Page updated: May 23, 2021
Author: Curtis Mobley
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Wave Variance Spectra: Theory

This page discusses the fundamentals of wave variance spectra, which will be needed for surface generation. Just as for the Fourier transforms discussed on the previous page, this is well known material. However, the entire business of wave spectra can be exceedingly confusing, and journal articles always assume that the reader already understands the underlying ideas and mathematics. It is therefore worthwhile to review the results that will be needed. A good reference for the development of wave spectra is Holthuijsen (2007), who is very careful in his notation and terminology. Banner (1990) and Massel (2011) provide review articles. The notation used below is chosen to conform to that used in Elfouhaily et al. (1997), which will often be referred to as ECKV, after the last letters of the authors’ names.

The figures of the following pages are illustrated using two commonly used wave variance spectra. The one-dimensional surfaces of the Spectra to Surfaces: 1D page use the Pierson-Moskowitz spectrum, and the two-dimensional surfaces of the Spectra to Surfaces: 2D and Time-Dependent Surfaces pages use the spectrum of Elfouhaily et al. (1997). After the introductory overview on this page, the next page presents these two specific wave spectra for later reference.

Wave Energy

Consider a sinusoidal wave of amplitude A and wavenumber ν. It can be shown (e.g., Hydrologic Optics (1976), vol. VI, page 72) that the total energy per unit horizontal area of sea surface of this wave, averaged over a wavelength, is

energy area = 1 2ρgA2 + 1 4τν2A2, (1)


ρ 103kgm3 is the density of water,
g 9.8ms2 is the acceleration of gravity,
τ 0.072Nm1 (at 25 deg C) is the surface tension of water.

The 1 2ρgA2 term is the sum of the kinetic and gravitational potential energy of the wave. (The potential energy is relative to the mean sea surface at z = 0.) The 1 4τν2A2 term is the energy required to stretch the level surface into a sinusoid, working against surface tension. It is easy to see that the units of these terms are Jm2.

The variance of a sinusoidal surface is the average over one wavelength Λ of the surface elevation squared (assuming that the mean surface is at zero):

var{z} 1 Λ0Λ A sin 2πx Λ 2dx = 1 2A2. (2)

Thus the energy per unit area of a sinusoidal wave also can be written in terms of its variance:

energy area = ρg + 1 2τν2 var{z}. (3)

Wave Variance Spectra

The fundamental quantity of interest for the description of sea surfaces is a spectral density function that tells how much of the surface wave elevation variance is contained in particular frequencies. Other quantities of interest, in particular a spectral density function that tells how much sea surface slope variance is contained in different frequencies, can be derived from the elevation variance spectral density.

Wave Elevation Variance Spectra

As just seen, the variance of the surface elevation is proportional to the amplitude squared. Equation (1) shows that the energy per unit horizontal area of a sinusoidal wave of a given amplitude and spatial frequency is also proportional to the amplitude squared, or to the variance of the surface elevation. For a sea surface containing waves of many different amplitudes and spatial frequencies, the total variance of the sea surface is the sum of the variances of in the individual waves (the variance of a sum of independent random variables is is the sum of the variances of the individual variables). Likewise, the total energy of the waves is the sum of the energies of the individual waves.

We thus have the following line of reasoning:

  • For a discrete sample z(r) of a zero-mean sea surface, the variance is
    var{z} = 1 N r=0N1|z(r)|2.

    (Division by N rather than N 1 in computing the sample variance is correct in this case, because it is assumed that the mean surface elevation is known and equal to 0.)

  • Parseval’s identity (Eq. (17) of the previous page),
    r=0N1|z(r)|2 = N u=0N1|(u)|2,
  • gives
    var{z} = u=0N1|(u)|2.
  • We therefore identify 𝒮(u) |(u)|2 as the discrete variance spectrum, with units of m2.

Such reasoning led Schuster (1897), Schuster (1898) to the seminal observation that Fourier transforms can be used to decompose the total variance contained in a signal into the variance contained in each frequency. Schuster’s original interest was in searching for what he call “hidden periodicities” in weather phenomena (as opposed to obvious periodicities such as daily or seasonal cycles). He soon applied the technique to looking for periodicities in earthquakes, sunspots, and other phenomena. Today, the computational speed of the FFT allows his method of analysis to be widely applied in science, engineering, economics and other fields where it is desired to know the energy or power of a signal as a function of spatial or temporal frequency. Regardless of what physical quantity is under consideration, the essence of a spectrum is that it gives the distribution of variance in that quantity as a function of frequency.

Because of the proportionality (3) of elevation variance to wave energy, 𝒮(u) is often loosely referred to as the discrete energy (or power) spectrum. To be precise, 𝒮(u) is not an energy variance spectrum unless it is multiplied by the factor ρg + 1 2τν2 to convert elevation units to energy units. (In other applications, the variance of some quantity such as voltage or current in an electrical circuit can be multiplied by an appropriate conversion factor to get the spectrum of energy or power.)

Care is required to formulate wave spectra for continuous variables. As in Eqs. (2) and (6) of the Wave Representations page, we can write the continuous surface as a sum of sinusoids, e.g.

z(x) = n=0z n(x) = n=0A n cos(knx + ϕn). (4)

As we saw in the Wave Energy section above, the variance of the sinusoid with frequency kn is 1 2An2. The waves of different frequencies are independent, so the total variance of the surface can be written as the sum of the variances of the individual waves:

var{z} = n=01 2An2. (5)

Now let Δkn be a frequency interval centered on frequency kn, whose sinusoid has amplitude An. We then define

𝒮(k = kn) lim Δkn01 2An2 Δkn . (6)

In this definition, keep in mind that each An is associated with a particular frequency kn, and that the limit operation holds for each value of n. We are thus defining a function of the spatial frequency, which becomes a continuous function of k as the bandwidth Δkn goes to zero.

The continuous function 𝒮(k) is called the omnidirectional elevation variance spectrum. “Omnidirectional” means that there is no reference direction (e.g., a direction of wave propagation relative to the wind direction) included in the quantity. This is the case if a wave record is made at a single point as a function of time: the waves go past and their elevations are recorded, but no information is obtained about the direction the waves are traveling. 𝒮(k) is also called the omnidirectional elevation spectrum for obvious reasons. As is to be expected, there is no uniformity of notation for this spectrum, but 𝒮 seems to be the most common symbol—and what is used in both Pierson and Moskowitz (1964) and Elfouhaily et al. (1997), to be discussed in the next sections—so that is what is used here. ( seems to be the second-most-common symbol and is used in Holthuijsen (2007).) Equation (6) shows that the units of 𝒮(k) are clearly m2(radm). Equations (5) and (6) show that integrating the omnidirectional variance spectrum over all frequencies gives the total elevation variance:

var{z} = z2 =0𝒮(k)dk.

(The equations above are written in terms of spatial angular frequency k, as used for surface generation, but the reasoning and functional form of Eq. (6) are the same for any other spatial or temporal frequency variable.)

We can repeat this process for two dimensions, starting with

z(x,y) = n=0 m=0A n,m cos[knx + my + ϕn,m]. (7)

Here k is associated with spatial frequencies in the x direction, and corresponds to frequencies in the y direction. This leads to a function

Ψ(k = kn, = m) lim Δkn0 lim Δm0 1 2An,m2 ΔknΔm. (8)

The frequency intervals in the x and y directions, Δkn and Δm, do not depend on the respective frequency indices n and m. That is, the frequency intervals are equally spaced, but they need not be the same in the x and y directions.

The notation in the last two equations is non-standard. The convention is to use kx for frequencies in the x direction (k above), and ky for frequencies in the y direction ( above). The Ψ(k,) of Eq. (8) is then written as Ψ(kx,ky). This leads to confusion in the subscripts, which can denote either frequency variables kx,ky, or specific discrete values kn,km. However, the Ψ(kx,ky) notation is standard in the literature, so that is what is used below.

Ψ(kx,ky) (i.e., Ψ(k,)) is the directional variance spectrum in Cartesian coordinates. Its units are clearly m2(radm)2. This spectrum is often called the “two-dimensional wavenumber spectrum,” and its arguments are often written in vector form, Ψ(k), where k = (kx,ky) denotes the location of the frequency point in the 2-D frequency plane.

Equation (8) is the conceptual definition of Ψ(k). In practice, if we have discrete measurements of the two-dimensional sea surface elevation at a given time, z(xr,ys) = z(r,s) = z(xrs), then the two-dimensional discrete Fourier transform of z(r,s) gives the two-dimensional amplitudes

(kuv) = [kx(u),ky(v)] = (u,v) = 𝒟{z(r,s)}.

Dividing by the discrete frequency bandwidths gives an estimate (called a 2-D periodigram) of the two-dimensional elevation variance spectral density

P(kuv) |(kuv)|2 ΔkuΔkv Ψ(kuv) ΔkuΔkv. (9)

The arguments of z(xrs), (kuv), and Ψ(kuv) show that these are discrete functions, whereas Ψ(k) denotes a spectral density function of the continuous variable k. The 2-D periodogram is an approximation of the 2-D variance spectral density, P(kuv) Ψ(k), where the symbol “” is used to denote “is an estimate of.” As Eq. (9) shows, the discrete function Ψ(kuv) has units of m2, whereas P(kuv) and Ψ(k) have units of m2(radm)2. A single periodogram contains statistical noise because it is computed from a single realization of a random sea surface. However, if many sets of observations are made, and the respective periodograms are averaged, then the noise tends to average out, and the average of the periodograms approaches the conceptual limit of the spectrum Ψ(k). Questions such as how many periodograms must be averaged to obtain a spectrum with a given level of uncertainty lie in the domain of spectrum estimation, which need not concern us here.

A 2-D spectrum depends on direction, i.e., on the direction of the (kx,ky) point in a two-dimensional frequency plane. Usually, the + x direction is chosen to be pointing downwind and, correspondingly, + kx represents the spatial frequencies of the waves propagating downwind. In this case, the angle φ = tan 1(k ykx) gives the direction relative to the wind direction. As Eq. (8) shows, the integral of Ψ(kx,ky) over all frequencies gives the variance of the two-dimensional surface:

var{z} = z2 =Ψ(k x,ky)dkxdky.

It is also common to define a directional spectrum in terms of polar coordinates given by the magnitude k and direction φ of the vector k. These are are related to kx,ky by

k = kx 2 + ky 2 φ = tan 1 ky kx

and inversely by

kx = k cos φ ky = k sin φ.

In this case we define

Ψ ̃(k,φ) lim Δk0 lim Δφ01 2An,m2 ΔkΔφ. (10)

This spectrum has units of m2[(radm)rad]. (The tilde notation is used here to distinguish this spectrum from the Ψ(k,φ) spectrum of Elfouhaily, which is defined on the next page. Some authors reserve the name “directional spectrum” for Ψ ̃(k,φ) and refer to Ψ(kx,ky) as the wavenumber spectrum.) As before, definition (10) shows that integrating Ψ ̃(k,φ) over k and φ gives the variance:

var{z} =002πΨ ̃(k,φ)dkdφ.

The ECKV directional spectrum given on the next page is specified in terms of polar coordinates k,φ. However, on the next pages we will need a spectrum in terms of Cartesian coordinates kx,ky for use in a rectangular FFT grid. The change of variables from polar to Cartesian coordinates is effected by the Jacobian

Ψ(kx,ky) = Ψ ̃(k,φ) (k,φ) (kx,ky) = Ψ ̃(k,φ) k kx k ky φ kx φ ky = Ψ ̃(k,φ)1 k. (11)

Note that the 1k factor converts the units of Ψ ̃(k,φ) into the units of Ψ(kx,ky).

In Eq. (4) on the next page, this last equation is partitioned as

Ψ(kx,ky) = 1 k𝒮(k)Φ(k,φ) Ψ(k,φ),[ECKV45] (12)

where 𝒮(k) is an omnidirectional spectrum and Φ(k,φ) is a nondimensional spreading function, which shows how waves of different frequencies propagate (or “spread out”) relative to the downwind direction at φ = 0. Labels such as [ECKV 45] refer to the corresponding equations in Elfouhaily et al. (1997). The spreading function by definition satisfies

02πΦ(k,φ)dφ = 1 (13)

for all k.

Equation (12) shows that to obtain the ECKV variance spectrum in Cartesian coordinates we need only evaluate the ECKV Ψ(k,φ) spectrum for the corresponding values of k and φ, i.e.

Ψ(kx,ky) = Ψ k = kx 2 + ky 2,φ = tan 1(k ykx) . (14)

Note in particular that there is no “extra” k factor involved in the conversion of Ψ(k,φ) to Ψ(kx,ky); both quantities have the same units. (The k factor seen in the differentials dkxdky = kdkdφ is cancelled by the 1k coming from the Jacobian as seen in Eq. 11.)

Integration of Eq. (12) over the respective (kx,ky) and (k,φ) frequency planes gives the variance as

z2 = Ψ(k x,ky)dkxdky = 002π1 k𝒮(k)Φ(k,φ)kdkdφ = 0𝒮(k)dk,[ECKV A2] (15)

after noting the normalization of Eq. (13). Thus the variance of the sea surface is still contained in the omnidirectional spectrum, even in the two-dimensional case. The omnidirectional spectrum 𝒮(k) is obtained from Ψ(k,φ) via

𝒮(k) =ππΨ(k,φ)kdφ.[ECKV A3]

Unfortunately, making measurements of 2-D sea surfaces is extremely difficult. There are very few such data sets—obtained, for example, by laser reflectance measurements (e.g., Huang et al. (2000))—and these do not cover the full range of spatial scales. Given the paucity of empirical 2-D wave data from which to develop 2-D variance spectra, the common procedure is to start with a 1-D or omnidirectional spectrum 𝒮(k) and add an angular spreading function Φ(k,φ) to distribute the wave energy over different directions relative to the downwind direction. In nature, most waves travel more or less downwind, a small amount of energy (i.e., variance) is contained in waves propagating in nearly cross-wind directions, and almost no energy is contained in waves that by some chance (such as the breaking of a larger wave generating smaller waves in all directions) might be propagating in upwind directions. The spreading function must capture this behavior. Although omnidirectional wave spectra are well grounded in observations, the choice of a spreading function is still something of a black art.

Wave Slope Variance Spectra

Now return to Eq. (4) and take the derivative to get the slope of the sea surface for the nth wave:

dzn(x) dx = Ankn sin(knx + ϕn).

As in Eq. (2), the variance of this slope is

var dzn dx 1 Λn0Λn Ankn sin(knx + ϕn) 2dx = 1 2An2k n2.

A limit operation corresponding to Eq. (6) gives

lim Δkn01 2An2k n2 Δkn = k2𝒮(k). (16)

The quantity k2𝒮(k) is the omnidirectional slope variance spectrum, usually called just the slope spectrum. The units of k2𝒮(k) are mrad. Integrating the slope spectrum over all frequencies gives the total variance σ2 of the sea surface slope:

σ2 var dz dx = dz dx2 =0k2𝒮(k)dk.

This variance is usually called the mean square slope or mss. The units of mss are rad2. Radians are nondimensional numbers, but the label of rad2 reminds us that we can think of the slope as an angle from the horizontal measured in radians.

Comment: There is a subtle inconsistency in the units of mean square slopes as seen in the literature. As obtained above from the slope variance spectrum, the mss has units of radians squared. However, as defined using a finite difference of a sea surface elevation sample z(xr), the slope of the surface between two sample points xr and xr+1 is

slope = Δz Δx = z(xr+1) z(xr) xr+1 xr .

This finite difference is a non-dimensional slope as defined in analytic geometry, and the corresponding mss is obtained by averaging the squares of the finite differences over all of the sample points. However, a mss computed from finite differences (ΔzΔx)2 is not the same as a mss with units of rad2 as computed from a slope spectrum. For example, a slope of ΔzΔx = 0.1 corresponds to a slope angle of tan 1(0.1) = 0.09966865rad. This is a negligible numerical difference for this slope, which is typical of actual sea surfaces, but there is a philosophical difference in a nondimensional slope as defined in analytic geometry and a slope defined as an angle with units of radians. The difference would not be negligible for large slopes: e.g., a slope of ΔzΔx = 1 corresponds to an angle of 0.78 rad (45 deg), not 1 rad. I have never seen any reference to this distinction in the literature, which seems to apply “mean square slope” to both forms of the surface slope. Perhaps a reader of this page can inform me of how this issue is resolved in the wave spectrum community when comparing theoretical mean square slopes with measured ones.

There is another way to view slope spectra. As we know from Eq. (4) of the Fourier transforms page, the 1-D surface elevation z(x) is related to the Fourier amplitude (k) by

z(x) = 1 2π(k)e+ikxdk.

Differentiating this equation with respect to x gives the 1D slope of the sea surface as

σ(x) dz(x) dx = 1 2π(k)ike+ikxdk.

This leads us to identify ik(k) as the Fourier amplitude corresponding to the sea surface slope. This gives us two ways to study the slope statistics of random sea surfaces, given the Fourier amplitude (k) (which we will learn to create from wave variance spectra in the following pages). The first way is to take the inverse Fourier transform of (k) to obtain z(x), and then to differentiate z(x) to get the slope. The second way is to take the inverse transform of ik(k) to get the slope directly, without ever creating the surface z(x) itself. These two processes will in general give different realizations of the surface slopes, but the slope statistics, e.g. the mean square slopes σ2, will be the same.

The corresponding relations for two dimensions are derived in the same fashion and lead to similar results. Assuming that the wind is blowing in the + x direction, the mean-square slope in the along-wind direction is given by either of

z(x,y) x 2 σ x2 mss x =k x2Ψ(k x,ky)dkxdky = ππk2 cos 2φΨ(k,φ)kdkdφ.[ECKVA4]

The corresponding equation for the cross-wind direction is

z(x,y) y 2 σ y2 mss y =k y2Ψ(k x,ky)dkxdky = ππk2 sin 2φΨ(k,φ)kdkdφ.[ECKVA5]

Recalling that variances of random variables add to get the total variance due to all variables gives the total mean square slope

mss = mssx + mssy = (k x2 + k y2)Ψ(k x,ky)dkxdky = ππk2Ψ(k,φ)kdkdφ = k2𝒮(k)dk.[ECKVA6]

Thus, even in the 2-D case, the total slope variance can be obtained from the omnidirectional slope spectrum.

Table 1 summarizes the spectral quantities used on the following pages.

Spectrum Name Symbols Units

1-D or omnidirectional
variance or elevation𝒮(k)m2(radm)
slope k2𝒮(k)mrad

2-D or directional
variance or elevationΨ(kx,ky), Ψ(k,φ)m2(radm)2
alongwind slope kx2Ψ(k x,ky),k2 cos 2φΨ(k,φ)rad2
crosswind slope ky2Ψ(k x,ky),k2 sin 2φΨ(k,φ)rad2
total slope (kx2 + k y2)Ψ(k x,ky),k2Ψ(k,φ)rad2

Table 1: Summary of wave variance spectral quantities.

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