5.1.1 Linear Wave Theory

Waves in the ocean are nonlinear, and do not exactly follow linear wave theory; nevertheless, linear wave theory is very helpful in understanding and assessing more complicated waves such as nonlinear, irregular, and random waves, which will be discussed further later in this chapter. Linear wave theory can also be applied to many engineering and science problems with reasonable accuracy. A detailed derivation of linear wave theory can be found in many standard wave mechanic textbooks. Here, we present the main assumptions, parameters, equations, and wave properties associated with linear wave theory.

The main assumptions of linear wave theory can be summarized as [1]:

• • The fluid is incompressible and inviscid.
• • The flow is irrotational.¹
• • The bed is horizontal with a constant depth; the bed is impermeable.
• • The wave amplitude is small.
• • A regular wave with a constant wave period is considered.
• • Coriolis effects are neglected.
• • Waves are long-crested: the hydrodynamic solution is provided for a 2D vertical plane.
• • The pressure at the water surface is constant.

Referring back to Chapter 2, we mentioned that the Euler equations represent an incompressible and inviscid flow as follows:

$\begin{array}{ll}\hfill \nabla \cdot \mathbf{u}& =0\hfill \end{array}$

$\begin{array}{ll}\hfill \frac{d\mathbf{u}}{dt}=\frac{\partial \mathbf{u}}{\partial t}+\mathbf{u}\cdot \nabla \mathbf{u}& =-\frac{1}{\rho }\nabla p-g\widehat{k}\hfill \end{array}$

The Euler equations are the governing equations of an ideal fluid. A particular case of interest is potential flow, in which the flow vorticity is also zero. The other term, which is commonly used in potential flow theory, is irrotational flow (i.e. zero vorticity). Irrotational flow is simply a flow in which the particles do not rotate. It can be shown that the rate of rotation of particles in a flow field is directly proportional to the curl (i.e. ∇×) of velocity

$\begin{array}{l}\hfill \overrightarrow{\omega }=\frac{1}{2}\nabla \times \mathbf{u}\end{array}$

where $\overrightarrow{\omega }$ is called vorticity (i.e. rotation vector); hence, irrotational flows have zero vorticity. Referring to Eq. (5.2), if we take the curl of this equation, the RHS will become zero, and this leads to

$\begin{array}{l}\hfill \frac{d(\nabla \times \mathbf{u})}{dt}=\frac{d\overrightarrow{\omega }}{dt}=0\end{array}$

In other words, the flow vorticity is constant for ideal flow. Therefore, by assuming that the flow is inviscid and incompressible, we cannot necessarily conclude that the flow is irrotational. Nevertheless, irrotational flow is a special case in this context, where vorticity is not only constant, but it is also zero (i.e. $\overrightarrow{\omega }=\nabla \times \mathbf{u}=0$). In hydrodynamics and wave mechanics, it is much easier to deal with incompressible, inviscid and irrotational flows, because a scalar potential function can satisfy the governing equations and provide us with the hydrodynamic flow field. This is referred to as potential flow theory. Through some mathematical manipulation, you can easily show that the curl of a gradient of any scalar function is always zero. In other words,

$\begin{array}{l}\hfill \nabla \times \nabla \Phi =0\text{and}\nabla \times \mathbf{u}=0\Rightarrow \mathbf{u}=\nabla \Phi \end{array}$

where Φ is called the potential function, which as mentioned is a scalar function. In potential flow theory, the flow velocity is the gradient of this potential function. The momentum equation is already satisfied because $\frac{d\overrightarrow{\omega }}{dt}=\frac{d0}{dt}=0$. For the continuity equation, we have

$\begin{array}{l}\hfill \nabla \cdot \mathbf{u}=0\Rightarrow \nabla \cdot \mathbf{u}=\nabla \cdot (\nabla \Phi )=0\Rightarrow {\nabla }^2\Phi =0\end{array}$

This is called the Laplace equation, and for a 2D wave field can be expanded as follows:

$\begin{array}{l}\hfill {\nabla }^2\Phi =\frac{{\partial }^2\Phi }{\partial x^2}+\frac{{\partial }^2\Phi }{\partial z^2}=0\end{array}$

In linear wave theory, it is assumed that the flow is inviscid, incompressible, and irrotational; therefore, a potential function, which satisfies the periodic boundary conditions, will be the solution.

Consider a sinusoidal progressive wave (Fig. 5.1). The basic parameters such as wave length (L), wave height (H), still water depth (d), and water surface elevation (η) are presented in this figure. For a progressive wave, it can be shown that the potential function is given by

$\begin{array}{l}\hfill \Phi =\frac{H}{2}\frac{g}{\sigma }\frac{coshk(d+z)}{coshkd}sin(kx-\sigma t)\end{array}$

where k is the wave number and σ is the wave angular frequency. Although we skipped the derivation of this equation, let us just show that the above solution satisfies the Laplace equation. Taking the derivatives of the potential function leads to

$\begin{array}{ll}\hfill \frac{{\partial }^2\Phi }{\partial x^2}& =-k^2\left[\frac{H}{2}\frac{g}{\sigma }\frac{coshk(d+z)}{coshkd}sin(kx-\sigma t)\right]\hfill \end{array}$

$\begin{array}{ll}\hfill \frac{{\partial }^2\Phi }{\partial z^2}& =k^2\left[\frac{H}{2}\frac{g}{\sigma }\frac{coshk(d+z)}{coshkd}sin(kx-\sigma t)\right]\hfill \end{array}$

$\begin{array}{ll}\hfill \Rightarrow & \frac{{\partial }^2\Phi }{\partial x^2}+\frac{{\partial }^2\Phi }{\partial z^2}=0\hfill \end{array}$

Also, the potential function (Eq. 5.8) is periodic. This is because $\Phi (x,z,t)=\Phi (x,z,t+T)=\Phi (x,z,t+\frac{2\pi }{\sigma })$, which is demonstrated as follows

$\begin{array}{l}\hfill sin\left(kx-\sigma \left[t+\frac{2\pi }{\sigma }\right]\right)=sin\left(kx-\sigma t-2\pi \right)=sin(kx-\sigma t)\end{array}$

Using linear wave theory, all of the hydrodynamic variables, including velocity, pressure, water surface elevation, and wave celerity, can be derived from the potential function and implementation of the boundary conditions. Table 5.1 summarizes the wave properties based on linear wave theory.

Table 5.1

Wave Properties Based on Linear Wave Theory

Wave Property	Equation
Velocity potential	$\Phi =\frac{H}{2}\frac{g}{\sigma }\frac{coshk(d+z)}{coshkd}sin(kx-\sigma t)$
Wave celerity	$C=\frac{L}{T}=\frac{\sigma }{k}$
Horizontal component of velocity	$u=\frac{\partial \Phi }{\partial x}=\frac{H}{2}\frac{gk}{\sigma }\frac{coshk(d+z)}{coshkd}cos(kx-\sigma t)$
Vertical component of velocity	$v=\frac{\partial \Phi }{\partial z}=\frac{H}{2}\frac{gk}{\sigma }\frac{sinhk(d+z)}{coshkd}sin(kx-\sigma t)$
Wave surface displacement	$\eta =\frac{H}{2}cos(kx-\sigma t)$
Pressure (hydrostatic + dynamic)	$p=-\rho gz+\rho g\frac{coshk(d+z)}{coshkd}\left(\frac{H}{2}cos(kx-\sigma t)\right)=p_{\mathrm{o}}+p_{\mathrm{D}}$
Horizontal acceleration	$a_x=\frac{\partial u}{\partial t}=\frac{gkH}{2}\frac{coshk(d+z)}{coshkd}sin(kx-\sigma t)$
Vertical acceleration	$a_z=\frac{\partial w}{\partial t}=-\frac{gkH}{2}\frac{sinhk(d+z)}{coshkd}cos(kx-\sigma t)$
Dispersion	$C^2=\frac{g}{k}tanhkd$

5.1.2 Relationship Between Wave Celerity, Wave Number, and Water Depth: The Dispersion Equation

Wave celerity is dependent on water depth and wave length, and important wave processes such as wave refraction can be explained by the dependence of wave celerity on depth (Section 5.2.2). The relation of wave celerity and water depth can be derived by implementing the (kinematic) free surface boundary condition, which implies that the vertical speed of water particles at the water surface is equal to the speed of the free surface (i.e. $v=\frac{d\eta }{dt}=\frac{\partial \eta }{\partial t}+u\frac{\partial \eta }{\partial t}$ at the water surface). Assuming a small amplitude wave, it results in

$\begin{array}{l}\hfill C^2=\frac{g}{k}tanhkd\Rightarrow {\sigma }^2=gktanhkd\end{array}$

which is called the dispersion equation, because waves of different celerities will be dispersed according to their wave length. Looking at Fig. 5.2, the hyperbolic tangent function ($tanhkd$) in the dispersion equation approaches 1 for large depth values (deep water waves) and kd for small values of depth (shallow water waves). Therefore,

$\begin{array}{ll}\hfill C^2& =\frac{g}{k}\Rightarrow {\sigma }^2=gk:\text{deep water waves}\hfill \end{array}$

$\begin{array}{ll}\hfill C^2& =\frac{g}{k}(kd)\Rightarrow C^2=gd:\text{shallow water waves}\hfill \end{array}$

Note that the shallow water wave approximation is nondispersive, because it does not relate the wave celerity to wave length.

5.1.3 Wave Energy and Wave Power

Assume that a regular wave, with a wave period T and wave height H, is propagating in constant water depth d. Power (the rate of energy conversion) is the product of velocity and force. In linear wave theory, the force is generated by pressure (i.e. F = pA). Referring to Table 5.1, pressure depends on z. Therefore, wave power can be evaluated by multiplying the pressure and velocity and integrating over depth; the wave power, or more specifically the average energy flux per unit width over a wave period, is given by

$\begin{array}{ll}\hfill P={\int }_0^T{\int }_{-d}^{\eta }{p}_{\mathrm{D}}udzdt& =\frac{1}{8}\rho g{H}^{2}C\left\{\frac{1}{2}\left(1+\frac{2kh}{sinh2kh}\right)\right\}=E{C}_{\mathrm{g}}=\text{cst}\hfill \end{array}$

$\begin{array}{ll}\hfill C_{\mathrm{g}}& =C\left\{\frac{1}{2}\left(1+\frac{2kh}{sinh2kh}\right)\right\}\hfill \end{array}$

where p_D is the dynamic pressure and u is the horizontal velocity given in Table 5.1. Note that wave power is conserved in linear wave theory (i.e. the RHS of Eq. (5.16) is constant).

In the above formulation, $E=\frac{1}{8}\rho g{H}^2$ is the total mechanical energy (i.e. potential and kinetic) of waves per unit surface area averaged over a wave period. In other words, wave power is simply the product of wave energy and wave group velocity. The group velocity C_g (which is dependent on the wave celerity C, wave number k, and water depth) is the speed of wave energy propagation. Referring to the dispersion equation, it is also defined as [1]

$\begin{array}{l}\hfill C_{\mathrm{g}}=\frac{\partial \sigma }{\partial k}=\frac{\partial }{\partial k}\sqrt{gktanh(kh)}\end{array}$

In realistic ocean environments, ocean currents, which are generated by tides, wind, or temperature, affect the propagation of wave energy. Tidal waves and wind waves can be regarded as long and short waves, which interact in various ways. The Doppler shift (which is explained in Chapter 7) is an example where the frequency of waves is influenced by currents:

$\begin{array}{l}\hfill \omega =\sigma +ku\to \frac{\partial \omega }{\partial k}=\frac{\partial \sigma }{\partial k}+u\end{array}$

where σ is the relative wave frequency (observed in a coordinate system moving with the same velocity as the ambient current), ω is the absolute wave frequency (observed in a fixed frame), and u is the ambient current velocity. The wave energy flux when we have ambient currents becomes

$\begin{array}{l}\hfill C_{\mathrm{g}}^{*}=C_{\mathrm{g}}+u\to E{C}_{\mathrm{g}}^{*}=E{C}_{\mathrm{g}}+uE\ne cst\end{array}$

where C_g* is the group velocity in the presence of ambient currents, EC_g is the wave energy transport by the group velocity, and uE is the wave energy transport by tidal currents. When waves propagate in the presence of currents, the wave energy flux is no longer conserved. This happens due to the exchange of energy between wave and current fields. To avoid this issue, wave models apply the conservation of the wave action (E/σ), rather than conservation of wave energy in their formulations (e.g. [2]). Nevertheless, the total energy flux due to waves and currents is conserved. Therefore, the conservation of energy, in a more general way, can be expressed as follows [3]

$\begin{array}{l}\hfill \left[E{C}_{\mathrm{g}}+Eu\right]+\left\{\frac{1}{2}\rho gh{u}^3\right\}+\left\{u\left(2\frac{C_{\mathrm{g}}}{C}-\frac{1}{2}\right)E\right\}=\text{cst}\end{array}$

It is clear that when ambient currents are zero, the earlier equations reduce to P = EC_g = cst.

5.1.4 Irregular Waves

So far, we have considered a wave that is characterized by a single frequency and direction (i.e. a monochromatic wave). Monochromatic waves can be generated in laboratories (wave tanks), but are rarely seen in the ocean. Waves, which have been generated by wind, are irregular and include many wave frequencies. As ‘wind waves’ travel over long distances, they are dispersed, because the wave celerity/speed is dependent on the frequency. These waves are called ‘swell waves’ and are more similar to regular waves.

Within the range of validity of linear wave theory (see the next section), we can superimpose several monochromatic/sine waves to construct a more realistic irregular wave. Conversely, we can decompose an irregular wave into a number of sinusoidal waves, which have different frequencies. Mathematically speaking, this is the concept of the Fourier series/integral, which is based on the principal that any function can be constructed by combining a set of simple sine waves. This is a powerful method, because we can also find the wave properties, such as pressure, acceleration, and energy, by combining/superimposing the hydrodynamic field of these individual sine waves.

Two main approaches exist to deal with irregular waves: statistical and spectral. In the statistical approach, statistical variables are calculated and used to represent irregular waves. For instance, parameters such as the average wave period, and the root mean square wave height, can be used to represent the wave period, and the wave height, respectively, for a realistic sea state. In this approach, the sea state is represented by a probability distribution function such as Rayleigh, and statistical parameters are derived from that distribution. The spectral method is based on the concept of the Fourier transform and is the basis of spectral wave models, such as SWAN [2], TOMAWAC [4], or STWAVE [5], which are commonly used for wave energy resource characterization. A detailed discussion about irregular waves is beyond the scope of this book and can be found in many other resources (e.g. [1, 6]). Here, basic concepts, with a focus on wave power, are presented.

As mentioned, we can combine several waves with a range of frequencies and phases, to construct an irregular wave, or vice versa. Mathematically speaking, this means

$\begin{array}{l}\hfill \eta (t)=\sum _{i=1}^N{a}_{i}sin({\sigma }_{i}t+g_i)\end{array}$

where a_i and g_i are the amplitude and phase of an individual sine wave, respectively, and η represents the time series of an irregular wave. The total energy of this irregular wave is found by adding up the energy of all sine waves as follows

$\begin{array}{l}\hfill E=\frac{1}{2}\rho g\sum _{i=1}^N{a}_i^2=\frac{1}{8}\rho g\sum _{i=1}^N{H}_i^2\end{array}$

Each individual sine wave has a different frequency, and different wave energy, which contributes to the total wave energy. Fig. 5.3 shows an example. An irregular wave has been decomposed into four sine waves, with frequencies of 0.1, 0.2, 0.3, and 0.4 Hz and varying amplitudes. The distribution of the amplitudes of these waves versus frequency is shown in this figure. The plot of the energy of these sine waves (or the square of the amplitude) with respect to the frequency represents the energy spectrum. In other words, the energy spectrum shows how the total energy of an irregular wave is distributed amongst various frequencies. If we use many sine waves to decompose the time series of an irregular wave, the difference between the frequencies of these individual sine waves approaches zero; this will lead to a continuous wave spectrum. A continuous energy density spectrum is defined as

$\begin{array}{l}\hfill E_{\mathrm{tot}}=E(\delta \sigma )+E(2\delta \sigma )+\cdots +E(\infty )={\int }_0^{\infty }E(\sigma )d\sigma \end{array}$

where E(σ) is called the ‘energy density spectrum’, because it is energy per unit angular frequency (J/rad). Alternatively, the ‘variance density spectrum’ (S(σ); m²/rad) can be used to represent the distribution of energy with respect to frequency. It is defined as,

$\begin{array}{l}\hfill E_{\mathrm{tot}}=\rho g{\int }_0^{\infty }S(\sigma )d\sigma =\rho g\sum _1^{N}S({\sigma }_i)\delta (\sigma )\to E(\sigma )=\rho gS(\sigma )\end{array}$

Considering Eqs (5.23), (5.25), we can see that the variance density spectrum represents the square of the amplitude of each individual sine wave:

$\begin{array}{l}\hfill \frac{1}{2}{a}_i^2=S({\sigma }_i)\delta \sigma \end{array}$

The moments of the variance density spectrum are frequently used to compute statistical wave properties. The rth moment (m_r) is defined as

$\begin{array}{l}\hfill m_r={\int }_0^{\infty }{\sigma }^{r}S(\sigma )d\sigma \end{array}$

For instance, the significant wave height H_mo, which is the mean of the highest one-third of waves in a record, is given by

$\begin{array}{l}\hfill H_{\mathrm{mo}}=4\sqrt{m_0}\end{array}$

5.1.5 Nonlinear Waves

Linear wave theory is the basis of many analytical methods and numerical models, which describe the properties and propagation of regular/irregular waves in the open ocean and coastal regions. Nevertheless, we should be aware of the range of validity of this theory, and wave processes which cannot be explained by this theory. Referring to Section 5.1.1, several assumptions were made to develop linear wave theory, which may not be valid in real-word applications. In particular, in the derivation of linear wave theory, it is assumed that the amplitude of the wave is small compared with the water depth and wave length. Fig. 5.4 shows the range of validity of linear wave theory. As you can see, for deep waters $\left(\text{e.g.}\frac{d}{g{T}^2}>10^{-3}\right)$ and small amplitude waves $\left(\text{e.g.}\frac{H}{g{T}^2}<10^{-3}\right)$, linear wave theory is valid; however, in shallow waters and for large amplitude waves, other wave theories such as Stokes and Cnoidal are more appropriate. Note that in certain depths, or for certain wave steepnesses, the wave is no longer stable and will break. This is discussed further in the following section, and is shown by breaking criteria on this figure (shallow and deep criteria).

Nonlinear Dispersion Equation

As we discussed, the linear dispersion equation relates the water depth and the wave frequency to the wave celerity (or wave length). If nonlinear effects are considered, the dispersion equation will also be dependent on the wave height. The nonlinear dispersion equation is given by [9]

$\begin{array}{l}\hfill {\sigma }^2=gktanhkd\left[1+{\left(k\frac{H}{2}\right)}^2\left(\frac{8+cosh4kd-2{tanh}^{2}kd}{8{sinh}^{4}kd}\right)\right]\end{array}$

  (5.45)

As can be seen, when kH/2 or the wave steepness is small, the previous equation approaches what we already saw in the linear wave theory (Eq. 5.13). Fig. 5.5 shows the comparison between the linear and nonlinear dispersion equations for two wave heights (2 and 3 m), assuming a water depth of 10 m. You can try to reproduce these plots by solving the linear and nonlinear dispersion equations.

5.2.1 Wave Shoaling

As water shoals, remembering that waves transport energy in the direction of wave advance, it is important to make use of the principal that wave energy flux per unit surface area, EC_g, is constant. This assumes that no energy is dissipated by the wave train, for example, by wave breaking or bottom friction, as it propagates from deep to shallow water. If we apply a subscript ‘0’ to represent deep water values, this implies that

$\begin{array}{l}\hfill E{C}_{\mathrm{g}}=E_0{C}_{g0}=\text{constant}\end{array}$

Therefore, if we have knowledge of ‘deep water’ wave properties, we can determine the corresponding wave properties in shoaling water.

Consider a horizontally 1D case, in which the wave crests are everywhere parallel to the depth contours (Fig. 5.6). As the waves propagate through water of gradually decreasing depth (shoaling water), their phase speed, and consequently their group velocity, reduces (Eq. 5.13), but their frequency tends to remain constant—in other words, waves do not catch one another up in the same way that they would if they were ‘dispersing’. By way of illustration, for a wave period T = 8 s we can calculate the wave properties shown in Fig. 5.6 and Table 5.2 between ‘deep’ and ‘shallow’ water.

Therefore, a fourfold reduction in wavelength and phase speed is accompanied by a halving of the group velocity. Since E = 1/2ρga², when C_g is halved, it follows that E is doubled (Eq. 5.46), and so wave amplitude increases by a factor of $\sqrt{2}$ (Fig. 5.6C). This is known as wave shoaling. Note that this illustration is for a 1D case (Fig. 5.6A), but for a more generalized case, another process can complicate wave transformation in shoaling water—refraction.

5.2.2 Wave Refraction

In the earlier shoaling example, the wave crests were parallel to the depth contours, and the wave propagated normal to the coastline. In most situations, waves propagate obliquely to the depth contours, and are observed to slowly change direction as they approach the coast. This gradual change in wave direction is known as refraction.

Examining Fig. 5.7, which shows wave crests propagating at an oblique angle to the depth contours, there is a variation in water depth experienced along each of the wave crests. Whereas further offshore the wave crest is propagating in relatively deep water, the part of the wave crest closer to the coast is propagating in relatively shallow water. Since wave celerity is related to water depth through the dispersion equation (5.13), this means that the portion of the wave crest in deeper water is travelling faster than the portion of the wave crest that is travelling in shallower water. As a result of this change in phase speed along each wave crest, in a given time interval the crest moves over a larger distance in deeper water than it does in shallower water, and so the waves tend to turn (i.e. refract) towards the depth contours as they propagate.

Variation of ψ (the angle of incidence of the wave orthogonal to the outward normal of the coastline; see Fig. 5.7) is governed by Snell’s law. This is the familiar law from studies of optics and, in the present context, expresses the constancy of wavenumber k in the direction parallel to the shore. In other words, the distance between successive crests measured parallel to the shoreline, L₁, remains constant as the waves travel into shallow water (Fig. 5.7). This condition may be written as

$\begin{array}{l}\hfill ksin\psi =k_{0}sin{\psi }_0=\mathrm{constant}\end{array}$

Based on Eq. (5.47), refraction diagrams can be generated for given offshore wave conditions and maps of bathymetry/topography. However, such analysis is based on the following assumptions [10]:

• • The wave energy between wave rays (orthogonals) remains constant.
• • The direction of wave advance is perpendicular to the wave crest.
• • The speed of the wave of a given period depends only on the water depth at that location.
• • Changes in bathymetry are gradual.
• • Waves are long-crested, constant-period, small-amplitude, and monochromatic.
• • Effects of currents, winds, and reflections from beaches, and underwater topographic features, are considered negligible.

A typical refraction diagram, for a wave of period T = 10 s, is shown for an irregular coastline (a headland and two bays) in Fig. 5.8. The wave orthogonals converge at the headland and diverge in the bays. Because wave energy between the wave orthogonals remains constant (see the earlier assumptions), this implies that the wave height will generally increase at the headland and reduce in the two bays.

As we saw in Eq. (5.46), wave energy flux is a constant, provided no energy is dissipated, for example, by wave breaking or friction. Now, in the more general case, this flux towards the coast is a constant per unit distance parallel to the shoreline

$\begin{array}{l}\hfill \overline{E}{C}_{\mathrm{g}}cos\psi =\mathrm{constant}\end{array}$

Therefore, the variation in wave amplitude may be expressed in terms of deep water reference values as follows

$\begin{array}{l}\hfill a=a_0{\left[\frac{C_{g0}}{C_{\mathrm{g}}}\right]}^{1/2}{\left[\frac{cos{\psi }_0}{cos\psi }\right]}^{1/2}\end{array}$

This result shows that, since ψ₀ > ψ in general, waves approaching a coast obliquely are amplified less than those approaching a coast normally (where ψ = ψ₀ = 0).

There are many WEC technologies (it has been estimated that there are over 1000 device patents), but they can mostly be grouped into one of five technology types:

• • Attenuator
• • Surface point absorber
• • Oscillating wave surge converter
• • Oscillating water column
• • Overtopping devices

The main features of these WEC types are summarized in Fig. 5.10.

Surface Point Absorber

Surface point absorbers are floating structures that absorb wave energy from all directions (i.e. in contrast to an attenuator device). Point absorbers have relatively small dimensions relative to typical wavelengths, and so have diameters of a few metres. Examples of point absorbers are WaveBob and PowerBuoy (Fig. 5.11).

5.4.2 Comparison Between WEC Technologies

Issues associated with WECs are survivability (because they will generally be sited in locations where the wave climate is, by its very nature, extreme), and the cost associated with designing the devices to survive such conditions. As an example of the scale of these devices, data extracted from Previsic [12] provides a comparison (Table 5.3).

The scale is immense—every kilowatt of rated power requires around 0.7 tonnes of steel.²

5.4.3 Basic Motions of WECs

An object, which is floating on the water surface, or submerged underwater, generally has six degrees of freedom—provided that no constraint prevents its motion. It can translate along or rotate around the three major axes. Assuming that waves are propagating along the x axis (Fig. 5.1), the three translational motions are called, surge (x), sway (y), and heave (z). The rotational motions are called rolling, pitching, and yawing around x, y, and z axes, respectively.

5.5.1 Theoretical, Technical, and Practical Resources

A wave resource assessment of the type presented in the previous section is what is known as a ‘theoretical resource’ assessment. A theoretical resource assessment estimates the annual average energy production for a specific wave (or tidal) resource, and is generally based on numerical modelling, with limited validation from historical data records. By contrast, the ‘technical’ resource is defined as the portion of the theoretical resource that can be captured using a specified technology. It therefore includes device characteristics (e.g. efficiency) and constraints such as interdevice spacings. The calculation of the technical wave resource requires access to a power matrix for the selected technology (e.g. Fig. 5.17). A comparison between the theoretical resource and the technical resource (based on a single Pelamis device) at Bernera in the west of Scotland is shown in Fig. 5.18. Although both theoretical and technical resources exhibit strong seasonal and interannual variability, the extremes in the theoretical resource (e.g. regularly over 500 kW/m) are capped by the device characteristics, rated at 750 kW. Out of interest, the capacity factor (see Section 1.5) for the Pelamis device at this location over a decade of simulation is 22%, which is fairly encouraging compared with other renewable technologies (e.g. Table 1.5).

Finally, the practical resource is the portion of the technical resource that is available after considering all constraints such as socioeconomic and environmental factors. The practical resource excludes regions of low power density, regions that are distant from electricity infrastructure, navigation channels, marine protected areas, etc.

5.5.2 Survivability and Maintenance

WECs operating on the water surface are exposed to storms and other extreme events. In particular, high and steep waves, especially breaking waves, are likely to damage WECs. Further, a particular WEC is designed to operate, and convert energy efficiently, over a limited range of conditions (e.g. see the Pelamis power matrix in Fig. 5.17), usually tuned to the most frequent wave conditions that occur at a particular site: extreme waves are not usually within this range. Therefore, it is necessary for most wave devices to have a survival mode; for example, a device could be sunk to a particular depth where wave orbital motion will be reduced (e.g. related to the wave length), or to relax the load on a turbine in an OWC device.

At the other end of the scale, prolonged periods of low wave energy provide opportunities for device maintenance. A site that is consistently energetic may be desirable from theoretical and technical resource perspectives, but is undesirable from a practical resource perspective.