Fundamental formulae for wave-energy conversion.

The time-average wave power that is absorbed from an incident wave by means of a wave-energy conversion (WEC) unit, or by an array of WEC units-i.e. oscillating immersed bodies and/or oscillating water columns (OWCs)-may be mathematically expressed in terms of the WEC units' complex oscillation amplitudes, or in terms of the generated outgoing (diffracted plus radiated) waves, or alternatively, in terms of the radiated waves alone. Following recent controversy, the corresponding three optional expressions are derived, compared and discussed in this paper. They all provide the correct time-average absorbed power. However, only the first-mentioned expression is applicable to quantify the instantaneous absorbed wave power and the associated reactive power. In this connection, new formulae are derived that relate the 'added-mass' matrix, as well as a couple of additional reactive radiation-parameter matrices, to the difference between kinetic energy and potential energy in the water surrounding the immersed oscillating WEC array. Further, a complex collective oscillation amplitude is introduced, which makes it possible to derive, by a very simple algebraic method, various simple expressions for the maximum time-average wave power that may be absorbed by the WEC array. The real-valued time-average absorbed power is illustrated as an axisymmetric paraboloid defined on the complex collective-amplitude plane. This is a simple illustration of the so-called 'fundamental theorem for wave power'. Finally, the paper also presents a new derivation that extends a recently published result on the direction-average maximum absorbed wave power to cases where the WEC array's radiation damping matrix may be singular and where the WEC array may contain OWCs in addition to oscillating bodies.

Introduction

For a general three-dimensional case, the basic linearized theory for conversion of ocean-wave energy by means of one oscillating body was developed in the mid-1970s [1-4]. The starting point was to consider power input as the product of the net wave force and the body's oscillation velocity. In addition, Newman [1], based on some reciprocity relations, discussed how the absorbed wave energy is related to wave interference in the far-field region. We may refer to this latter point of view as global, as opposed to local point of view (span class="Chemical">pan class="Chemical">LPV), which corresponds to the physical process taking place at the immersed oscillating body's wave-interacting surface, i.e. its wetted surface. One purpose of this paspan>per is to compare these two points of view. They are connected through the principle of energy conservation, as well as through a few additional reciprocity relations.

Budal & Falnes [5, p. 478] despan class="Chemical">cribed, qualitatively, the global point of view (GPV) as follows: ‘a secondary, ring-shaped, outgoing wave is generated, which interferes with the incoming wave in such a way that the resulting transmitted wave carries with it less energy than the incoming wave does’. Subsequently, Budal [6] applied this principle, quantitatively, to discuss wave-energy absorption by an array of oscillating bodies. Some years later, Farley [7] applied a far-field wave-interference analysis to wave-energy conversion by flexible rafts. Contrary to Budal, Farley did not dispan class="Chemical">criminate between two types of outgoing waves, namely diffracted and radiated waves.

After Budal's pioneering work on arrays, Evans [8] and Falnes [9], independently, analysed wave-energy absorption as taking place at the array's wave-interacting surfaces (the bodies' wetted surfaces). Later, this study was extended to include also oscillating pan class="Chemical">water columns (OWCs) in the wave-absorbing array by Falnes & McIver [10] and, independently, by Fernandes [11]. This was also an extension of previous mathematical analyses developed by Falcão & Sarmento [12], Sarmento & Falcão [13] and by Evans [14] for wave-power absorption by OWCs.

Newman [1] presented a review of previously known pan class="Chemical">waterspan>-wave reciprocity relations, as well as a few new ones. These reciprocity relations are derived by application of Green's theorem to velocity-potential theory for surface waves on pan class="Chemical">water, which is assumed to be an ideal fluid. Concerning a couple of the presented relations, Newman [1, §7] admitted that they ‘are not physically related to each other in any obvious manner’. Apparently, the application of these relations has caused some controversy recently [15,16] over the question of whether it is the forward (down-wave) or backward (up-wave) radiation that matters. A reason for the controversy may be the existence of at least two versions of what has been called ‘the fundamental theorem for wave power’ [17]. Hopefully, this paper will assist in clarifying the matters discussed.

For any wave-energy converter (WEC) array of oscillating bodies, Wolgamot et al. [18] showed that the direction-average maximum absorption width equals N times the wavelength divided by 2π, on the condition that the array's N×N radiation-damping matrix is non-singular, where N is the array's total number of used oscillating-body modes. In this pa<span class="Chemical">per, we generalize this result to cases where the radiation-damping matrix may be singular and where the WEC array may contain OWC units in addition to oscillating bodies. The mathematical details are given in appendix A.

A further subject of this pa<span class="Chemical">per is the relationship between the reactive radiation-parameter matrices and the reactive power, which is related to the kinetic–potential energy difference in the pan class="Chemical">water that surrounds the WEC array. It is found that some of the equations which were presented nearly three decades ago by Falnes & McIver [10] need to be corrected.

Throughout this paper, we shall assume that deviation from equilibrium is sufficiently small to make linear theory applicable. We choose a coordinate system with the z-axis pointing upwards, where the z=0 plane coincides with the mean free surface. We may use Cartesian or polar horizontal coordinates. They are related by . Except for some introductory time-domain consideration, we shall assume an incident, monochromatic, plane wave, for which the wave elevation has a complex amplitude (where a time-varying factor ei is suppressed). The corresponding incident wave power level (incident wave power transport per unit width of the wave front) is where A is the complex wave elevation amplitude of the (undisturbed) incident wave at the origin (x,y)=(0,0). The incident wave propagates at an angle β relative to the x-axis. Moreover, k=ω/vp is the angular repetency (wavenumber), ω the angular frequency and vp the wave's phase velocity. Finally, the wave's group velocity is vg, the water density is ρ and the acceleration of gravity is g. Observe that Jw equals the group velocity multiplied by the propagating incident wave's time-average energy per unit of horizontal sea surface. Half of this energy is potential energy related to water being lifted against gravity from wave troughs to wave crests, while the remaining half is kinetic energy associated with the water's oscillating velocity. However, for a situation where a purely propagating wave, as given by (2.1), interferes with a wave propagating in a different direction, then the surface densities of kinetic energy and potential energy may be different, as discussed in some detail in appendix B.

Wave-energy absorption at immersed wave-energy converter boundaries

Concerning absorption of wave energy by means of an immersed oscillating body, the instantaneous, as well as the time-average, power absorbed from the wave may be quantified as a product of the net wave force and the velocity of the body. This approach was used, for example, by Budal & Falnes [3] and Evans [2]. There are two contributions to this wave force: firstly, the excitation force, in consequence of the existence of the immersed body, and, secondly, the radiation force, in consequence of the oscillation of the immersed body. The first force contribution is linearly related to the incident wave but independent of the body's motion, while the second force contribution is not explicitly related to the incident wave but linearly related to the body's motion. Assuming, for simplicity, that the immersed WEC body is oscillating in only one mode—mode i, say—of its six possible modes (degrees of freedom), then we shall denote the two wave force contributions by Fe,(t) for the excitation force, and by Fr,(t) for the radiation force. In the case of a monochromatic wave and harmonic oscillation with angular frequency ω, we denote the complex amplitudes of the two wave-force contributions by res<span class="Chemical">pectively, where u is the complex velocity amplitude for oscillation mode i. The complex proportionality coefficients fe,=fe,(β,ω), i.e. the excitation-force coefficient, and Z=Z(ω), i.e. the radiation impedance, as well as the latter's real and imaginary parts, R=R(ω), i.e. the radiation resistance, and X=ωm=X(ω)=ωm(ω), i.e. the radiation reactance, are functions of ω. The coefficient m is called ‘added mass’ although it may be negative in exceptional cases [19]! Moreover, the coefficient fe, also depends on β, the angle of wave incidence.

Observe that, in terms of complex amplitudes, the radiation-force Fr, has two components, an active component and a reactive one, which are in phase with the velocity u and the acceleration iωu, respectively. By inverse Fourier transformation, where products in the frequency domain correspond to convolutions in the time domain, we may find a corresponding decomposition of the general, time-domain, radiation force [20]: As shown below, only the active force component contributes to the net time-averaged energy transfer, while the reactive force component serves temporary energy exchange between differently sized stores of kinetic energy and potential energy.

To provide the desired immersed-body motion, the WEC unit needs to be equipped with a machinery for control and power take-off (PTO). This provides an additional force Fpto,(t), with corresponding complex amplitude Fpto, for the monochromatic-wave case. Then we may write the equation of motion, in complex-amplitude representation, as where m is the mass of the immersed body and c its hydrostatic stiffness coefficient. We have also introduced a coefficient rloss, to represent linear power loss. Introducing the body's excursion from equilibrium position s(t)—thus —we may, in time-domain representation, write the equation of motion as In contrast to the frequency-domain model, for the time-domain model we may include possible additional nonlinear forces in the Fpto,(t) term of (3.5).

Our next task will be to find an expression for the time-average power Pa absorbed by the PTO. For this purpose, we multiply through (3.5) by , and rearrange terms. We then find where the overbar denotes averaging over a time interval that is sufficiently long to make the contribution from reactive force components negligible. For periodic waves and oscillations, it is sufficient to average over one period. Note that the reactive-force component—the second one of the two l.h.s. terms of (3.5)—does not contribute to the time-averaged absorbed wave power, as given in (3.6). In relation to u(t), also Fe,(t) has a reactive part Fe,(t), for which .

The product of the total reactive force and the velocity is the instantaneous reactive power, namely, where is the time derivative of the sum of the body's kinetic energy and potential energy, and where (d/dt)W(t) is the time derivative of the sum of kinetic energy and potential energy of the pan class="Chemical">water surrounding the body. At instants when u(t)=0, there is no kinetic energy, and at instants when s(t)=0, there is no potential energy. Except for conditions of resonance, the r.h.s. of (3.7) does not vanish at all instants. In general, the PTO machinery has to cope with reactive forces and reactive power, because of unequal magnitudes of the kinetic and potential energy stores. The r.h.s.—and hence also the l.h.s.—of (3.7) has, however, a vanishing time average.

For a sinusoidal oscillation with complex velocity amplitude , we have and . Using this, we find which, together with (3.7), explicitly shows how the reactive power is directly related to the difference between the maximum values of kinetic energy and potential energy. In analogy with (3.8), (d/dt)W(t) is related to such an energy difference associated with the pan class="Chemical">waterspan> surrounding an array of immersed WEC units. This matter is discussed in §6.2,6.3 and more extensively in appendix B.

In the remaining part of this <span class="Chemical">paper, we consider only a monochromatic wave and the corresponding sinusoidal oscillation of immersed WEC units. Without considering the details of the PTO machinery, we shall rather consider the WEC units' complex oscillation amplitudes—for instance u—to be independent variables, and a goal of our analysis is to find their optimum values corresponding to the incident wave as given by (2.1). Then the complex-amplitude version of (3.6) is where the asterisk (*) denotes complex conjugate. Assuming ideal conditions, we set rloss,=0 in the following.

Moreover, we shall find it convenient to make the following substitutions: where the non-negative quantity Pr represents the radiated wave power (caused by any forced oscillation of the immersed body). Although the introduced complex quantity might, in general, have any arbitrary phase angle δ in the interval −π<δ≤π, we shall find it convenient that it is chosen to have the same phase angle as A*E*(β). Then A*E*(β)/U is a real positive quantity, which, notably, is independent of the complex velocity amplitude u.

We may now simplify (3.9) for the time-averaged absorbed wave power to where is the ‘excitation power’. An important motivation behind the substitution of by E(β) and by |U|2 is that the above (3.11) and (3.12) as well as the following (3.13)–(3.17) are applicable also for a WEC array consisting of several WEC units—oscillating bodies and/or OWCs—provided the parameters E(β) and |U|2 are properly redefined, as explained later in this paper (see (6.20) and (7.1) and (7.2)). For this reason, we propose the terms ‘collective excitation-power coefficient’ and ‘collective oscillation amplitude’ for the complex quantities E(β) and U, respectively.

The usefulness of introducing the quantity U is that (3.11) may be rewritten as from which we, simply by inspection, see that the first term equals the maximum possible absorbed power, provided the last term vanishes, that is, if the quantities U and E(β) have optimum values U0 and E0(β) that satisfy the optimum condition Hence, at optimum, the three terms on the r.h.s. of (3.11) have the same, real and non-negative, magnitude. The last one of the three terms is the optimum radiated power. It follows that we have several different alternative expressions for the maximum absorbed power, e.g. We may consider this series of alternative mathematical expressions as reciprocity relations for the maximum absorbed power. For instance, the maximum absorbed power <span class="Chemical">Pa,MAX equals the optimum radiated power Pr,OPT=|U0|2. In (3.11), the last term, the radiated-power term Pr=|U|2, appears to be a power-loss term, but it should, rather, be considered as a necessity, because the radiated wave is needed to extract power from—that is, to interfere destructively with—the incident wave.

As we have chosen A*E*(β)/U to be a real positive quantity, which is independent of u and, therefore, also of U, we have where we have made use of the optimum condition (3.14). In general, we shall consider U to be an inde<span class="Chemical">pendent complex oscillation-state variable, while the optimum value U0(β) is real and positive, because we have chosen U to have the same phase angle as A*E*(β) has. According to (3.16), and AE(β)=U0U*. If we insert this into (3.11) and also use (3.15), we obtain the simple equation which, for a fixed value Pa<Pa,MAX, corresponds to the equation of a circle of radius centred at U0(β) in the complex U plane. Equation (3.17) may be illustrated as an axisymmetric paraboloid in a diagram where a pan class="Disease">vertical real Pa axis is erected on a horizontal complex U plane, as shown in figure 1.

Figure 1.

The wave-power ‘island’, illustrating (3.17). Absorbed wave power Pa as a function of the complex collective oscillation amplitude U=Re{U}+ i Im{U}=|U| ei, where the phase is chosen such that AE/U* is a real positive quantity, and where |U| is given by (3.10) for the one-mode oscillating-body case, and by (6.20) for the case of a general WEC array. The largest possible absorbed wave power Pa,MAX is indicated by a star on the top of the axisymmetric paraboloid, and U0 is the optimum collective oscillation amplitude. Colour changes indicate levels where Pa/Pa,MAX equals 0, , and . (a) Side view, (b) top view and (c) inclined view.

The wave-power ‘island’, illustrating (3.17). Absorbed wave power Pa as a function of the complex collective oscillation amplitude U=Re{U}+ i Im{U}=|U| ei, where the phase is chosen such that AE/U* is a real positive quantity, and where |U| is given by (3.10) for the one-mode oscillating-body case, and by (6.20) for the case of a general WEC array. The largest possible absorbed wave power <span class="Chemical">Pa,MAX is indicated by a star on the top of the axisymmetric paraboloid, and U0 is the optimum collective oscillation amplitude. Colour changes indicate levels where Pa/Pa,MAX equals 0, , and . (a) Side view, (b) top view and (c) inclined view.

Assuming that the PTO machinery of the oscillating WEC body contains sufficient control equipment to achieve the desired oscillation, we may consider the complex velocity amplitude u, as well as U, to be an independent variable. However, the optimum value u, as well as U0, depends on the excitation force fe,(β)A, and is, consequently, dependent on the incident-wave parameters A and β. For an optimum oscillation velocity u=u(β), say, corresponding to maximum absorbed wave power Pa=Pa,MAX—cf. e.g. (3.14) and (3.15)—we have optimum collective parameters E(β)=E0(β) and U=U0. Note that |U0|2, as well as all the other alternative expressions given in (3.15) for the maximum absorbed power, depends implicitly on β, and, moreover, it is proportional to |A|2—remembering that u(β), in contrast to an independent variable u, is proportional to A.

Destructive far-field wave interference

Excluding waves and oscillations of general time variation, but considering only monochromatic waves and corresponding harmonical oscillations, we may, alternatively, calculate the time-average absorbed power by analysing wave interference in the far-field region, that is, many wavelengths away from the immersed WEC body, or more generally, WEC array. We then have to assume that the wave propagation takes place in span class="Chemical">pan class="Chemical">water that we may consider to be an ideal loss-free fluid. In the following, we shall apply such a GPV, and then compaspan>re the results with the above pan class="Chemical">LPV results.

If we assume that, within a localized area near our chosen origin (x,y,z)=(0,0,0), the body—or, more generally, a WEC array—is installed, but not oscillating, then an incident plane, monochromatic wave, with wave elevation , as given by (2.1), produces a diffracted wave, for which the wave elevation has a complex amplitude ηd=ηd(r,θ), say. (This may also include diffraction effects of possible reefs and rocks.) Next, imagine that the immersed body, or the array, is performing forced oscillations with no incident wave, that is, with A=0. Then a wave will be radiated from the body, or the array. Let ηr=ηr(r,θ) denote the complex elevation amplitude of this radiated wave.

When there is an incident wave, and the immersed WEC array is oscillating, then the complex amplitude of the wave elevation is η=η0+ηg=η0+ηd+ηr. We have here introduced ηg=ηd+ηr as the complex elevation amplitude of the total ‘outgoing’ wave. Depending on the geometrical details of the array, the outgoing waves may have a complicated mathematical structure near the array, in the so-called near-field region. We shall, however, only need far-field mathematical details, which, in general, are asymptotically valid several wavelengths away from the WEC array. The far-field diffracted/radiated/outgoing wave elevations may thus be expressed in the form where the complex functions <span class="Chemical">Cd(θ), pan class="Chemical">Cr(θ) and Cg(θ) are the far-field coefficients for the diffracted wave, the radiated wave and the outgoing wave, respectively. It is convenient to express these coefficients in terms of the so-called Kochin functions Note that the diffracted wave is linearly related to A, the complex amplitude of the incident-wave elevation amplitude at the origin, r=0, while the radiated wave is linearly related to all WEC units' oscillation amplitudes.

WEC arrays will be discussed in §6. At present, we shall consider the simpler case of only one immersed, single-mode oscillating, body. Introducing complex Kochin function coefficients of proportionality by corresponding lower case symbols, we may write the Kochin functions as for the diffracted wave and the radiated wave, respectively. The total generated wave's Kochin function is Note that for an optimum oscillation vector u=u(β), there corresponds optimum Kochin functions Hr(θ)=Hr0(θ) and Hg(θ)=Hg0(θ), which depend, implicitly, also on β. In particular, u(β) and, thus, Hr0(θ) are linearly related to the excitation force Fe,(β)=fe,(β)A. However, the coefficient h(θ) does not depend on β, in contrast to the coefficient hd(θ), which depends implicitly on β, since the diffracted wave is a response to the incident wave.

In correspondence with our derivation of (3.9) for the wave power absorbed by an immersed oscillating body as the product of the net wave force and the body's oscillation velocity, Newman [1, §10] expressed the power Pa absorbed by an oscillating immersed body as an integral over the body's wetted surface, where the integrand is the hydrodynamic pressure multiplied by the normal component of the fluid velocity. Then, applying Green's theorem, he expressed <span class="Chemical">Pa as an integral over an, envisaged, cylindrical control surface in the far-field region, a surface that encloses the immersed body and all pan class="Chemical">water between the body's wetted surface and the control surface. In this way, Newman [1, eqns 58 and 59] moved from the pan class="Chemical">LPV to the GPV, and expressed the absorbed wave power in terms of Kochin functions.

Accordingly, following Newman, we may write the time-averaged absorbed wave power as where Pi=I(β)+I*(β) is the ‘input power’ and is the (non-negative) total ‘outgoing power’. Here vp=ω/k and vg=dω/dk are the phase velocity and the group velocity, respectively. Moreover, and we may write the input power as

An approach corresponding to (4.5)–(4.8) has been applied by Farley [7,15] and Rainey [17]. Their approach shows the physical details of wave-interference energy removal in the far-field region. By wave interference in the far-field region, wave-energy removal takes place where the outgoing wave ηg travels in the same direction as the incident wave η0, that is, for direction θ coinciding with the incident-wave direction β. As there is no energy exchange between two plane waves propagating in different directions, there is no contribution to far-field wave-energy removal by the outgoing wave in directions where θ≠β (or, more precisely, outside a small θ interval around θ=β, an interval that tends to zero as ).

Noting that (4.5) has a similar mathematical structure as (3.11), it might appear that (3.14)–(3.17) are valid also if we replace the span class="Chemical">pan class="Chemical">LPV paspan>rameters Pe, Pr, E(β) and U by the GPV parameters Pi, Pg, I(β)/A and G, respectively. However, note that the optimum LPV parameters are more directly related to the optimum WEC body oscillations than the GPV ones are. If we compare LPV equations (3.10)–(3.12) with GPV equations (4.5)–(4.8), we may note that Pe is proportional to A and linearly related also to the WEC body's oscillation amplitude, while Pr is quadratically related to this amplitude, but independent of A. By contrast, Pi, as well as Pg, is related in a more complicated way to A and to the WEC body's oscillation amplitude. Equations (3.14)–(3.17) therefore do not apply for the GPV parameters.

We may mitigate this drawback by rearranging the GPV equations (4.5)–(4.8) as follows. Firstly, we observe that if the single-mode oscillating WEC body does not oscillate, i.e. u=0, then no wave energy is being absorbed, i.e. Pa=0. Moreover, the radiated wave's Kochin function vanishes, i.e. Hr(θ)=0. Then the GPV equations (4.4)–(4.8) agree with the following reciprocity relation for the diffracted wave's Kochin function [1, eqn 33]: Secondly, from the same GPV equations (4.4)–(4.8), we then find, for the oscillating-body case (i.e. u≠0), that the power <span class="Chemical">Pa, which is removed by the far-field wave interference, is as given by the pan class="Chemical">LPV(!) equations (3.10)–(3.12), but now with collective parameters |U|2 and E(β) expressed in terms of far-field quantities, namely, and Note that |U|2, in contrast to |G|2, is independent of the wave amplitude A, and quadratic in the WEC body's oscillation amplitude. Further, E(β)A, in contrast to I(β), is linearly related to the oscillation amplitude, and proportional to the incident wave amplitude A. Equations (3.10) and (4.10) present two different expressions for the radiated power |U|2. Physically, this means that the power which is radiated from the WEC body's wave-interacting surface into a lossless fluid equals the power that is associated with the radiated wave in the far-field region of the fluid.

From a physical point of view, what may be observed in the far-field region is a superposition of the plane incident wave and the outgoing wave. As observed in the far-field region, one may not know whether the outgoing wave originates from one single-mode oscillating body or from an array consisting of many WEC units. For this reason, all equations in the present section are valid for this latter WEC system, provided the two-factor product hu that ap<span class="Chemical">pears in (4.3) is generalized to a sum of such products, one product for each of the WEC array's oscillating modes. Details are given in §6.

Relationships between radiated and diffracted waves

Among many pan class="Chemical">waterspan>-wave reciprocity relations, there are two relations, which relate diffraction and radiation parameters, and about which Newman [1, eqns 45 and 48] remarked that the corresponding two involved physical problems that ‘are not physically related to each other in any obvious manner’. Newman used these two relations to convert the formula for absorbed wave power from the version of (4.5) to the version of (3.11), but with the collective parameters |U|2 and E(β) expressed solely in terms of radiation Kochin function coefficients, that is, without the diffraction Kochin function—which we still need to eliminate from (4.11).

The first one of the above-mentioned two reciprocity relations is the Haskind relation [21,22], which relates the excitation force Fe,(β)=fe,(β)A to the radiated wave's Kochin function Hr(θ)|=h(β + π)u, namely,

The second one is a relation between Hd(θ) and h(θ), a relation which Newman [1, eqn 61] used to simplify (4.11) to where we have introduced the ‘adjoint companion’, of the radiated wave's Kochin function Hr(θ)—cf. (4.3). The complex conjugation star on the complex velocity amplitude u in (5.3) corresponds, in time domain, to time-reversed motion.

In this way, Newman succeeded to eliminate, mathematically, the diffracted wave's Kochin function Hd=hdA, that appears for example in (4.5)–(4.8) and in (4.11). Referring to (3.10), we may, however, arrive at the same result (5.2) without referring to the second one of the above-mentioned two, not very obvious, reciprocity relations. The result simply follows by applying the Haskind relation (5.1) to the excitation-force coefficient fe,(β) in (3.10).

Although reciprocity relations between diffraction and radiation parameters connect different physical problems ‘which are not physically related to each other in any obvious manner’, as admitted by Newman [1, §7], the Haskind relation (5.1) may be supported by the following physical argument. Imagine a non-symmetric WEC, e.g. the well-known nodding-duck device [23], which is installed with an optimum orientation to absorb waves arriving from west, thus incident waves pro<span class="Chemical">pagating eastwards. If, in a case with no incident wave, the device is performing forced oscillations, the device will primarily radiate waves propagating westwards. Thus, the addition of an angle π in the argument on the r.h.s. of (5.1) seems reasonable. Moreover, it is reasonable that the excitation-force coefficient fe,(β) of the incident-wave problem (diffraction problem) is proportional to the radiation-ability coefficient h(β+π) of the forced-oscillation problem (radiation problem). Admittedly, however, the second one of the two above-mentioned reciprocity relations, which directly connects (4.11) and (5.2), is less obvious from a physical point of view, namely the reciprocity relation presented by Newman [1, eqn 48]:

Earlier, the LPVspan> quantities |U|2 and E(β) appearing in (3.11) were given by the two equations (3.10). However, when we now have, alternatively, expressed E(β) by equation (5.2) and |U|2 by equation (4.10), which are far-field, or global, equations, this corresponds to a mixed, or hybrid, global–local point of view (GLPV), because we have now expressed the LPV parameters |U|2 and E(β) in terms of radiation Kochin functions, which are far-field parameters. Although it is not easy to give the GLPV version a direct physical interpretation, it has the advantage that it may be a basis for several reciprocity relations [1] and, moreover, also for certain mathematical derivations below, as exemplified later on in this paper; see §6.1,6.2.

Generalization to wave-energy converter arrays

We consider a case of wave-energy absorption by an array of immersed oscillating rigid bodies and of OWCs, as indicated in figure 2. Let us assume that the number of wave-interacting oscillators is N=Nu+Np, where Np is the number of OWCs and Nu is the number of used body modes, whose number may be up to six times the number of bodies. The oscillation state and the excitation due to an incident plane wave may be despan class="Chemical">crspan>ibed by N-dimensional column vectors v and x, respectively, where Here we have introduced two Nu-dimensional column vectors u=[u1 u2 u3 ⋯ u]T and Fe=[Fe,1 Fe,2 Fe,3 ⋯ Fe,]T, where u and F are the complex amplitudes of the oscillation velocity and of the excitation force for rigid-body oscillation mode i. Correspondingly, we have introduced two Np-dimensional column vectors p=[p1 p2 p3 ⋯ p]T and Q=[Qe,1 Qe,2 Qe,3 ⋯ Qe,]T, where p and Q are the complex amplitudes of the oscillating dynamic air pressure and of the excitation volume flow for OWC i. We may think of v and x as vectors in an N-dimensional complex space. The superspan class="Chemical">cript ‘T’ denotes the transpose of a matrix, and the complex conjugate transpose of a matrix is correspondingly denoted by the dagger symbol (†).

Figure 2.

Wave-interacting objects inside an envisaged (control) surface , chosen as a cylindrical surface r=const. Two floating bodies are indicated, as well as two OWCs, one in a floating structure, the other in a fixed (bottom-standing) structure. This figure is reproduced from Falnes & Hals [24].

Applying linear theory, we have also introduced the following Nu- and Np-dimensional vectors: the complex vectorial proportionality excitation vector coefficients fe=fe(β) and qe=qe(β), respectively. The complex excitation vector x=x(β), acting on the WEC array, depends on the angle β of wave incidence, and it is proportional to the complex elevation amplitude A of the undisturbed incident wave at the origin (x,y)=(0,0). We shall, however, here consider the complex oscillation vector v to be an independent variable, assuming that we have an ideal machinery for PTO and motion control.

Global point of view

For this, rather general, WEC array, we may extend the second equation of (4.3) to write the radiated wave's Kochin function as We may then generalize Hr correspondingly in, for example (4.4), (4.6)–(4.8), (4.10) and (4.11). Moreover, the Haskind relation (5.1) is generalized to for i=1,2,3,…,N. The adjoint radiation Kochin function (5.3) is generalized to [24] and (5.2) to Note that this latter equation may be considered as a generalized Haskind relation for the collective excitation-power coefficient E(β). Moreover, on the basis of the general equations (4.11) and (6.5), we easily see that (5.4)—the least obvious one of the reciprocity relations presented by Newman [1, eqn 48]—is still valid with the general radiation Kochin function Hr, as given by (6.2). Equations (4.11) and (6.5) provide two different mathematical relations between the pan class="Chemical">LPVspan> collective excitation-power coefficient E(β) and the N GPV Kochin function coefficients h for the radiated wave. Equation (4.10), with (6.2), provides a mathematical relation between these h coefficients and the pan class="Chemical">LPV collective amplitude |U|, and also the corresponding complex amplitude U if we remember that we have chosen the phase of U to equal the phase of A*E*(β), in accordance with (3.16).

Let us next consider the optimum case for maximum absorbed power. Algebraic procedures for determining the optimum value v0=[v10 v20 v30 ⋯ v]T of the complex oscillation-state vector v are treated in more detail in appendix A. Correspondingly, according to (6.2), there exists an optimum Kochin function for the radiated wave. Note that, even if we, in general, consider v to be an independent variable, the optimum value v0=v0(β), as well as the β-dependent optimum Kochin function Hr0(θ), is linearly related to the incident wave amplitude. From the optimum condition (3.14), we have , which, in combination with (4.10) and (4.11), gives the condition which the optimum radiated wave's Kochin function Hr0(θ) needs to satisfy. Combining this condition with the reciprocity relation (5.4)—see also (6.4)—yields That this is real and positive corresponds to the radiated wave having optimum phase. If we choose A to be real and positive, then also has to be real and positive.

In (3.15), we presented several different expressions for the maximum power Pa that is possible to be absorbed by the WEC array. We shall find it convenient to add also the following expressions: Applying the last one of the fractions shown in the pan class="Chemical">LPV equations (6.9), and then inserting from the GPV equations (4.10) and (6.5), we get where Jw is the wave-power level, as given by (2.2), and da≡Pa/Jw is the ‘absorption width’. Moreover, we have introduced the—at optimum—gain function which is an extension of a formula presented, independently, by Newman [25] and by Evans [8] for the single-body, one-mode case. It is remarkable that we here have been able to express the maximum absorbed power in terms of optimum far-field Kochin functions for the radiated wave only. It should be emphasized that this gain function Gg0(β) applies only to the optimum case for maximum absorbed wave power.

The Haskind relation (5.2) and the collective Haskind relation (6.5), as well as the optimum gain function as given in (6.11), indicate that it is important for a WEC system to possess the ability to radiate a wave propagating in a direction opposite to the direction of the incident wave. This ability is represented, quantitatively, by the N coefficients h(β+π) in (6.5). If N=1, we see from (4.3) and (5.3) that, with i=1, we have . However, with (4.3) and (5.3) generalized to (6.2) and (6.4), we should note that, in general, for N≥2. For instance, referring to (6.2)–(6.4) for N=2, we have, for any θ, including θ=β+π, that , which, in general, deviates from zero for arbitrary as well as for optimum values of the complex velocity amplitudes v1 and v2. For the circularly oscillating Evans Cylinder [2], we may, as shown below in §7.1—see (7.9)—replace by |Hr0(β)|2 in the numerator of (6.11). A corresponding replacement may be made if diffraction is negligible or, otherwise, in cases where the integral on the r.h.s. of (5.4) vanishes when Hr(θ)=Hr0(θ).

Concerning the GPV discussion, presented in §§4 and 5, for the case of one single-mode body WEC unit, we have, so far, here in §6, extended results to the case of an array of WEC units. Our next task will be to generalize some of the pan class="Chemical">LPVspan> matter discussed in §3.

Local point of view

For a single-mode oscillating body, the complex amplitude of two wave-force contributions, the excitation force Fe, and the radiation force Fr,, are given by (3.1). For our WEC array, the excitation vector x, introduced by the second equation of (6.1), is an extension of Fe,, while [10] is an extension of Fr,, where Z and Y are the Nu×Nu radiation-impedance matrix for the oscillating bodies and the Np×Np radiation-admittance matrix for the OWCs, res<span class="Chemical">pectively. These matrices are symmetric, that is, ZT=Z and YT=Y. The Nu×Np matrix H represents hydrodynamic coupling between the oscillating bodies and the OWCs, which compose the WEC array. It is convenient to split these complex matrices into real and imaginary parts: where the radiation resistance matrix R, the radiation reactance matrix X, the radiation conductance matrix G, the radiation susceptance matrix B, as well as the matrices C and J, are real. All these matrices are frequency dependent. Further comments concerning these matrices are given in appendix B; see (B 24)–(B 28) and related text.

Let us now, for the WEC array, extend (3.1) and (3.2), where we defined the radiation force Fr, and split it into active and reactive components. The extension reads where We may note that which means that the radiation-damping matrix D is hermitian. Also the matrix (Dreactive/i) is hermitian. From this it follows that, for any N-dimensional complex column vector v, the scalar matrix products v†Dv and v†Dreactivev are real and purely imaginary, respectively. (We may observe that the matrices D and (Dreactive/i) are real and symmetric if the WEC array contains no OWCs or no oscillating bodies, that is, in cases where Dcomplete=Z=R+iX or Dcomplete=Y=G+iB, res<span class="Chemical">pectively.)

If we premultiply (6.14) by −v†/2, we get the ‘complex radiated power’ where the last term, the reactive-power term, v†Dreactivev/2 is purely imaginary, while the first term, the radiated-power term, v†Dv/2≡Pr is real and non-negative—see (6.21). Moreover, if we premultiply by v†/2 the excitation vector x, defined by equation (6.1), we get the ‘complex excitation power’ We may note that the imaginary part represents reactive power (see (B 49)).

For any oscillation vector v=[u −p]T, the time-average wave power absorbed by the array is Pa=Pe−Pr, where the ‘excitation power’ Pe and the radiated power Pr are given by [10] We may express this in the form of (3.11) provided we define the collective excitation-power coefficient E(β) and the collective oscillation amplitude U by which is an extension of (3.10). We still choose the phase angle of U such as to make A*E*(β)/U a real and positive quantity.

For a case with no incident wave, x=0 (which means that Pe=0), energy conservation requires that the absorbed wave power cannot be positive. Thus, for all possible finite oscillation-state vectors v, we have Thus, in general, the radiation damping matrix D is positive semidefinite. It is singular in cases when its determinant vanishes, |D|=0. Otherwise, it is positive definite, v†Dv>0.

It is well known [10] that the maximum wave power that can be absorbed by the array is where v0=v0(β) is an optimum value of the oscillation-state vector v that has to satisfy the optimum condition

By manipulating (6.16), (6.19), (6.22) and (6.23), we can show that For a fixed value of the absorbed wave power Pa, where <span class="Chemical">Pa<Pa,MAX, equation (6.24) represents an ‘ellipsoid’ in the complex N-dimensional v space, —but reduced to an r-dimensional v space, , in cases where the radiation damping matrix D is singular and of rank rPa=0 runs through for example points v=0 and v=2v0. The degenerate ‘ellipsoid’ that corresponds to Pa=Pa,MAX is just one point, which represents the (unconstrained) optimum situation. Choosing smaller Pa results in inpan class="Chemical">creased size of the ‘ellipsoid’. If N=1, then the ‘ellipsoid’ simplifies to a circle in the complex v1 plane. Then, as v10=x1/2D11, we may, from the general equation (6.24), derive |v1/v10−1|2=(Pa,MAX−Pa)8D11/|x1|2=1−Pa/Pa,MAX. Note that a similar simple circle equation may be derived for cases where the radiation damping matrix D is singular and of rank r=1, although N≥2; as exemplified by (7.25), for an axisymmetric system [26, eqn 37].

Considering how the absorbed power Pa varies with v, the relationship (6.24) may be thought of as a ‘<span class="Chemical">paraboloid’ in the complex N-dimensional v space, . The top point of this ‘paraboloid’ corresponds to the optimum, (v0,Pa,MAX). Here, N should be replaced by r in cases where the radiation matrix D is singular.

The simple equation (3.17), which for a fixed Pa represents a circle in the complex U plane, can be shown to be equivalent to (6.24) above, which represents an ‘ellipsoid’ in the complex v s<span class="Chemical">pace, by making use of (3.16), (6.20) and (6.23). Starting from (3.17), we have noting that the collective excitation-power coefficient E(β) is a scalar, and thus equals its own transpose, and pan class="Disease">recalling the hermitian property (6.16) of the radiation-damping matrix D.

The proof (6.25) also serves to demonstrate that, with the generalizations (6.20), equations (3.11)–(3.17) are valid not only for a single, one-mode oscillating body, but even for an array consisting of several WEC units—oscillating bodies, as well as OWCs. In particular, equation (3.17), as illustrated in figure 1, is applicable even to the general case of wave energy absorption by an array of oscillating bodies as well as OWCs. The involved physical quantities refer to the array objects' wave-interacting surfaces. Thus, if inverse Fourier transformation is applied to, for example (3.11) and (6.20), they may be applied to analyse the WEC array's wave-power absorption also in the case of non-sinusoidal time variation.

As long as we have not taken any equation of motion into account, we may here consider the components v of the vector v to be independent variables—assuming that the WEC array contains sufficient control equipment to achieve the desired oscillations. As in the case of a single WEC unit oscillating in one mode (cf. §3), however, all optimum values v, and thus the optimum column vector v0, depend on the excitation vector x(β), and are, consequently, dependent on the incident-wave parameters A and β.

Reactive radiation parameters

Let us now return to consider the reactive power corresponding to the radiated wave in a forced oscillation case, that is, without any incident wave. The imaginary part of the complex r.h.s. term in (6.17) is where we, in the last step, made use of (B 25), which applies for a case with no incident wave. Here, T−V is the time-average difference between kinetic energy and potential energy of the span class="Chemical">pan class="Chemical">water surrounding the WEC array, or more precisely, the paspan>n class="Chemical">water in the near-field region. (Note that, in the far-field region, such an energy difference averages to zero.) Details are discussed in appendix B.

For the case of a single-mode one-body WEC unit, (3.8) corresponds to a time-domain analogue of the frequency-domain equation (6.26), but with energy difference in the mechanical oscillating system itself, rather than in the surrounding pan class="Chemical">waterspan>. For this one-mode case, (6.26) simplifies to where m=m(ω)=X(ω)/ω is the so-called ‘added mass’, a term which may appear confusing in particular cases when it shows up to be negative, that is, in potential-energy-dominating cases where T−V <0 [19]. From (3.8), we may conclude that the time-average difference between kinetic energy and potential energy of the oscillating body itself is (m−c/ω2)|u|2/4, which is positive only if . Resonance may occur for angular frequencies ω=ω0, which satisfy the equation . At resonance, the WEC unit's PTO machinery need not exchange reactive power with the oscillating system.

The values of m or, more generally, of Dreactive at infinite frequency are important for time-domain models. It is well known that the elements of matrix m=X/ω, in general, tend to finite, non-zero, constants at infinite frequency. The infinite-frequency behaviours of the other two matrices which make up the matrix Dreactive are less well known, although Evans & Porter [27] observed that the radiation susceptance matrix B is zero at infinite frequency and Kurniawan et al. [28] reported that the real part, C, of the radiation coupling matrix has elements which tend to finite, non-zero, constants at infinite frequency. A physical explanation for these behaviours is given in the following.

Consider first an array of oscillating bodies with no OWCs, where one of the bodies is forced to oscillate harmonically with a unit velocity corresponding to mode i, in the absence of incident waves, while the other bodies are held fixed. As the oscillation frequency is inpan class="Chemical">crspan>eased to infinity, the acceleration also inpan class="Chemical">creases to infinity. The force required to move the body will necessarily also be infinite. There is therefore sufficient force to accelerate the fluid, which on the wetted body surface needs to move with the same velocity as the body. As the potential energy is zero in this limiting case of infinite frequency (as there are no radiated waves at infinite frequency), while the kinetic energy is positive (as the velocity of the fluid is finite), is necessarily positive, according to (6.27). From (B 1), it also follows that is positive definite and that the off-diagonal elements of , i.e. , are generally non-zero.

Next, consider an array of OWCs with no oscillating bodies, where an oscillating finite pressure is applied on the internal free surface of OWC i, in the absence of incident waves, while the other OWCs are open to the atmosphere. As the oscillation frequency is inspan class="Chemical">pan class="Chemical">creased to infinity, the force on the free surface remains finite since the pressure is finite. There is therefore paspan>n class="Disease">insufficient force to accelerate the fluid, and hence the kinetic energy of the fluid is zero. Since the potential energy is also zero at infinite frequency, must be zero according to (6.26). It follows that all are also zero.

The fact that has non-zero elements may be explained by pan class="Disease">recallingspan> that, in an array of oscillating bodies and OWCs, the radiation coupling coefficient H relates the velocity of rigid-body oscillation mode j to the resulting volume flow apan class="Chemical">cross the internal free surface of OWC i, when it is open to the atmosphere. Since the fluid is assumed to be incompressible, we cannot avoid pan class="Chemical">creating a volume flow by moving the body, even at infinite frequency.

Two-mode wave-energy converter examples

In agreement with (6.20) we may, for the case of N=2 oscillation modes, write the collective excitation-power coefficient as where the complex wave excitation variables x and the complex oscillation amplitudes v are defined by (6.1). Corresponding to the two r.h.s. terms in (7.1), we may, with reference to (6.4) and (6.5), note that also the adjoint radiation Kochin function is, for these examples, composed of two terms. Moreover, the complex collective oscillation amplitude U is determined, firstly, by the phase requirement that A*E*(β)/U is a real and positive quantity, and secondly, by a modulus (magnitude) requirement that where the diagonal entries D11 and D22 are real, and non-negative. Further, according to the general relation D=D†. Moreover, as is evident from (6.13)–(6.15) and associated text, the off-diagonal entries are either purely imaginary, and thus D21=−D12 in the case of one body mode and one OWC mode, or real, and thus D21=D12 otherwise.

According to (6.23), the column vector v0=v0(β), of the optimum complex oscillation amplitudes, has to satisfy the algebraic equation Dv0=x(β)/2. This optimum vector v0 determines the optimum collective parameters E0(β) and U0. Referring to (3.15), the corresponding maximum absorbed power may be expressed as, for example . Considering v=[v1 v2]T as an inde<span class="Chemical">pendent variable in a two-dimensional complex space , the relationship see (6.24), represents a ‘paraboloid’ in , where the top point corresponds to the optimum, v=v0 and Pa=Pa,MAX. See further discussion in §§7.1 and in 8.3.

In the last one of the following three 2-mode examples, §7.1–7.3, which are discussed later, the WEC consists of one OWC and one single-mode oscillating body. Then we set v1=u and v2=p. Moreover, the radiation damping matrix is complex and hermitian, D†=D. In the first two 2-mode examples, only oscillating bodies are involved, and then we set v1=u1 and v2=u2. Moreover, the radiation damping matrix is real and symmetric, D=R=R. In the first example, with one symmetric body in heave and surge, the matrix is diagonal, that is R21=R12=0. Then there is no hydrodynamical coupling between the two modes. In the second example, R21=R12≠0, and, moreover, R22=R11, as we have, for convenience, chosen two equal bodies oscillating in the heave mode, only.

One symmetric body in heave and surge

We shall consider an example with only one immersed body, which has two vertical symmetry planes, one per<span class="Chemical">pendicular to the x-axis and one to the y-axis. This body is assumed to oscillate in just N=2 modes, surge (i=1) and heave (i=2), with complex velocity amplitudes v1=u1 and v2=u2, and excitation forces x1(β)=fe,1(β)A and x2(β)=fe,2(β)A, respectively. The radiated-wave Kochin coefficient is antisymmetric, h1(β+π)=−h1(β), for surge, and symmetric, h2(β+π)=h2(β), for heave. Because of the body symmetry, the radiation damping matrix is diagonal, i.e. D=R=diag(R11,R22); thus there is no hydrodynamical coupling between the two modes, i.e. R12=R21=0. One or both of the diagonal matrix elements R11 and R22 for the body may become zero for certain frequencies but are otherwise positive. Let us, however, restrict the following discussion to sufficiently low frequencies to ensure that R11 and R22 are never zero, but only positive. Then the radiation damping matrix D=R is non-singular in the frequency interval of interest.

According to (7.1) and (7.2) we now have, for this symmetric-body example, It is interesting to note that the last equation here contains two terms, in contrast to the four terms in (7.2). Thus, for this example, (7.4) appears simply as a two-term extension of (3.10). Consequently, because there is no hydrodynamic coupling between the surge and heave modes, the maximum absorbed power may, in agreement with the alternative equations (3.15), be written simply as where and u(β) is the optimum value of u. Here, in the last step, we made use of the Haskind relation (5.1) or (6.3). Using (6.5), we note that Pa,MAX=AE0(β), which is one of the alternative expressions in (3.15).

For this symmetric body, where the radiation-resistance matrix is diagonal, that is R=diag(R11,R22), the last line in (7.3) vanishes, and thus For a fixed value of the absorbed wave power Pa, where Pa<Pa,MAX, this equation represents an ‘ellipsoid surface’ in the complex two-dimensional u space, . The centre of the ‘ellipsoid’ is at the point u=u0. The elliptical semi-axes are for i=1,2. Considering how the absorbed power Pa varies with u, the relationship (7.7) may be thought of as a ‘paraboloid surface’ in the complex two-dimensional u space, . The top point of this ‘paraboloid’ corresponds to the optimum, (u0,Pa,MAX). If, for a fixed value of u2, the ‘paraboloid’ is projected onto the complex u1 plane, this projection corresponds to the axisymmetric surface illustrated in figure 1, but with Pa,MAX−Pa now replaced by Pa,MAX−Pa−R22|u20−u2|2/2, where the last term vanishes if u2=u20. Figure 3, on the other hand, indicates what the ‘paraboloid’ looks like, if projected onto a real plane spanned by the Re{u1/[fe,1(β)A]} and Re{u2/[fe,2(β)A]} axes. Graphical illustrations of absorbed wave power, similar to figures 1 and 3, were previously presented by Evans [29].

Figure 3.

Illustration of equation (7.7) surface cross sections corresponding to Im{u1/fe,1(β)A}=0 and Im{u2/fe,2(β)A}=0. The largest possible absorbed wave power Pa,MAX is indicated by a star on the top of the paraboloid, and colour changes indicate levels where Pa/Pa,MAX equals 0, , and . (a) Side view. The upper parabola and the lower parabola are cross sections, of the paraboloid, in the planes Re{u2/fe,2(β)A}=u20/fe,2(β)A and Re{u2/fe,2A}=0, respectively. (b) Top view. The four ellipses indicated by colour changes are, in order of decreasing size, cross sections of the ellipsoids that correspond to Pa/Pa,MAX equalling 0, , and , respectively.

Illustration of equation (7.7) surface crspan>oss sections corresponding to Im{u1/fe,1(β)A}=0 and Im{u2/fe,2(β)A}=0. The largest possible absorbed wave power Pa,MAX is indicated by a star on the top of the paraboloid, and colour changes indicate levels where Pa/Pa,MAX equals 0, , and . (a) Side view. The upper parabola and the lower parabola are cross sections, of the paraboloid, in the planes Re{u2/fe,2(β)A}=u20/fe,2(β)A and Re{u2/fe,2A}=0, respectively. (b) Top view. The four ellipses indicated by colour changes are, in order of decreasing size, cross sections of the ellipsoids that correspond to Pa/Pa,MAX equalling 0, , and , respectively.

A particular case of a symmetric body, for which (7.5)–(7.7) apply, is an axisymmetric body, which was analysed by Newman [1, §10], who found for this case.

Another particular case of a surging and heaving symmetric body is the famous Evans Cylinder [2]. It is a two-dimensional WEC device, a horizontal circular cylinder, which is submerged below the free pan class="Chemical">water surface. Let the cylinder axis be aligned in the y-direction, and let the incident wave propaspan>gate in the positive x-direction, that is β=0. For this submerged cylinder, there is no reflected wave, that is, no wave diffraction in the up-wave direction. This means that the Kochin function for diffraction, as introduced by (4.1)–(4.3), vanishes in the up-wave direction. Hence, it follows from the principle of conservation of energy, that the transmitted wave has the same amplitude |A| as the incident wave. Consequently, a non-zero diffracted-wave Kochin function coefficient in the down-wave direction cannot contribute to the amplitude, but only the phase of the transmitted wave.

Another feature of the Evans Cylinder is that the Kochin function coefficients h, for radiation, as introduced by (6.2), have the property that h1(β)=ih2(β). Thus if we choose u2=iu1, which corresponds to a circularly polarized oscillation in the clockwise direction if the x-axis is pointing to the right, then Hr(0)=Hr(β)=h1(β)u1+h2(β)u2=(−ii+1)h2(β)u2=2h2(β)u2, while Hr(π)=Hr(β + π)=h1(β + π)u1+h2(β+π)u2= −h1(β)u1+h2(β)u2=(ii+1)h2(β)u2=0. With this circularly polarized oscillation, the radiated waves due to surge and heave are equally large, and they cancel each other in the up-wave direction, but add together constructively in the down-wave direction. Thus, there is neither wave diffraction nor wave radiation in the up-wave direction. All incident wave energy will be absorbed by the Evans Cylinder, provided the circularly polarized oscillation has an optimum amplitude and an optimum phase, in such a way that the down-wave radiated wave exactly cancels the above-mentioned transmitted wave. Half of the incident wave energy is absorbed by each of the two modes, surge and heave.

For the adjoint Kochin function, as defined by (6.4), we now have corresponding expressions, , and . We found, above, that Hr(β)=Hr(0)=2h2(β)u2. Thus, we have if the Evans Cylinder has a circularly polarized oscillation.

Two equal heaving bodies

Let us consider a system of two equal, semisubmerged, axisymmetric bodies with their vertical symmetry axes located at horizontal positions (x,y)=(∓d/2,0). We shall assume that they are oscillating in the heave mode only. With this assumption, the excitation-force vector is of the form x(β)=Fe(β)=[Fe,1(β) Fe,2(β)]T. Further, the radiation damping matrix may be written as Note that the diagonal entry Rd is positive, while the off-diagonal entry Rc, which represents hydrodynamical coupling between the two bodies, may be positive or negative, depending on the distance d between the two bodies.

As explained in appendix A.2, we may assume that the matrix R is non-singular, and hence . According to (A 10), (A 14), (A 19) and (A 21)–(A 23), the maximum wave power absorbed by the two optimally heaving bodies is and the two bodies' optimum complex velocity amplitudes u10 and u20 satisfy We observe that (7.11) has a main algebraic structure similar to that of (7.5) and (7.6), which concern example §7.1, where the resistance-damping matrix is diagonal, R=diag(R11,R22). Before we, in ap<span class="Chemical">pendix A, derived (7.11) and (7.12) we carried out a similarity transformation in order to diagonalize our given radiation damping matrix (7.10); see the similarity-transforming equations (A 5)–(A 11).

From a wave-body-interaction point of view, it is interesting to note that the first r.h.s. term in (7.11) and the first equation of (7.12) correspond to a sub-optimum situation when the two, equal, heaving bodies cooperate as a source-mode (monopole) radiator, that is, when the constraint u2=u1 is applied. Then the two bodies are constrained to heave with equal amplitudes and equal phases. By contrast, the last r.h.s. term in (7.11) and the last equation of (7.12) correspond to a sub-optimum situation when the two bodies are constrained to coo<span class="Chemical">perate as a pan class="Chemical">dipole-mode radiator, that is, when the constraint u2=−u1 is applied. In general, (7.11) and (7.12) may be considered to quantify the optimum situation for this combined monopole–pan class="Chemical">dipole wave-absorbing system.

If the maximum radius of each body is sufficiently small, say less than of a wavelength, it may be considered as a point absorber, for which the heave excitation force Fe is dominated by the Froude–Krylov force, and the diffraction force may be neglected. If, moreover, the centre-to-centre distance d between the two bodies is large in comparison with the maximum body radius, then where F0=σρgπ[a(0)]2A. Here a(0) is each body's span class="Chemical">pan class="Chemical">water-plane radius, and A is the complex amplitude of the incident-wave elevation at the chosen origin (x,y)=(0,0). Further, σ≤1 is a factor that corrects for the diminishing of hydrodynamic pressure with distance below the paspan>n class="Chemical">water surface. (In many cases of practical interest, this correction factor may be approximated to σ≈1.) For this point-absorber case, the entries in the radiation-resistance matrix R in (7.10) are approximately given by [9, eqns 43–44] (see also (A 8)) where J0 denotes the Bessel function of the first kind and zero order. We observe that the matrix R is non-singular, and moreover, are positive, since −10.

Using formulae (7.11)–(7.15), we find and Note that, in general, this maximum absorbed power is not equally divided between these two bodies [9, eqn 51].

We may note from the point-absorber approximation (7.13) that, since F0/A is real, for i=1,2. Correspondingly, we then find from (6.2)–(6.4) that and, similarly, . Thus, for the two considered heaving bodies, we have here explicitly demonstrated that the term containing the integral on the r.h.s. of (5.4) is, as expected, negligible in the point-absorber limit, because diffraction effects are then negligible.

One single-mode body and one oscillating water column

We consider one single floating body that contains one OWC, and we make the simplifying assumption that only one rigid-body oscillating mode is involved. It could be, for instance, a pan class="Chemical">BBDBspan> device structure [30], in a case where the OWC-containing body is restricted to oscillate in the pitch mode only. Otherwise, we shall also discuss an axisymmetric system where the rigid-body structure is restricted to oscillate in the heave mode only.

With this example, the two N-dimensional column vectors v and x, as well as the N×N radiation damping matrix D, introduced by (6.1), as well as (6.15), reduce to the following two two-dimensional vectors: as well as the 2×2 matrix respectively.

In order to determine the maximum absorbed power and the corresponding optimum oscillation, it is convenient to apply similarity transformation as shown in some detail in appendix A. For the present example, the eigenvalues λ1 and λ2 of the radiation damping matrix (7.19) are solutions of the second-degree algebraic equation |D−λI|=λ2−(R+G)λ+RG−J2=0. Thus, λ1 and λ2 are given by The corresponding two eigenvectors, which satisfy (A 2) and (A 4), are

In terms of similarity transformed excitation amplitudes x′(β) and corresponding optimum oscillation amplitudes v′(β), both of which are given below, the maximum absorbed power may, according to (A 10) and (A 11), be written as corresponding to the optimum condition We note that the main algebraic structure is similar here and in (7.5)–(7.6) and (7.11)–(7.12). According to (A 4)–(A 6), the similarity transformed complex amplitudes are given by x′=[x′1 x′2]T=S†x and v′=[v′1 v′2]T=S†v, where S=[e1 e2] is the similarity transforming matrix—see (A 5).

By means of the similarity transformation, (7.3) may be simplified to see also (A 15). We note that (7.24) has an algebraic structure, as well as a geometrical interpretation, similar to that of (7.7).

For the particular case of a heaving axisymmetric body that contains an axisymmetric OWC, we have J2=RG, and thus, from (7.20), we see that λ1=R+G and λ2=0, which means that matrix D, in this case, is singular and of rank r=1 [10, eqn 73]. In this case, there is only one term in the sum on the r.h.s. of (7.24), which simplifies to which represents a circle in the complex v′1 plane. The centre of the circle is at v′1=v′10(β)=x′1(β)/(2λ1)=x′1(β)/(2R+2G), and the radius is . While figure 3 may serve to illustrate (7.24), figure 1 is more relevant as an illustration of (7.25). Because of the singularity of the radiation damping matrix, the similarity-transformed variable v′2 is irrelevant, and may have any arbitrary value, without influencing the absorbed power. The physical reason for the singularity is that both modes, the heaving-body mode and the OWC mode can radiate only isotropic outgoing waves. To realize maximum absorbed wave power, the optimum isotropically radiated wave may be realized by any optimum combined wave radiation from the axisymmetric OWC and the heaving axisymmetric body. The transformed oscillation v′2 corresponds to a situation where the heave mode and the OWC mode cancel each other's radiated waves in the far-field region.

Discussion

In this section, we first compare two versions of the so-called ‘fundamental theorem of wave power’ [17]. We shall discuss, secondly, the direction-averaged maximum absorbed wave power for an array of WEC units, and also, thirdly, the physical interpretation of the absorbed-wave-power surfaces. Finally, we shall comment on a disputed formula applied to the optimum performance of the Evans Cylinder.

The ‘fundamental theorem of wave power’

In this paper, by considering the physical process of wave-power absorption at the wetted surface of an oscillating immersed body, and, more generally, at a WEC array's wave-interacting surfaces, we derived, in §§3 and 6.2, respectively, an LPV version of the ‘fundamental theorem of wave power’, equation (3.11) : Pa=AE(β)+A*E*(β)−|U|2. Moreover, we presented, in §4, a GPV version, equation (4.5): Pa=I(β)+I*(β)−|G|2, where I(β)+I*(β) is the wave-power input through an envisaged surface enclosing all WEC units, and |G|2 is the outgoing wave power through the same envisaged surface, which, for mathematical convenience, is chosen in the far-field region of the generated waves. In §§5 and 6.1 we introduced a mixed, or hybrid, GLPV version, where the LPV parameters |U|2 and E(β), by means of (4.10), (5.2) and (6.5), are expressed in terms of global far-field quantities.

For these versions of the ‘fundamental theorem of wave power’, the r.h.s. has three terms, the sum of two complex conjugate terms minus a real, non-negative, term. The third term of the pan class="Chemical">LPVspan> version—including the Gpan class="Chemical">LPV version—contrary to the GPV version, is quadratically dependent on the oscillation amplitudes, but independent of the incident wave amplitude, while the first and second terms are linear in both kinds of amplitudes. For the pan class="Chemical">LPV version, the first two terms, the excitation power, Pe=AE(β)+A*E*(β), represent the gross power input from the incident wave, while the third term, Pr=|U|2, is the necessary, outward-propagating, radiated power.

With the pan class="Chemical">LPVspan>/Gpan class="Chemical">LPV and GPV versions, the third terms |U|2 and |G|2, which represent energy associated with the radiated waves and the outgoing waves, as given by (4.10) and (4.6), respectively, should be considered as a necessity rather than a power loss. In order to absorb wave energy, it is necessary, firstly, to have wave-diffracting WEC units immersed in the sea, and, secondly, to let the WEC units oscillate and thus produce radiated waves, which interfere destructively with the incident wave. The WEC units need to oscillate in order to receive wave energy.

Before comparing the span class="Chemical">pan class="Chemical">LPV/Gpaspan>n class="Chemical">LPV and GPV versions applied to a point absorber, let us consider a two-dimensional 100% absorbing WEC unit, such as an optimally run Evans Cylinder [2] or a hinged oscillating flap in the down-wave end of a wave channel, a flap that we may consider as an ideal nodding-duck device [23]. For these examples, the LPV equation (3.15) shows that the optimum values of the excitation power Pe and the radiated power Pr correspond to 200% and 100%, respectively. For the GPV version, which does not discriminate between radiated waves and diffracted waves, the optimum values of the input power, Pi=I(β)+I*(β) and the outgoing power Pg=|G|2 correspond to 100% and 0%, respectively. (Note that for a real nodding-duck WEC that absorbs less than 100%, the optimum outgoing power is not zero, and the optimum input power is larger than the maximum absorbed power.) As we shall see below, the two versions show a less drastic difference when applied to a point absorber.

In a case where the WEC array is not oscillating, there is no absorbed wave power, Pa|=0, and also no radiated power, Pr|=0. Then it follows from (3.11) that Pe|=0, and, moreover, from (4.5) that Pi|=Pg|≡Pd, where Pd is the outgoing power associated with the diffracted wave alone. From (4.8), we may note that Pi|=Pd=ρvpvgRe{Hd(β)A*}≥0.

In cases of rather weak diffraction, as with a wave-power-absorbing very small point absorber, Pd may be negligibly small. We may note that, if Hd(θ) is small for all θ (including θ=β), then the r.h.s. of (4.9) is small of second order. Thus, in cases of very weak diffraction, Hd(β) is, approximately, purely imaginary, if we choose A to be real. This matter has been discussed in more detail by Farley [15]. By com<span class="Chemical">paring (4.7) with (4.11) and (4.6) with (4.10), we observe that, for cases where the diffracted wave is negligible compared to the radiated wave, I(β)≈E(β)A and |G|2≈|U|2. Thus, for such cases, there is no great difference between corresponding terms of the pan class="Chemical">LPV/Gpan class="Chemical">LPV and GPV versions of the ‘fundamental theorem of wave power’.

In §3, oscillations, wave forces, power and energy were quantitatively discussed in the time domain, but elsewhere, in this pa<span class="Chemical">per, only in the frequency domain. In the case of non-sinusoidal waves, it may be desirable to carry out analyses in the time domain. In this situation, a time-domain type of the ‘fundamental theorem of wave power’ may be desirable. This type should correspond to an inverse Fourier transform of the pan class="Chemical">LPV version derived in §6.2—or §3 for the one-mode case. It should neither be the GPV version nor the Gpan class="Chemical">LPV version, which are derived and discussed in §§4, 5 and 6.1. These versions cannot represent the instantaneous power absorbed by the WEC, but only the time-average power. With a time-domain analysis, also the reactive-power terms—see §3 and §6.2—need to be taken into account.

Direction-averaged maximum absorbed wave power

For the case of only one immersed WEC unit oscillating in a single mode i=1, we have, in agreement with (4.3) and (5.3), that Hr0(θ)=h1(θ)v10(β), and that . Then the optimum gain function (6.11) simplifies to Since the Kochin function coefficient h1(θ) is a function of geometry and mode of motion, this means that the optimum gain function Gg0 for this case depends on geometry and mode of motion only, and not on the WEC velocity. However, to maximize power absorption, the WEC unit needs to move with optimum amplitude and phase. For an isotropically radiating system, such as a heaving axisymmetric body, the optimum gain function is Gg0=1 and inde<span class="Chemical">pendent of the wave-incident angle β. Then, (6.10) gives the maximum absorption width da,MAX≡Pa,MAX/Jw=Gg0/k=1/k, a well-known result since the mid-1970s.

From (8.1), we find that the direction-averaged optimum gain function is as averaged over all directions of wave incidence, a result reported by Fitzgerald & Thomas [31]. However, in some singular cases, we may find that Gg0,average=0. For instance, any axisymmetric body oscillating only in the yaw mode can, in an ideal fluid, neither radiate nor absorb wave energy, for any frequency. Then Hr(θ)≡0. For a floating semi-submersible platform, as well as for a floating bottle-shaped axisymmetric body that has a relatively small pan class="Chemical">water-plane area, the heave excitation force vanishes at a certain frequency [32, p. 77]. Hence, according to the Haskind relation (5.1), h1(β+π)=0, and then Gg0(β)=0 at this paspan>rticular frequency.

For a general WEC array oscillating in N modes, with N≥2, it is not convenient to apply (6.11) to determine Gg0,average. By means of another mathematical procedure, involving similarity transformation, as applied in appendix A, it is found that Gg0,average, in general, equals the rank r of the radiation damping matrix D—see (A 17). In cases where this matrix is non-singular, the rank of the matrix equals its dimensionality N. Thus, in general, the direction-averaged value of the optimum gain function is equal to an integer in the interval 0≤Gg0,average≤N. For instance, an immersed body may oscillate in N=6 different modes. However, if the body has a vertical axis of symmetry, then Gg0,average=3 in general, or less in exceptional cases [1, §10]. These results extend the findings of Wolgamot et al. [18], who found the result Gg0(β)=N for cases where the, general, hermitian radiation-damping matrix D s<span class="Chemical">pecializes to a, non-singular, real radiation-resistance matrix.

We considered in §7.2 an array consisting of two heaving point absorbers, and we derived formula (7.17) for the maximum absorbed power. To find the direction-averaged maximum absorbed power, we need to integrate from β=0 to β=2π. Since this integral equals J0(kd) (e.g. [33, formula 9.1.18, p. 360]) we see, from (7.17) and the first equation of (7.14), that the direction-average of the absorbed power is |F0|2/(4R0)=2Jw/k. This result was, according to (A 1) or (A 17), to be expected for this non-singular system of two heaving bodies.

Absorbed-wave-power surfaces

The absorbed wave power relative to its maximum may be expressed as two equally general functions of the WEC oscillation amplitudes relative to the optimum amplitudes. The first of these expressions is given in (3.17), which may be illustrated as an axisymmetric paraboloid on the complex collective-amplitude U plane (figure 1). The second expression is given in (6.24), which may be thought of as a ‘<span class="Chemical">paraboloid’ in the complex N-dimensional v space, (see figure 3 for an example with N=2).

In popular terms, it might be useful to think of this ‘paraboloid’, for <span class="Chemical">Pa>0, as a single ‘mountain island’ in a ‘world’, where there is otherwise, for Pa<0, only an infinite ‘ocean’ (cf. figures 1 and 3). The top of the absorbed-power ‘mountain’ corresponds to optimum, and the ‘shore’ of the ‘island’ to Pa = 0. This ‘mountain’ top can be reached only if no technical or practical constraint prevents the required complex amplitudes v from being realized for all i=1,2,3,…,N.

For practical reasons, it may not be possible to realize the optimum condition v=v0. Note that all components of the excitation vector x(β), and hence also of the optimum oscillation amplitude vector v0(β), are proportional to the complex amplitude A of the wave elevation of the undisturbed incident wave, and that Pa,MAX is proportional to A*A=|A|2. As oscillation amplitudes cannot exceed their design-s<span class="Chemical">pecified limits, it will not be possible to realize the despan class="Chemical">cribed optimum situation, if the amplitude of the incident wave exceeds a certain pan class="Chemical">critical value. With such constraints, or for other technical reasons preventing realization of the optimum condition (6.23), the practical, constrained-case, maximum absorbed power Pa,max will be less than the ideal Pa,MAX. (In such a case, it will not be practically possible to ‘climb’ to the ‘top’ of the above envisaged ‘mountain island’.)

A disputed 1979 formula

For the optimum case, the maximum absorption width may be expressed as according to (6.9)–(6.11). The single-mode version of this formula was derived by Newman [25] and, independently, by Evans [8]. For the case of more than one oscillation mode, we have found it necessary to introduce the adjoint Kochin function in the numerator (see (5.3) and (6.4)).

It should be emphasized that formula (8.3) applies only to the optimum case, as it is based on the fact that the optimum radiated power is equal to the maximum absorbed wave power. Neither any wave force nor the incident wave amplitude is explicitly present in (8.3). However, as each component v of the optimum oscillation vector is proportional to the incident wave amplitude, the numerator, as well as the denominator, of fraction (8.3) is proportional to the square of the incident wave amplitude.

The controversy [15,16]—concerning the numerator in formula (8.3)—is mainly related to the Evans Cylinder, which we, in the last three paragraphs of §7.1, discussed in some detail. Let this cylinder be aligned <span class="Chemical">perpendicular to the incident wave direction. At optimum oscillation, this submerged horizontal cylinder absorbs all incident wave energy. Then the optimum radiated wave has to propagate only down-wave in order to cancel the transmitted wave, as there is no reflected wave to cancel up-wave. In agreement with this physical observation, it is reassuring to observe that we, according to (7.9), which is valid for the Evans Cylinder, may replace by |Hr0(β)| in the numerator of (8.3). Such a replacement seems to resolve the controversy, because Hr0(β)=h1(β)u10+h2(β)u20 is the Kochin function for the down-wave radiated wave when optimum wave-power absorption is actually taking place, that is, when the rotating Evans Cylinder's surge and heave modes' complex velocity amplitudes have their optimum values, u10 and u20, respectively. In comparison, is the adjoint Kochin function, which corresponds to a wave that is radiated in the opposite direction if the rotating Evans Cylinder is oscillating with opposite sense of rotation. Observe that to replace u by corresponds to time reversal, since (ei)*=ei.

It seems that we have to include the relation (8.3) among the reciprocity relations between physical quantities about which Newman [1, §7] expressed that they ‘are not physically related to each other in any obvious manner’.

Conclusion

After the petroleum span class="Chemical">pan class="Chemical">crisis in 1973, the basic theory for primary wave-energy conversion was developed during the mid- and late 1970s and the early 1980s. Different versions—the paspan>n class="Chemical">LPV version (cf. §§3 and 6.2), the GLPV version (cf. §§5 and 6.1) and the GPV version (cf. §4)—of the so-called ‘fundamental theorem of wave power’ have given rise to some controversy even during recent years. Comparative discussion of these different versions has been presented in §8.1. The GLPV version, in particular, is mathematically convenient when proving some useful reciprocity relations, as applied, for instance, by Newman [1]. It is, however, difficult to give a physical interpretation of some of these relations and of the GLPV version. This may be the cause of recent controversy concerning the GLPV version. All of these versions provide, however, the correct value of the time-average absorbed wave power. Hopefully, the discussion in §8.4 helps to do away with some of this controversy.

The pan class="Chemical">LPVspan>, the GPV and the Gpan class="Chemical">LPV versions express, respectively, the absorbed wave power Pa in terms of the WEC units' complex oscillation amplitudes, in terms of the outgoing (diffracted plus radiated) wave, and in terms of the radiated wave alone. For mathematical convenience, the outgoing and radiated waves in the far-field region are considered, explicitly.

For a general WEC array consisting of oscillating immersed bodies and OWCs, we have found it convenient to introduce complex collective parameters, the collective oscillation amplitude U and the collective excitation-power coefficient E(β) (see (3.10) and (6.20)). Then it is, even for a WEC array, a rather simple algebraic exercise to derive expressions for the maximum absorbed wave power <span class="Chemical">Pa,MAX and the corresponding optimum values U0 and E0(β) (see (3.11)–(3.15)). Moreover, we may illustrate the dependence of the absorbed wave power Pa versus U as an axisymmetric paraboloid; cf. (3.17) and figure 1. For an N-mode WEC array, we may, in greater detail than figure 1, consider the real-valued Pa as represented by a paraboloid in an N-dimensional complex v space . pan class="Chemical">Cross sections of such a paraboloid are, as an example, illustrated in figure 3. Mathematically, the mentioned paraboloids are represented by rather simple mathematical expressions (3.17) and (6.24), which may be considered as alternative variants of the pan class="Chemical">LPV version of the ‘fundamental theorem of wave power’.

In contrast to the GPV version and the GLPVspan> version, only the LPV version is applicable for the purpose of quantifying the instantaneous absorbed wave power. Then it is necessary to take also the reactive power into account. When deriving the LPV version (cf. §§3 and 6.2), we also discussed the reactive power that is associated with wave-power absorption. In appendix B, we have derived expressions that relate the ‘added-mass’ matrix, as well as a couple of additional reactive radiation-parameter matrices, to the difference between kinetic energy and potential energy in the water surrounding the immersed oscillating WEC array. To the best of the authors' knowledge, some of these derived relations are new results (e.g. (B 25), (B 27) and (B 28)). In appendix B.5, we have also derived new relations concerning reactive power associated with the incident wave.

In appendix A, we applied similarity transformation of the radiation damping matrix to derive a formula for the direction-average maximum absorbed wave power <span class="Chemical">Pa,. Correspondingly, as discussed in §8.2, we found that the direction-average value Gg0,average of the optimum gain function Gg0(β)—defined by (6.11)—equals an integer in the interval 0≤Gg0,average≤N, where N is the WEC array's number of modes of oscillation (number of degrees of motion). Only when the radiation damping matrix is non-singular, we have Gg0,average=N, as derived by Wolgamot et al. [18, eqn 21] for an N-mode WEC array consisting of oscillating bodies only. Thus, our result is an extension of theirs, to WEC arrays that may contain OWCs and also may have a singular radiation damping matrix. In general, Gg0,average equals the rank of this N×N matrix.