Waves and the Concept of a Wave Spectrum - Revision history

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

2012-12-09T20:17:48Z

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

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	== Calculating The Wave Spectrum ==		== Calculating The Wave Spectrum ==
	The ~~digital~~ Fourier transform <math>Z_n</math> of a wave record <math>\zeta_j</math> is:		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>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>		<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		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).		[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).

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

2012-12-09T19:18:42Z

Introduction: The asterisk should be superscripted

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

	<center><math>S(nf)=~~Z_nZ_n~~*</math></center>		<center><math>S(nf)=Z_n Z_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.		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.		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.

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

2012-12-09T19:13:34Z

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

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	== Introduction ==		== 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.		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		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:
	[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>

	~~<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		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> 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>		<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.		<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		These equations can be simplified using

	<center><math>\exp(2{\pi}inft)=\cos(2{\pi}nft)+i\sin(2{\pi}nft)\,</math></center>		<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:		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>
			<center><math>\zeta(t)=\sum_{n=-\infty}^\infty Z_n\,\exp^{i2{\pi}nft}</math></center>

	where		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>		<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]		<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:		The spectrum <math>S(f)</math> of <math>\zeta(t)</math> is:
	<center><math>S(nf)=Z_nZ_n*</math></center>
			<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.		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.

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	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.		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		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]].
	[[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.		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.
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	[[Image:Fig16-1s.jpg]]		[[Image:Fig16-1s.jpg]]
	</center>		</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]]).		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 ==		== Sampling the Sea Surface ==

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

2012-12-09T19:08:36Z

Introduction: Added parenthesis around arguments for sin and cos

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	(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:		(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\~~cos2~~{\pi}nft+b_n\~~sin2~~{\pi}nft) \,\!</math></center>		<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		where
	<Center><math> a_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\~~cos2~~{\pi}nft\mathrm{d}t,\qquad (n=0,1,2,...)</math></center>		<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)\~~sin2~~{\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		<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.		[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.

Adi Kurniawan: /* Introduction */

2009-03-17T05:36:49Z

Introduction

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	The spectrum <math>S(f)</math> of <math>\zeta(t)</math> is:		The spectrum <math>S(f)</math> of <math>\zeta(t)</math> is:
	<center><math>S(nf)=Z_nZ_n*</math></center>		<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 ~~describing~~ the computation of ocean wave spectra.		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.		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.

Adi Kurniawan: /* Introduction */

2009-03-17T05:35:42Z

Introduction

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	These equations can be simplified using		These equations can be simplified using
	<center><math>\exp(2{\pi}inft)=\cos(2{\pi}nft)+i\sin(2{\pi}nft)\,</math></center>		<center><math>\exp(2{\pi}inft)=\cos(2{\pi}nft)+i\sin(2{\pi}nft)\,</math></center>
	where <math>i =</math> ~~square root of (-1)~~. The equations for the Fourier Series then become:		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>		<center><math>\zeta(t)=\sum_{n=-\infty}^\infty Z_n\,\exp^{i2{\pi}nft}</math></center>
	where		where

Adi Kurniawan at 05:12, 17 March 2009

2009-03-17T05:12:41Z

← Older revision		Revision as of 05:12, 17 March 2009
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	= Introduction =		== 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.		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.

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	[[Linear Theory of Ocean Surface Waves]]).		[[Linear Theory of Ocean Surface Waves]]).

	= Sampling the Sea Surface =		== 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.		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.
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	where <math>T = N\Delta</math> is the length of the time series, and <math>f</math> is the frequency in Hertz.		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 =		== Calculating The Wave Spectrum ==
	The digital Fourier transform <math>Z_n</math> of a wave record <math>\zeta_j</math> is:		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>Z_n=\frac{1}{N}\sum_{j=0}^{N-1}\,\zeta_j\,\exp[-i2{\pi}jn/N]</math></center>
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	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.		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 =		== Summary ==
	We can summarize the calculation of a spectrum into the following steps:		We can summarize the calculation of a spectrum into the following steps:

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	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.		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=		== 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]].		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:Geophysics]]

Adi Kurniawan: /* Introduction */

2008-12-23T05:25:37Z

Introduction

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	<center><math>\zeta(t)=\sum_{n=-\infty}^\infty Z_n\,\exp^{i2{\pi}nft}</math></center>		<center><math>\zeta(t)=\sum_{n=-\infty}^\infty Z_n\,\exp^{i2{\pi}nft}</math></center>
	where		where
	<center><math>Z_n=\frac{1}{T}\int_{-T/2}^{T/2}\zeta(t)\,exp^{-i2{\pi}nft}\,dt,		<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>		\qquad(n=0,1,2,...)</math></center>
	<math>Z_n</math> is called the complex Fourier series of <math>\zeta(t)</math>. This		<math>Z_n</math> is called the complex Fourier series of <math>\zeta(t)</math>. This

Adi Kurniawan: /* Introduction */

2008-12-23T05:24:56Z

Introduction

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	<center><math>\zeta(t) = \frac{a_0}{2}+\sum_{n=1}^\infty (a_n\cos2{\pi}nft+b_n\sin2{\pi}nft) \,\!</math></center>		<center><math>\zeta(t) = \frac{a_0}{2}+\sum_{n=1}^\infty (a_n\cos2{\pi}nft+b_n\sin2{\pi}nft) \,\!</math></center>
	where		where
	<Center><math> a_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\cos2{\pi}nft\~~text~~{d}t,\qquad (n=0,1,2,...)</math></center>		<Center><math> a_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\cos2{\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)\sin2{\pi}nft\~~text~~{d}t,\qquad(n=0,1,2,...)</math></center>		<Center><math>b_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\sin2{\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		<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.		[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.

Adi Kurniawan: /* Introduction */

2008-12-23T05:24:26Z

Introduction

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	<center><math>\zeta(t) = \frac{a_0}{2}+\sum_{n=1}^\infty (a_n\cos2{\pi}nft+b_n\sin2{\pi}nft) \,\!</math></center>		<center><math>\zeta(t) = \frac{a_0}{2}+\sum_{n=1}^\infty (a_n\cos2{\pi}nft+b_n\sin2{\pi}nft) \,\!</math></center>
	where		where
	<Center><math> a_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\cos2{\pi}nft{d}t,\qquad (n=0,1,2,...)</math></center>		<Center><math> a_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\cos2{\pi}nft\text{d}t,\qquad (n=0,1,2,...)</math></center>
	<Center><math>b_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\sin2{\pi}nft{d}t,\qquad(n=0,1,2,...)</math></center>		<Center><math>b_n=\frac{2}{T}\int_{-T/2}^{T/2}\zeta(t)\sin2{\pi}nft\text{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		<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.		[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.