## Abstract

For a weak point source or dipole, or a small body operating as either, we show that the power from a wave energy converter (WEC) is the product of the particle velocity in the waves, and the wave force (suitably defined). There is a thus a strong analogy with a wind or tidal turbine, where the power is the product of the fluid velocity through the turbine, and the force on it. As a first approximation, the cost of a structure is controlled by the force it has to carry, which governs its strength, and the distance it has to be carried, which governs its size. Thus, WECs are at a disadvantage compared with wind and tidal turbines because the fluid velocities are lower, and hence the forces are higher. On the other hand, the distances involved are lower. As with turbines, the implication is also that a WEC must make the most of its force-carrying ability—ideally, to carry its maximum force all the time, the ‘100% sweating WEC’. It must be able to limit the wave force on it in larger waves, ultimately becoming near-transparent to them in the survival condition—just like a turbine in extreme conditions, which can stop and feather its blades. A turbine of any force rating can achieve its maximum force in low wind speeds, if its diameter is sufficiently large. This is not possible with a simple monopole or dipole WEC, however, because of the ‘*nλ*/2*π*’ capture width limits. To achieve reasonable ‘sweating’ in typical wave climates, the force is limited to about 1 MN for a monopole device, or 2 MN for a dipole. The conclusion is that the future of wave energy is in devices that are not simple monopoles or dipoles, but multi-body devices or other shapes equivalent to arrays.

## 1. Introduction

Wave energy is one of the great unsolved engineering problems of our time—is it inherently uneconomic, or can wave energy be extracted economically? This paper looks at the key theoretical features of the problem, to try and reveal why the problem is difficult, and by what means it might be solved. Attention is restricted to devices in an infinite ocean, of any depth—devices mounted in sea walls are excluded, because the number of sites is limited.

## 2. The relationship between power and force: monopole and dipole cases

The ‘fundamental theorem of wave power’, which is implicit in the work of Falnes and his contemporaries (the result was derived by several authors independently, as described elsewhere in this Theme Issue—its derivation is given for completeness in appendix A), relates the power from a wave energy converter (WEC) to the amplitude of the incident and ‘produced’ waves. By produced wave, we mean everything apart from the incident wave. In the usual terminology [1], pp. 287–288, the produced wave would be considered as the sum of the diffracted (scattered) wave produced around the fixed WEC, and the additional radiated waves produced by its motion—but we have no need of that additional breakdown here.

We take the amplitude of the incident wave in the usual way as *R*{*a*e^{i(kx−ωt)}}, where *a* is the complex wave amplitude, *ω* is the wave frequency (=2*π*/period), *k* is the wavenumber (=2*π*/wavelength) and *x* is the distance measured from the WEC, in the direction of wave propagation. The complex amplitude of the far-produced waves can be expressed in polar coordinates (*R*, *θ* taking *R*=0 when *x*=0, and *θ*=0 as the positive *x*-axis) as
2.1for large *R*, where *f*(*θ*) is some function of *θ* alone. The wave amplitude has this simple form at large *R* because the produced waves will ultimately decay inversely as the square root of the distance *R* from the centre of the device, by energy conservation. The fundamental theorem of wave power then states that the power absorbed by the WEC is
2.2Here *ρ* is the density of water, *g* is the acceleration owing to gravity, ℜ denotes the real part, the overbar denotes the complex conjugate and the underbar denotes the average value over all *θ*. The first term in the brackets is the power input term, which depends just on the value of *f*(*θ*) at *θ*=0 (the ‘forward scattering’), and the second is the power-loss term, which represents the power lost in all directions by wave radiation.

The simplest case is a point source of (complex) mass flow rate *q*, close to the surface. This generates a velocity potential of *q*/(−4*πρr*) a small distance *r* away; see Lighthill [2], §1.4, who describes the equivalent acoustic case. The mass of water displaced is *q*/(−i*ω*); so the point source feels a (complex) buoyancy force *F*=*qg*/(−i*ω*). We can thus write the velocity potential as (−i*ωF*/*g*)/(−4*πρr*). The relation between this near field and the far field is given in Linton & McIver [3], eqn B.67 (with as in eqn B.77). Compared with Linton & McIver [3], we have introduced a multiplier (−i*ωF*/*g*)/(−4*πρ*), and to evaluate *f*(*θ*) in (2.1) we must also obtain the wave amplitude by taking the velocity potential on the surface and adding an additional multiplier i*ω*/*g*. We thus obtain overall
2.3The next simplest case is a horizontal point dipole aligned with the wave direction, and also close to the surface. This can be envisaged as a point source of mass flow rate *q* at *x*=0, and another of mass flow rate −*q* at *x*=−*l*, where *l* is some small distance that we reduce with *l*(d*q*/d*t*)=*l*(−i*ω*)*q* being held constant (see [2], §1.5). The force *F* felt by the dipole from the water, in the direction of positive *x*, is then *l*i*ωq* (see [2], p. 30). Thus for the wave amplitude in the far field, we can apply eqn B.67 of Linton & McIver [3] to each source, with a multiplier *F*/(*liω*) instead of (−i*ωF*/*g*). We thus obtain
2.4These expressions (2.3) and (2.4) for *f*(*θ*) can now be inserted into (2.2) to obtain the power as
2.5At the location of the source or dipole, the particle velocity in the incident wave is −i*ωa* upwards and *ωa* in the direction of positive *x*. Thus, we reach the important conclusion that the power input term is simply the product of the wave force on the device, and the particle velocity in the incident wave, in the direction of the force—and, of course, the two need to be in phase to maximize the power. The second (radiation loss) term rises with the square of the force on the device, and so will ultimately dominate if the force is increased. It is considered in appendix A, which shows that for maximum power it should be half the energy input term, and gives the well-known result, owing to Falnes and his contemporaries, that the capture widths are then 1/*k* and 2/*k*, respectively, for a simple source and a dipole (better known as the ‘*λ*/2*π*’ and ‘2*λ*/2*π*’ limits, where *λ*=wavelength).

In terms of the economics of wave power, however, it is the dependence of the power input term on the wave force and the particle velocity that is significant. This matter is considered in more detail in §§4 and 5.

## 3. Small-body theory

These arguments reduce to a much simpler physical form when we consider the special case of the weak source or dipole produced by a small body. For a source, the body is oscillated vertically on the water surface like a buoy; for a dipole, it is oscillated horizontally beneath the water surface like a small submarine. In both cases, the produced wave amplitude *f*(*θ*) falls as the size of the body is reduced, so that, for a small body, the second (power loss) term in (2.2) (and thus the second term in (2.5)) vanishes in comparison with the first (power input) term.

The disturbance to the incident wave falls too, and we can accordingly calculate the force on the body (and into the notional power take-off connected to the sea-bed) simply from the properties of the incident wave. For the source case, we can calculate the wave force by Archimedes' principle, from the varying immersed volume, as shown in figure 1. There is an additional wave force that the buoy feels from the acceleration in the incident wave, owing to its finite volume (unlike the point source and dipole of the previous section, which have zero volume), but this is negligible in comparison because, in the linear theory, this acceleration is small compared with *g*.

If the buoy motion relative to the incident wave surface is as shown, then, from Archimedes' principle, the varying buoyancy force is , where *B* is the water-plane area, so the power is
3.1The mean power is maximum when *φ*=90^{°} (we assume that this can be engineered, e.g. by a servomechanism, as well as the relative motion amplitude *a*), and is then *ρgBbωa*/2 or *ωa*/2 times the amplitude *ρgBb* of the varying buoyancy force. Exactly as we obtained in (2.5). Note that the additional velocity *ωb* of the buoy is immaterial—it can be much larger than that of the water particles, but that is irrelevant.

A dipole device, such as a small bottom-mounted flap, with the power take-off at the sea-bed hinge, is shown in idealized form in figure 2.

This time, we cannot ignore the wave force that the body feels from the acceleration in the incident wave, owing to its finite volume. It is the force being transmitted by the body to the power take-off that is relevant. When this is zero, the body moves back and forth like a water particle (assuming it is neutrally buoyant), i.e. with the wave amplitude *a*, as shown. If the body has an additional horizontal motion as shown, the force transmitted to the power take-off will be minus the additional mass–acceleration, i.e. , where *M* is the body's mass plus added mass. Thus, the power is
3.2The mean power is maximum when *φ*=90^{°} (we assume this can be engineered, as before), and is then *Mω*^{2}*bωa*/2 or *ωa*/2 times the amplitude *Mω*^{2}*b* of the force transmitted to the power take-off. Again exactly as we obtained in (2.5). And again the additional velocity *ωb* of the body is irrelevant.

## 4. Analogy with turbines: economics of wave energy converters compared with wind and tidal turbines

For turbines, the equivalent of the ‘fundamental theorem of wave power’ is the single-streamtube theory of a turbine in an open flow, given for completeness in appendix B. It relates the downstream force *F* on a turbine of area *A* to the velocity (*V* −*δV*) through the turbine, compared with free-stream velocity *V* , thus
4.1Hence, the turbine power is
4.2This is strongly analogous to (2.5). As there, the first term is a power input term that is the product of the free-stream fluid velocity and the force on the device. And the second term is a power-loss term (here caused by the flow diverting around the turbine) that involves the square of the force on the device, and so will ultimately dominate if the force is increased. And again, the well-known result, given in appendix B, is when the force is optimized for maximum power. For a turbine, this is the Betz limit, at which *δV* =*V*/3.

As a first approximation, the cost of any structure is related to the force it has to carry (the ‘load’), and the distance it has to be carried (‘the length of the loadpath’, to continue the structural engineering jargon). Thus, we see from (2.5) and (4.2) that a WEC is at a disadvantage compared with a wind or tidal turbine, because particle velocities in waves are on average less than those in tidal streams (the high velocities seen in breaking wave crests are misleading—average velocities are much lower) and much less than wind velocities (the lower density of air is immaterial, because it affects power and force equally).

On the other hand, the distance over which the force is to be carried is typically less on a WEC than on a tidal turbine. And much less than on a wind turbine, where the force on the blades has to be carried back to the hub and then down to the ground, a distance of typically 100 m or so.

## 5. Further analogy with wind turbines: need for wave energy converters to be nonlinear to be economic

To optimize the economics of a wind turbine, we do not operate at the Betz limit in strong winds because this would require a very strong and costly turbine, capable of withstanding very large forces *F*. Instead, we shed power above a certain modest wind speed (the rated speed), and in theory should hold the force *F* at its maximum value, so that the power will continue to rise, albeit less rapidly. In practice, wind turbines rarely do this because either the torque would have to increase (which would mean a more expensive gearbox) or the tip speed would have to increase, which would generate unacceptable aerodynamic noise. However, it remains a theoretical ideal to ‘sweat the assets’ in the accountants' jargon, and operate at maximum force *F* whenever possible. The ideal would be a ‘100% sweating turbine' with a very low rated wind speed, above which it operated at its maximum force *F*, thereby achieving this maximum force *F* (almost) all the time.

The same reasoning applies to WECs, where again cost is sensitive to the maximum wave force *F* that the WEC can withstand. This time the wave height is varying very rapidly, from wave to wave. A ‘100% sweating WEC’ would be a nonlinear device that continuously adjusted itself to maintain the wave force *F* constant despite the varying wave height. The archetypal such device is the latching buoy, proposed 25 years ago by Budal and Falnes, as described elsewhere in this Theme Issue (figure 3).

In still water, the buoy is 50 per cent immersed, and connected to the sea-bed by a vertical line under tension (the buoy is light), which serves as a power take-off as it is reeled in and out. On the front of a wave, it is fully immersed (the buoy is small) and the line is at maximum tension as it reels out with the rising buoy, generating power. On the back of a wave, the buoy is at zero immersion, and no power is needed to reel the line back in. Whatever the wave height or period, the force on the line is at its maximum—it is a ‘100% sweating WEC’. Because of the relation (2.3) between produced waves and force, the produced waves are limited also; so the buoy is becoming more and more transparent to the waves as the wave height increases. Ultimately, in the survival condition, it is almost completely transparent to them, and can in fact become completely transparent by ceasing to produce power altogether. This is obviously highly desirable from the survivability point of view, and is just like a wind turbine, which can stop and feather its blades in extreme conditions.

Note that the nonlinear operating principle of the buoy maintains the force always in phase with the vertical velocity, as required by (2.5) for maximum power. This is the feature originally highlighted by Budal and Falnes, and is much superior to a buoy operating linearly, which can only achieve this optimum phase when at its resonant frequency. For a small buoy, the bandwidth of this resonance will be very small. (From eqn 173 of Newman [1], p. 304, the half-power bandwidth of a small buoy of optimum capture width 1/*k* is *Bk*^{2}.)

## 6. The size limit on monopole and dipole wave energy converters

We have seen that, for a ‘100% sweating WEC’, we need the maximum wave force *F* to be reached in small waves; just as with a ‘100% sweating wind turbine’, we need the rated wind speed to be low. In principle, a wind turbine can be given a very low rated wind speed, without reducing the wind force *F* (and thus the power), by making the turbine diameter very large.

This is not true with a simple monopole or dipole WEC, however—we cannot keep increasing the size to maintain *F*, because of the ‘*nλ*/2*π*’ limits mentioned at the end of §2. To see what the practical limit on *F* is, we consider a typical wave climate, say that west of the Hebrides, which is given in table 1 below.

For the present purposes, we can assume that the spectra are all narrow-banded, so that *ω* is fixed at 2*π*/*Te*. And, from the Rayleigh distribution, the mean value of the wave amplitude *a* is 5/4×r.m.s. amplitude=5/16×significant wave height. For each cell in table 1, we can thus evaluate the mean value of the particle velocity amplitude *ωa* and so calculate the power from a ‘100% sweating buoy’ that is sweating at a force of 1 MN, say (so the swept volume *Bb* is 100 m^{3}). Multiplying by the probability of occurrence of the cell gives the contribution to the annual average power, as shown in table 2. The cells making contributions above 3 kW, totalling 80 per cent of the annual average power, are shaded grey.

It may be seen that significant wave heights below 1 m do not make an important contribution to annual average power, although they are relatively common—the buoy is still sweating 100 per cent then, but the wave height is too low to generate much power. And significant wave heights above 6.5 m do not make an important contribution either—not because the buoy is not generating lots of power, but because these wave heights are too rare.

The final step in the argument is to consider the capture width (i.e. power/(power in waves, per unit crest length)) in the cells. This is given in table 3 below, which shows the same ‘important’ cells, highlighted in grey, as in table 2.

We immediately see that in a significant number of cells, where the figures are shown in italics, the capture width in wavelengths is greater than the theoretical limit of (2*π*)^{−1} for a monopole. We thus conclude that monopole WECs which aspire to ‘100% sweating’ status cannot have maximum force ratings much above 1 MN.

Exactly the same arguments apply to dipole WECs. The only difference is that they will face the dipole capture width limit of 2*λ*/2*π* rather than the monopole limit of *λ*/2*π*. Thus, their maximum force rating will be limited to about 2 MN rather than 1 MN.

## 7. Implications for the future of wave energy

The problem with small WECs is that, like small ships, they tend to be uneconomic because of their operational costs—an annual routine maintenance visit, for example, can cost as much as on a much larger WEC (the size of the maintenance man is the same!), but is supported by much less revenue. As pointed out by Jefferys [5], it is the *net* revenue that counts—if the operational cost is greater than the gross revenue, the device will be uneconomic even if its capital cost is zero.

The implication is that WECs cannot be single monopole or dipole devices, but must be combinations of many small ‘highly sweating’ monopoles or dipoles, into a single maintainable unit.

Several of the WECs described elsewhere in these proceedings follow this recipe. Pelamis is clearly a line of small monopoles, over a wavelength long (see [6]), and has a force limit set hydraulically, low enough that it is reached for a significant portion of the time. Oyster also has a hydraulic force limit, is long enough to be more than a single dipole in short waves, and is intended as one component of a line of devices (parallel to the wave crests, rather than 90^{°} to them like Pelamis) sharing the same hydraulic machinery. Both devices are almost completely transparent to the waves if they release hydraulic pressure, which ensures excellent survivability. Anaconda is like Pelamis—if operated with a nonlinear distributed power take-off that limits the pressure excursions (and thus the bulge amplitude), it is a line distribution of ‘100% sweating’ monopoles. If the power take-off can limit the pressure excursion to zero, the survivability is also excellent.

## Appendix A. The fundamental theorem of wave power

This result can be readily derived, as in Rainey [6], by considering the energy flux (power) in the far field . This is the product of the wave pressure and the wave velocity, integrated over a fixed cylindrical control surface where *R* has some large value *R*_{C}. The product of wave velocity and hydrostatic pressure does not count because it is purely oscillatory, and the flux of kinetic energy does not count because it is of higher order in wave height. The mean value of the product of any two complex oscillatory variables {*a*e^{−iωt}} and {*b*e^{−iωt}} is ; so if the incident and produced complex wave pressures are *p*_{I} and *p*_{P}, and their complex velocities into the control surface are *v*_{I} and *v*_{P}, the mean energy flux into the control surface is
A1Here, the variables are taken at the still-water position, the integration over the depth *z* having yielded the factor 1/(2*k*), because of the exponential decay e^{kz} of both pressure and velocity. We now observe that:

— the

*p*_{I}*v*_{I}term must integrate to zero because there is zero mean energy flux from the incident waves alone,— the

*p*_{P}*v*_{P}term must always give an energy loss because the produced waves are leaving the system, and— the cross terms must, therefore, be the source of any energy input.

Dealing with the energy-loss term first, and taking the pressure and velocity from the wave amplitudes given in §2, this is readily integrated as
A2the underbar denoting the average value over all *θ*, as in (2.2). The energy input from the cross terms is
A3The energy input evidently fluctuates rapidly around the control surface, because of the e term, with the significant peak being at *θ*=0, i.e. downwave of the WEC. This is because at that point the produced waves are travelling in the same direction as the incident waves, so that the total energy flux is proportional to the square of their combined amplitude. Depending on their relative phase, this is either greater or less than the sum of their individual amplitudes squared, so requiring a substantial cross term. At *θ*=±*π*, on the other hand, the integrand is zero because the waves are travelling in opposite directions, and their total energy flux is thus equal to the sum of their individual fluxes (as is well known, and can be seen from (A1) given the opposite directions of the velocities under wave crests). The required integral can be found exactly by Stokes’ method of stationary phase [7, art. 241]. Only the energy flux around *θ*=0 is significant to the integral, and (A3) becomes
A4Thus, the net power production will be (A4)+(A2), i.e.
A5Given |*f*(*θ*)|, the energy-loss term (A2) will be fixed, but the energy input term (A4) will vary with the phase of *f*(0)—evidently for maximum power input we require *f*(0) to have the phase e^{−iπ/4}.

The well-known results are the special cases when the WEC is a point source (monopole) or a dipole. Here *f*(*θ*)=*ca* and , where *c* is a constant representing the source or dipole strength. Assuming we maintain the optimum phase relationship just described, the power will then be
A6

We can easily find the optimum values of *c* as 1 and 2, respectively, which gives the powers as
A7

Because the power in a water wave is *ρgωa*^{2}/(4*k*) per unit crest length, the capture widths come to 1/*k* and 2/*k*, respectively.

It should also be noted that (A5) is not the only candidate for the title ‘the fundamental theorem of wave power’—some authors give prominence to a further development of (A5), owing to Newman, which eliminates the diffracted (scattered) waves. See Mei [8], where (A5) is eqn 9.15, and Newman's further result is eqn 9.26. At first sight, this further result appears to depend on the wave radiation in the opposite direction, i.e. *f*(*π*) rather than *f*(0), but on closer inspection it may be seen that the amplitude of the relevant term (where *V* _{α} are the modal velocities, and *f*_{α}(*θ*) is *f*(*θ*) for unit velocity in each mode) is equal to
A8

So the wave radiation amplitudes are those produced by WEC motions of A9in other words by the motions that would be seen in a cine film of the actual motions, played backwards. Thus, for example, the Bristol Cylinder [9] would rotate in the opposite direction to reality, and accordingly radiate waves in the opposite direction. This is a relatively sophisticated viewpoint, not pursued in this paper.

## Appendix B. The single-streamtube theory of free-stream turbines

In this simple classical theory, the flow through the turbine of area *A* is imagined as a single streamtube, in which the flow velocity far upstream is the free-stream velocity *V* , and slows down (with the streamtube widening accordingly) to *V* −*δV* as it passes through the turbine, and the n slows down further (with further widening of the streamtube) to *V* −*δ***V* far downstream. Applying Bernoulli's theorem separately to the upstream and downstream halves of the streamtube, and noting that the pressure far upstream and downstream must equal the free-stream pressure (because the streamtube is parallel-sided there), we conclude that the pressure difference Δ*p* across the turbine must be
B1We can also calculate the force *F*=*A*Δ*p* on the turbine by considering a control volume bounded by two planes far upstream and downstream (both perpendicular to the velocity *V*) and elsewhere by distant streamlines. The pressure on this bounding surface is everywhere the same; so the force on the turbine must be attributable to the deficit in the momentum flux, which is
B2

By equating the two expressions for the force *F*, we conclude that *δ***V* =2*δV* , i.e. the velocity deficit far downstream is twice the velocity deficit at the turbine. We can thus write the force on the turbine in terms of the velocity deficit *δV* there alone,
B3This is the result required in §4. The well-known result, however, is the value of *δV* for maximum power—the Betz limit. For this, we consider the power *F*(*V* −*δV*), which comes to
B4This can easily be shown to have a maximum when *δV* =*V*/3.

## Footnotes

One contribution of 18 to a Theo Murphy Meeting Issue ‘The peaks and troughs of wave energy: the dreams and the reality’.

- This journal is © 2011 The Royal Society