Spectral description of wind waves

Wind generated waves have irregular wave heights and periods, caused by the irregular nature of wind. Due to this irregular nature, the sea surface is continually varying, which means that a deterministic approach to describe the sea surface is not feasible. On the other hand, statistical properties of the surface, like average wave height, wave periods and directions, appear to vary slowly in time and space, compared to typical wave periods and wave lengths.


The surface elevation of waves in the ocean, at any location and any time, can be seen as the sum of a large number of harmonic waves, each of which has been generated by turbulent wind in different places and times. They are therefore statistically independent in their origin. According to linear wave theory, they remain independent during their journey across the ocean. Under these conditions, the sea surface elevation on a time scale of hundreds of characterstic wave periods is sufficiently well described as a stationary, Gaussian process. The sea surface elevation in one point as a function of time can be described as

  $$\eta (t) = \sum_i a_i \cos (\sigma_i t + {\alpha}_i)$$ (2.1)

with $\eta$ the sea surface elevation, $a_i$ the random amplitude of the $i^{\rm th}$ wave component, $\sigma_i$ the relative radian or circular frequency of the $i^{\rm th}$ wave component in the presence of the ambient current (equals the absolute radian frequency $\omega$ when no ambient current is present) and ${\alpha}_i$ the random phase of the $i^{\rm th}$ wave component. This is called the random-phase model (Holthuijsen, 2007). Note that the random variables $a_i$ and ${\alpha}_i$ are characterized by their probability density functions; the amplitude of each wave component is Rayleigh distributed and the phase of each component is uniformly distributed between 0 and 2$\pi$.


In the presence of the ambient current, it is assumed that it is uniform with respect to the vertical co-ordinate and the changes in the mean flow within a wave length are so small that they affect only negligibly the dispersion relation. The absolute radian frequency $\omega$ then equals the sum of the relative radian frequency $\sigma$ and the inner product of the wave number and ambient current velocity vectors, as follows

  $$\omega = \sigma + \vec{k} \cdot \vec{u}$$ (2.2)

which is the usual Doppler shift. Here, $\vec{k} = (k_x,k_y)$ and for linear waves the relative frequency is given by

  $$\sigma = \sqrt{g\vert\vec{k}\vert \tanh (\vert\vec{k}\vert d)}$$ (2.3)

where $g$ is the acceleration of gravity and $d$ is the water depth.


Ocean waves are chaotic and a description in the time domain is rather limited. Alternatively, many manipulations are more readily described and understood with the variance density spectrum, which is the Fourier transform of the auto-covariance function of the sea surface elevation

  $$E'(f) = \int_{-\infty}^{+\infty} C(\tau) e^{-2\pi i f \tau} d\tau$$ (2.4)

with $f = \sigma/2\pi$ the frequency (in Hz) and

  $$C(\tau) = < \eta(t) \eta(t+\tau) >$$ (2.5)

where $C(\tau)$ is the auto-covariance function, $<\cdot>$ represents ensemble average of a random variable, $\tau$ is the time lag and $\eta(t)$, $\eta(t+\tau)$ describe two random processes of sea surface elevation.


For a stationary wave condition, it is conventional to consider a spectrum $E(f)$ different from the above one, as follows

  $$E(f) = 2 E'(f) \quad \mbox{for} \, \, f \geq 0 \quad \mbox{and} \, \, E(f) = 0 \quad \mbox{for} \, \, f < 0$$ (2.6)

The description of the wave field through the defined variance density spectrum $E(f)$ is called spectral description of water waves. This description is complete in a statistical sense under the assumption that the sea surface is considered as a stationary, Gaussian random process.


The variance of the sea surface elevation is given by

  $$<\eta^2> = C(0) = \int_{0}^{+\infty} E(f) df$$ (2.7)

which indicates that the spectrum distributes the variance over frequencies. $E(f)$ should therefore be interpreted as a variance density. The dimension of $E(f)$ is m$^2$/Hz if the surface elevation is given in meters and the frequency in Hz.


The variance $<\eta^2>$ is closely linked to the total energy $E_{\rm tot}$ of the waves per unit surface area, as follows

  $$E_{\rm tot} = \frac{1}{2} \rho_w g < \eta^2 >$$ (2.8)

with $\rho_w$ the water density. The terms variance and energy density spectrum will therefore be used indiscriminately in this document (however, see Zijlema, 2021).


In many wave problems it is not sufficient to define the energy density as a function of frequency alone. It is mostly required to distribute the wave energy over directions as well. This spectrum, which distributes the wave energy over frequencies $f$ and directions $\theta$, is denoted by $E(f,\theta)$. Again, this spectrum is assumed to provide a complete spectral description of the wave field if this field is statistically quasi-homogeneous (and stationary Gaussian), which especially holds for broad-banded directional waves (i.e. wind sea). Section 2.7 discusses the extension of this description to include the statistical inhomogeneity of the wave field (e.g. due to wave interference patterns).


As the total energy density at a frequency $f$ is distributed over the directions $\theta$ in $E(f,\theta)$, it follows that

  $$E(f) = \int_{0}^{2\pi} E(f,\theta) d \theta$$ (2.9)

The energy density spectra $E(f)$ and $E(f,\theta)$ are depicted in Figure 2.1.

Figure 2.1: Illustrations of 1D and 2D wave spectra. (Reproduced from Holthuijsen (2007) with permission of Cambridge University Press.)

Based on the energy density spectrum, the integral wave parameters can be obtained. These parameters can be expressed in terms of the so-called $n-$th moment of the energy density spectrum

  $$m_n = \int_{0}^{\infty} f^n E(f) df$$ (2.10)

So, the variance of the sea surface elevation is given by $m_0 = <\eta^2>$. Well-known parameters are the significant wave height

  $$H_s = 4 \sqrt{m_0}$$ (2.11)

and some wave periods

  $$T_{m01} = \frac{m_0}{m_1}\, , \quad T_{m02} = \sqrt{\frac{m_0}{m_2}}\, , \quad T_{m-10} = \frac{m_{-1}}{m_0}$$ (2.12)

In SWAN, the energy density spectrum $E(\sigma,\theta)$ is generally used. On a larger scale the spectral energy density function $E(\sigma,\theta)$ becomes a function of space and time, that is, $E(\vec{x}, t; \sigma,\theta)$ and wave dynamics should be considered to determine the evolution of the spectrum in space and time.