7.15c
7.15d
Corresponding to the forcing , the responses can be treated similarly. Let the solution be of the form
7.16a
7.16b
The coefficients , *j*=1,2,3,4 can be obtained by replacing *K*_{S} with −*K*_{S} in , i.e.
7.17
It is now easy to see that
7.18a
7.18b
are the solutions of the radiation problem, satisfying equation (6.7) as well as the boundary conditions (7.5). We have also confirmed these formulas by direct numerical solution using the Finite Volume method.

### (b) Freely floating buoys

In the limiting case of freely floating buoys, λ_{g}′ = 0, so that . The evolution equations for β^{±} are no longer coupled. After omitting the factor e^{−iΩT}, we simply get
7.19
7.20
These decoupled first-order ordinary differential equations have solutions of the form
7.21
7.22
with constants *C*_{1} and *C*_{2} to be determined. Note that
7.23
after using the fact that the scattering wave number is given by , so that
This gives the total potential
7.24a
and thus
7.24b
where *η*_{0} is defined in a similar way as ζ_{0}. Matching directly the total potential *ϕ*_{0}+*ϕ*_{0} with the potential outside the array, we find
7.25
so
7.26
Hence the buoys move with the same amplitude and phase as the surrounding free surface and to the leading order a sparse array of small freely floating buoys does not affect the incoming wave. This result is expected as the consequence of negligible buoy inertia.

## 8. The combined effects of scattering, radiation and extraction

The combined effects of scattering, radiation and energy extraction on the free surface and buoy displacements are presented in figures 8 and 9. Two representative values of the detuning are chosen here: one outside the band gap of the scattering problem (*Ω* = −*Ω*_{0}) and one inside (*Ω* = *Ω*_{0}). One sees in figure 9 that the effect of the length is minor. In contrast, the effect of the extraction force shown in figure 8 has much stronger influence, as shown for the the two limits of fixed and free buoys. The displacements of the free surface and the buoys decrease faster through the array for frequencies inside the band gap, and as the extraction rate increases. Understandably, the displacement of the buoy is always smaller than that of the water surface. Note that there is no resonance.

In order to characterize the waves outside of the array, let us introduce transmission and reflection coefficients for the complete problem denoted, respectively, by and , such that the free surface in the open water regions is given by
8.1a
8.1b
In terms of the scattering and radiation problems studied previously, the coefficients are given by
8.2a
8.2b
Analytical formulas can be obtained from the results §4 and §7. While scattering is negligible for freely floating buoys when there is no power extracted as shown in §7*b*, for moving and energy-extracting buoys is close to 1 when the detuning frequency is within the band gap 0 ≤ *Ω* ≤ 2*Ω*_{0}, as shown in figures 10 and 11. In particular, reflection increases with the energy extraction rate λ_{g}′ and with the length of the array, and is of course the greatest for fixed buoys, which is equivalent to . In comparison with fixed buoys, the detuning corresponding to the maximum reflection is shifted slightly from *Ω* = *Ω*_{0} towards *Ω* = 0.

## 9. Energy extraction

Now the flow and buoy displacements are known. The period-averaged rate of energy extracted by the *j*th buoy is given, in physical form, by:
9.1
The total energy extracted by *N* buoys is therefore
9.2
Since *N*≫1, the above series can be approximated by an integral:
9.3
Dividing by the energy flux rate of the incident wave across the width *d*: , the efficiency of power absorption is found in terms of the buoy displacement:
9.4

As a check, the absorbed energy can also be calculated from the difference of the incoming and outgoing energy flux rates at . From the radiation and scattering coefficients, we can find 9.5 The electronic supplementary material includes the mathematical proof which shows that equations (9.4) and (9.5) are both the same.

Figure 12 shows how the extracted energy varies with the detuning, the length of the array and the energy extraction rate. First note that when the detuning frequency is outside the band gap of the pure scattering problem (0<*Ω*/*Ω*_{0}<2), the energy extraction efficiency tends to the same constant value predicted by the approximate value given by equation (3.12) based on the Froude–Krylov approximation. The good agreement shows the robustness of the asymptotic theory here despite its intended realm in the small neighbourhood of Bragg resonance.

It is clear that within the band gap, the extraction efficiency is substantially reduced by Bragg resonance. The reduction is the greatest around *Ω* = 0 (perfect tuning) and is larger for wider arrays (figure 12*a*) and greater extraction rate (figure 12*b*).

Although the maximum efficiency over most of the detuning frequencies is approximately equal to which increases with the length of the array, it is interesting that shorter arrays can yield more energy for small detuning, as shown in figure 13.

## 10. Conclusions

We have developed an asymptotic theory for wave interaction with a periodic array of small buoys in order to gain a better understanding of its potential as a wave-power farm.

We showed that fixed cylindrical buoys of small dimensions produce a scattering effect of the same order of magnitude as vertical piles of the same radius but extending across the entire depth. It can be easily shown that the results derived here can be extended to small buoys of arbitrary geometry. In particular, we have shown that hemi-spherical buoys have the same influence as circular cylindrical buoys. We have extended the work of Li & Mei (2007*a*) and deduced analytically the frequency band gap within which one-dimensional wave propagation is inhibited.

By multiple-scale analysis we have further solved the radiation problem of an array of movable buoys partially constrained by energy absorbing devices. The absorber is modelled as a damping force proportional to the velocity of the buoy. We have shown that Bragg resonance reduces the potential for energy extraction, somewhat similar to viscous damping in wall boundary layers studied recently by Tabaei & Mei (2009). While the present theory is designed only for the immediate vicinity of Bragg resonance, it agrees with the approximate theory valid far outside the band gap. Therefore, it may be a practical tool for analysing random incident waves with a broad frequency band. Modifications for shorter waves satisfying the Bragg condition *k**d* = *n**π* appears straightforward. For practical design of arrays of energy converters with complex phase control of the power-takeoff system, numerical techniques would be necessary.

Future extensions may include oblique incidence on a wide array. In this case, the wave physics of multiple scattering is two-dimensional. Approximation used here can be extended as in Li & Mei (2007*a*), Tabaei & Mei (2009) for piles.

## Acknowledgements

This research has been supported by a grant from MASDAR Institute of Science and Technology in the program of MIT-Abu Dhabi Alliance. C.C.M. also received partial support from US -Israel Bi-National Science Foundation and from Earth Systems Initiative at MIT.

## Footnotes

↵† Present address: Laboratoire d’Hydrodynamique, CNRS–Ecole Polytechnique, 91128 Palaiseau, France.

- Received August 29, 2009.
- Accepted September 1, 2009.

- © 2009 The Royal Society