Maximum Wave-power Absorption

7/30/2019 Maximum Wave-power Absorption

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arXiv:1112.3494v5

[physics.flu-dyn]14Nov2012

Maximum wave-power absorption

by attenuating line absorbers

under volume constraints

Paul Stansell, David J. Pizer

Pelamis Wave Power Ltd., 31 Bath Road, Edinburgh,

UK, EH6 7AH

Abstract

This work investigates the consequences of imposing a volume constraint on the maximum power

that can be absorbed from progressive regular incident waves by an attenuating line absorber heav-ing in a travelling wave mode. Under assumptions of linear theory an equation for the maximumabsorbed power is derived in terms of two dimensionless independent variables representing thelength and the half-swept volume of the line absorber. The equation gives the well-known resultfor a point absorber wave energy converter in the limit of zero length and it gives Budals upperbound in the limit of zero volume. The equation shows that the maximum power absorbed by aheaving point absorber is limited regardless of its volume, while for a heaving line absorber whoselength tends to infinity the maximum power is proportional to its swept volume, with no limit.Power limits arise for line absorbers of practical lengths and volumes but they can be multiples ofthose achieved for point absorbers of similar volumes. This conclusion has profound implicationsfor the scaling and economics of wave energy converters.

Keywords: Wave power absorption, attenuating line absorbers, point absorbers, volume

constraints.

Corresponding authorEmail addresses: [email protected]

(Paul Stansell), [email protected] (DavidJ. Pizer)

Preprint submitted to Elsevier 15th November 2012
http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5http://arxiv.org/abs/1112.3494v5


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1. Introduction

A large variety of wave energy converters(WECs) have been proposed, with a few demon-strated at full scale. This paper is concernedwith the following two categories of buoyancy-driven heaving WECs: the archetypal point ab-sorber which is a single semi-submerged floatwith horizontal dimensions a that are small com-pared to the wavelength of incident wave; andthe archetypal attenuating line absorber whichis a continuous line of semi-submerged floatswith its long dimension orientated in the dir-ection of travel of the incident wave, its width asmall compared to the wavelength and its length

l of the order of a wavelength.It is well established from linear water-wave

theory that the upper limit of the capture widthof a heaving point absorber is /2 (see, forexample, Evans [1], Mei [2], Newman [3]). As increases, however, reaching a capture widthof /2 requires an increasingly large sweptvolume. Of more practical significance are theresults of Evans [4] and Pizer [5] that accountfor a constraint on the motionand, therefore,the swept volumeof the float.

Farley [6] provides a theory for the capturewidth of an attenuating line absorber oscillat-

ing in a travelling wave mode, but he does notincorporate a motion constraint. Newman [7]does consider a motion constraint but his the-ory is restricted to modes of motion that haveuniform temporal phase and so he does not solveexplicitly for the more practical travelling wavemode. The present work follows the approachesof Farley [6] and Rainey [8] in considering thetravelling wave mode response, and it derivesthe maximum capture width with a motion con-straint to limit the maximum swept volume ofthe device.

The method calculates the capture width byconsidering the radiated and diffracted wavesfrom a line composed of a continuum of pointwave sources and their interaction with the in-cident wave in the far-field. The motion con-straint is applied by matching the far-field ra-diated wave amplitude with the swept volumeof a line segment using the relative motion hy-pothesis. Example calculations are provided forpoint and line absorbers of different volumes inwave conditions of practical interest. The effectsof variations in wavelength and wave height arealso illustrated.

2. Background Theory

2.1. Coordinate systems

The coordinate systems and related notationare described here. Let denote an iner-tial reference frame described by the Cartesiancoordinates (x,y,z) (x1, x2, x3) with basis(ex, ey, ez) (e1, e2, e3) and by the cylindricalpolar coordinates (r,,z) with basis (er, e, ez).The origins of these coordinate systems coin-cide, the z-axis is directed vertically upward, theplane of the undisturbed free surface is at z = 0and the fluid bottom is at z = h. When = 0the unit vectors er and ex are equal. For anyvector n, the following notations are equivalent:n n /x /n n, where is the

gradient operator and can take values from{1, 2, 3} or {x,y,z}.

2.2. Governing equations

Consider a rigid body floating on the surfaceof an inviscid, irrotational, incompressible fluidand interacting with a plane incident wave. Allmotions are assumed to be time-harmonic withangular frequency . It is assumed that thebody-induced perturbations are small so thatthe linearised theory of the interaction of waterwaves and structures can be applied and higher-

order effects can be neglected.Under these assumptions the fluid is describedby either its real velocity potential, , or bythe time-independent complex amplitude of thevelocity potential, , where

=

eit

. (1)

The velocity potential, , must satisfy theLaplace equation,

2 = 0, h < z < 0. (2)

In the linearised approximation for small amp-litude waves and constant atmospheric pressureat z = 0, the boundary conditions at the seabed and the free surface are

z= 0, z = h, (3)

2 + g

z= 0, z = 0, (4)

where g is the acceleration due to gravity.The boundary condition on the wetted sur-

face, denoted by SB, of a heaving body is ob-tained by equating the normal velocity of the

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body to the normal velocity of the fluid on SB.The result is

n =

i n

zAz

on SB

, (5)

where n is the unit vector in the direction of thenormal pointing into the body at any point onSB, nz is the component of n in the directionof ez, and Az is the complex amplitude of theoscillatory heave motion of the body about itsmean position.

2.3. Extraction of wave energy

The time averaged power, denoted by P,transferred from the fluid to the body is ob-tained by integrating the time average of thefluid pressure, p, times the fluid velocity, /n,over SB . This may be expressed as1

P =

SB

1

T

T0

p

ndt

dS, (6)

where T = 2/ is the period of oscillation.Note that P is the time average of the rate ofchange of work done on the body so that a pos-itive value of P represents energy absorbed bythe body from the fluid.

The pressure field in the fluid is given by thelinearised Bernoulli equation which can be writ-ten as

t

+ gz + p

= p0

, (7)

where p0 is the pressure at z = 0, g is the ac-celeration due to gravity and is the fluid dens-ity. In the absence of a wave it is assumed that

p = p0 is constant at z = 0.Substituting from (1) and p from (7) into

the integrand in (6), it can be shown that

1

T

T0

p

ndt =

i

4

n

n

, (8)

where is the complex conjugate of and use

has been made of the fact that the integrals ofthe oscillatory terms are zero. Substituting (8)into (6) gives

P =i

4

SB

n

n

dS. (9)

1Within the linear theory applied here, it is accept-able to integrate over the time average of the position ofthe surface of the body as integrating over the instant-aneous position of the surface of the body adds at mosta second-order term to the time average of the absorbedpower.

Application of Greens second identity to theintegral in (9), and using the homogeneousboundary conditions (3) and (4), gives the in-

tegral over the bodys surface in terms of an in-tegral over a far-field control surface. The resultis

SB

n

n

dS

+

SC

n

n

dS = 0, (10)

where SC is a cylindrical control surface whoseaxis is in the direction ez and which encirclesthe body. Substituting (10) into (9) converts theintegral over the bodys surface into an integralover the far-field control surface, and it followsthat

P = i

4

SC

n

n

dS. (11)

Forthcoming algebra is simplified by substitut-ing

n

n 2i

n

into (11) to give2

P = 2

SC

ndS

. (12)

This equation expresses the power, P, trans-ferred from the fluid to the body in terms ofthe total power entering the control volumebounded by the control surface, SC.

2.4. Linear decomposition of velocity potential

According to the linearised theory of the in-teraction of water waves with heaving bodies,the total velocity potential of the fluid may beapproximated by

= A0 0 + A0 d + Az z, (13)

where: 0 and d are the velocity potentialsper unit complex amplitude for the incident anddiffracted waves respectively; A0 is the complexamplitude of the incident wave; z is the ve-locity potential per unit complex amplitude ofthe oscillating heave motion of the rigid-body;

2Note that (11) and (12) appear as unnumbered equa-tions at the top of page 320 in Mei [ 9].

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and Az is the complex amplitude of the oscillat-ing heave-mode of the body as introduced pre-viously in (5). Each of the velocity potentials

0 , d and z satisfy the Laplace equation (2),the sea bed condition (3), and the free-surfacecondition (4). Furthermore, d and z satisfythe Sommerfield radiation condition at large dis-tances from the body.

In this treatment the incident wave is a planewave travelling in the positive x-direction witha velocity potential given by

0 = A00 = iA0g

cosh(k(z + h))

cosh(kh)eikx,

(14)and a dispersion relationship given by

2 = gk tanh(kh), (15)

where is the waves angular frequency and kis its wavenumber.

The linearised boundary conditions on SB areobtained by substituting from (13) into theboundary condition (5). Setting Az = 0 gives

n d = n 0 on SB , (16)

whilst setting A0 = 0 gives

n z = i nz on SB. (17)

2.5. Relative motion hypothesis

This section summarises the application of therelative motion hypothesis and matched asymp-totic expansions used by McIver [10] to give theleading order relationship between the near andfar-field expressions of d and z . Note thatthe finite-depth analysis of McIver [10, 9 and10] is not applicable to the problem here as itapplies to cases in which the body dimensionsare of the same order as the water depth. Wetherefore proceed with the finite-depth gener-alisations of the McIvers infinite depth results[10, 7].

It is assumed that SB is centred at the ori-gin of the coordinate system and that its extentis small compared with the wavelength of theincident wave. Under these assumptions it isconsistent with the analysis presented here toapproximate 0 on SB by using just the termsup to first-order in ka in the Taylor expansion of0 from (14) with respect to the dimensionlessvariables kx and kz about the origin x = y = 0.That is,

0 = ig

1 + k (ix + z tanh(kh)) + O(ka)2

.

(18)

This is the finite-depth equivalent to (158) in[10]. Equation (18) can be substituted into thebody boundary conditions (16), (17), and the

equivalent surge boundary condition for x, toshow that in the vicinity of the body d may beexpressed as a linear combination of x and z ,that is,

d ix

tanh(kh) z on SB.

This is the finite depth equivalent to near-fieldrelationship given by (160) in [10].

The far-field relationship between the dif-fraction and radiation potentials is obtainedby McIver [10, 7] using matched asymptotic

analysis. He shows that for an axisymmetricsurface-piercing body the far-field velocity po-tential of radiated waves arising from heave mo-tions is proportional to (ka)2, whereas that fromsurge motions is proportional to (ka)3. In par-ticular, using McIvers [10] far-field expressions(132) and (161), it is clear that the leading or-der terms in the far-field diffracted and radiatedwaves are related by d z. Thus, whenperforming the integral over the far-field controlsurface SC in (12) it is consistent with the ap-proximations used here to write

d z on SC. (19)

3. Heaving attenuating line absorber

3.1. Capture width for unlimited volume

The slender body approximation is used toderive an expression for the capture width ofan attenuating line absorber, as in Newman [7].This approximation gives an expression for thefar-field radiated velocity potential at a distancer from a slender body. For a body of lengthl and beam b it states that, provided b

and l r, an approximation for the far-fieldradiated velocity potential at r can be obtainedby integrating over the contributions from theinfinitesimal elements of waterplane area of thebody, b l where l . Denoting by n,z thevelocity potential from the heave motions of thenth infinitesimal element, it follows that the far-field velocity potential due to the heave motionsof all the infinitesimal elements comprising theslender body is given by

z = n,z. (20)

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To express each n,z, consider the onlysource of waves in a fluid to be Az amplitudeheave oscillations of the nth infinitesimal ele-

ment of the body. Let (r

n,

n, z

) be the polarcoordinates in the coordinate system for whichthe infinitesimal element is centred at rn = 0.The velocity potential per unit complex amp-litude of this infinitesimal element at a large dis-tance rn from the centre of the body is given by

3

n,z(r

n,

n, z)

g

Hz(

n)

2

kr n

1/2

cosh(k(z + h))

cosh(kh)eikr

ni/4, krn , (21)

where Hz(

n

) is the Kochin function [for adefinition, see 9], which is a dimensionless func-tion describing the angular dependency of theradiated wave amplitude of the heave motion inthe limit krn . Generally, for a heavingbody with infinitesimal waterplane area Sw,the Kochin function is independent of and isgiven by Hz =

12

k2 Sw [see A3 on page 22 of10]. Thus, for the infinitesimal rigid elementscomprising a slender body of constant beam,Sw = b l, it follows that

Hz =1

2k2bl. (22)

Here, the assumption is made that the water-plane area of the element is not a function ofthe depth of submergence of that element. Thisassumption is strictly valid either for infinites-imal relative motions or for vertical wall-sidedelements or both.

To apply the slender body approximation in(20), let the coordinates in of the centre of thenth infinitesimal rigid-body element be x = xnalong the line z = y = 0 such that l/2 xn l/2 and let the length of the element bel = xn. Denote by

n,z(x,y,z; xn) = An,z n,z(x,y,z; xn) (23)

the far-field velocity potential at (x,y,z) in due to heave oscillations of the infinitesimalsource at x = xn. Let n denote the coordinatesystem with the same Cartesian basis vectorsas but with its origin shifted to (xn, 0, 0) as

3See, for example, McIver [10], in particular seeEqs. (10) , (143) and the unnumbered equation immedi-ately following (143) which is the Kochin function, de-noted by A3 in McIvers notation.

measured from . Thus, in n the nth infin-itesimal element is at (xn, y

n, z

n) = (0, 0, 0) inCartesian coordinates or, equivalently, at rn = 0

in polar coordinates (r

n,

n, z

n). The Cartesiancoordinate transformations between n and are

xn = x xn, (24)

yn = y, (25)

zn = z. (26)

Substituting (24)(26) into rn =

x2n + y2n

1/2,

expanding, and using r2 = x2 + y2 and x =r cos , it can be shown that

r

n = r xn cos + Ox2n

r

. (27)

Substituting (22), (27) and l = xn into(21) and neglecting terms of order O(x2n/r) andhigher, n,z(rn,

n, z) is transformed from then reference frame to the reference frameyielding

n,z(r,,z; xn) gk2b xn

(2kr)1/2

cosh(k(z + h))

cosh(kh)eikri/4eikxn cos ,

kr . (28)

For an attenuating line absorber composed ofheaving elements oscillating with relative phasesthat give the dynamics of a travelling wave mov-ing in the positive x-direction along the lengthof the attenuator, the complex amplitude can bewritten as the function ofxn given by

An,z(xn) = Az eikxn , (29)

where Az is the complex amplitude of the heave-mode oscillation of the element located at the

origin of the reference frame. Substituting(29) into (23), it follows that

n,z(r,,z; xn) = Az eikxn n,z(r,,z; xn).

(30)From (20), (28) and (30), z(r,,z) can be eval-uated as

z(r,,z) gk2Az bl

(2kr)1/2

cosh(k(z + h))

cosh(kh)

j0

kl

2(1 cos )

eikri/4, kr , (31)

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Figure 1: Schematic of the surface elevation of the waveradiated from a line of heaving elements oscillating onotherwise still water. The line has a length of l/ = 2and is oscillating in a travelling wave mode with velocitypotential given by (31).

where j0 is the zeroth-order spherical Besselfunction of the first kind (also known as the

sine cardinal or sinc function). All of the-dependency in (31) is within the sphericalBessel function. The heave-mode oscillationshave wavelength and are travelling in the ex-direction along the length, l, of the line ab-sorber. Figure 1 shows a schematic of the -dependency of the surface elevation of the radi-ated wave, given by = g1 z/t|z=0, forz in (31) with l/ = 2. The concentration ofwaves in the ex-direction resulting from thesephased motions is clearly visible in the figure.

For a given amplitude Az and waterplane areabl, and since j0(0) = 1, it can be seen from (31)

that in the limit kr the ratio of z fora line absorber of length l to that of a pointabsorber is simply j0

kl2 (1 cos )

. Since the

surface elevation of the radiated waves is pro-portional to z at z = 0, the ratio of the surfaceelevation of a line absorber to that of a pointabsorber is also j0

kl2 (1 cos )

. Graphs of

j0kl2 (1 cos )

for a point absorber and four

line absorbers of different lengths are shown inFigure 2. From this figure it is clear that thepoint absorber radiates waves in all directionswith the same amplitude, whereas the line ab-sorbers radiate at the same amplitude as thepoint absorber in the = 0 or ex-direction, butthey radiate at reduced amplitudes in other dir-ections. Also evident is that the longer the lineabsorber, the narrower the beam of radiatedwaves and the less the energy radiated.

An expression for the full velocity potential,, resulting from heave motions and wave dif-fraction in an incident wave is obtained by sub-stituting the relative motion hypothesis (19)into (13) to give

A0 0 + Ar z , (32)

1.0 0.5 0.0 0.5 1.0

0.2

0.0

0.2

0.4

0.6

0.8

1.0

j0kl1

cos

2

pointl = 0.5l = 1l = 2l = 4

Figure 2: Plots of the function j0

kl

2(1 cos )

. This

function describes the -dependency of the surface eleva-tion of the radiated waves and that of the velocity poten-tial, z, from (31). The -dependency plots for a pointabsorber and four line absorbers of different lengths l/are shown.

where the relative complex amplitude is definedby

Ar Az A0. (33)

Substituting (32) into the integrand in the ex-pression for absorbed power in (12) and expand-ing gives

n= |A0|

2 00n

+ A0Arz0n

+ A0Ar0zn

+ |Ar|2 z

zn

. (34)

The incident wave will contribute no net energyflux through the control surface SC, therefore

SC 0 0/ndS = 0. With this observationthe term proportional to |A0|2 in (34) can beignored and the integral in (12) can be writtenas

SC

ndS =

SC

A0Ar0

zn

+A0Arz0n

+ |Ar|2 z

zn

dS. (35)

The terms on the right-hand side of (35) areevaluated in cylindrical polar coordinates in Ap-pendix A. The result shows that P in (12) can

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be written as

P =

2

A0Ar A0Ar cggkbli |Ar|

2 cggk3 b2l2 I(kl)

, (36)

where I(kl) is given by4

I(kl) 1

2

j20

kl

2(1 cos )

d, (37)

=4

3cos(kl) J0(kl)

+2

3kl(2kl sin(kl) cos(kl)) J1(kl),

(38)

and where J0 and J1 are the zeroth and first-order Bessel functions of the first kind. Since

A0Ar A0Ar = 2i

A0Ar

,

it follows that

P =1

2cggkbl

2

A0Ar

|Ar|2 k2bl I(kl)

.

(39)In this equation the first term, which can bepositive or negative depending on the relativephase ofA0 and Ar, gives the energy transferredfrom the fluid to the body. From the point ofview of a near-field perspective, it represents thefluid pressure on the body times the velocity ofthe body; from the point of view of a far-fieldperspective, it represents the interference of theincident and radiated waves in the far-field. Thesecond term, which is always negative, gives thepower lost due to energy radiated away from thebody.

The mean power absorbed by the body, P in(39), is equal to the mean power extracted fromthe incident wave. The mean power extractedfrom a wave can be written as the product of

the mean energy flux in the wave, denoted byJ, and the power capture width, denoted by w,that is

P = w J. (40)

The mean energy flux of the plane incident wavedescribed by (14) can be written as

J =1

2cgg |A0|

2 , (41)

4The analytical expression in (38) for the integral in(37) is given by Farley [1982, Eq. (20)].

where cg is the group velocity given by

cg =d

dk

=

2k1 + 2kh

sinh(2kh) . (42)

Substituting (39) and (41) into (40) and rearran-ging gives the power capture width as

w =2k bl

|A0|2

A0Ar

k3 b2l2

|Ar|2

|A0|2 I(kl). (43)

This capture width can now be optimised withrespect to the phase and amplitude ofAr. Sincethe second term on the right-hand side of (43) ispositive and independent of phase, to optimise wwith respect to phase requires the maximisationof

A0Ar

= |A0| |Ar| sin r (44)

with respect to r = arg

A0Ar

= arg(A0) arg(Ar), which is the relative phase between A0and Ar. Maximisation requires that sin r = 1,implying that r = /2 so that Ar lags A0 by/2 radians and the response velocity (whichis proportional to nz) is in phase with theexcitation force (which is proportional to 0).Substituting r = /2 into (44), and the resultinto (43), gives

w = 2k bl|A0| |Ar|

|A0|2 k3 b2l2

|Ar|2

|A0|2I(kl). (45)

The first term in this equation, which is alwayspositive, accounts for the net power entering thecontrol volume bounded by SC due to the inter-action of the incident and radiated waves; thesecond term, which is always negative, accountsfor the energy leaving the control volume due tothe radiated waves.

To optimise (45) with respect to the relativeamplitude |Ar| consider

dw

d |Ar|

= 2k bl|A0|

|A0|2 2k

3 b2l2|Ar|

|A0|2 I(kl) = 0,

the solution of which is

|Ar| =1

k2 bl I(kl)|A0| . (46)

Substituting |Ar| from (46) into the first andsecond terms on the right-hand side of (45) givesthe optimal amplitude condition that the radi-ated power represented by the second term isexactly half of the absorbed power representedby the first term. This, together with the op-timal phase condition that Ar lags A0 by /2

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radians corresponds to the familiar condition ofimpedance matching to maximise the energytransfer between a source and a load.

The value of |Ar| in (46) is not limited by thevolume of the device, and since k is typically re-latively small the value of |Ar | can be very largein comparison with |A0|. Because of this, when|Ar| from (46) is substituted back into (45) itgives the optimum capture width for a deviceof unconstrained motion, or equivalently, unlim-ited volume, as

w =1

k I(kl)if |Ar| is unlimited. (47)

This is identical to the result reported by Farley

[6] (see Eq. (18) and the appendix of that work).In the limit that the length of the line ab-

sorber tends to zero, by (A.12) in Appendix A,limkl0 I(kl) 1 and therefore w = 1/k, whichagrees with the result obtained for the capturewidth of an unconstrained heaving point ab-sorbed by, for example, Evans [1] and Newman[3].

At this point a physical interpretation canbe given as to why higher capture widths areachievable from line absorbers than from pointabsorbers: for a line absorber the energy beingcarried away by radiated waves can be made ar-bitrarily small while maintaining the amplitude,|Ar|, of the radiated wave in the = 0 directionthat is necessary to remove energy from the in-cident wave by destructive interference. This in-terpretation is clear from consideration of (45)in the light of (37) and Figure 2. Obviously,to increase the capture width the energy lossgiven by the second term on the right-hand sideof (45) should be minimised while maintaininga high value for the energy absorption given bythe first term on the right-hand side. For a pointabsorber I(kl) = 1, so |Ar| is the only para-

meter available that is independent of the incid-ent wave and whose value can be adjusted tomaximise w. The magnitude of|Ar| needs to beincreased to increase the energy absorption fromthe first term in (45), which is linear in |Ar|; butif |Ar| is made too large the energy loss fromthe second term in (45), which is quadratic in|Ar|, dominates. For a line absorber, however,the added degree-of-freedom of length can beused to reduce the spread of the radiated waves,as shown in Figure 2. This reduces the mag-nitude ofI(kl) while keeping a large magnitudefor |Ar|, so achieving a high energy absorption

from the first term on the right-hand side of (45)and a low energy loss from the second.

3.2. Capture width for limited volume

In the preceding section no constraints wereplaced on the maximum value of the relat-ive amplitude, |Ar|, and the maximum capturewidth for a line absorber of unlimited volumewas derived as (47). In this section a motionconstraint is placed on |Ar| in (45). In the lin-ear theory presented here such a constraint on|Ar| may be equated to a limit on the maximumswept volume of the device. This constraintmay be a consequence of the limited volume ofa device, or it may be a consequence of other

engineering or control constraints that limit therange of heave motion. Assume for simplicitythat the depth, D, and the beam, b, of eachvertically-wall-sided element are independent ofits position along the length, l, of the line ab-sorber. Here, D is the vertical difference inposition between the minimum and maximumsubmergences of each element of the line ab-sorber. If the element is symmetric about theplane z = 0 the draft and the freeboard are equaland their sum is equal to D. In this case halfthe maximum swept volume of the line absorberis given by

V = bl D2

,

and the maximum value of |Ar| is

max(|Ar|) =D

2=

V

bl. (48)

Substituting |Ar| from (46) into (48) gives thevolume limiting condition in terms of |A0| as

max(|A0|) = k2V I(kl). (49)

Thus, for a limited volume device the maximum

|Ar| is reached when |A0| > k2V I(kl) and themaximum capture width is given by substitutingthe maximum of |Ar| from (48) into (45). Thisgives

w =kV

|A0|

2

k2V

|A0|I(kl)

if |A0| > k

2V I(kl).

When |A0| k2V I(kl) the optimum capturewidth is still given by (47). Thus, the maximumcapture width of a volume-limited heaving line

8


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attenuator can be written as

w =

1

k I(kl)

, |A0| k2V I(kl),

kV

|A0|

2

k2V

|A0|I(kl)

, |A0| > k

2V I(kl),

(50)where an analytic expression for I(kl) is givenby (38).

4. Dimensionless capture width and ab-

sorbed power

It is instructive to formulate the results of 3

in dimensionless units. The dimensionless inde-pendent variables of device length and devicehalf-swept volume, and the dimensionless de-pendent variable of capture width, are definedby

l kl, (51)

V k2V

|A0|, (52)

w kw. (53)

Since capture width and absorbed power are re-

lated by (40), the definition of w

in (53) sug-gests a non-dimensional power defined by

P k P

J, (54)

where J is given by (41). From (40), (53) and(54) is follows that the dimensionless variablesw and P are equivalent. Substituting (51)(53) into (50) gives the dimensionless capturewidth and power as

w P =

1

I(l

)

, V I(l) 1,

V(2 V I(l)), V I(l) < 1,(55)

where I(l) = I(kl) is the integral given by (38).The forms of the functions I(l) and 1/I(l) areshown in Figure 3. In these plots l/2 = l/ isused as the dimensionless unit on the abscissaas these units correspond to multiples of thewavelength. Shown in the right-hand side plotin Figure 3 is the line V = 1/I(l) which marksthe boundary between the limited and unlimitedvolume regimes.

The point absorber limit is obtained by sub-stituting liml0 I(l

) 1 into (55) to give

liml0

w 1, V

1,

2VV2, V < 1.(56)

The term in (56) which applies when V 1,that is w = 1, is the well known result for thetheoretical maximum capture width of a heav-ing point absorber of unlimited volume. WhenV 1 the term, which is first-order in volume,dominates and the capture width can be writ-ten as w = 2V, or, equivalently, P/V = 2.This corresponds to Budals upper bound5 onthe maximum capture width, or power per unit

volume, of a finite-volume heaving point ab-sorber [see 11]. The term that is second-orderin volume is a modification arising from the in-clusion of energy loss due to radiated waves.

The infinite length line absorber limit is ob-tained by substituting liml I(l

) 0 into(55) to give

liml

w 2V for all V. (57)

This can also be written as P/V = 2 for allV. Comparing (56) and (57) shows that fora heaving point absorber the maximum cap-

ture width is limited to w

= 1, no matterhow great the volume; for a line absorber inthe limit infinite length, however, the maximumcapture width scales linearly with volume, thatis w = 2V.

Plots of dimensionless capture width (or ab-sorbed power) from (55) are shown in Figure 4.The two plots give different illustrations of theeffect on w of changing the volume and lengthof a device. The solid line in the left-hand plotin Figure 4 shows that increasing volume aboveV = 1 gives no increase in w for a pointabsorber; the dashed lines show that, for line

absorbers, increasing V

above 1 can increasew, and that the increase is greater for a longerdevice. In fact, the dotted line marks wherew = 2V which shows that the maximum di-mensionless capture widths for an infinite lengthline absorber is equal to its dimensionless swept

5Budals result is generally quoted as the upperbound on the absorbed power per unit volume given by

P

VT=

1

4g |A0| ,

where VT = 2V is the total swept volume of the body.

9


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0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

l* 2

Il*

0 1 2 3 4

0

2

4

6

8

l* 2

V*

Unlimited volume regime

Limited volume regime

V* = 1 I

Figure 3: The dimensionless functions I(kl) = I(l) and V = 1/I(l).

volume. The fine dotted line marks where w =V which is the line of transition between theunlimited volume regime below the line and thelimited volume regime above it. It is clear fromthis plot that the minimum half-swept volumerequired for a point absorber to achieve its max-imum capture width is V = 1, and for a line ab-sorber it is V = w. Note that V = 1 is equi-valent to V/2 |A0| = 1/42 0.025, which aidsin interpreting the dimensionless volume as itstates that the minimum half-swept volume fora point absorber to achieve its maximum cap-ture width is about 2.5% of the natural measureof volume given by the square of the wavelengthtimes the amplitude of the wave, that is, 2 |A0|.

The right-hand plot in Figure 4 shows thegains in w that can be achieved by stretch-ing out different fixed-volume point absorbersto line absorbers of different lengths. The points

where the lines terminate at l

= 0 represent thedimensionless capture widths of point absorbersof the appropriate volumes. There is little dif-ference in w between point and line absorberswhen V < 0.5, that is, well within the volume-limited regime of the point absorber. For theunlimited volume regime of V 1 the capturewidth of the point absorber is at its theoreticalmaximum of w = 1 and the lines for V = 1,2 and 4 all converge to w = 1 as l 0. Forfinite length line absorbers, however, significantincreases in capture width are achievable abovew = 1, as shown in the figure and in the ex-

half-swept length, capture-volume, V l/2 = l/ width, w

1 0 11 1 1.6841 2 1.7822 0 12 1 2.7352 2 3.1273 0 13 1 3.1543 2 4.036

unlimited 0 1unlimited 1 3.162unlimited 2 4.583

Table 1: Examples of maximum dimensionless capturewidths for different dimensionless volumes and lengths.

amples given in Table 1.

Plots of dimensionless absorbed powers (orcapture widths) per unit dimensionless volumeare shown in Figure 5. In the left-hand plotin Figure 5 the fine dotted line marks the lineof P/V = 1, which is the line of trans-ition between the limited volume regime abovethe line and unlimited volume regime belowit. In this plot the limited-volume dimension-less powers converge to P/V = 2 at V =0 for all device lengths l. This maximumvalue of P/V = 2 corresponds to Budals up-per bound expressed in the dimensional unitsof (51)(54). Budals upper bound applies to

10


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0 2 4 6 8

0

1

2

3

4

5

6

7

V*

w*

point

l* 2 = 0.5l* 2 = 1l* 2 = 2l* 2 = 4w* = V*w* = 2V*

0 1 2 3 4

0

1

2

3

4

5

6

7

l* 2

w*

V* = 0.25

V* = 0.5V* = 1V* = 2V* = 4Unlimited vol.

Figure 4: Dimensionless capture width, w, from (55) as a function of dimensionless half-swept volume, V, anddimensionless device length, l.

all wave energy absorbers, however, point ab-sorbers can only reach Budals upper bound inthe zero-volume limit, whereas line absorber inthe infinite-length limit can reach Budals upperbound for any volume. This is evident from theinfinite-length limit of (55) giving P/V = 2which is independent of volume. From the left-

hand plot in Figure 5 it is clear that for smallvolume devices there is no significant advantageto using an attenuating line absorber comparedto a point absorber. For larger volume devices,however, the same plot shows how the powerper unit volume decreases less rapidly for a lineabsorber than for an equivalent volume pointabsorber, and the rate of decrease of power perunit volume is less for longer line absorbers (be-ing zero in the limit of an infinitely long lineabsorber). In fact, by dividing volume of asingle point absorber with V = 1 into eversmaller volume point absorbers the maximumpower per unit volume that can be achieved goesfrom P/V = 1 to P/V = 2 and, therefore,the maximum absorbed power is doubled. Thissevere drop-off in P/V with increasing V forpoint absorbers is an illustration of the small isbeautiful argument made by Falnes [11].

The right-hand plot in Figure 5 gives analternative illustration of how, if efficiency ismeasured in terms of absorbed power per unitvolume, for all device lengths, l, devices ofsmaller volume always have greater efficiencies.Note that in this plot the dotted line starting at

half-sweptvolume, length,V (m3) l (m) WEC of similar size

240 0 OPT PB150 [12]940 0 WaveBob [13]350 120 Pelamis FSP [14]790 180 Pelamis P2 [15]

1700 210 possible future Pelamis

Table 2: Selection of WEC sizes chosen to represent real-istic production sizes based on estimates of the sizes ofdevices already built or recently proposed.

l = 0 and P/V = 2 represents the power perunit volume in the unlimited volume regime forthe case V = 0.5. The corresponding line forV = 0.25 is off the scale.

5. Application to realistically sized

WECs

The advantage of formulating the results indimensionless units is that the number of in-dependent variables is reduced from four (k, l,V, and |A0|) to just two (l and V). In di-mensionless units, however, it is not immedi-ately obvious where a particular incident waveand WEC combination will occur on the graphs.In this section we consider the five realistic-ally sized WECs described in Table 2. It isstressed that the results in the following fig-ures are not representative of the actual capture

11


12/17

0 2 4 6 8

0.0

0.5

1.0

1.5

2.0

V*

P*V*

point

l* 2 = 0.5l* 2 = 1l* 2 = 2l* 2 = 4Unlimited vol.P* V* = 1

0 1 2 3 4

0.0

0.5

1.0

1.5

2.0

l* 2

P*V*

V* = 0.25V* = 0.5V* = 1V* = 2V* = 4Unlimited vol.P* V* = 1

Figure 5: Dimensionless power per unit dimensionless volume, P/V, from (55) as a function of dimensionlesshalf-swept volume, V, and dimensionless device length, l.

widths or absorbed powers of these devices, butonly that devices of these approximate volumesand lengths have already been built or may bebuilt in the near future.

Figure 6 shows the points of maximum dimen-sionless capture width plotted against dimen-sionless volume for the sizes of WECs given inTable 2 operating in the four incident waves with(H, T) equal to (2m, 8s), (2m, 10s), (3m, 8s) and(3m, 10s). These four incident waves broadlycorrespond to those that would occur close tothe centres of the available energy resource whenweighted by annual average occurrence. Theyhave incident power in ratios of 1, 1.25, 2.25and 2.81 respectively. Also, shown in Figure 6are contours of constant dimensionless length.The set of four plots show how the dimension-less volume and maximum capture width of aparticular WEC depends on the wave height and

period of the incident wave. The plots show thatthe largest volume point absorber appears belowthe line w = V and is, therefore, operatingin its unlimited volume regime and unnecessar-ily large for all but the (3m, 10s) incident wave.The largest volume line-absorber, however, isonly unnecessarily large for the (2m, 8s) incidentwave, even though it is nearly twice the volumeof the largest point absorber. The smallest pointabsorber has its maximum capture width lim-ited by its small volume in all but the (2m, 8s)incident wave. The mid-sized line absorber hasless volume than the large point absorber, yet

its maximum capture width is about four timeslarger in the (2m, 8s) wave, three times larger inthe (3m, 8s) wave, twice as large in the (2m, 10s)wave, and one and half times as large in the(3m, 10s) wave. For further clarity, Figure 7shows points of dimensional maximum absorbedpower plotted against dimensional volume for

the size of WECs given in Table 2 operating inthe four incident waves as used in Figure 6. Aset of conveniently spaced contours of constantlength are also shown in these plots. This setof four plots shows how the maximum absorbedpower of a particular size of WEC depends onthe wave height and period of the incident wave.

6. Conclusions

The theoretical maximum absorbed powerand capture width of a limited volume attenu-

ating line absorber heaving in a travelling wavemode in the presence of progressive regular in-cident wave has been derived in the frequencydomain using the linearised theory of the inter-action of water waves and structures. The res-ults are presented in dimensionless form whichhas the advantage of reducing the number of de-pendent variables from four to just two: dimen-sionless length, l, and dimensionless half-sweptvolume, V. In the zero-length limit the res-ults for the limited volume line absorber reduceto those for a point absorber; and in the zero-volume limit they reduce to the result expressed

12


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0 2 4 6 8

0

1

2

3

4

5

V*

w*

V (m3), l (m)

240, 0940, 0350, 120790, 1801700, 210

H = 2m, T = 8s

0 2 4 6 8

0

1

2

3

4

5

V*

w*

point

l* 2 = 0.5l* 2 = 1l* 2 = 2l* 2 = 4w* = V*

H = 2m, T = 10s

0 2 4 6 8

0

1

2

3

4

5

V*

w*

H = 3m, T = 8s

0 2 4 6 8

0

1

2

3

4

5

V*

w*

H = 3m, T = 10s

Figure 6: Dimensionless capture width, w, from (55) as a function of dimensionless half-swept volume, V, forWECs in Table 2 operating in four different incident wave defined by their (H, T).

by Budals upper bound. It is shown that, indimensionless units, the maximum dimension-

less capture width of a point absorber is w = 1and that the smallest volume required to achievethis capture width is V = 1. The dimensionlesscapture width and volume both depend on theincident wave: a value of w = 1 correspondsto a width of about 16% of the wavelength; avalue of V = 1 corresponds to about 2.5% ofthe rectangular volume given by the square ofthe wavelength times the wave amplitude. In-creasing the volume of a point absorber beyondV = 1 gives no increase in capture width. Foran attenuating line absorber, however, in the

limit of infinite length the maximum capturewidth is w = 2V. Thus, there is no limit

to the capture width of a line absorber providedit has sufficient volume and length. It is shownthat a particular fixed-length line absorber willhave a maximum capture width of w > 1 andthat the smallest volume required to achieve thiscapture width is V = w .

This has profound implications for the eco-nomics of power generation from wave energyconverters. Even though small volume pointabsorbers are more efficient than larger volumepoint absorbers in terms of power absorbed perunit swept-volume of device, engineering limit-

13


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0 500 1000 1500 2000

0

1

2

3

4

5

6

7

V (m3)

P

MW

V (m3), l (m)

240, 0940, 0350, 120790, 1801700, 210

H = 2m, T = 8s

0 500 1000 1500 2000

0

1

2

3

4

5

6

7

V (m3)

P

MW

point

l = 50 ml = 100 ml = 200 ml = 400 m

H = 2m, T = 10s

0 500 1000 1500 2000

0

1

2

3

4

5

6

7

V (m3)

P

MW

H = 3m, T = 8s

0 500 1000 1500 2000

0

1

2

3

4

5

6

7

V (m3)

P

MW

H = 3m, T = 10s

Figure 7: Absorbed power, P, from (55) as a function of half-swept volume, V, for WECs with sizes described inTable 2 operating in four different (H, T) incident waves.

ations make installation and operation of verylarge numbers of very small devices excessively

costly and uneconomic. A point absorber has alimit on its capture width and usable volume,whereas, in theory, line absorbers can be in-definitely scaled-up in volume and length togive unlimited capture widths. This theoret-ical advantage of line absorbers over point ab-sorbers applies to realistically sized wave en-ergy converters currently in operation. Thus,line absorbers can be progressively scaled-upin volume and length to give increasingly largecapture widths while retaining dimensions com-patible with cost-effective engineering. For ex-

ample, a line absorber with a volume equal tothe maximum usable volume of a point absorber

and a length equal to twice the wavelength ofthe incident wave can absorb nearly 80% morepower than a point absorber of the same volume.Doubling this line absorbers volume gives overthree times the absorbed power of the point ab-sorber, and tripling it gives over four times theabsorbed power. Thus, it is clear that by in-stalling fewer larger line absorbers valuable eco-nomies of scale can be achieved from line ab-sorbers that are not achievable from point ab-sorbers.

14


15/17

Appendix A. Integrals by the method of

stationary phase

The terms on the right-hand side of (35) areevaluated in cylindrical polar coordinates as fol-lows. Substituting x = r cos into (14), 0 iswritten as

0 = ig

cosh(k(z + h))

cosh(kh)eikr cos . (A.1)

Also, using (14) along with nx = cos and nz =0 on the control surface SC, and the observationthat 0 is not a function ofy, it follows that

0n

=gk

cosh(k(z + h))

cosh(kh)cos eikr cos ,

on SC. (A.2)

Next, using

n = er on SC,

and the gradient operator in cylindrical coordin-

ates

= er

r+ e

1

r

+ ez

z,

it follows that

n z =zn

zr

on SC.

Evaluating z/n by differentiating (31) withrespect to r, it can be shown that

zn

igk3bl

(2kr)1/2cosh(k(z + h))

cosh(kh)

j0

kl

2(1 cos )

eikr+i/4 on SC,

(A.3)

where j0 is the zeroth spherical Bessel functionof the first kind and, since kr on SC, onlythe highest order terms in kr have been retained.

Combining (31), (A.1), (A.2) and (A.3), thethree integrands on the right-hand side of (35)

can now be simplified and written as

0z

n

g2k3bl

2 (2kr)1/2 cosh(k(z + h))

cosh(kh)2

j0

kl

2(1 cos )

eikr(1cos )+i/4,

(A.4)

z0n

g2k3bl

2 (2kr)1/2

cosh(k(z + h))

cosh(kh)

2

cos j0

kl

2(1 cos )

eikr(1cos )i/4,

(A.5)

zzn

ik

2kr gk2bl

cosh(k(z + h))

cosh(kh) 2

j20

kl

2(1 cos )

. (A.6)

In cylindrical polar coordinates the integralover SC on the left-hand of (35) is expressed as

SC

ndS =

0h

nrddz. (A.7)

Each of (A.4)(A.6) has the same z-dependency.Integrating out this z-dependency gives, aftersimplification and substitution of the dispersionrelationship from (15),0

h

cosh2(k(z + h))

cosh2(kh)dz =

cggk

, (A.8)

where cg is the group velocity given by (42). Theintegrals over of (A.4)(A.6) are evaluated us-ing the method of stationary phase, which statesthat for real-valued smooth functions f and g,when f(0) = 0 and K ,

g() eiKf() d

g(0) 2

Kf(0) ei(Kf(0)+/4),

with f(0) = 0, K . (A.9)

Substituting K = kr, g(x) = j0kl2 (1 cos )

,

f(x) = cos 1 and 0 = 0 into (A.9) gives theintegral of (A.4) over as

j0

kl

2(1 cos )

eikr(1cos ) d

2

krei/4. (A.10)

15


16/17

Substituting K = kr, g(x) =cos j0

kl2 (1 cos )

, f(x) = 1 cos ,

and 0 = 0 into (A.9) gives the integral over

of (A.5) as

cos j0

kl

2(1 cos )

eikr(1cos ) d

2

krei/4. (A.11)

For convenience of notation the integral of (A.6)over is denoted by I(kl) as defined by the in-tegral in (37). As stated by Farley [6, Eq. (20)],I(kl) can be expressed analytically as in (38) interms of first-order Bessel functions of the firstkind, J0 and J1. Since the kl 0 limits of the

Bessel functions are

limkl0

J0(kl) = 1,

limkl0

J1(kl)

kl=

1

2,

the kl 0 limit of (38) is

limkl0

I(kl) 1. (A.12)

Equations (A.8), (A.10), (A.11) and (37) arenow used, along with (A.7), to evaluate the in-tegrals of (A.4)(A.6) over SC. The results are

0h

0 zn

rddz cggkbl

,0h

z0n

rddz cggkbl

,

0h

zzn

rddz icggk

3b2l2

I(kl).

Substituting these expressions into (35) and theresult into (12) gives (36).

References

[1] D. V. Evans, A theory for wave-power ab-sorption by oscillating bodies, Journal ofFluid Mechanics 77 (1976) 125.

[2] C. C. Mei, Power extraction from waterwaves, Journal of Ship Research 20 (2)(1976) 6366.

[3] J. N. Newman, The Interaction of Sta-tionary Vessels with Regular Waves, in:Proceedings of the 11th Symposium onNaval Hydrodynamics, Mechanical Engin-eering Publications Limited, London, UK,491501, 1976.

[4] D. V. Evans, Maximum wave-power ab-sorption under motion constraints, AppliedOcean Research 3 (1981) 200.

[5] D. J. Pizer, Maximum wave-power absorp-tion of point absorbers under motion con-straints, Applied Ocean Research 15 (1993)227234.

[6] F. J. M. Farley, Wave energy conversion byflexible resonant rafts, Applied Ocean Re-search 4 (1) (1982) 5763.

[7] J. N. Newman, Absorption of wave energyby elongated bodies, Applied Ocean Re-search 4 (1979) 189196.

[8] R. C. T. Rainey, The Pelamis wave energyconverter: it may be jolly good in practice,but will it work in theory?, in: Proceedingsof the 16th International Workshop on Wa-ter Waves and Floating Bodies, Hiroshima,Japan, 16, 2001.

[9] C. C. Mei, The applied dynamics of oceansurface waves, Wiley-Interscience, JohnWiley & Sons, Inc., iSBN 0-471-06407-6,1989.

[10] P. McIver, Low-frequency asymptotics ofhydrodynamic forces on fixed and float-

ing structures, in: M. Rahman (Ed.),Waves Engineering, Computational Mech-anics Publications, 149, 1994.

[11] J. Falnes, Small is beautiful: How to makewave energy economic, in: European waveenergy symposium, Edinburgh, Scotland,367372, 1993.

[12] O. P. T. Inc., URL http://www.oceanpowertechnologies.com/pb150.

htm, a 50% submerged torus of aboutdiameter 11m and height 2.5m, 2011.

[13] W. Ltd., URL http://wavebob.com/key-features/, a 50% submerged torus ofabout diameter 20m and height 8m, 2011.

[14] Pelamis Wave Power Ltd., URLhttp://www.pelamiswave.com/

our-technology/development-history,a 30% submerged line absorber of length120m and diameter 3.5m, 2011.

[15] Pelamis Wave Power Ltd., URLhttp://www.pelamiswave.com/

our-technology/the-p2-pelamis, a

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35% submerged line absorber of length180m and diameter 4m, 2011.

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