WikiWaves - User contributions [en]

Ocean-Wave Spectra

2012-12-09T20:48:28Z

Kri: /* JONSWAP Spectrum */ Why substitute x for F?? F is a variable just like x.

== Introduction ==

Ocean waves are produced by the wind. The faster the wind, the longer the wind blows, and the bigger the area over which the wind blows, the bigger the waves. In designing ships or offshore structures we wish to know the biggest waves produced by a given wind speed. Suppose the wind blows at 20m/s for many days over a large area of the North Atlantic. What will be the spectrum of ocean waves at the downwind side of the area?

It is important to realise that the spectra presented in the section are attempts to describe the ocean wave spectra
in very special conditions, namely the conditions after a wind with constant velocity has been blowing for a long
time. A typical ocean wave spectrum wil be much more complicated and variable. For example it may have two
peaks, one from distance swell and the other generated by the local wind.

The concept of a wave spectrum can be quite abstract and is described in [[Waves and the Concept of a Wave Spectrum]]

== Pierson-Moskowitz Spectrum ==
Various idealized spectra are used to answer the question in oceanography and ocean engineering. Perhaps the simplest is that proposed by [[Pierson and Moskowitz 1964]]. They assumed that if the wind blew steadily for a long time over a large area, the waves would come into equilibrium with the wind. This is the concept of a '''fully developed sea''' (a sea produced by winds blowing steadily over hundreds of miles for several days).Here, a long time is roughly ten-thousand wave periods, and a "large area" is roughly five-thousand wave-lengths on a side.
<center>
[[Image:Fig16-7s.jpg]]
</center>
<center>
Figure 1 Wave spectra of a fully developed sea for different wind speeds according to [[Moskowitz 1964]].
</center>

To obtain a spectrum of a fully developed sea, they used measurements of waves made by accelerometers on British weather ships in the North Atlantic. First, they selected wave data for times when the wind had blown steadily for long times over large areas of the North Atlantic. Then they calculated the wave spectra for various wind speeds, and they found that the spectra were of the form (Figure 1):
<center>
<math> S(\omega) = \frac{\alpha g^2}{\omega^5}\exp\left(-\beta\left(\frac{\omega_0}{\omega}\right)^4\right) </math>
</center>

where <math> \omega = 2\pi f</math>, <math>f</math> is the wave frequency in Hertz, <math> \alpha = 8.1\times 10^{-3}</math>, <math> \beta = 0.74 </math>, <math>\omega_0 = g / U_{19.5}</math> and <math>U_{19.5}</math> is the wind speed at a height of 19.5m above the sea surface, the height of the anemometers on the weather ships used by [[Pierson and Moskowitz 1964]].

For most air flow over the sea the atmospheric boundary layer has nearly neutral stability, and

<center>
<math>U_{19.5}\approx 1.026 U_{10}</math>
</center>

assuming a drag coefficient of <math> 1.3 \times 10^{-3}</math>.

The frequency of the peak of the Pierson-Moskowitz spectrum is calculated by solving <math>dS / d\omega = 0</math> for <math>\omega_p</math>, to obtain

<center>
<math>\omega_p = 0.877 g / U_{19.5}\,\!</math>
</center>

The speed of waves at the peak is calculated from (16.10), which gives:

<center><math>c_p=\frac{g}{\omega_p}=1.14U_{19.5}\approx 1.17U_{10}</math></center>

Hence waves with frequency <math>\omega_p</math> travel 14% faster than the wind at a height of 19.5m or 17% faster than the wind at a height of 10m. This poses a difficult problem: How can the wind produce waves traveling faster than the wind? We will return to the problem after we discuss the JONSWAP spectrum and the influence of nonlinear interactions among wind-generated waves.

The significant wave-height is calculated from the integral of <math>S\mbox{ }(\omega)</math> to obtain:

<center>
<math>\left \langle \zeta^2\right \rangle=\int_{0}^{\infty}S(\omega)\mbox{ }\mathrm{d}\omega=2.74 \times 10^{-3}\frac{(U_{10.5})^4}{g^2}</math>
</center>

Remembering that <math>H_{1/3} = 4\mbox{ }<\mbox{ }\zeta^2 \mbox{ }>\mbox{ }^{1/2}</math>, the significant wave-height calculated from the Pierson-Moskowitz spectrum is:

<center>
<math>H_{1/3}=0.21\frac{(U_{19.5})^2}{g}\approx0.22\frac{(U_{10})^2}{g}</math>
</center>

Practical wave analysis of uses the frequency, <math> f </math>, instead of the angular frequency <math> \omega </math>. If we have a frequency spectrum, <math> S(\omega) </math>, then the corresponding frequency spectrum, <math> S'(f) </math>, will be

<center><math> S'(f) = S(2\pi f) \frac{\mathrm{d}\omega}{\mathrm{d}f} = 2\pi S(2\pi f) </math></center>
(the factor of <math> 2\pi </math> insures that the area under the curve remains the same).

In the following we omit the ' symbol. For practical reasons, it has also become standard to relate the variables to the main sea state parameters rather than wind speed, and to use slightly different values
so that the Pierson-Moskowitz spectrum can be expressed as

<center>
<math>
S(f) = \alpha g^2(2\pi)^{-4}f^{-5}\exp\left(-\frac{5}{4}\left(\frac{f_m}{f}\right)^4\right)
</math>
</center>
where <math> \alpha </math> is Phillips constant and <math> f_m </math> is the peak frequency. Remember that
these expressions are an approximation to the real sea spectrum.

Figure 2 gives significant wave-heights and periods calculated from the Pierson-Moskowitz spectrum.

<center>
[[Image:Fig16-8s.jpg]]
</center>
<center>
Figure 2 Significant wave-height and period at the peak of the spectrum of a fully developed sea calculated from the Pierson-Moskowitz spectrum.
</center>

== JONSWAP Spectrum ==
[[Hasselmann et al. 1973]], after analyzing data collected during the Joint North Sea Wave Observation Project JONSWAP, found that the wave spectrum is never fully developed. It continues to develop through non-linear, wave-wave interactions even for very long times and distances. Hence an extra and somewhat artificial factor was added to the Pierson-Moskowitz spectrum in order to improve the fit to their measurements. The JONSWAP spectrum is thus a Pierson-Moskowitz spectrum multiplied by an extra peak enhancement factor
<math> \gamma^r </math>

<center><math>S_j(\omega)=\frac{\alpha g^2}{\omega^5}\exp\left[-\frac{5}{4}\left(\frac{\omega_p}{\omega}\right)^4\right]\gamma^r</math></center>

<center><math>r=\exp\left[-\frac{(\omega-\omega_p)^2}{2\sigma^2\omega_p^2}\right]</math></center>

<center>
[[Image:Fig16-9s.jpg]]
</center>
<center>
Figure 3 Wave spectra of a developing sea for different fetches according to [[Hasselmann et al. 1973]].
</center>

Figure 3 show wave spectra as mearured from {[Hasselmann et al. 1973]] and Figure 4 shows a comparison
of the JONSWAP and Pierson-Moskowitz spectra.
<center>
[[Image:SpSj.jpg]]
</center>
<center>
Figure 4 The JONSWAP and Pierson-Moskowitz spectra.
</center>

Wave data collected during the JONSWAP experiment were used to determine the values for the constants in the
above equations:

::::::::::::<math>\alpha=0.076\left(\frac{U_{10}^2}{F\mbox{ }g}\right)^{0.22}</math>

::::::::::::<math>\omega_p=22\left(\frac{g^2}{U_{10}F}\right)^{1/3}</math>

::::::::::::<math>\gamma = 3.3 \,\!</math>

::::::::::::<math> \sigma = \begin{cases} 0.07 & \omega \le \omega_p \\ 0.09 & \omega > \omega_p \end{cases} </math>

where <math>F</math> is the distance from a lee shore, called the fetch, or the distance over which the wind blows with constant velocity.

The energy of the waves increases with fetch:

<center><math>\left \langle \zeta^2\right \rangle =1.67 \times 10^{-7} \frac{(U_{10})^2}{g}F.</math></center>

The JONSWAP spectrum is similar to the Pierson-Moskowitz spectrum except that waves continues to grow with distance (or time) as specified by the a term, and the peak in the spectrum is more pronounced, as specified by the g term. The latter turns out to be particularly important because it leads to enhanced non-linear interactions and a spectrum that changes in time according to the theory of [[Hasselmann 1966]].

== Generation of Waves by Wind ==
We have seen in the last few paragraphs that waves are related to the wind. We have, however, put off until now just how they are generated by the wind. Suppose we begin with a mirror-smooth sea (Beaufort Number 0). What happens if the wind suddenly begins to blow steadily at say 8m/s? Three different physical processes begin:

# The turbulence in the wind produces random pressure fluctuations at the sea surface, which produces small waves with wavelengths of a few centimeters ([[Phillips 1957]]).
# Next, the wind acts on the small waves, causing them to become larger. Wind blowing over the wave produces pressure differences along the wave profile causing the wave to grow. The process is unstable because, as the wave gets bigger, the pressure differences get bigger, and the wave grows faster. The instability causes the wave to grow exponentially ([[Miles 1957]]).
# Finally, the waves begin to interact among themselves to produce longer waves ([[Hasselmann et al. 1973]]). The interaction transfers wave energy from short waves generated by Miles mechanism to waves with frequencies slightly lower than the frequency of waves at the peak of the spectrum. Eventually, this leads to waves going faster than the wind, as noted by Pierson and Moskowitz.

==Acknowledgement==

The material in this page has been derived from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by [[Robert Stewart]].

[[Category:Geophysics]]

Ocean-Wave Spectra

2012-12-09T20:37:16Z

Kri: /* JONSWAP Spectrum */ This is incorrect (alpha is defined differently for the two different spectra)

== Introduction ==

Ocean waves are produced by the wind. The faster the wind, the longer the wind blows, and the bigger the area over which the wind blows, the bigger the waves. In designing ships or offshore structures we wish to know the biggest waves produced by a given wind speed. Suppose the wind blows at 20m/s for many days over a large area of the North Atlantic. What will be the spectrum of ocean waves at the downwind side of the area?

It is important to realise that the spectra presented in the section are attempts to describe the ocean wave spectra
in very special conditions, namely the conditions after a wind with constant velocity has been blowing for a long
time. A typical ocean wave spectrum wil be much more complicated and variable. For example it may have two
peaks, one from distance swell and the other generated by the local wind.

The concept of a wave spectrum can be quite abstract and is described in [[Waves and the Concept of a Wave Spectrum]]

== Pierson-Moskowitz Spectrum ==
Various idealized spectra are used to answer the question in oceanography and ocean engineering. Perhaps the simplest is that proposed by [[Pierson and Moskowitz 1964]]. They assumed that if the wind blew steadily for a long time over a large area, the waves would come into equilibrium with the wind. This is the concept of a '''fully developed sea''' (a sea produced by winds blowing steadily over hundreds of miles for several days).Here, a long time is roughly ten-thousand wave periods, and a "large area" is roughly five-thousand wave-lengths on a side.
<center>
[[Image:Fig16-7s.jpg]]
</center>
<center>
Figure 1 Wave spectra of a fully developed sea for different wind speeds according to [[Moskowitz 1964]].
</center>

To obtain a spectrum of a fully developed sea, they used measurements of waves made by accelerometers on British weather ships in the North Atlantic. First, they selected wave data for times when the wind had blown steadily for long times over large areas of the North Atlantic. Then they calculated the wave spectra for various wind speeds, and they found that the spectra were of the form (Figure 1):
<center>
<math> S(\omega) = \frac{\alpha g^2}{\omega^5}\exp\left(-\beta\left(\frac{\omega_0}{\omega}\right)^4\right) </math>
</center>

where <math> \omega = 2\pi f</math>, <math>f</math> is the wave frequency in Hertz, <math> \alpha = 8.1\times 10^{-3}</math>, <math> \beta = 0.74 </math>, <math>\omega_0 = g / U_{19.5}</math> and <math>U_{19.5}</math> is the wind speed at a height of 19.5m above the sea surface, the height of the anemometers on the weather ships used by [[Pierson and Moskowitz 1964]].

For most air flow over the sea the atmospheric boundary layer has nearly neutral stability, and

<center>
<math>U_{19.5}\approx 1.026 U_{10}</math>
</center>

assuming a drag coefficient of <math> 1.3 \times 10^{-3}</math>.

The frequency of the peak of the Pierson-Moskowitz spectrum is calculated by solving <math>dS / d\omega = 0</math> for <math>\omega_p</math>, to obtain

<center>
<math>\omega_p = 0.877 g / U_{19.5}\,\!</math>
</center>

The speed of waves at the peak is calculated from (16.10), which gives:

<center><math>c_p=\frac{g}{\omega_p}=1.14U_{19.5}\approx 1.17U_{10}</math></center>

Hence waves with frequency <math>\omega_p</math> travel 14% faster than the wind at a height of 19.5m or 17% faster than the wind at a height of 10m. This poses a difficult problem: How can the wind produce waves traveling faster than the wind? We will return to the problem after we discuss the JONSWAP spectrum and the influence of nonlinear interactions among wind-generated waves.

The significant wave-height is calculated from the integral of <math>S\mbox{ }(\omega)</math> to obtain:

<center>
<math>\left \langle \zeta^2\right \rangle=\int_{0}^{\infty}S(\omega)\mbox{ }\mathrm{d}\omega=2.74 \times 10^{-3}\frac{(U_{10.5})^4}{g^2}</math>
</center>

Remembering that <math>H_{1/3} = 4\mbox{ }<\mbox{ }\zeta^2 \mbox{ }>\mbox{ }^{1/2}</math>, the significant wave-height calculated from the Pierson-Moskowitz spectrum is:

<center>
<math>H_{1/3}=0.21\frac{(U_{19.5})^2}{g}\approx0.22\frac{(U_{10})^2}{g}</math>
</center>

Practical wave analysis of uses the frequency, <math> f </math>, instead of the angular frequency <math> \omega </math>. If we have a frequency spectrum, <math> S(\omega) </math>, then the corresponding frequency spectrum, <math> S'(f) </math>, will be

<center><math> S'(f) = S(2\pi f) \frac{\mathrm{d}\omega}{\mathrm{d}f} = 2\pi S(2\pi f) </math></center>
(the factor of <math> 2\pi </math> insures that the area under the curve remains the same).

In the following we omit the ' symbol. For practical reasons, it has also become standard to relate the variables to the main sea state parameters rather than wind speed, and to use slightly different values
so that the Pierson-Moskowitz spectrum can be expressed as

<center>
<math>
S(f) = \alpha g^2(2\pi)^{-4}f^{-5}\exp\left(-\frac{5}{4}\left(\frac{f_m}{f}\right)^4\right)
</math>
</center>
where <math> \alpha </math> is Phillips constant and <math> f_m </math> is the peak frequency. Remember that
these expressions are an approximation to the real sea spectrum.

Figure 2 gives significant wave-heights and periods calculated from the Pierson-Moskowitz spectrum.

<center>
[[Image:Fig16-8s.jpg]]
</center>
<center>
Figure 2 Significant wave-height and period at the peak of the spectrum of a fully developed sea calculated from the Pierson-Moskowitz spectrum.
</center>

== JONSWAP Spectrum ==
[[Hasselmann et al. 1973]], after analyzing data collected during the Joint North Sea Wave Observation Project JONSWAP, found that the wave spectrum is never fully developed. It continues to develop through non-linear, wave-wave interactions even for very long times and distances. Hence an extra and somewhat artificial factor was added to the Pierson-Moskowitz spectrum in order to improve the fit to their measurements. The JONSWAP spectrum is thus a Pierson-Moskowitz spectrum multiplied by an extra peak enhancement factor
<math> \gamma^r </math>

<center><math>S_j(\omega)=\frac{\alpha g^2}{\omega^5}\exp\left[-\frac{5}{4}\left(\frac{\omega_p}{\omega}\right)^4\right]\gamma^r</math></center>

<center><math>r=\exp\left[-\frac{(\omega-\omega_p)^2}{2\sigma^2\omega_p^2}\right]</math></center>

<center>
[[Image:Fig16-9s.jpg]]
</center>
<center>
Figure 3 Wave spectra of a developing sea for different fetches according to [[Hasselmann et al. 1973]].
</center>

Figure 3 show wave spectra as mearured from {[Hasselmann et al. 1973]] and Figure 4 shows a comparison
of the JONSWAP and Pierson-Moskowitz spectra.
<center>
[[Image:SpSj.jpg]]
</center>
<center>
Figure 4 The JONSWAP and Pierson-Moskowitz spectra.
</center>

Wave data collected during the JONSWAP experiment were used to determine the values for the constants in the
above equations:

::::::::::::<math>\alpha=0.076\left(\frac{U_{10}^2}{F\mbox{ }g}\right)^{0.22}</math>

::::::::::::<math>\omega_p=22\left(\frac{g^2}{U_{10}F}\right)^{1/3}</math>

::::::::::::<math>\gamma = 3.3 \,\!</math>

::::::::::::<math> \sigma = \begin{cases} 0.07 & \omega \le \omega_p \\ 0.09 & \omega > \omega_p \end{cases} </math>

where <math>F</math> is the distance from a lee shore, called the fetch, or the distance over which the wind blows with constant velocity.

The energy of the waves increases with fetch:

<center><math>\left \langle \zeta^2\right \rangle =1.67 \times 10^{-7} \frac{(U_{10})^2}{g}x</math></center>

where x is fetch.

The JONSWAP spectrum is similar to the Pierson-Moskowitz spectrum except that waves continues to grow with distance (or time) as specified by the a term, and the peak in the spectrum is more pronounced, as specified by the g term. The latter turns out to be particularly important because it leads to enhanced non-linear interactions and a spectrum that changes in time according to the theory of [[Hasselmann 1966]].

== Generation of Waves by Wind ==
We have seen in the last few paragraphs that waves are related to the wind. We have, however, put off until now just how they are generated by the wind. Suppose we begin with a mirror-smooth sea (Beaufort Number 0). What happens if the wind suddenly begins to blow steadily at say 8m/s? Three different physical processes begin:

# The turbulence in the wind produces random pressure fluctuations at the sea surface, which produces small waves with wavelengths of a few centimeters ([[Phillips 1957]]).
# Next, the wind acts on the small waves, causing them to become larger. Wind blowing over the wave produces pressure differences along the wave profile causing the wave to grow. The process is unstable because, as the wave gets bigger, the pressure differences get bigger, and the wave grows faster. The instability causes the wave to grow exponentially ([[Miles 1957]]).
# Finally, the waves begin to interact among themselves to produce longer waves ([[Hasselmann et al. 1973]]). The interaction transfers wave energy from short waves generated by Miles mechanism to waves with frequencies slightly lower than the frequency of waves at the peak of the spectrum. Eventually, this leads to waves going faster than the wind, as noted by Pierson and Moskowitz.

==Acknowledgement==

The material in this page has been derived from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by [[Robert Stewart]].

[[Category:Geophysics]]

Waves and the Concept of a Wave Spectrum

2012-12-09T20:17:48Z

Kri: /* Calculating The Wave Spectrum */ corrected index and "digital" -> "discrete"

== Introduction ==

If we look out to sea, we notice that waves on the sea surface are not simple sinusoids. The surface appears to be composed of random waves of various lengths and periods. How can we describe this surface? The simple answer is, Not very easily. We can however, with some simplifications, come close to describing the surface. The simplifications lead to the concept of the spectrum of ocean waves. The spectrum gives the distribution of wave energy among different wave frequencies of wave-lengths on the sea surface.

The concept of a spectrum is based on work by [http://en.wikipedia.org/wiki/Joseph_Fourier Joseph Fourier] (1768 - 1830), who showed that almost any function <math>\zeta(t)</math> (or <math>\zeta(x)</math> if you like), can be represented over the interval <math>-T/2 < t < T/2</math> as the sum of an infinite series of sine and cosine functions with harmonic wave frequencies:

<center><math>\zeta(t) = \frac{a_0}{2}+\sum_{n=1}^\infty (a_n\cos(2{\pi}nft)+b_n\sin(2{\pi}nft)) \,\!</math></center>

where

<Center><math> a_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\cos(2{\pi}nft)\mathrm{d}t,\qquad (n=0,1,2,...)</math></center>

<Center><math>b_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\sin(2{\pi}nft)\mathrm{d}t,\qquad(n=0,1,2,...)</math></center>

<math>f = 1/T</math> is the fundamental frequency, and <math>nf</math> are harmonics of the fundamental frequency. This form of <math>\zeta(t)</math> is called a [http://en.wikipedia.org/wiki/Fourier_series Fourier series]. Notice that <math>a_0</math> is the mean value of <math>\zeta(t)</math> over the interval.

These equations can be simplified using

<center><math>\exp(2{\pi}inft)=\cos(2{\pi}nft)+i\sin(2{\pi}nft)\,</math></center>

where <math>i = \sqrt{-1}</math>. The equations for the Fourier Series then become:

<center><math>\zeta(t)=\sum_{n=-\infty}^\infty Z_n\,\exp^{i2{\pi}nft}</math></center>

where

<center><math>Z_n=\frac{1}{T}\int_{-T/2}^{T/2}\zeta(t)\,\exp^{-i2{\pi}nft}\,\mathrm{d}t, \qquad(n=0,1,2,...)</math></center>

<math>Z_n</math> is called the complex Fourier series of <math>\zeta(t)</math>. This form is preferable and is the basis of algorithms like the [http://en.wikipedia.org/wiki/Fast_Fourier_Transform Fast Fourier Transform]

The spectrum <math>S(f)</math> of <math>\zeta(t)</math> is:

<center><math>S(nf)=Z_n Z_n^*</math></center>

where <math>Z_n^*</math> is the complex conjugate of <math>Z_n</math>. We will use these forms for the Fourier series and spectra when we describe the computation of ocean wave spectra.

We can expand the idea of a Fourier series to include series that represent surfaces <math>\zeta(x,y)</math> using similar techniques. Thus, any surface can be represented as an infinite series of sine and cosine functions oriented in all possible directions.

Now, let's apply these ideas to the sea surface. Suppose for a moment that the sea surface were frozen in time. Using the Fourier expansion, the frozen surface can be represented as an infinite series of sine and cosine functions of different wave numbers oriented in all possible directions. If we unfreeze the surface and let it evolve in time, we can represent the sea surface as an infinite series of sine and cosine functions of different wave-lengths moving in all directions. Because wave-lengths and wave frequencies are related through the dispersion relation, we can also represent the sea surface as an infinite sum of sine and cosine functions of different frequencies moving in all directions.

Note in our discussion of Fourier series that we assume the coefficients <math>(a_n,\mbox{ }b_n,\mbox{ } Z_n)</math> are constant. For times of perhaps an hour, and distances of perhaps tens of kilometers, the waves on the sea surface are sufficiently fixed that the assumption is true. Furthermore, non-linear interactions among waves are very weak. Therefore, we can represent a local sea surface by a linear super-position of real, sine waves having many different wave-lengths or frequencies and different phases traveling in many different directions. The Fourier series in not just a convenient mathematical expression, it states that the sea surface is really, truly composed of sine waves, each one propagating according to the equations in [[Linear Theory of Ocean Surface Waves]].

The concept of the sea surface being composed of independent waves can be carried further. Suppose I throw a rock into a calm ocean, and it makes a big splash. According to Fourier, the splash can be represented as a superposition of cosine waves all of nearly zero phase so the waves add up to a big splash at the origin. Furthermore, each individual Fourier wave then begins to travel away from the splash. The longest waves travel fastest, and eventually, far from the splash, the sea consists of a dispersed train of waves with the longest waves further from the splash and the shortest waves closest. This is exactly what we see in Figure 1. The storm makes the splash, and the waves disperse as seen in the figure.

<center>
[[Image:Fig16-1s.jpg]]
</center>

Figure 1 Contours of wave energy on a frequency-time plot calculated from spectra of waves measured by pressure gauges offshore of southern California. The ridges of high wave energy show the arrival of dispersed wave trains from distant storms. The slope of the ridge is inversely proportional to distance to the storm. D is distance in degrees, <math>\theta</math> is direction of arrival of waves at California. From [[Munk et al. 1963]]. (this figure is discussed in detail in [[Linear Theory of Ocean Surface Waves]]).

== Sampling the Sea Surface ==

Calculating the Fourier series that represents the sea surface is perhaps impossible. It requires that we measure the height of the sea surface <math>\zeta(x,\,y,\,t)</math> everywhere in an area perhaps ten kilometers on a side for perhaps an hour. So, let's simplify. Suppose we install a wave staff somewhere in the ocean and record the height of the sea surface as a function of time <math>\zeta(t)</math>. We would obtain a record like that in Figure 2. All waves on the sea surface will be measured, but we will know nothing about the direction of the waves. This is a much more practical measurement, and it will give the frequency spectrum of the waves on the sea surface.

<center>
[[Image:Fig16-2s.jpg]]
</center>
Figure 2. A short record of wave amplitude measured by a wave buoy in the North Atlantic.

Working with a trace of wave-height on say a piece of paper is difficult, so let's digitize the output of the wave staff to obtain
<center><math>\begin{matrix}\zeta_j\equiv\zeta(t_j), & \qquad & t_j \equiv j\triangle \\ \ & \qquad & j=0,1,2,...,N-1 \end{matrix}</math></center>
where <math>\Delta</math> is the time interval between the samples, and <math>N</math> is the total number of samples. The length <math>T</math> of the record is <math>T = N\Delta</math>. Figure 3 shows the first 20 seconds of wave-height from Figure 2 digitized at intervals of
<math>\Delta</math> = 0.32 s.
<center>
[[Image:Fig16-3s.jpg]]
</center>
Figure 3 The first 20 seconds of digitized data from Figure 2.
<math>\Delta</math> = 0.32s.

Notice that <math>\zeta_j</math> is not the same as <math>\zeta(t)</math>.We have absolutely no information about the height of the sea surface between samples. Thus we have converted from an infinite set of numbers which describes <math>\zeta(t)</math> to a finite set of numbers which describe <math>\zeta_j</math>. By converting from a continuous function to a digitized function, we have given up an infinite amount of information about the surface.

The sampling interval <math>\Delta</math> defines a [http://en.wikipedia.org/wiki/Nyquist_frequency Nyquist critical frequency] ([[Press et al. 1992]]: 494)
<math>N_y = 1/(2\Delta)</math>.

The Nyquist critical frequency is important for two related, but distinct, reasons. One is good news, the other is bad news. First the good news. It is the remarkable fact known as the [http://en.wikipedia.org/wiki/Sampling_theorem sampling theorem]: If a continuous function <math>z(t)</math> , sampled at an interval <math>\Delta</math>, happens to be bandwidth limited to frequencies smaller in magnitude than <math>N_y</math>,
i.e. if <math>S(f) = 0</math> for all <math>|f| > N_y</math>, then the function <math>z(t)</math> is completely determined by its samples <math>z_j</math>. This is a remarkable theorem for many reasons, among them that it shows that the "information content" of a bandwidth limited function is, in some sense, infinitely smaller than that of a general continuous function.

Now the bad news. The bad news concerns the effect of sampling a continuous function that is not bandwidth limited to less than the Nyquist critical frequency. In that case, it turns out that all of the power spectral density that lies outside the frequency range -Ny < nf < Ny is spuriously moved into that range. This phenomenon is called aliasing. Any frequency component outside of the range (-Ny, Ny ) is aliased (falsely translated) into that range by the very act of discrete sampling... There is little that you can do to remove aliased power once you have discretely sampled a signal. The way to overcome aliasing is to (i) know the natural bandwidth limit of the signal - or else enforce a known limit by analog filtering of the continuous signal, and then (ii) sample at a rate sufficiently rapid to give at least two points per cycle of the highest frequency present.

Figure 4 illustrates the aliasing problem. Notice how a high frequency signal is aliased into a lower frequency if the higher frequency is above the critical frequency. Fortunately, we can can easily avoid the problem: (i) use instruments that do not respond to very short, high frequency waves if we are interested in the bigger waves; and (ii) chose <math>\Delta</math> small enough that we lose little useful information. In the example shown in Figure 3, there are no waves in the signal to be digitized with frequencies higher than <math>N_y</math> = 1.5625 Hz.
<center>
[[Image:Fig16-4s.jpg]]</center>
Figure 4 Sampling a 4 Hz sine wave (heavy Line) every 0.2 s aliases the frequency to 1 Hz (light line) The critical frequency is <math>1/(2 \times 0.2</math> s) = 2.5 Hz, which is less than 4 Hz.

Let's summarize. Digitized signals from a wave staff cannot be used to study waves with frequencies above the Nyquist critical frequency. Nor can the signal be used to study waves with frequencies less than the fundamental frequency determined by the duration T of the wave record. The digitized wave record contains information about waves in the frequency range:
<center><math>\frac{1}{T}<f<\frac{1}{2\triangle}</math></center>
where <math>T = N\Delta</math> is the length of the time series, and <math>f</math> is the frequency in Hertz.

== Calculating The Wave Spectrum ==
The discrete Fourier transform <math>Z_n</math> of a wave record <math>\zeta_j</math> is:
<center><math>Z_n=\frac{1}{N}\sum_{j=0}^{N-1}\,\zeta_j\,\exp[-i2{\pi}jn/N]</math></center>
<center><math>\zeta_j=\sum_{n=0}^{N-1}\,Z_j\,\exp[i2{\pi}jn/N]\,</math></center>
for <math>j \,=\, 0,\, 1, \dots,\, N\, -\, 1; \,n\, = \,0, \,1, \dots, \,N \,- \,1</math>. These equations can be summed very quickly using the
[http://en.wikipedia.org/wiki/Fast_fourier_transform Fast Fourier Transform], especially if <math>N</math> is a power of 2 ([[Cooley, Lewis, and Welch 1970]]; [[Press et al. 1992]]: 542).

The simple spectrum <math>S_n</math> of <math>\zeta_j</math>, which is called the periodogram, is:
<Center><math>S_n=\frac{1}{N^2}[|Z_n| ^2+|Z_{N-n}|^2];\qquad n=1,2,...,(N/2-1)</math></Center>
<center><math>S_0=\frac{1}{N^2}|Z_0|^2</math></center>
<Center><math>S_{N/2}=\frac{1}{N^2}|Z_{N/2}|^2</math></center>
where <math>S_N</math> is normalized such that:
<center><math>\sum_{j=0}^{N-1}\,|\zeta_j|^2=\sum_{n=0}^{N/1}S_n</math></center>
thus the variance of <math>\zeta_j</math> is the sum of the <math>(N/2\, + \,1)</math> terms in the periodogram. Note, the terms of <math>S_n</math> above the frequency <math>(N/2)</math> are symmetric about that frequency. Figure 5 shows the periodogram of the time series shown in Figure 2.
<center>
[[Image:Fig16-5s.jpg]]
</center>
Figure 5 The periodogram calculated from the first 164s of data from Figure 2. The Nyquist frequency is 1.5625Hz.

The periodogram is a very noisy function. The variance of each point is equal to the expected value at the point. By averaging together 10-30 periodograms we can reduce the uncertainty in the value at each frequency. The averaged periodogram is called the spectrum of the wave-height (Figure 6). It gives the distribution of the variance of '''sea-surface''' (the height the sea surface would be if there were no waves) height at the wave staff as a function of frequency. Because wave energy is proportional to the variance the spectrum is called the energy spectrum or the wave-height spectrum. Typically three hours of wave staff data are used to compute a spectrum of wave-height.
<center>
[[Image:Fig16-6s.jpg]]
</center>
Figure 6 The spectrum of waves calculated from 11 minutes of data shown in Figure 2 by averaging four periodograms to reduce uncertainty in the spectral values. Spectral values below 0.04Hz are in error due to noise.

== Summary ==
We can summarize the calculation of a spectrum into the following steps:

# Digitize a segment of wave-height data to obtain useful limits according to out equations. For example, use 1024 samples from 8.53 minutes of data sampled at the rate of 2 samples/second.
# Calculate the digital, fast Fourier transform <math>Z_n</math> of the time series.
# Calculate the periodogram <math>S_n</math> from the sum of the squares of the real and imaginary parts of the Fourier transform.
# Repeat to produce M = 20 periodograms.
# Average the 20 periodograms to produce an averaged spectrum <math>S_M</math>.
# <math>S_M</math> has values that are [http://en.wikipedia.org/wiki/Chi-square_distribution <math>\chi^2</math> distributed] with <math>2M</math> degrees of freedom.

This outline of the calculation of a spectrum ignores many details. For more complete information see, for example, [[Percival and Walden 1993]], [[Press et al. 1992]], [[Oppenheim and Schafer 1975]], or other texts on digital signal processing.

== Acknowledgement ==

The material in this page has come from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by [[Robert Stewart]].

[[Category:Geophysics]]

Waves and the Concept of a Wave Spectrum

2012-12-09T19:18:42Z

Kri: /* Introduction */ The asterisk should be superscripted

== Introduction ==

If we look out to sea, we notice that waves on the sea surface are not simple sinusoids. The surface appears to be composed of random waves of various lengths and periods. How can we describe this surface? The simple answer is, Not very easily. We can however, with some simplifications, come close to describing the surface. The simplifications lead to the concept of the spectrum of ocean waves. The spectrum gives the distribution of wave energy among different wave frequencies of wave-lengths on the sea surface.

The concept of a spectrum is based on work by [http://en.wikipedia.org/wiki/Joseph_Fourier Joseph Fourier] (1768 - 1830), who showed that almost any function <math>\zeta(t)</math> (or <math>\zeta(x)</math> if you like), can be represented over the interval <math>-T/2 < t < T/2</math> as the sum of an infinite series of sine and cosine functions with harmonic wave frequencies:

<center><math>\zeta(t) = \frac{a_0}{2}+\sum_{n=1}^\infty (a_n\cos(2{\pi}nft)+b_n\sin(2{\pi}nft)) \,\!</math></center>

where

<Center><math> a_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\cos(2{\pi}nft)\mathrm{d}t,\qquad (n=0,1,2,...)</math></center>

<Center><math>b_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\sin(2{\pi}nft)\mathrm{d}t,\qquad(n=0,1,2,...)</math></center>

<math>f = 1/T</math> is the fundamental frequency, and <math>nf</math> are harmonics of the fundamental frequency. This form of <math>\zeta(t)</math> is called a [http://en.wikipedia.org/wiki/Fourier_series Fourier series]. Notice that <math>a_0</math> is the mean value of <math>\zeta(t)</math> over the interval.

These equations can be simplified using

<center><math>\exp(2{\pi}inft)=\cos(2{\pi}nft)+i\sin(2{\pi}nft)\,</math></center>

where <math>i = \sqrt{-1}</math>. The equations for the Fourier Series then become:

<center><math>\zeta(t)=\sum_{n=-\infty}^\infty Z_n\,\exp^{i2{\pi}nft}</math></center>

where

<center><math>Z_n=\frac{1}{T}\int_{-T/2}^{T/2}\zeta(t)\,\exp^{-i2{\pi}nft}\,\mathrm{d}t, \qquad(n=0,1,2,...)</math></center>

<math>Z_n</math> is called the complex Fourier series of <math>\zeta(t)</math>. This form is preferable and is the basis of algorithms like the [http://en.wikipedia.org/wiki/Fast_Fourier_Transform Fast Fourier Transform]

The spectrum <math>S(f)</math> of <math>\zeta(t)</math> is:

<center><math>S(nf)=Z_n Z_n^*</math></center>

where <math>Z_n^*</math> is the complex conjugate of <math>Z_n</math>. We will use these forms for the Fourier series and spectra when we describe the computation of ocean wave spectra.

We can expand the idea of a Fourier series to include series that represent surfaces <math>\zeta(x,y)</math> using similar techniques. Thus, any surface can be represented as an infinite series of sine and cosine functions oriented in all possible directions.

Now, let's apply these ideas to the sea surface. Suppose for a moment that the sea surface were frozen in time. Using the Fourier expansion, the frozen surface can be represented as an infinite series of sine and cosine functions of different wave numbers oriented in all possible directions. If we unfreeze the surface and let it evolve in time, we can represent the sea surface as an infinite series of sine and cosine functions of different wave-lengths moving in all directions. Because wave-lengths and wave frequencies are related through the dispersion relation, we can also represent the sea surface as an infinite sum of sine and cosine functions of different frequencies moving in all directions.

Note in our discussion of Fourier series that we assume the coefficients <math>(a_n,\mbox{ }b_n,\mbox{ } Z_n)</math> are constant. For times of perhaps an hour, and distances of perhaps tens of kilometers, the waves on the sea surface are sufficiently fixed that the assumption is true. Furthermore, non-linear interactions among waves are very weak. Therefore, we can represent a local sea surface by a linear super-position of real, sine waves having many different wave-lengths or frequencies and different phases traveling in many different directions. The Fourier series in not just a convenient mathematical expression, it states that the sea surface is really, truly composed of sine waves, each one propagating according to the equations in [[Linear Theory of Ocean Surface Waves]].

The concept of the sea surface being composed of independent waves can be carried further. Suppose I throw a rock into a calm ocean, and it makes a big splash. According to Fourier, the splash can be represented as a superposition of cosine waves all of nearly zero phase so the waves add up to a big splash at the origin. Furthermore, each individual Fourier wave then begins to travel away from the splash. The longest waves travel fastest, and eventually, far from the splash, the sea consists of a dispersed train of waves with the longest waves further from the splash and the shortest waves closest. This is exactly what we see in Figure 1. The storm makes the splash, and the waves disperse as seen in the figure.

<center>
[[Image:Fig16-1s.jpg]]
</center>

Figure 1 Contours of wave energy on a frequency-time plot calculated from spectra of waves measured by pressure gauges offshore of southern California. The ridges of high wave energy show the arrival of dispersed wave trains from distant storms. The slope of the ridge is inversely proportional to distance to the storm. D is distance in degrees, <math>\theta</math> is direction of arrival of waves at California. From [[Munk et al. 1963]]. (this figure is discussed in detail in [[Linear Theory of Ocean Surface Waves]]).

== Sampling the Sea Surface ==

Calculating the Fourier series that represents the sea surface is perhaps impossible. It requires that we measure the height of the sea surface <math>\zeta(x,\,y,\,t)</math> everywhere in an area perhaps ten kilometers on a side for perhaps an hour. So, let's simplify. Suppose we install a wave staff somewhere in the ocean and record the height of the sea surface as a function of time <math>\zeta(t)</math>. We would obtain a record like that in Figure 2. All waves on the sea surface will be measured, but we will know nothing about the direction of the waves. This is a much more practical measurement, and it will give the frequency spectrum of the waves on the sea surface.

<center>
[[Image:Fig16-2s.jpg]]
</center>
Figure 2. A short record of wave amplitude measured by a wave buoy in the North Atlantic.

Working with a trace of wave-height on say a piece of paper is difficult, so let's digitize the output of the wave staff to obtain
<center><math>\begin{matrix}\zeta_j\equiv\zeta(t_j), & \qquad & t_j \equiv j\triangle \\ \ & \qquad & j=0,1,2,...,N-1 \end{matrix}</math></center>
where <math>\Delta</math> is the time interval between the samples, and <math>N</math> is the total number of samples. The length <math>T</math> of the record is <math>T = N\Delta</math>. Figure 3 shows the first 20 seconds of wave-height from Figure 2 digitized at intervals of
<math>\Delta</math> = 0.32 s.
<center>
[[Image:Fig16-3s.jpg]]
</center>
Figure 3 The first 20 seconds of digitized data from Figure 2.
<math>\Delta</math> = 0.32s.

Notice that <math>\zeta_j</math> is not the same as <math>\zeta(t)</math>.We have absolutely no information about the height of the sea surface between samples. Thus we have converted from an infinite set of numbers which describes <math>\zeta(t)</math> to a finite set of numbers which describe <math>\zeta_j</math>. By converting from a continuous function to a digitized function, we have given up an infinite amount of information about the surface.

The sampling interval <math>\Delta</math> defines a [http://en.wikipedia.org/wiki/Nyquist_frequency Nyquist critical frequency] ([[Press et al. 1992]]: 494)
<math>N_y = 1/(2\Delta)</math>.

The Nyquist critical frequency is important for two related, but distinct, reasons. One is good news, the other is bad news. First the good news. It is the remarkable fact known as the [http://en.wikipedia.org/wiki/Sampling_theorem sampling theorem]: If a continuous function <math>z(t)</math> , sampled at an interval <math>\Delta</math>, happens to be bandwidth limited to frequencies smaller in magnitude than <math>N_y</math>,
i.e. if <math>S(f) = 0</math> for all <math>|f| > N_y</math>, then the function <math>z(t)</math> is completely determined by its samples <math>z_j</math>. This is a remarkable theorem for many reasons, among them that it shows that the "information content" of a bandwidth limited function is, in some sense, infinitely smaller than that of a general continuous function.

Now the bad news. The bad news concerns the effect of sampling a continuous function that is not bandwidth limited to less than the Nyquist critical frequency. In that case, it turns out that all of the power spectral density that lies outside the frequency range -Ny < nf < Ny is spuriously moved into that range. This phenomenon is called aliasing. Any frequency component outside of the range (-Ny, Ny ) is aliased (falsely translated) into that range by the very act of discrete sampling... There is little that you can do to remove aliased power once you have discretely sampled a signal. The way to overcome aliasing is to (i) know the natural bandwidth limit of the signal - or else enforce a known limit by analog filtering of the continuous signal, and then (ii) sample at a rate sufficiently rapid to give at least two points per cycle of the highest frequency present.

Figure 4 illustrates the aliasing problem. Notice how a high frequency signal is aliased into a lower frequency if the higher frequency is above the critical frequency. Fortunately, we can can easily avoid the problem: (i) use instruments that do not respond to very short, high frequency waves if we are interested in the bigger waves; and (ii) chose <math>\Delta</math> small enough that we lose little useful information. In the example shown in Figure 3, there are no waves in the signal to be digitized with frequencies higher than <math>N_y</math> = 1.5625 Hz.
<center>
[[Image:Fig16-4s.jpg]]</center>
Figure 4 Sampling a 4 Hz sine wave (heavy Line) every 0.2 s aliases the frequency to 1 Hz (light line) The critical frequency is <math>1/(2 \times 0.2</math> s) = 2.5 Hz, which is less than 4 Hz.

Let's summarize. Digitized signals from a wave staff cannot be used to study waves with frequencies above the Nyquist critical frequency. Nor can the signal be used to study waves with frequencies less than the fundamental frequency determined by the duration T of the wave record. The digitized wave record contains information about waves in the frequency range:
<center><math>\frac{1}{T}<f<\frac{1}{2\triangle}</math></center>
where <math>T = N\Delta</math> is the length of the time series, and <math>f</math> is the frequency in Hertz.

== Calculating The Wave Spectrum ==
The digital Fourier transform <math>Z_n</math> of a wave record <math>\zeta_j</math> is:
<center><math>Z_n=\frac{1}{N}\sum_{j=0}^{N-1}\,\zeta_j\,\exp[-i2{\pi}jn/N]</math></center>
<center><math>\zeta_n=\sum_{n=0}^{N-1}\,Z_j\,\exp[i2{\pi}jn/N]\,</math></center>
for <math>j \,=\, 0,\, 1, \dots,\, N\, -\, 1; \,n\, = \,0, \,1, \dots, \,N \,- \,1</math>. These equations can be summed very quickly using the
[http://en.wikipedia.org/wiki/Fast_fourier_transform Fast Fourier Transform], especially if <math>N</math> is a power of 2 ([[Cooley, Lewis, and Welch 1970]]; [[Press et al. 1992]]: 542).

The simple spectrum <math>S_n</math> of <math>\zeta_j</math>, which is called the periodogram, is:
<Center><math>S_n=\frac{1}{N^2}[|Z_n| ^2+|Z_{N-n}|^2];\qquad n=1,2,...,(N/2-1)</math></Center>
<center><math>S_0=\frac{1}{N^2}|Z_0|^2</math></center>
<Center><math>S_{N/2}=\frac{1}{N^2}|Z_{N/2}|^2</math></center>
where <math>S_N</math> is normalized such that:
<center><math>\sum_{j=0}^{N-1}\,|\zeta_j|^2=\sum_{n=0}^{N/1}S_n</math></center>
thus the variance of <math>\zeta_j</math> is the sum of the <math>(N/2\, + \,1)</math> terms in the periodogram. Note, the terms of <math>S_n</math> above the frequency <math>(N/2)</math> are symmetric about that frequency. Figure 5 shows the periodogram of the time series shown in Figure 2.
<center>
[[Image:Fig16-5s.jpg]]
</center>
Figure 5 The periodogram calculated from the first 164s of data from Figure 2. The Nyquist frequency is 1.5625Hz.

The periodogram is a very noisy function. The variance of each point is equal to the expected value at the point. By averaging together 10-30 periodograms we can reduce the uncertainty in the value at each frequency. The averaged periodogram is called the spectrum of the wave-height (Figure 6). It gives the distribution of the variance of '''sea-surface''' (the height the sea surface would be if there were no waves) height at the wave staff as a function of frequency. Because wave energy is proportional to the variance the spectrum is called the energy spectrum or the wave-height spectrum. Typically three hours of wave staff data are used to compute a spectrum of wave-height.
<center>
[[Image:Fig16-6s.jpg]]
</center>
Figure 6 The spectrum of waves calculated from 11 minutes of data shown in Figure 2 by averaging four periodograms to reduce uncertainty in the spectral values. Spectral values below 0.04Hz are in error due to noise.

== Summary ==
We can summarize the calculation of a spectrum into the following steps:

# Digitize a segment of wave-height data to obtain useful limits according to out equations. For example, use 1024 samples from 8.53 minutes of data sampled at the rate of 2 samples/second.
# Calculate the digital, fast Fourier transform <math>Z_n</math> of the time series.
# Calculate the periodogram <math>S_n</math> from the sum of the squares of the real and imaginary parts of the Fourier transform.
# Repeat to produce M = 20 periodograms.
# Average the 20 periodograms to produce an averaged spectrum <math>S_M</math>.
# <math>S_M</math> has values that are [http://en.wikipedia.org/wiki/Chi-square_distribution <math>\chi^2</math> distributed] with <math>2M</math> degrees of freedom.

This outline of the calculation of a spectrum ignores many details. For more complete information see, for example, [[Percival and Walden 1993]], [[Press et al. 1992]], [[Oppenheim and Schafer 1975]], or other texts on digital signal processing.

== Acknowledgement ==

The material in this page has come from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by [[Robert Stewart]].

[[Category:Geophysics]]

Waves and the Concept of a Wave Spectrum

2012-12-09T19:13:34Z

Kri: /* Introduction */ Added blank lines between paragraphs and removed new lines within the paragraphs

== Introduction ==

If we look out to sea, we notice that waves on the sea surface are not simple sinusoids. The surface appears to be composed of random waves of various lengths and periods. How can we describe this surface? The simple answer is, Not very easily. We can however, with some simplifications, come close to describing the surface. The simplifications lead to the concept of the spectrum of ocean waves. The spectrum gives the distribution of wave energy among different wave frequencies of wave-lengths on the sea surface.

The concept of a spectrum is based on work by [http://en.wikipedia.org/wiki/Joseph_Fourier Joseph Fourier] (1768 - 1830), who showed that almost any function <math>\zeta(t)</math> (or <math>\zeta(x)</math> if you like), can be represented over the interval <math>-T/2 < t < T/2</math> as the sum of an infinite series of sine and cosine functions with harmonic wave frequencies:

<center><math>\zeta(t) = \frac{a_0}{2}+\sum_{n=1}^\infty (a_n\cos(2{\pi}nft)+b_n\sin(2{\pi}nft)) \,\!</math></center>

where

<Center><math> a_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\cos(2{\pi}nft)\mathrm{d}t,\qquad (n=0,1,2,...)</math></center>

<Center><math>b_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\sin(2{\pi}nft)\mathrm{d}t,\qquad(n=0,1,2,...)</math></center>

<math>f = 1/T</math> is the fundamental frequency, and <math>nf</math> are harmonics of the fundamental frequency. This form of <math>\zeta(t)</math> is called a [http://en.wikipedia.org/wiki/Fourier_series Fourier series]. Notice that <math>a_0</math> is the mean value of <math>\zeta(t)</math> over the interval.

These equations can be simplified using

<center><math>\exp(2{\pi}inft)=\cos(2{\pi}nft)+i\sin(2{\pi}nft)\,</math></center>

where <math>i = \sqrt{-1}</math>. The equations for the Fourier Series then become:

<center><math>\zeta(t)=\sum_{n=-\infty}^\infty Z_n\,\exp^{i2{\pi}nft}</math></center>

where

<center><math>Z_n=\frac{1}{T}\int_{-T/2}^{T/2}\zeta(t)\,\exp^{-i2{\pi}nft}\,\mathrm{d}t, \qquad(n=0,1,2,...)</math></center>

<math>Z_n</math> is called the complex Fourier series of <math>\zeta(t)</math>. This form is preferable and is the basis of algorithms like the [http://en.wikipedia.org/wiki/Fast_Fourier_Transform Fast Fourier Transform]

The spectrum <math>S(f)</math> of <math>\zeta(t)</math> is:

<center><math>S(nf)=Z_nZ_n*</math></center>

where <math>Z*</math> is the complex conjugate of <math>Z</math>. We will use these forms for the Fourier series and spectra when we describe the computation of ocean wave spectra.

We can expand the idea of a Fourier series to include series that represent surfaces <math>\zeta(x,y)</math> using similar techniques. Thus, any surface can be represented as an infinite series of sine and cosine functions oriented in all possible directions.

Now, let's apply these ideas to the sea surface. Suppose for a moment that the sea surface were frozen in time. Using the Fourier expansion, the frozen surface can be represented as an infinite series of sine and cosine functions of different wave numbers oriented in all possible directions. If we unfreeze the surface and let it evolve in time, we can represent the sea surface as an infinite series of sine and cosine functions of different wave-lengths moving in all directions. Because wave-lengths and wave frequencies are related through the dispersion relation, we can also represent the sea surface as an infinite sum of sine and cosine functions of different frequencies moving in all directions.

Note in our discussion of Fourier series that we assume the coefficients <math>(a_n,\mbox{ }b_n,\mbox{ } Z_n)</math> are constant. For times of perhaps an hour, and distances of perhaps tens of kilometers, the waves on the sea surface are sufficiently fixed that the assumption is true. Furthermore, non-linear interactions among waves are very weak. Therefore, we can represent a local sea surface by a linear super-position of real, sine waves having many different wave-lengths or frequencies and different phases traveling in many different directions. The Fourier series in not just a convenient mathematical expression, it states that the sea surface is really, truly composed of sine waves, each one propagating according to the equations in [[Linear Theory of Ocean Surface Waves]].

The concept of the sea surface being composed of independent waves can be carried further. Suppose I throw a rock into a calm ocean, and it makes a big splash. According to Fourier, the splash can be represented as a superposition of cosine waves all of nearly zero phase so the waves add up to a big splash at the origin. Furthermore, each individual Fourier wave then begins to travel away from the splash. The longest waves travel fastest, and eventually, far from the splash, the sea consists of a dispersed train of waves with the longest waves further from the splash and the shortest waves closest. This is exactly what we see in Figure 1. The storm makes the splash, and the waves disperse as seen in the figure.

<center>
[[Image:Fig16-1s.jpg]]
</center>

Figure 1 Contours of wave energy on a frequency-time plot calculated from spectra of waves measured by pressure gauges offshore of southern California. The ridges of high wave energy show the arrival of dispersed wave trains from distant storms. The slope of the ridge is inversely proportional to distance to the storm. D is distance in degrees, <math>\theta</math> is direction of arrival of waves at California. From [[Munk et al. 1963]]. (this figure is discussed in detail in [[Linear Theory of Ocean Surface Waves]]).

== Sampling the Sea Surface ==

Calculating the Fourier series that represents the sea surface is perhaps impossible. It requires that we measure the height of the sea surface <math>\zeta(x,\,y,\,t)</math> everywhere in an area perhaps ten kilometers on a side for perhaps an hour. So, let's simplify. Suppose we install a wave staff somewhere in the ocean and record the height of the sea surface as a function of time <math>\zeta(t)</math>. We would obtain a record like that in Figure 2. All waves on the sea surface will be measured, but we will know nothing about the direction of the waves. This is a much more practical measurement, and it will give the frequency spectrum of the waves on the sea surface.

<center>
[[Image:Fig16-2s.jpg]]
</center>
Figure 2. A short record of wave amplitude measured by a wave buoy in the North Atlantic.

Working with a trace of wave-height on say a piece of paper is difficult, so let's digitize the output of the wave staff to obtain
<center><math>\begin{matrix}\zeta_j\equiv\zeta(t_j), & \qquad & t_j \equiv j\triangle \\ \ & \qquad & j=0,1,2,...,N-1 \end{matrix}</math></center>
where <math>\Delta</math> is the time interval between the samples, and <math>N</math> is the total number of samples. The length <math>T</math> of the record is <math>T = N\Delta</math>. Figure 3 shows the first 20 seconds of wave-height from Figure 2 digitized at intervals of
<math>\Delta</math> = 0.32 s.
<center>
[[Image:Fig16-3s.jpg]]
</center>
Figure 3 The first 20 seconds of digitized data from Figure 2.
<math>\Delta</math> = 0.32s.

Notice that <math>\zeta_j</math> is not the same as <math>\zeta(t)</math>.We have absolutely no information about the height of the sea surface between samples. Thus we have converted from an infinite set of numbers which describes <math>\zeta(t)</math> to a finite set of numbers which describe <math>\zeta_j</math>. By converting from a continuous function to a digitized function, we have given up an infinite amount of information about the surface.

The sampling interval <math>\Delta</math> defines a [http://en.wikipedia.org/wiki/Nyquist_frequency Nyquist critical frequency] ([[Press et al. 1992]]: 494)
<math>N_y = 1/(2\Delta)</math>.

The Nyquist critical frequency is important for two related, but distinct, reasons. One is good news, the other is bad news. First the good news. It is the remarkable fact known as the [http://en.wikipedia.org/wiki/Sampling_theorem sampling theorem]: If a continuous function <math>z(t)</math> , sampled at an interval <math>\Delta</math>, happens to be bandwidth limited to frequencies smaller in magnitude than <math>N_y</math>,
i.e. if <math>S(f) = 0</math> for all <math>|f| > N_y</math>, then the function <math>z(t)</math> is completely determined by its samples <math>z_j</math>. This is a remarkable theorem for many reasons, among them that it shows that the "information content" of a bandwidth limited function is, in some sense, infinitely smaller than that of a general continuous function.

Now the bad news. The bad news concerns the effect of sampling a continuous function that is not bandwidth limited to less than the Nyquist critical frequency. In that case, it turns out that all of the power spectral density that lies outside the frequency range -Ny < nf < Ny is spuriously moved into that range. This phenomenon is called aliasing. Any frequency component outside of the range (-Ny, Ny ) is aliased (falsely translated) into that range by the very act of discrete sampling... There is little that you can do to remove aliased power once you have discretely sampled a signal. The way to overcome aliasing is to (i) know the natural bandwidth limit of the signal - or else enforce a known limit by analog filtering of the continuous signal, and then (ii) sample at a rate sufficiently rapid to give at least two points per cycle of the highest frequency present.

Figure 4 illustrates the aliasing problem. Notice how a high frequency signal is aliased into a lower frequency if the higher frequency is above the critical frequency. Fortunately, we can can easily avoid the problem: (i) use instruments that do not respond to very short, high frequency waves if we are interested in the bigger waves; and (ii) chose <math>\Delta</math> small enough that we lose little useful information. In the example shown in Figure 3, there are no waves in the signal to be digitized with frequencies higher than <math>N_y</math> = 1.5625 Hz.
<center>
[[Image:Fig16-4s.jpg]]</center>
Figure 4 Sampling a 4 Hz sine wave (heavy Line) every 0.2 s aliases the frequency to 1 Hz (light line) The critical frequency is <math>1/(2 \times 0.2</math> s) = 2.5 Hz, which is less than 4 Hz.

Let's summarize. Digitized signals from a wave staff cannot be used to study waves with frequencies above the Nyquist critical frequency. Nor can the signal be used to study waves with frequencies less than the fundamental frequency determined by the duration T of the wave record. The digitized wave record contains information about waves in the frequency range:
<center><math>\frac{1}{T}<f<\frac{1}{2\triangle}</math></center>
where <math>T = N\Delta</math> is the length of the time series, and <math>f</math> is the frequency in Hertz.

== Calculating The Wave Spectrum ==
The digital Fourier transform <math>Z_n</math> of a wave record <math>\zeta_j</math> is:
<center><math>Z_n=\frac{1}{N}\sum_{j=0}^{N-1}\,\zeta_j\,\exp[-i2{\pi}jn/N]</math></center>
<center><math>\zeta_n=\sum_{n=0}^{N-1}\,Z_j\,\exp[i2{\pi}jn/N]\,</math></center>
for <math>j \,=\, 0,\, 1, \dots,\, N\, -\, 1; \,n\, = \,0, \,1, \dots, \,N \,- \,1</math>. These equations can be summed very quickly using the
[http://en.wikipedia.org/wiki/Fast_fourier_transform Fast Fourier Transform], especially if <math>N</math> is a power of 2 ([[Cooley, Lewis, and Welch 1970]]; [[Press et al. 1992]]: 542).

The simple spectrum <math>S_n</math> of <math>\zeta_j</math>, which is called the periodogram, is:
<Center><math>S_n=\frac{1}{N^2}[|Z_n| ^2+|Z_{N-n}|^2];\qquad n=1,2,...,(N/2-1)</math></Center>
<center><math>S_0=\frac{1}{N^2}|Z_0|^2</math></center>
<Center><math>S_{N/2}=\frac{1}{N^2}|Z_{N/2}|^2</math></center>
where <math>S_N</math> is normalized such that:
<center><math>\sum_{j=0}^{N-1}\,|\zeta_j|^2=\sum_{n=0}^{N/1}S_n</math></center>
thus the variance of <math>\zeta_j</math> is the sum of the <math>(N/2\, + \,1)</math> terms in the periodogram. Note, the terms of <math>S_n</math> above the frequency <math>(N/2)</math> are symmetric about that frequency. Figure 5 shows the periodogram of the time series shown in Figure 2.
<center>
[[Image:Fig16-5s.jpg]]
</center>
Figure 5 The periodogram calculated from the first 164s of data from Figure 2. The Nyquist frequency is 1.5625Hz.

The periodogram is a very noisy function. The variance of each point is equal to the expected value at the point. By averaging together 10-30 periodograms we can reduce the uncertainty in the value at each frequency. The averaged periodogram is called the spectrum of the wave-height (Figure 6). It gives the distribution of the variance of '''sea-surface''' (the height the sea surface would be if there were no waves) height at the wave staff as a function of frequency. Because wave energy is proportional to the variance the spectrum is called the energy spectrum or the wave-height spectrum. Typically three hours of wave staff data are used to compute a spectrum of wave-height.
<center>
[[Image:Fig16-6s.jpg]]
</center>
Figure 6 The spectrum of waves calculated from 11 minutes of data shown in Figure 2 by averaging four periodograms to reduce uncertainty in the spectral values. Spectral values below 0.04Hz are in error due to noise.

== Summary ==
We can summarize the calculation of a spectrum into the following steps:

# Digitize a segment of wave-height data to obtain useful limits according to out equations. For example, use 1024 samples from 8.53 minutes of data sampled at the rate of 2 samples/second.
# Calculate the digital, fast Fourier transform <math>Z_n</math> of the time series.
# Calculate the periodogram <math>S_n</math> from the sum of the squares of the real and imaginary parts of the Fourier transform.
# Repeat to produce M = 20 periodograms.
# Average the 20 periodograms to produce an averaged spectrum <math>S_M</math>.
# <math>S_M</math> has values that are [http://en.wikipedia.org/wiki/Chi-square_distribution <math>\chi^2</math> distributed] with <math>2M</math> degrees of freedom.

This outline of the calculation of a spectrum ignores many details. For more complete information see, for example, [[Percival and Walden 1993]], [[Press et al. 1992]], [[Oppenheim and Schafer 1975]], or other texts on digital signal processing.

== Acknowledgement ==

The material in this page has come from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by [[Robert Stewart]].

[[Category:Geophysics]]

Waves and the Concept of a Wave Spectrum

2012-12-09T19:08:36Z

Kri: /* Introduction */ Added parenthesis around arguments for sin and cos

== Introduction ==
If we look out to sea, we notice that waves on the sea surface are not simple sinusoids. The surface appears to be composed of random waves of various lengths and periods. How can we describe this surface? The simple answer is, Not very easily. We can however, with some simplifications, come close to describing the surface. The simplifications lead to the concept of the spectrum of ocean waves. The spectrum gives the distribution of wave energy among different wave frequencies of wave-lengths on the sea surface.

The concept of a spectrum is based on work by
[http://en.wikipedia.org/wiki/Joseph_Fourier Joseph Fourier]
(1768 - 1830), who showed that almost any function <math>\zeta(t)</math> (or <math>\zeta(x)</math> if you like), can be represented over the interval <math>-T/2 < t < T/2</math> as the sum of an infinite series of sine and cosine functions with harmonic wave frequencies:

<center><math>\zeta(t) = \frac{a_0}{2}+\sum_{n=1}^\infty (a_n\cos(2{\pi}nft)+b_n\sin(2{\pi}nft)) \,\!</math></center>
where
<Center><math> a_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\cos(2{\pi}nft)\mathrm{d}t,\qquad (n=0,1,2,...)</math></center>
<Center><math>b_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\sin(2{\pi}nft)\mathrm{d}t,\qquad(n=0,1,2,...)</math></center>
<math>f = 1/T</math> is the fundamental frequency, and <math>nf</math> are harmonics of the fundamental frequency. This form of <math>\zeta(t)</math> is called a
[http://en.wikipedia.org/wiki/Fourier_series Fourier series]. Notice that <math>a_0</math> is the mean value of <math>\zeta(t)</math> over the interval.

These equations can be simplified using
<center><math>\exp(2{\pi}inft)=\cos(2{\pi}nft)+i\sin(2{\pi}nft)\,</math></center>
where <math>i = \sqrt{-1}</math>. The equations for the Fourier Series then become:
<center><math>\zeta(t)=\sum_{n=-\infty}^\infty Z_n\,\exp^{i2{\pi}nft}</math></center>
where
<center><math>Z_n=\frac{1}{T}\int_{-T/2}^{T/2}\zeta(t)\,\exp^{-i2{\pi}nft}\,\mathrm{d}t,
\qquad(n=0,1,2,...)</math></center>
<math>Z_n</math> is called the complex Fourier series of <math>\zeta(t)</math>. This
form is preferable and is the basis of algorithms like the [http://en.wikipedia.org/wiki/Fast_Fourier_Transform Fast Fourier Transform]

The spectrum <math>S(f)</math> of <math>\zeta(t)</math> is:
<center><math>S(nf)=Z_nZ_n*</math></center>
where <math>Z*</math> is the complex conjugate of <math>Z</math>. We will use these forms for the Fourier series and spectra when we describe the computation of ocean wave spectra.

We can expand the idea of a Fourier series to include series that represent surfaces <math>\zeta(x,y)</math> using similar techniques. Thus, any surface can be represented as an infinite series of sine and cosine functions oriented in all possible directions.

Now, let's apply these ideas to the sea surface. Suppose for a moment that the sea surface were frozen in time. Using the Fourier expansion, the frozen surface can be represented as an infinite series of sine and cosine functions of different wave numbers oriented in all possible directions. If we unfreeze the surface and let it evolve in time, we can represent the sea surface as an infinite series of sine and cosine functions of different wave-lengths moving in all directions. Because wave-lengths and wave frequencies are related through the dispersion relation, we can also represent the sea surface as an infinite sum of sine and cosine functions of different frequencies moving in all directions.

Note in our discussion of Fourier series that we assume the coefficients <math>(a_n,\mbox{ }b_n,\mbox{ } Z_n)</math> are constant. For times of perhaps an hour, and distances of perhaps tens of kilometers, the waves on the sea surface are sufficiently fixed that the assumption is true. Furthermore, non-linear interactions among waves are very weak. Therefore, we can represent a local sea surface by a linear super-position of real, sine waves having many different wave-lengths or frequencies and different phases traveling in many different directions. The Fourier series in not just a convenient mathematical expression, it states that the sea surface is really, truly composed of sine waves, each one propagating according to the equations in
[[Linear Theory of Ocean Surface Waves]].

The concept of the sea surface being composed of independent waves can be carried further. Suppose I throw a rock into a calm ocean, and it makes a big splash. According to Fourier, the splash can be represented as a superposition of cosine waves all of nearly zero phase so the waves add up to a big splash at the origin. Furthermore, each individual Fourier wave then begins to travel away from the splash. The longest waves travel fastest, and eventually, far from the splash, the sea consists of a dispersed train of waves with the longest waves further from the splash and the shortest waves closest. This is exactly what we see in Figure 1. The storm makes the splash, and the waves disperse as seen in the figure.

<center>
[[Image:Fig16-1s.jpg]]
</center>
Figure 1 Contours of wave energy on a frequency-time plot calculated from spectra of waves measured by pressure gauges offshore of southern California. The ridges of high wave energy show the arrival of dispersed wave trains from distant storms. The slope of the ridge is inversely proportional to distance to the storm. D is distance in degrees, <math>\theta</math> is direction of arrival of waves at California. From [[Munk et al. 1963]]. (this figure is discussed in detail in
[[Linear Theory of Ocean Surface Waves]]).

== Sampling the Sea Surface ==

Calculating the Fourier series that represents the sea surface is perhaps impossible. It requires that we measure the height of the sea surface <math>\zeta(x,\,y,\,t)</math> everywhere in an area perhaps ten kilometers on a side for perhaps an hour. So, let's simplify. Suppose we install a wave staff somewhere in the ocean and record the height of the sea surface as a function of time <math>\zeta(t)</math>. We would obtain a record like that in Figure 2. All waves on the sea surface will be measured, but we will know nothing about the direction of the waves. This is a much more practical measurement, and it will give the frequency spectrum of the waves on the sea surface.

<center>
[[Image:Fig16-2s.jpg]]
</center>
Figure 2. A short record of wave amplitude measured by a wave buoy in the North Atlantic.

Working with a trace of wave-height on say a piece of paper is difficult, so let's digitize the output of the wave staff to obtain
<center><math>\begin{matrix}\zeta_j\equiv\zeta(t_j), & \qquad & t_j \equiv j\triangle \\ \ & \qquad & j=0,1,2,...,N-1 \end{matrix}</math></center>
where <math>\Delta</math> is the time interval between the samples, and <math>N</math> is the total number of samples. The length <math>T</math> of the record is <math>T = N\Delta</math>. Figure 3 shows the first 20 seconds of wave-height from Figure 2 digitized at intervals of
<math>\Delta</math> = 0.32 s.
<center>
[[Image:Fig16-3s.jpg]]
</center>
Figure 3 The first 20 seconds of digitized data from Figure 2.
<math>\Delta</math> = 0.32s.

Notice that <math>\zeta_j</math> is not the same as <math>\zeta(t)</math>.We have absolutely no information about the height of the sea surface between samples. Thus we have converted from an infinite set of numbers which describes <math>\zeta(t)</math> to a finite set of numbers which describe <math>\zeta_j</math>. By converting from a continuous function to a digitized function, we have given up an infinite amount of information about the surface.

The sampling interval <math>\Delta</math> defines a [http://en.wikipedia.org/wiki/Nyquist_frequency Nyquist critical frequency] ([[Press et al. 1992]]: 494)
<math>N_y = 1/(2\Delta)</math>.

The Nyquist critical frequency is important for two related, but distinct, reasons. One is good news, the other is bad news. First the good news. It is the remarkable fact known as the [http://en.wikipedia.org/wiki/Sampling_theorem sampling theorem]: If a continuous function <math>z(t)</math> , sampled at an interval <math>\Delta</math>, happens to be bandwidth limited to frequencies smaller in magnitude than <math>N_y</math>,
i.e. if <math>S(f) = 0</math> for all <math>|f| > N_y</math>, then the function <math>z(t)</math> is completely determined by its samples <math>z_j</math>. This is a remarkable theorem for many reasons, among them that it shows that the "information content" of a bandwidth limited function is, in some sense, infinitely smaller than that of a general continuous function.

Now the bad news. The bad news concerns the effect of sampling a continuous function that is not bandwidth limited to less than the Nyquist critical frequency. In that case, it turns out that all of the power spectral density that lies outside the frequency range -Ny < nf < Ny is spuriously moved into that range. This phenomenon is called aliasing. Any frequency component outside of the range (-Ny, Ny ) is aliased (falsely translated) into that range by the very act of discrete sampling... There is little that you can do to remove aliased power once you have discretely sampled a signal. The way to overcome aliasing is to (i) know the natural bandwidth limit of the signal - or else enforce a known limit by analog filtering of the continuous signal, and then (ii) sample at a rate sufficiently rapid to give at least two points per cycle of the highest frequency present.

Figure 4 illustrates the aliasing problem. Notice how a high frequency signal is aliased into a lower frequency if the higher frequency is above the critical frequency. Fortunately, we can can easily avoid the problem: (i) use instruments that do not respond to very short, high frequency waves if we are interested in the bigger waves; and (ii) chose <math>\Delta</math> small enough that we lose little useful information. In the example shown in Figure 3, there are no waves in the signal to be digitized with frequencies higher than <math>N_y</math> = 1.5625 Hz.
<center>
[[Image:Fig16-4s.jpg]]</center>
Figure 4 Sampling a 4 Hz sine wave (heavy Line) every 0.2 s aliases the frequency to 1 Hz (light line) The critical frequency is <math>1/(2 \times 0.2</math> s) = 2.5 Hz, which is less than 4 Hz.

Let's summarize. Digitized signals from a wave staff cannot be used to study waves with frequencies above the Nyquist critical frequency. Nor can the signal be used to study waves with frequencies less than the fundamental frequency determined by the duration T of the wave record. The digitized wave record contains information about waves in the frequency range:
<center><math>\frac{1}{T}<f<\frac{1}{2\triangle}</math></center>
where <math>T = N\Delta</math> is the length of the time series, and <math>f</math> is the frequency in Hertz.

== Calculating The Wave Spectrum ==
The digital Fourier transform <math>Z_n</math> of a wave record <math>\zeta_j</math> is:
<center><math>Z_n=\frac{1}{N}\sum_{j=0}^{N-1}\,\zeta_j\,\exp[-i2{\pi}jn/N]</math></center>
<center><math>\zeta_n=\sum_{n=0}^{N-1}\,Z_j\,\exp[i2{\pi}jn/N]\,</math></center>
for <math>j \,=\, 0,\, 1, \dots,\, N\, -\, 1; \,n\, = \,0, \,1, \dots, \,N \,- \,1</math>. These equations can be summed very quickly using the
[http://en.wikipedia.org/wiki/Fast_fourier_transform Fast Fourier Transform], especially if <math>N</math> is a power of 2 ([[Cooley, Lewis, and Welch 1970]]; [[Press et al. 1992]]: 542).

The simple spectrum <math>S_n</math> of <math>\zeta_j</math>, which is called the periodogram, is:
<Center><math>S_n=\frac{1}{N^2}[|Z_n| ^2+|Z_{N-n}|^2];\qquad n=1,2,...,(N/2-1)</math></Center>
<center><math>S_0=\frac{1}{N^2}|Z_0|^2</math></center>
<Center><math>S_{N/2}=\frac{1}{N^2}|Z_{N/2}|^2</math></center>
where <math>S_N</math> is normalized such that:
<center><math>\sum_{j=0}^{N-1}\,|\zeta_j|^2=\sum_{n=0}^{N/1}S_n</math></center>
thus the variance of <math>\zeta_j</math> is the sum of the <math>(N/2\, + \,1)</math> terms in the periodogram. Note, the terms of <math>S_n</math> above the frequency <math>(N/2)</math> are symmetric about that frequency. Figure 5 shows the periodogram of the time series shown in Figure 2.
<center>
[[Image:Fig16-5s.jpg]]
</center>
Figure 5 The periodogram calculated from the first 164s of data from Figure 2. The Nyquist frequency is 1.5625Hz.

The periodogram is a very noisy function. The variance of each point is equal to the expected value at the point. By averaging together 10-30 periodograms we can reduce the uncertainty in the value at each frequency. The averaged periodogram is called the spectrum of the wave-height (Figure 6). It gives the distribution of the variance of '''sea-surface''' (the height the sea surface would be if there were no waves) height at the wave staff as a function of frequency. Because wave energy is proportional to the variance the spectrum is called the energy spectrum or the wave-height spectrum. Typically three hours of wave staff data are used to compute a spectrum of wave-height.
<center>
[[Image:Fig16-6s.jpg]]
</center>
Figure 6 The spectrum of waves calculated from 11 minutes of data shown in Figure 2 by averaging four periodograms to reduce uncertainty in the spectral values. Spectral values below 0.04Hz are in error due to noise.

== Summary ==
We can summarize the calculation of a spectrum into the following steps:

# Digitize a segment of wave-height data to obtain useful limits according to out equations. For example, use 1024 samples from 8.53 minutes of data sampled at the rate of 2 samples/second.
# Calculate the digital, fast Fourier transform <math>Z_n</math> of the time series.
# Calculate the periodogram <math>S_n</math> from the sum of the squares of the real and imaginary parts of the Fourier transform.
# Repeat to produce M = 20 periodograms.
# Average the 20 periodograms to produce an averaged spectrum <math>S_M</math>.
# <math>S_M</math> has values that are [http://en.wikipedia.org/wiki/Chi-square_distribution <math>\chi^2</math> distributed] with <math>2M</math> degrees of freedom.

This outline of the calculation of a spectrum ignores many details. For more complete information see, for example, [[Percival and Walden 1993]], [[Press et al. 1992]], [[Oppenheim and Schafer 1975]], or other texts on digital signal processing.

== Acknowledgement ==

The material in this page has come from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by [[Robert Stewart]].

[[Category:Geophysics]]

Linear Theory of Ocean Surface Waves

2010-12-11T16:15:03Z

Kri: /* Group Velocity */ Framed image

{{complete pages}}

== Introduction ==
Looking out to sea from the shore, we can see waves on the sea surface. Looking carefully, we notice the waves are undulations of the sea surface with a height of around a meter, where height is the vertical distance between the bottom of a trough and the top of a nearby crest. The wavelength, which we might take to be the distance between prominent crests, is around 50m - 100m. Watching the waves for a few minutes, we notice that wave-height and wave-length are not constant. The heights vary randomly in time and space, and the statistical properties of the waves, such as the mean height averaged for a few hundred waves, change from day to day. These prominent offshore waves are generated by wind. Sometimes the local wind generates the waves, other times distant storms generate waves which ultimately reach the coast. For example, waves breaking on the Southern California coast on a summer day may come from vast storms offshore of Antarctica 10,000km away.

If we watch closely for a long time, we notice that sea level changes from hour to hour. Over a period of a day, sea level increases and decreases relative to a point on the shore by about a meter. The slow rise and fall of sea level is due to the tides, another type of wave on the sea surface. Tides have wavelengths of thousands of kilometers, and they are generated by the slow, very small changes in gravity due to the motion of the sun and the moon relative to Earth.

Surface waves are inherently nonlinear: The solution of the equations of motion depends on the surface boundary conditions, but the surface boundary conditions are the waves we wish to calculate. How can we proceed?

We begin by assuming that the amplitude of waves on the water surface is infinitely small so the surface is almost exactly a plane. To simplify the mathematics, we can also assume that the flow is 2-dimensional with waves traveling in the x-direction. We also assume that the Coriolis force and viscosity can be neglected. If we retain rotation, we get Kelvin waves.

With these assumptions, the sea-surface elevation <math> \zeta\ </math> of a wave traveling in the <math>x</math> direction is:

<center>
<math> \zeta\ = a \sin(kx - \omega t) \,\!</math>
</center>

with

<center>
<math> \omega = 2 \pi\,f = \frac{2\pi}{T}; \qquad k = \frac{2\pi}{\lambda} \,\!</math>
</center>

where <math>\omega</math> is wave frequency in radians per second, <math>f</math> is the wave frequency in Hertz (Hz), <math>k</math> is wave number, <math>T</math> is wave period, <math>\lambda</math> is wave-length, and where we assume, as stated above, that <math>k a = O (0)</math>.

The wave period <math>T</math> is the time it takes two successive wave crests or troughs to pass a fixed point. The wave-length <math>\lambda</math> is the distance between two successive wave crests or troughs at a fixed time.

= Dispersion Relation =
Wave frequency <math>\omega</math> is related to wave number <math>k</math> by the
[[Dispersion Relation for a Free Surface|dispersion relation]] ([[Lamb 1932]] §228):

<center>
<math>\omega^2 = gk\tanh(kh) \,\!</math>
</center>

where <math>h</math> is the water depth and <math>g</math> is the acceleration of gravity.

Two approximations are especially useful.

# [[Infinite Depth|Deep-water]] approximation is valid if the water depth <math>h</math> is much greater than the wave-length <math>L</math>. In this case, <math>h</math> >> <math>\lambda</math>, <math>kh</math> >> 1, and <math>\tanh (kh) = 1</math>.
# [[:Category:Shallow Depth|Shallow-water]] approximation is valid if the water depth is much less than a wavelength. In this case, <math>h << \lambda</math>, <math>kh</math> << 1, and <math>\tanh (kh) = kh</math>.

For these two limits of water depth compared with wavelength the dispersion relation reduces to:
<math>\omega^2 = g k \,\!</math> for <math>h > \lambda / 4 </math> for the Deep-water dispersion relation and
<math>\omega^2 = g k^2 h</math>, <math>h < \lambda / 11</math> for the Shallow-water dispersion relation.

The stated limits for <math>h / \lambda</math> give a dispersion relation accurate within 10%. Because many wave properties can be measured with accuracies of 5-10%, the approximations are useful for calculating wave properties. Later we will learn to calculate wave properties as the waves propagate from deep to shallow water.

== Phase Velocity ==
The phase velocity c is the speed at which a particular phase of the wave propagates, for example, the speed of propagation of the wave crest. In one wave period <math>T</math> the crest advances one wave-length \lambda and the phase speed is <math>c = \lambda / T = \omega / k</math>. Thus, the definition of phase speed is:

<center> <math>c \equiv \frac{\omega}{k} \,\!</math> </center>

The direction of propagation is perpendicular to the wave crest and toward the positive <math>x-</math>direction. The deep- and shallow-water approximations for the dispersion relation give:

<center> <math>c = \sqrt{\frac{g}{k}} = \frac{g}{\omega} \qquad \qquad </math> Deep-water phase velocity </center>

<center> <math>c = \sqrt{gh} \qquad \qquad </math> Shallow-water phase velocity </center>

The approximations are accurate to about 5% for limits stated above.

In deep water, the phase speed depends on wave-length or wave frequency. Longer waves travel faster. Thus, deep-water waves are said to be dispersive. In shallow water, the phase speed is independent of the wave; it depends only on the depth of the water. Shallow-water waves are non-dispersive.

== Group Velocity ==
The concept of group velocity <math>c_g</math> is fundamental for understanding the propagation of linear and nonlinear waves. First, it is the velocity at which a group of waves travels across the ocean. More importantly, it is also the propagation velocity of wave energy. [[Whitham 1974]] ( §1.3 and §11.6) gives a clear derivation of the concept and the fundamental equation.

The definition of group velocity in two dimensions is:

<center> <math> c_g \equiv \frac{\partial \omega}{\partial k} \,\!</math> </center>

Using the approximations for the dispersion relation:

<center> <math> c_g = \frac {g}{2 \omega} = \frac {c}{2} \,\!</math> Deep-water group velocity </center>

<center> <math> c_g = \sqrt {gh} = c </math> Shallow-water group velocity </center>

For ocean-surface waves, the direction of propagation is perpendicular to the wave crests in the positive x direction. In the more general case of other types of waves, such as [http://en.wikipedia.org/wiki/Kelvin_Wave Kelvin] and [http://en.wikipedia.org/wiki/Rossby_wave Rossby] waves, the group velocity is not necessarily in the direction perpendicular to wave crests.

Notice that a group of deep-water waves moves at half the phase speed of the waves making up the group. How can this happen? If we could watch closely a group of waves crossing the sea, we would see waves crests appear at the back of the wave train, move through the train, and disappear at the leading edge of the group.

Each wave crest moves at twice the speed of the group. Do real ocean waves move in groups governed by the dispersion relation? Yes. [[Munk et al. 1963]] in a remarkable series of experiments in the 1960s showed that ocean waves propagating over great distances are dispersive, and that the dispersion could be used to track storms. They recorded waves for many days using an array of three pressure gauges just offshore of San Clemente Island, 60 miles due west of San Diego, California. Wave spectra were calculated for each day's data. (The concept of a spectra is discussed below.) From the spectra, the amplitudes and frequencies of the low-frequency waves and the propagation direction of the waves were calculated. Finally, they plotted contours of wave energy on a frequency-time diagram (Figure 1).

[[image:Fig16-1s.jpg|frame|center|Figure 1 Contours of wave energy on a frequency-time plot calculated from spectra of waves measured by pressure gauges offshore of southern California. The ridges of high wave energy show the arrival of dispersed wave trains from distant storms. The slope of the ridge is inversely proportional to distance to the storm. D is distance in degrees, <math>\theta</math> is direction of arrival of waves at California. From [[Munk et al. 1963]].]]

To understand the figure, consider a distant storm that produces waves of many frequencies. The lowest-frequency waves (smallest w) travel the fastest and they arrive before other, higher-frequency waves. The further away the storm, the longer the delay between arrivals of waves of different frequencies. The ridges of high wave energy seen in the Figure are produced by individual storm <math> \theta </math>. The slope of the ridge gives the distance to the storm in degrees <math> \Delta </math> along a great circle; and the phase information from the array gives the angle to the storm. The two angles give the storm's location relative to San Clemente. Thus waves arriving from 15 to 18 September produce a ridge indicating the storm was 115° away at an angle of 205° which is south of new Zealand near Antarctica.

The locations of the storms producing the waves recorded from June through October 1959 were compared with the location of storms plotted on weather maps and in most cases the two agreed well.

== Wave Energy ==

The wave energy density <math>E</math> in Joules per square meter is related to the variance of sea-surface displacement <math> \zeta </math> by:

<center> <math>E = p_w \ g < \zeta^2 > \,\! </math> </center>

where <math>p_w</math> is water density, <math>g</math> is gravity, and the brackets denote a time average. Note that this formula requires that there is quasi steady state so that the average kinetic and potential energies are equal and is only valid for linear waves. Although the formula in theory will give different energy densities for different locations (e.g. for a standing wave there will be locations where the displacement, hence also the energy density, will always be zero), it will in practice give a good result which doesn't vary much from location to location.

[[image:Fig16-2s.jpg|frame|center|Figure 2. A short record of wave amplitude measured by a wave buoy in the North Atlantic.]]

= Significant Wave-Height =
What do we mean by wave-height? If we look at a wind-driven sea, we see waves of various heights. Some are much larger than most, others are much smaller (Figure 2). A practical definition that is often used is the height of the highest 1/3 of the waves, <math>H_{1/3}</math>. The height is computed as follows: measure wave-height for a few minutes, pick out say 120 wave crests and record their heights. Pick the 40 largest waves and calculate the average height of the 40 values. This is <math>H_{1/3}</math> for the wave record.

The concept of significant wave-height was developed during the World War II as part of a project to forecast ocean wave-heights and periods. [[Wiegel 1964]]: p. 198 reports that work at the Scripps Institution of Oceanography showed

''... wave-height estimated by observers corresponds to the average of the highest 20 to 40 per cent of waves... Originally, the term significant wave-height was attached to the average of these observations, the highest 30 percent of the waves, but has evolved to become the average of the highest one-third of the waves, (designated <math>HS</math> or <math>H_{1/3}</math>)''

More recently, significant wave-height is calculated from measured wave displacement. If the sea contains a narrow range of wave frequencies, <math>H_{1/3}</math> is related to the standard deviation of sea-surface displacement ([[NAS 1963]]: 22; [[Hoffman and Karst 1975]])

<center> <math>H_{1/3} = 4 < \zeta^2 >^{1/2} \,\!</math> </center>

where <math>< \zeta^2 >^{1/2}</math> is the standard deviation of surface displacement. This relationship is much more useful, and it is now the accepted way to calculate wave-height from wave measurements

== Acknowledgment ==

The material in this page has come from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by Robert Stewart.

[[Category:Geophysics]][[Category:Linear Water-Wave Theory]]

Linear Theory of Ocean Surface Waves

2010-12-11T16:13:43Z

Kri: /* Wave Energy */ clarified

{{complete pages}}

== Introduction ==
Looking out to sea from the shore, we can see waves on the sea surface. Looking carefully, we notice the waves are undulations of the sea surface with a height of around a meter, where height is the vertical distance between the bottom of a trough and the top of a nearby crest. The wavelength, which we might take to be the distance between prominent crests, is around 50m - 100m. Watching the waves for a few minutes, we notice that wave-height and wave-length are not constant. The heights vary randomly in time and space, and the statistical properties of the waves, such as the mean height averaged for a few hundred waves, change from day to day. These prominent offshore waves are generated by wind. Sometimes the local wind generates the waves, other times distant storms generate waves which ultimately reach the coast. For example, waves breaking on the Southern California coast on a summer day may come from vast storms offshore of Antarctica 10,000km away.

If we watch closely for a long time, we notice that sea level changes from hour to hour. Over a period of a day, sea level increases and decreases relative to a point on the shore by about a meter. The slow rise and fall of sea level is due to the tides, another type of wave on the sea surface. Tides have wavelengths of thousands of kilometers, and they are generated by the slow, very small changes in gravity due to the motion of the sun and the moon relative to Earth.

Surface waves are inherently nonlinear: The solution of the equations of motion depends on the surface boundary conditions, but the surface boundary conditions are the waves we wish to calculate. How can we proceed?

We begin by assuming that the amplitude of waves on the water surface is infinitely small so the surface is almost exactly a plane. To simplify the mathematics, we can also assume that the flow is 2-dimensional with waves traveling in the x-direction. We also assume that the Coriolis force and viscosity can be neglected. If we retain rotation, we get Kelvin waves.

With these assumptions, the sea-surface elevation <math> \zeta\ </math> of a wave traveling in the <math>x</math> direction is:

<center>
<math> \zeta\ = a \sin(kx - \omega t) \,\!</math>
</center>

with

<center>
<math> \omega = 2 \pi\,f = \frac{2\pi}{T}; \qquad k = \frac{2\pi}{\lambda} \,\!</math>
</center>

where <math>\omega</math> is wave frequency in radians per second, <math>f</math> is the wave frequency in Hertz (Hz), <math>k</math> is wave number, <math>T</math> is wave period, <math>\lambda</math> is wave-length, and where we assume, as stated above, that <math>k a = O (0)</math>.

The wave period <math>T</math> is the time it takes two successive wave crests or troughs to pass a fixed point. The wave-length <math>\lambda</math> is the distance between two successive wave crests or troughs at a fixed time.

= Dispersion Relation =
Wave frequency <math>\omega</math> is related to wave number <math>k</math> by the
[[Dispersion Relation for a Free Surface|dispersion relation]] ([[Lamb 1932]] §228):

<center>
<math>\omega^2 = gk\tanh(kh) \,\!</math>
</center>

where <math>h</math> is the water depth and <math>g</math> is the acceleration of gravity.

Two approximations are especially useful.

# [[Infinite Depth|Deep-water]] approximation is valid if the water depth <math>h</math> is much greater than the wave-length <math>L</math>. In this case, <math>h</math> >> <math>\lambda</math>, <math>kh</math> >> 1, and <math>\tanh (kh) = 1</math>.
# [[:Category:Shallow Depth|Shallow-water]] approximation is valid if the water depth is much less than a wavelength. In this case, <math>h << \lambda</math>, <math>kh</math> << 1, and <math>\tanh (kh) = kh</math>.

For these two limits of water depth compared with wavelength the dispersion relation reduces to:
<math>\omega^2 = g k \,\!</math> for <math>h > \lambda / 4 </math> for the Deep-water dispersion relation and
<math>\omega^2 = g k^2 h</math>, <math>h < \lambda / 11</math> for the Shallow-water dispersion relation.

The stated limits for <math>h / \lambda</math> give a dispersion relation accurate within 10%. Because many wave properties can be measured with accuracies of 5-10%, the approximations are useful for calculating wave properties. Later we will learn to calculate wave properties as the waves propagate from deep to shallow water.

== Phase Velocity ==
The phase velocity c is the speed at which a particular phase of the wave propagates, for example, the speed of propagation of the wave crest. In one wave period <math>T</math> the crest advances one wave-length \lambda and the phase speed is <math>c = \lambda / T = \omega / k</math>. Thus, the definition of phase speed is:

<center> <math>c \equiv \frac{\omega}{k} \,\!</math> </center>

The direction of propagation is perpendicular to the wave crest and toward the positive <math>x-</math>direction. The deep- and shallow-water approximations for the dispersion relation give:

<center> <math>c = \sqrt{\frac{g}{k}} = \frac{g}{\omega} \qquad \qquad </math> Deep-water phase velocity </center>

<center> <math>c = \sqrt{gh} \qquad \qquad </math> Shallow-water phase velocity </center>

The approximations are accurate to about 5% for limits stated above.

In deep water, the phase speed depends on wave-length or wave frequency. Longer waves travel faster. Thus, deep-water waves are said to be dispersive. In shallow water, the phase speed is independent of the wave; it depends only on the depth of the water. Shallow-water waves are non-dispersive.

== Group Velocity ==
The concept of group velocity <math>c_g</math> is fundamental for understanding the propagation of linear and nonlinear waves. First, it is the velocity at which a group of waves travels across the ocean. More importantly, it is also the propagation velocity of wave energy. [[Whitham 1974]] ( §1.3 and §11.6) gives a clear derivation of the concept and the fundamental equation.

The definition of group velocity in two dimensions is:

<center> <math> c_g \equiv \frac{\partial \omega}{\partial k} \,\!</math> </center>

Using the approximations for the dispersion relation:

<center> <math> c_g = \frac {g}{2 \omega} = \frac {c}{2} \,\!</math> Deep-water group velocity </center>

<center> <math> c_g = \sqrt {gh} = c </math> Shallow-water group velocity </center>

For ocean-surface waves, the direction of propagation is perpendicular to the wave crests in the positive x direction. In the more general case of other types of waves, such as [http://en.wikipedia.org/wiki/Kelvin_Wave Kelvin] and [http://en.wikipedia.org/wiki/Rossby_wave Rossby] waves, the group velocity is not necessarily in the direction perpendicular to wave crests.

Notice that a group of deep-water waves moves at half the phase speed of the waves making up the group. How can this happen? If we could watch closely a group of waves crossing the sea, we would see waves crests appear at the back of the wave train, move through the train, and disappear at the leading edge of the group.

Each wave crest moves at twice the speed of the group. Do real ocean waves move in groups governed by the dispersion relation? Yes. [[Munk et al. 1963]] in a remarkable series of experiments in the 1960s showed that ocean waves propagating over great distances are dispersive, and that the dispersion could be used to track storms. They recorded waves for many days using an array of three pressure gauges just offshore of San Clemente Island, 60 miles due west of San Diego, California. Wave spectra were calculated for each day's data. (The concept of a spectra is discussed below.) From the spectra, the amplitudes and frequencies of the low-frequency waves and the propagation direction of the waves were calculated. Finally, they plotted contours of wave energy on a frequency-time diagram (Figure 1).
<center>
[[image:Fig16-1s.jpg]]
</center>
Figure 1 Contours of wave energy on a frequency-time plot calculated from spectra of waves measured by pressure gauges offshore of southern California. The ridges of high wave energy show the arrival of dispersed wave trains from distant storms. The slope of the ridge is inversely proportional to distance to the storm. D is distance in degrees, <math>\theta</math> is direction of arrival of waves at California. From [[Munk et al. 1963]].

To understand the figure, consider a distant storm that produces waves of many frequencies. The lowest-frequency waves (smallest w) travel the fastest and they arrive before other, higher-frequency waves. The further away the storm, the longer the delay between arrivals of waves of different frequencies. The ridges of high wave energy seen in the Figure are produced by individual storm <math> \theta </math>. The slope of the ridge gives the distance to the storm in degrees <math> \Delta </math> along a great circle; and the phase information from the array gives the angle to the storm. The two angles give the storm's location relative to San Clemente. Thus waves arriving from 15 to 18 September produce a ridge indicating the storm was 115° away at an angle of 205° which is south of new Zealand near Antarctica.

The locations of the storms producing the waves recorded from June through October 1959 were compared with the location of storms plotted on weather maps and in most cases the two agreed well.

== Wave Energy ==

The wave energy density <math>E</math> in Joules per square meter is related to the variance of sea-surface displacement <math> \zeta </math> by:

<center> <math>E = p_w \ g < \zeta^2 > \,\! </math> </center>

where <math>p_w</math> is water density, <math>g</math> is gravity, and the brackets denote a time average. Note that this formula requires that there is quasi steady state so that the average kinetic and potential energies are equal and is only valid for linear waves. Although the formula in theory will give different energy densities for different locations (e.g. for a standing wave there will be locations where the displacement, hence also the energy density, will always be zero), it will in practice give a good result which doesn't vary much from location to location.

[[image:Fig16-2s.jpg|frame|center|Figure 2. A short record of wave amplitude measured by a wave buoy in the North Atlantic.]]

= Significant Wave-Height =
What do we mean by wave-height? If we look at a wind-driven sea, we see waves of various heights. Some are much larger than most, others are much smaller (Figure 2). A practical definition that is often used is the height of the highest 1/3 of the waves, <math>H_{1/3}</math>. The height is computed as follows: measure wave-height for a few minutes, pick out say 120 wave crests and record their heights. Pick the 40 largest waves and calculate the average height of the 40 values. This is <math>H_{1/3}</math> for the wave record.

The concept of significant wave-height was developed during the World War II as part of a project to forecast ocean wave-heights and periods. [[Wiegel 1964]]: p. 198 reports that work at the Scripps Institution of Oceanography showed

''... wave-height estimated by observers corresponds to the average of the highest 20 to 40 per cent of waves... Originally, the term significant wave-height was attached to the average of these observations, the highest 30 percent of the waves, but has evolved to become the average of the highest one-third of the waves, (designated <math>HS</math> or <math>H_{1/3}</math>)''

More recently, significant wave-height is calculated from measured wave displacement. If the sea contains a narrow range of wave frequencies, <math>H_{1/3}</math> is related to the standard deviation of sea-surface displacement ([[NAS 1963]]: 22; [[Hoffman and Karst 1975]])

<center> <math>H_{1/3} = 4 < \zeta^2 >^{1/2} \,\!</math> </center>

where <math>< \zeta^2 >^{1/2}</math> is the standard deviation of surface displacement. This relationship is much more useful, and it is now the accepted way to calculate wave-height from wave measurements

== Acknowledgment ==

The material in this page has come from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by Robert Stewart.

[[Category:Geophysics]][[Category:Linear Water-Wave Theory]]

Talk:Linear Theory of Ocean Surface Waves

2010-12-11T15:56:13Z

Kri: /* Equation for water energy incorrect */ Ok now I saw it

== Equation for water energy incorrect ==

Hi, one thing that is stated in the article is that the energy of ocean waves in Joule per square meter is given by the equation

<math>E = p_w \ g < \zeta^2 ></math>

This is clearly wrong, considering that the ocean can contain standing waves (i.e. two waves with the same amplitude and wavelength moving in the opposite directions). There will be moments when the displacement for a standing wave is zero everywhere and the formula would evaluate to zero. However, the waves still contain energy because of the kinetic energy that now is because of the motion in the water. The formula does indeed calculate an energy, but it is only the potential energy of the waves and does not contain the kinetic energy. --[[User:Kri|Kri]] 22:11, 15 October 2010 (UTC)

Your point is not correct because for linear waves there is a change from potential to kinetic energy and there is a theory that states that these are equal over one period. The averaging is with respect to time. However, the point is very subtle because we have assumed some kind of stationarity. I have tried to include this in the explanation.

written by Meylan 22:02, 17 October 2010 (UTC)

:That's exactly what I wrote but in other words, there is a change from potential to kinetic energy, but the formula does ''not'' include the kinetic energy, at least not if the angle brackets only denotes an average over a large surface (it doesn't really say over what the average is taken).

:You say that they are equal over one period. What is it that you mean are equal? The time integral of the two forms of energies over one period? And how long is one period when you have a superposition of multiple waves with different wavelengths? It is written in the article that the average kinetic and potential energies (I guess over a large spatial distribution and over a long period of time) have to be equal, but this is true even if the state is composed of multiple wavelengths.

:I can agree with the equation if the angle brackets denotes the average both over a large surface and over a long period of time, since the expression for a stationary wave has nodal points (where the displacement is zero) both in space and in time, but I think it should be more clear in the article what the brackets denote (I'm still not sure). --[[User:Kri|Kri]] 15:48, 11 December 2010 (UTC)

::Ok, whit a second look I saw that the brackets denoted a time average, but I will do some additional clarifications in the article. --[[User:Kri|Kri]] 15:56, 11 December 2010 (UTC)

Linear Theory of Ocean Surface Waves

2010-12-11T15:51:58Z

Kri: /* Wave Energy */ + space average

{{complete pages}}

== Introduction ==
Looking out to sea from the shore, we can see waves on the sea surface. Looking carefully, we notice the waves are undulations of the sea surface with a height of around a meter, where height is the vertical distance between the bottom of a trough and the top of a nearby crest. The wavelength, which we might take to be the distance between prominent crests, is around 50m - 100m. Watching the waves for a few minutes, we notice that wave-height and wave-length are not constant. The heights vary randomly in time and space, and the statistical properties of the waves, such as the mean height averaged for a few hundred waves, change from day to day. These prominent offshore waves are generated by wind. Sometimes the local wind generates the waves, other times distant storms generate waves which ultimately reach the coast. For example, waves breaking on the Southern California coast on a summer day may come from vast storms offshore of Antarctica 10,000km away.

If we watch closely for a long time, we notice that sea level changes from hour to hour. Over a period of a day, sea level increases and decreases relative to a point on the shore by about a meter. The slow rise and fall of sea level is due to the tides, another type of wave on the sea surface. Tides have wavelengths of thousands of kilometers, and they are generated by the slow, very small changes in gravity due to the motion of the sun and the moon relative to Earth.

Surface waves are inherently nonlinear: The solution of the equations of motion depends on the surface boundary conditions, but the surface boundary conditions are the waves we wish to calculate. How can we proceed?

We begin by assuming that the amplitude of waves on the water surface is infinitely small so the surface is almost exactly a plane. To simplify the mathematics, we can also assume that the flow is 2-dimensional with waves traveling in the x-direction. We also assume that the Coriolis force and viscosity can be neglected. If we retain rotation, we get Kelvin waves.

With these assumptions, the sea-surface elevation <math> \zeta\ </math> of a wave traveling in the <math>x</math> direction is:

<center>
<math> \zeta\ = a \sin(kx - \omega t) \,\!</math>
</center>

with

<center>
<math> \omega = 2 \pi\,f = \frac{2\pi}{T}; \qquad k = \frac{2\pi}{\lambda} \,\!</math>
</center>

where <math>\omega</math> is wave frequency in radians per second, <math>f</math> is the wave frequency in Hertz (Hz), <math>k</math> is wave number, <math>T</math> is wave period, <math>\lambda</math> is wave-length, and where we assume, as stated above, that <math>k a = O (0)</math>.

The wave period <math>T</math> is the time it takes two successive wave crests or troughs to pass a fixed point. The wave-length <math>\lambda</math> is the distance between two successive wave crests or troughs at a fixed time.

= Dispersion Relation =
Wave frequency <math>\omega</math> is related to wave number <math>k</math> by the
[[Dispersion Relation for a Free Surface|dispersion relation]] ([[Lamb 1932]] §228):

<center>
<math>\omega^2 = gk\tanh(kh) \,\!</math>
</center>

where <math>h</math> is the water depth and <math>g</math> is the acceleration of gravity.

Two approximations are especially useful.

# [[Infinite Depth|Deep-water]] approximation is valid if the water depth <math>h</math> is much greater than the wave-length <math>L</math>. In this case, <math>h</math> >> <math>\lambda</math>, <math>kh</math> >> 1, and <math>\tanh (kh) = 1</math>.
# [[:Category:Shallow Depth|Shallow-water]] approximation is valid if the water depth is much less than a wavelength. In this case, <math>h << \lambda</math>, <math>kh</math> << 1, and <math>\tanh (kh) = kh</math>.

For these two limits of water depth compared with wavelength the dispersion relation reduces to:
<math>\omega^2 = g k \,\!</math> for <math>h > \lambda / 4 </math> for the Deep-water dispersion relation and
<math>\omega^2 = g k^2 h</math>, <math>h < \lambda / 11</math> for the Shallow-water dispersion relation.

The stated limits for <math>h / \lambda</math> give a dispersion relation accurate within 10%. Because many wave properties can be measured with accuracies of 5-10%, the approximations are useful for calculating wave properties. Later we will learn to calculate wave properties as the waves propagate from deep to shallow water.

== Phase Velocity ==
The phase velocity c is the speed at which a particular phase of the wave propagates, for example, the speed of propagation of the wave crest. In one wave period <math>T</math> the crest advances one wave-length \lambda and the phase speed is <math>c = \lambda / T = \omega / k</math>. Thus, the definition of phase speed is:

<center> <math>c \equiv \frac{\omega}{k} \,\!</math> </center>

The direction of propagation is perpendicular to the wave crest and toward the positive <math>x-</math>direction. The deep- and shallow-water approximations for the dispersion relation give:

<center> <math>c = \sqrt{\frac{g}{k}} = \frac{g}{\omega} \qquad \qquad </math> Deep-water phase velocity </center>

<center> <math>c = \sqrt{gh} \qquad \qquad </math> Shallow-water phase velocity </center>

The approximations are accurate to about 5% for limits stated above.

In deep water, the phase speed depends on wave-length or wave frequency. Longer waves travel faster. Thus, deep-water waves are said to be dispersive. In shallow water, the phase speed is independent of the wave; it depends only on the depth of the water. Shallow-water waves are non-dispersive.

== Group Velocity ==
The concept of group velocity <math>c_g</math> is fundamental for understanding the propagation of linear and nonlinear waves. First, it is the velocity at which a group of waves travels across the ocean. More importantly, it is also the propagation velocity of wave energy. [[Whitham 1974]] ( §1.3 and §11.6) gives a clear derivation of the concept and the fundamental equation.

The definition of group velocity in two dimensions is:

<center> <math> c_g \equiv \frac{\partial \omega}{\partial k} \,\!</math> </center>

Using the approximations for the dispersion relation:

<center> <math> c_g = \frac {g}{2 \omega} = \frac {c}{2} \,\!</math> Deep-water group velocity </center>

<center> <math> c_g = \sqrt {gh} = c </math> Shallow-water group velocity </center>

For ocean-surface waves, the direction of propagation is perpendicular to the wave crests in the positive x direction. In the more general case of other types of waves, such as [http://en.wikipedia.org/wiki/Kelvin_Wave Kelvin] and [http://en.wikipedia.org/wiki/Rossby_wave Rossby] waves, the group velocity is not necessarily in the direction perpendicular to wave crests.

Notice that a group of deep-water waves moves at half the phase speed of the waves making up the group. How can this happen? If we could watch closely a group of waves crossing the sea, we would see waves crests appear at the back of the wave train, move through the train, and disappear at the leading edge of the group.

Each wave crest moves at twice the speed of the group. Do real ocean waves move in groups governed by the dispersion relation? Yes. [[Munk et al. 1963]] in a remarkable series of experiments in the 1960s showed that ocean waves propagating over great distances are dispersive, and that the dispersion could be used to track storms. They recorded waves for many days using an array of three pressure gauges just offshore of San Clemente Island, 60 miles due west of San Diego, California. Wave spectra were calculated for each day's data. (The concept of a spectra is discussed below.) From the spectra, the amplitudes and frequencies of the low-frequency waves and the propagation direction of the waves were calculated. Finally, they plotted contours of wave energy on a frequency-time diagram (Figure 1).
<center>
[[image:Fig16-1s.jpg]]
</center>
Figure 1 Contours of wave energy on a frequency-time plot calculated from spectra of waves measured by pressure gauges offshore of southern California. The ridges of high wave energy show the arrival of dispersed wave trains from distant storms. The slope of the ridge is inversely proportional to distance to the storm. D is distance in degrees, <math>\theta</math> is direction of arrival of waves at California. From [[Munk et al. 1963]].

To understand the figure, consider a distant storm that produces waves of many frequencies. The lowest-frequency waves (smallest w) travel the fastest and they arrive before other, higher-frequency waves. The further away the storm, the longer the delay between arrivals of waves of different frequencies. The ridges of high wave energy seen in the Figure are produced by individual storm <math> \theta </math>. The slope of the ridge gives the distance to the storm in degrees <math> \Delta </math> along a great circle; and the phase information from the array gives the angle to the storm. The two angles give the storm's location relative to San Clemente. Thus waves arriving from 15 to 18 September produce a ridge indicating the storm was 115° away at an angle of 205° which is south of new Zealand near Antarctica.

The locations of the storms producing the waves recorded from June through October 1959 were compared with the location of storms plotted on weather maps and in most cases the two agreed well.

== Wave Energy ==

Wave energy <math>E</math> in Joules per square meter is related to the variance of sea-surface displacement <math> \zeta </math> by:

<center> <math>E = p_w \ g < \zeta^2 > \,\! </math> </center>

where <math>p_w</math> is water density, <math>g</math> is gravity, and the brackets denote a space and time average.
Not that this formula requires that there is quasi steady state so that the average kinetic and potential energies
are equal and is only valid for linear waves.
<center>
[[image:Fig16-2s.jpg]]
</center>
FIgure 2. A short record of wave amplitude measured by a wave buoy in the North Atlantic.

= Significant Wave-Height =
What do we mean by wave-height? If we look at a wind-driven sea, we see waves of various heights. Some are much larger than most, others are much smaller (Figure 2). A practical definition that is often used is the height of the highest 1/3 of the waves, <math>H_{1/3}</math>. The height is computed as follows: measure wave-height for a few minutes, pick out say 120 wave crests and record their heights. Pick the 40 largest waves and calculate the average height of the 40 values. This is <math>H_{1/3}</math> for the wave record.

The concept of significant wave-height was developed during the World War II as part of a project to forecast ocean wave-heights and periods. [[Wiegel 1964]]: p. 198 reports that work at the Scripps Institution of Oceanography showed

''... wave-height estimated by observers corresponds to the average of the highest 20 to 40 per cent of waves... Originally, the term significant wave-height was attached to the average of these observations, the highest 30 percent of the waves, but has evolved to become the average of the highest one-third of the waves, (designated <math>HS</math> or <math>H_{1/3}</math>)''

More recently, significant wave-height is calculated from measured wave displacement. If the sea contains a narrow range of wave frequencies, <math>H_{1/3}</math> is related to the standard deviation of sea-surface displacement ([[NAS 1963]]: 22; [[Hoffman and Karst 1975]])

<center> <math>H_{1/3} = 4 < \zeta^2 >^{1/2} \,\!</math> </center>

where <math>< \zeta^2 >^{1/2}</math> is the standard deviation of surface displacement. This relationship is much more useful, and it is now the accepted way to calculate wave-height from wave measurements

== Acknowledgment ==

The material in this page has come from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by Robert Stewart.

[[Category:Geophysics]][[Category:Linear Water-Wave Theory]]

Talk:Linear Theory of Ocean Surface Waves

2010-12-11T15:48:14Z

Kri: I can agree with the equation if the angle brackets denotes the average both over a large surface and over a long period of time

Linear Theory of Ocean Surface Waves

2010-10-17T00:08:41Z

Kri: /* Wave Energy */ I will let this comment be there for so long, untill anyone clears out the error or explains on the talk page why the equation is not wrong.

{{complete pages}}

== Introduction ==
Looking out to sea from the shore, we can see waves on the sea surface. Looking carefully, we notice the waves are undulations of the sea surface with a height of around a meter, where height is the vertical distance between the bottom of a trough and the top of a nearby crest. The wavelength, which we might take to be the distance between prominent crests, is around 50m - 100m. Watching the waves for a few minutes, we notice that wave-height and wave-length are not constant. The heights vary randomly in time and space, and the statistical properties of the waves, such as the mean height averaged for a few hundred waves, change from day to day. These prominent offshore waves are generated by wind. Sometimes the local wind generates the waves, other times distant storms generate waves which ultimately reach the coast. For example, waves breaking on the Southern California coast on a summer day may come from vast storms offshore of Antarctica 10,000km away.

If we watch closely for a long time, we notice that sea level changes from hour to hour. Over a period of a day, sea level increases and decreases relative to a point on the shore by about a meter. The slow rise and fall of sea level is due to the tides, another type of wave on the sea surface. Tides have wavelengths of thousands of kilometers, and they are generated by the slow, very small changes in gravity due to the motion of the sun and the moon relative to Earth.

Surface waves are inherently nonlinear: The solution of the equations of motion depends on the surface boundary conditions, but the surface boundary conditions are the waves we wish to calculate. How can we proceed?

We begin by assuming that the amplitude of waves on the water surface is infinitely small so the surface is almost exactly a plane. To simplify the mathematics, we can also assume that the flow is 2-dimensional with waves traveling in the x-direction. We also assume that the Coriolis force and viscosity can be neglected. If we retain rotation, we get Kelvin waves.

With these assumptions, the sea-surface elevation <math> \zeta\ </math> of a wave traveling in the <math>x</math> direction is:

<center>
<math> \zeta\ = a \sin(kx - \omega t) \,\!</math>
</center>

with

<center>
<math> \omega = 2 \pi\,f = \frac{2\pi}{T}; \qquad k = \frac{2\pi}{\lambda} \,\!</math>
</center>

where <math>\omega</math> is wave frequency in radians per second, <math>f</math> is the wave frequency in Hertz (Hz), <math>k</math> is wave number, <math>T</math> is wave period, <math>\lambda</math> is wave-length, and where we assume, as stated above, that <math>k a = O (0)</math>.

The wave period <math>T</math> is the time it takes two successive wave crests or troughs to pass a fixed point. The wave-length <math>\lambda</math> is the distance between two successive wave crests or troughs at a fixed time.

= Dispersion Relation =
Wave frequency <math>\omega</math> is related to wave number <math>k</math> by the
[[Dispersion Relation for a Free Surface|dispersion relation]] ([[Lamb 1932]] §228):

<center>
<math>\omega^2 = gk\tanh(kh) \,\!</math>
</center>

where <math>h</math> is the water depth and <math>g</math> is the acceleration of gravity.

Two approximations are especially useful.

# [[Infinite Depth|Deep-water]] approximation is valid if the water depth <math>h</math> is much greater than the wave-length <math>L</math>. In this case, <math>h</math> >> <math>\lambda</math>, <math>kh</math> >> 1, and <math>\tanh (kh) = 1</math>.
# [[:Category:Shallow Depth|Shallow-water]] approximation is valid if the water depth is much less than a wavelength. In this case, <math>h << \lambda</math>, <math>kh</math> << 1, and <math>\tanh (kh) = kh</math>.

For these two limits of water depth compared with wavelength the dispersion relation reduces to:
<math>\omega^2 = g k \,\!</math> for <math>h > \lambda / 4 </math> for the Deep-water dispersion relation and
<math>\omega^2 = g k^2 h</math>, <math>h < \lambda / 11</math> for the Shallow-water dispersion relation.

The stated limits for <math>h / \lambda</math> give a dispersion relation accurate within 10%. Because many wave properties can be measured with accuracies of 5-10%, the approximations are useful for calculating wave properties. Later we will learn to calculate wave properties as the waves propagate from deep to shallow water.

== Phase Velocity ==
The phase velocity c is the speed at which a particular phase of the wave propagates, for example, the speed of propagation of the wave crest. In one wave period <math>T</math> the crest advances one wave-length \lambda and the phase speed is <math>c = \lambda / T = \omega / k</math>. Thus, the definition of phase speed is:

<center> <math>c \equiv \frac{\omega}{k} \,\!</math> </center>

The direction of propagation is perpendicular to the wave crest and toward the positive <math>x-</math>direction. The deep- and shallow-water approximations for the dispersion relation give:

<center> <math>c = \sqrt{\frac{g}{k}} = \frac{g}{\omega} \qquad \qquad </math> Deep-water phase velocity </center>

<center> <math>c = \sqrt{gh} \qquad \qquad </math> Shallow-water phase velocity </center>

The approximations are accurate to about 5% for limits stated above.

In deep water, the phase speed depends on wave-length or wave frequency. Longer waves travel faster. Thus, deep-water waves are said to be dispersive. In shallow water, the phase speed is independent of the wave; it depends only on the depth of the water. Shallow-water waves are non-dispersive.

== Group Velocity ==
The concept of group velocity <math>c_g</math> is fundamental for understanding the propagation of linear and nonlinear waves. First, it is the velocity at which a group of waves travels across the ocean. More importantly, it is also the propagation velocity of wave energy. [[Whitham 1974]] ( §1.3 and §11.6) gives a clear derivation of the concept and the fundamental equation.

The definition of group velocity in two dimensions is:

<center> <math> c_g \equiv \frac{\partial \omega}{\partial k} \,\!</math> </center>

Using the approximations for the dispersion relation:

<center> <math> c_g = \frac {g}{2 \omega} = \frac {c}{2} \,\!</math> Deep-water group velocity </center>

<center> <math> c_g = \sqrt {gh} = c </math> Shallow-water group velocity </center>

For ocean-surface waves, the direction of propagation is perpendicular to the wave crests in the positive x direction. In the more general case of other types of waves, such as [http://en.wikipedia.org/wiki/Kelvin_Wave Kelvin] and [http://en.wikipedia.org/wiki/Rossby_wave Rossby] waves, the group velocity is not necessarily in the direction perpendicular to wave crests.

Notice that a group of deep-water waves moves at half the phase speed of the waves making up the group. How can this happen? If we could watch closely a group of waves crossing the sea, we would see waves crests appear at the back of the wave train, move through the train, and disappear at the leading edge of the group.

Each wave crest moves at twice the speed of the group. Do real ocean waves move in groups governed by the dispersion relation? Yes. [[Munk et al. 1963]] in a remarkable series of experiments in the 1960s showed that ocean waves propagating over great distances are dispersive, and that the dispersion could be used to track storms. They recorded waves for many days using an array of three pressure gauges just offshore of San Clemente Island, 60 miles due west of San Diego, California. Wave spectra were calculated for each day's data. (The concept of a spectra is discussed below.) From the spectra, the amplitudes and frequencies of the low-frequency waves and the propagation direction of the waves were calculated. Finally, they plotted contours of wave energy on a frequency-time diagram (Figure 1).
<center>
[[image:Fig16-1s.jpg]]
</center>
Figure 1 Contours of wave energy on a frequency-time plot calculated from spectra of waves measured by pressure gauges offshore of southern California. The ridges of high wave energy show the arrival of dispersed wave trains from distant storms. The slope of the ridge is inversely proportional to distance to the storm. D is distance in degrees, <math>\theta</math> is direction of arrival of waves at California. From [[Munk et al. 1963]].

To understand the figure, consider a distant storm that produces waves of many frequencies. The lowest-frequency waves (smallest w) travel the fastest and they arrive before other, higher-frequency waves. The further away the storm, the longer the delay between arrivals of waves of different frequencies. The ridges of high wave energy seen in the Figure are produced by individual storm <math> \theta </math>. The slope of the ridge gives the distance to the storm in degrees <math> \Delta </math> along a great circle; and the phase information from the array gives the angle to the storm. The two angles give the storm's location relative to San Clemente. Thus waves arriving from 15 to 18 September produce a ridge indicating the storm was 115° away at an angle of 205° which is south of new Zealand near Antarctica.

The locations of the storms producing the waves recorded from June through October 1959 were compared with the location of storms plotted on weather maps and in most cases the two agreed well.

== Wave Energy ==

Wave energy <math>E</math> in Joules per square meter is related to the variance of sea-surface displacement <math> \zeta </math> by:

<center> <math>E = p_w \ g < \zeta^2 > \,\! </math> (this formula in incorrect, see [[Talk:Linear Theory of Ocean Surface Waves#Equation for water energy incorrect|talk page]])</center>

where <math>p_w</math> is water density, <math>g</math> is gravity, and the brackets denote a time or space average.
<center>
[[image:Fig16-2s.jpg]]
</center>
FIgure 2. A short record of wave amplitude measured by a wave buoy in the North Atlantic.

= Significant Wave-Height =
What do we mean by wave-height? If we look at a wind-driven sea, we see waves of various heights. Some are much larger than most, others are much smaller (Figure 2). A practical definition that is often used is the height of the highest 1/3 of the waves, <math>H_{1/3}</math>. The height is computed as follows: measure wave-height for a few minutes, pick out say 120 wave crests and record their heights. Pick the 40 largest waves and calculate the average height of the 40 values. This is <math>H_{1/3}</math> for the wave record.

The concept of significant wave-height was developed during the World War II as part of a project to forecast ocean wave-heights and periods. [[Wiegel 1964]]: p. 198 reports that work at the Scripps Institution of Oceanography showed

''... wave-height estimated by observers corresponds to the average of the highest 20 to 40 per cent of waves... Originally, the term significant wave-height was attached to the average of these observations, the highest 30 percent of the waves, but has evolved to become the average of the highest one-third of the waves, (designated <math>HS</math> or <math>H_{1/3}</math>)''

More recently, significant wave-height is calculated from measured wave displacement. If the sea contains a narrow range of wave frequencies, <math>H_{1/3}</math> is related to the standard deviation of sea-surface displacement ([[NAS 1963]]: 22; [[Hoffman and Karst 1975]])

<center> <math>H_{1/3} = 4 < \zeta^2 >^{1/2} \,\!</math> </center>

where <math>< \zeta^2 >^{1/2}</math> is the standard deviation of surface displacement. This relationship is much more useful, and it is now the accepted way to calculate wave-height from wave measurements

== Acknowledgment ==

The material in this page has come from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by Robert Stewart.

[[Category:Geophysics]][[Category:Linear Water-Wave Theory]]

Talk:Linear Theory of Ocean Surface Waves

2010-10-15T22:11:48Z

Kri: /* Equation for water energy incorrect */

Talk:Linear Theory of Ocean Surface Waves

2010-10-15T22:11:06Z

Kri: Equation for water energy incorrect