Fourier Series (Basic) JavaScript Simulation Applet HTML5

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Fourier Coefficients

Fourier coefficients

The Fourier series of a periodic function f(x) with period x = 2π is of the form

f(x) = a0 /2 + Σ (ancos(nx) + bnsin(nx)); n=1,2,3....∞

To calculate the coefficients of the series, one starts with the following assumed identities:

∫f(x)cos(mx)dx = ∫cos(mx) Σ(a0/2 + Σ ancos(nx) + bnsin(nx))dx

∫f(x)sin(mx)dx = ∫sin(mx) Σ(a0/2 + Σ ancos(nx) + bnsin(nx))dx

where one integrates over one base period (m = 1).

Suppressing constants, the following types of integral are to be evaluated, summed over index n:

With m = 1,2,3...∞ and n = 1,2,3...∞: order of the harmonic (fundamental m, n = 1)

cos (mx)

sin (mx)

cos (mx) * (a*cos (nx) + b*sin (nx))

sin (mx) * (a*cos (nx) + b*sin(nx))

All integrals are zero except of those few where the indices are identical: m = n and the function types are the same (sine or cosine). Therefore every sum for a specific index n has only one member and the coefficients can easily be derived from the reduced equations as: 

a0 = 2/T∫f(t) dt

an= 2/T∫cos(nx) f(t) dt

bn = 2/T∫sin(nx) f(t) dt

This simulation demonstrates the different types of functions and their integral.

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Description

Operation of the simulation

A ComboBox holds a list of all function combinations described on the Fourier Coefficients page. When one is selected it is displayed in red. The antiderivative is calculated for the fundamental period and drawn as a blue curve. Its end value at x = 2pi is the definite integral over one fundamental period, which is needed for the calculation of the coefficients. The integration process is slowed down to visualize more clearly the consequence of changes in parameters or indices.

Some of the selectable functions contain parameters a and b which can be changed continuously by sliders a/ b . Two other sliders m/n select the critical indices m and n as real numbers between 1 and 10.

Parameters and indices are maintained when functions are changed. Integration is started automatically at any change as long as the selection Integral remains active.

By means of sliders a and b scaling of the ordinate can be adjusted to the specific function. They also allow phase shifting of functions.

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Experiments

E1: Choose cosx in the comboBox. It will be calculated and displayed in red. Activate the Integral check box. The integration process will begin with the initial value of the function at x = 0 and will progress in red to the end of the fundamental period x = 2 π. Reflect why the end value and hence the definite integral over the interval [0, 2pi] is zero for integer n.

E2: Change index n with the slider and watch the integral curve. Reflect again why the definite integral is always zero for integer n.

E3: Choose sinx and verify the experiments for it.

E4: Choose asinx + bcosnx and assure yourself by varying a, b, n that the superposition is always a simple, phase shifted periodical, whose definite integral is zero for integer n.

E5: Choose cosx * sinx and assure yourself that the definite integral is always zero for integer n.

E6: Choose the remaining "mixed" functions and assure yourself that the definite integral is non zero only when both terms are of the same type and have identical indices.

E7: Integrate some of the functions analytically and verify the experimental findings.

E8: Conclude in general which characteristics of the functions sine and cosine are the base of your results.

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Authors

This file was created by Dieter Roess November 2008

This simulation is part of

“Learning and Teaching Mathematics using Simulations

– Plus 2000 Examples from Physics”

ISBN 978-3-11-025005-3, Walter de Gruyter GmbH & Co. KG

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Translations

Code Language Translator Run

Credits

Dieter Roess - WEH- Foundation; Tan Wei Chiong; Loo Kang Wee

Overview:

This document provides a briefing on the "Fourier Series (Basic) JavaScript Simulation Applet HTML5" resource available on the Open Educational Resources / Open Source Physics @ Singapore website. This resource offers an interactive simulation designed to help users understand the fundamental concepts of Fourier series and the calculation of Fourier coefficients. The simulation allows users to explore different periodic functions and visualize the process of integration used to determine these coefficients.

Main Themes and Important Ideas/Facts:

  • Fourier Series Decomposition: The core concept highlighted is that a periodic function f(x) with a period of 2Ï€ can be represented as a sum of sine and cosine functions, known as the Fourier series:
  • "The Fourier Series is a series that decomposes any periodic curves into a sum of sines and cosines." The general form of the Fourier series is given as: f(x) = a0 /2 + Σ (ancos(nx) + bn sin(nx)); n=1,2,3....∞*
  • Fourier Coefficients: The resource focuses on how to calculate the coefficients a0, an, and bn that determine the amplitude of each sine and cosine component in the series. The formulas for these coefficients are provided:
  • a0 = 2/T ∫f(t) dt an= 2/T ∫ cos(nx) f(t) dt bn = 2/T ∫ sin(nx) f(t) dt where the integration is performed over one base period T (which is 2Ï€ in this specific context).
  • Integral Identities: The derivation of the coefficient formulas relies on specific integral identities involving products of sine and cosine functions. The resource points out that:
  • "All integrals are zero except of those few where the indices are identical: m = n and the function types are the same (sine or cosine)." This orthogonality property of sine and cosine functions is crucial for isolating and calculating individual coefficients.
  • JavaScript Simulation Applet: The resource utilizes an interactive JavaScript simulation to visually demonstrate these concepts. The applet features:
  • A ComboBox containing various predefined function combinations.
  • A red curve displaying the selected function over the fundamental period.
  • A blue curve representing the antiderivative of the function, illustrating the integration process.
  • Sliders (a/b and m/n) to adjust parameters and indices of the functions.
  • A "Integral" checkbox to activate the visualization of the integration.
  • The end value of the blue curve at x = 2Ï€ represents the definite integral over one period.
  • Operation of the Simulation: Users can select different functions, modify their parameters and indices using sliders, and observe how these changes affect the function and its integral. The slowed-down integration process helps visualize the accumulation of area under the curve.
  • Experiments: The resource suggests a series of experiments (E1-E8) for users to explore the relationship between different functions and their definite integrals over the fundamental period. These experiments encourage users to:
  • Observe that the definite integral of cosx and sinx (and their scaled and phase-shifted combinations) over [0, 2Ï€] is zero for integer n.
  • Verify that the definite integral of "mixed" functions (e.g., cosx * sinx) is typically zero unless the sine and cosine terms have the same type and identical indices.
  • Analytically integrate some functions to confirm the experimental findings.
  • Reflect on the properties of sine and cosine functions that lead to these results.
  • Learning Tool: The simulation is presented as a tool for learning and teaching mathematics and physics, particularly for understanding the decomposition of periodic functions. The "For Teachers" section explicitly states:
  • "The Fourier Series is a series that decomposes any periodic curves into a sum of sines and cosines." It encourages teachers and students to experiment with the parameters and observe the impact on the wave and its integral.
  • Open Educational Resource: The resource is part of the "Open Educational Resources / Open Source Physics @ Singapore" initiative and utilizes the EasyJavaScriptSimulations Library. The content is licensed under a Creative Commons Attribution-Share Alike 4.0 Singapore License, promoting sharing and adaptation for non-commercial purposes.

Key Quotes:

  • "The Fourier Series is a series that decomposes any periodic curves into a sum of sines and cosines." (For Teachers section)
  • f(x) = a0 /2 + Σ (ancos(nx) + bn sin(nx)); n=1,2,3....∞* (Fourier series formula)
  • a0 = 2/T ∫f(t) dt, an= 2/T ∫ cos(nx) f(t) dt, bn = 2/T ∫ sin(nx) f(t) dt (Fourier coefficient formulas)
  • "All integrals are zero except of those few where the indices are identical: m = n and the function types are the same (sine or cosine)." (Fourier Coefficients section)
  • "A ComboBox holds a list of all function combinations described on the Fourier Coefficients page. When one is selected it is displayed in red. The antiderivative is calculated for the fundamental period and drawn as a blue curve. Its end value at x = 2pi is the definite integral over one fundamental period..." (Operation of the simulation section)

Potential Uses and Implications:

  • Educational Tool: This simulation provides a valuable interactive resource for students learning about Fourier series and integral calculus. The visual representation of the integration process and the ability to manipulate parameters can enhance understanding.
  • Teacher Resource: Educators can use this applet for demonstrations in class, assign experiments to students, and encourage active learning of these mathematical concepts.
  • Self-Learning: Individuals interested in exploring Fourier analysis can use this simulation for self-study and gaining intuition about the underlying principles.
  • Foundation for Further Learning: Understanding Fourier series is fundamental in various fields like signal processing, image analysis, physics (wave phenomena, quantum mechanics), and engineering. This basic simulation can serve as a stepping stone for more advanced topics.

Conclusion:

The "Fourier Series (Basic) JavaScript Simulation Applet HTML5" is a well-designed and accessible educational tool that effectively illustrates the fundamental concepts of Fourier series and the calculation of Fourier coefficients through interactive visualization. The provided experiments encourage active learning and deeper understanding of the properties of sine and cosine functions in the context of integration. This resource aligns with the goals of open educational resources by providing a valuable learning tool under a Creative Commons license.

 

Fourier Series Study Guide

Key Concepts

  • Periodic Function: A function that repeats its values at regular intervals. The smallest positive interval for which this repetition occurs is called the fundamental period (T). In this context, the period is given as \(2\pi\).
  • Fourier Series: A representation of a periodic function as an infinite sum of sine and cosine waves. The general form for a function \(f(x)\) with period \(2\pi\) is: \(f(x) = a_0 /2 + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))\)
  • Fourier Coefficients: The coefficients \(a_0\), \(a_n\), and \(b_n\) that determine the amplitude of each sine and cosine component in the Fourier series. They are calculated using integrals of the function \(f(x)\) multiplied by cosine or sine functions over one period.
  • \(a_0 = 2/T \int_{0}^{T} f(t) dt\)
  • \(a_n = 2/T \int_{0}^{T} f(t) \cos(nx) dt\)
  • \(b_n = 2/T \int_{0}^{T} f(t) \sin(nx) dt\)
  • Harmonics: The individual sine and cosine terms in the Fourier series are called harmonics. The index \(n\) represents the order of the harmonic; \(n=1\) is the fundamental harmonic, \(n=2\) is the second harmonic, and so on.
  • Definite Integral over a Period: The net area under the curve of a function over one complete period. This value is crucial for calculating the \(a_0\) coefficient, which is related to the average value of the function over the period.
  • Orthogonality of Sine and Cosine: The property that the integral of the product of certain combinations of sine and cosine functions over a period is zero, unless the frequencies (represented by the indices \(m\) and \(n\)) are the same and the functions are of the same type (both sine or both cosine). This property is fundamental to isolating and calculating the Fourier coefficients.
  • Superposition: The principle that for linear systems, the total response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. In the context of Fourier series, a complex periodic function can be seen as the superposition of simpler sine and cosine waves.
  • Phase Shift: A horizontal shift of a periodic function. In the context of \(a \sin(nx) + b \cos(nx)\), different values of \(a\) and \(b\) can result in a sine or cosine wave with a different phase shift.

Quiz

  1. What is the fundamental principle behind the Fourier series representation of a periodic function?
  2. In the formula for the Fourier series of a function \(f(x)\) with period \(2\pi\), what do the coefficients \(a_n\) and \(b_n\) represent?
  3. Explain the significance of integrating over one base period when calculating Fourier coefficients.
  4. According to the source, under what specific condition is the integral of the product of sine and cosine functions (e.g., \(\int \cos(mx) \sin(nx) dx\)) non-zero over a period?
  5. What does the \(a_0\) coefficient in the Fourier series represent in relation to the function \(f(x)\)?
  6. Describe how the JavaScript simulation applet visualizes the definite integral of a selected function over one fundamental period.
  7. What is the purpose of the sliders labeled 'a/b' in the simulation, and how do they affect the displayed function?
  8. Based on the experiments described, what general observation can be made about the definite integral of simple sine and cosine functions (like \(\cos(nx)\) or \(\sin(nx)\) where \(n\) is an integer) over the interval \([0, 2\pi]\)?
  9. How does the simulation demonstrate the concept of superposition using the selectable functions?
  10. What general characteristic of sine and cosine functions is identified in Experiment E8 as being the basis for the observed integral results?

Quiz Answer Key

  1. The fundamental principle behind Fourier series is to decompose any periodic function into an infinite sum of simpler sine and cosine waves of different frequencies and amplitudes. This allows for the analysis of complex periodic signals in terms of their basic harmonic components.
  2. The coefficients \(a_n\) represent the amplitude of the cosine terms at different harmonic frequencies (\(n\)), while the coefficients \(b_n\) represent the amplitude of the sine terms at those same frequencies. Together, they determine the contribution of each harmonic to the overall Fourier series.
  3. Integrating over one base period is significant because it allows us to isolate the individual harmonic components of the periodic function due to the orthogonality properties of sine and cosine functions over that period. This ensures that we can uniquely determine each Fourier coefficient.
  4. According to the source, the integral of the product of sine and cosine functions over a period is non-zero only when both terms are of the same type (both sine or both cosine) and have identical indices (i.e., \(m = n\)).
  5. The \(a_0\) coefficient (specifically \(a_0/2\)) represents the average value (DC component) of the function \(f(x)\) over one period. It is proportional to the definite integral of the function over that period.
  6. The simulation calculates the antiderivative of the selected function for the fundamental period and displays it as a blue curve. The end value of this blue curve at \(x = 2\pi\) visually represents the definite integral of the function over one fundamental period \([0, 2\pi]\).
  7. The sliders labeled 'a/b' control the parameters \(a\) and \(b\) in some of the selectable functions, allowing for scaling of the ordinate (vertical axis) and phase shifting of the functions. This enables the user to observe how these parameters affect the shape and position of the periodic wave.
  8. The experiments demonstrate that the definite integral of simple sine and cosine functions, where \(n\) is a non-zero integer, over the interval \([0, 2\pi]\) is always zero. This is due to the symmetry of these functions about the x-axis over a complete number of cycles.
  9. The simulation allows users to choose functions like \(a\sin(nx) + b\cos(nx)\), which are a direct superposition (sum) of a sine and a cosine wave. By varying \(a\), \(b\), and \(n\), users can observe how the combination of these simpler periodic functions results in a new, phase-shifted periodic wave.
  10. Experiment E8 suggests that the key characteristics of sine and cosine functions leading to these integral results are their periodicity and their symmetry about the x-axis over integer multiples of their fundamental period.

Essay Format Questions

  1. Discuss the fundamental principles behind Fourier series and explain why they are a powerful tool for analyzing periodic phenomena in various fields, referencing the concepts presented in the provided source.
  2. Explain the role of the Fourier coefficients (\(a_0\), \(a_n\), \(b_n\)) in constructing the Fourier series representation of a periodic function. How are these coefficients derived, and what mathematical property of sine and cosine functions makes this derivation possible?
  3. Describe the functionality of the JavaScript simulation applet in visualizing Fourier series concepts. Discuss how the interactive features, such as the ComboBox and sliders, aid in understanding the relationship between functions, their integrals, and the underlying harmonic components.
  4. Based on the experimental observations outlined in the source (E1-E6), analyze the conditions under which the definite integral of various combinations of sine and cosine functions over a fundamental period is zero or non-zero. What conclusions can be drawn about the orthogonality of these functions?
  5. Consider the pedagogical value of the Fourier Series (Basic) JavaScript Simulation Applet HTML5 for students learning about Fourier series. How do the interactive elements and experiments facilitate a deeper understanding of the mathematical concepts involved, and what are the potential benefits of using such simulations in mathematics and physics education?

Glossary of Key Terms

  • Periodic Function: A function \(f(x)\) that satisfies the condition \(f(x + T) = f(x)\) for all \(x\) in its domain, where \(T\) is a non-zero constant called the period. The smallest positive such \(T\) is the fundamental period.
  • Fourier Series: An infinite series representation of a periodic function \(f(x)\) as a sum of weighted sine and cosine functions with frequencies that are integer multiples of the fundamental frequency of the periodic function.
  • Fourier Coefficients (\(a_0, a_n, b_n\)): The numerical values that determine the amplitude and phase of each sine and cosine component in the Fourier series. They are calculated through integration of the periodic function multiplied by sine or cosine functions over one period.
  • Harmonic: A component sine or cosine wave in a Fourier series, whose frequency is an integer multiple of the fundamental frequency. The \(n\)-th harmonic corresponds to the terms involving \(\cos(nx)\) and \(\sin(nx)\).
  • Fundamental Period (T): The smallest positive interval over which a periodic function completes one full cycle. In the context of the source, \(T = 2\pi\).
  • Definite Integral: The signed area between the graph of a function and the x-axis over a specified interval. In the context of Fourier coefficients, the definite integral over one period is crucial.
  • Orthogonality: A property of a set of functions where the integral of the product of any two distinct functions over a given interval is zero. Sine and cosine functions with different integer multiples of the fundamental frequency are orthogonal over a period.
  • Superposition: The principle that the combined effect of multiple inputs or components in a linear system is the sum of their individual effects. In Fourier series, a periodic function is expressed as the superposition of sine and cosine waves.
  • Phase Shift: A horizontal translation of a wave or periodic function, representing a change in the starting point of the cycle.
  • Antiderivative: A function whose derivative is the original function. In the simulation, the antiderivative is calculated and its value at the end of the period gives the definite integral.

Sample Learning Goals

[text]

For Teachers

The Fourier Series is a series that decomposes any periodic curves into a sum of sines and cosines.

In this simulation, you are instead given several functions with multiple parameters a, b, m, n to select from and you can adjust them with either the sliders or the fields provided. The appearance of the periodic wave will change accordingly.

There is also a red checkbox labeled "Show Integral" that when checked, does exactly what it says.

The integral is shown in red, and the value of the integral curve at a point denotes the net area under the curve from 0 to that point. Do play around with the parameters and see how it affects the curve.

Research

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Video

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 Version:

  1. http://weelookang.blogspot.sg/2016/02/vector-addition-b-c-model-with.html improved version with joseph chua's inputs
  2. http://weelookang.blogspot.sg/2014/10/vector-addition-model.html original simulation by lookang

Other Resources

[text]

What is a Fourier Series and what is its purpose?

A Fourier Series is a mathematical way to represent any periodic function as a sum of simple sine and cosine waves. Its purpose is to decompose complex periodic signals into their fundamental frequency components and their harmonics, making them easier to analyze and understand.

What is the fundamental period in the context of Fourier Series?

The fundamental period (denoted as T or 2Ï€ in the provided text) is the smallest repeating interval of a periodic function. The Fourier analysis focuses on this base period to determine the constituent sine and cosine waves that make up the function.

What are Fourier coefficients and how are they calculated?

Fourier coefficients (a₀, a<0xE2><0x82><0x99>, b<0xE2><0x82><0x99>) are the amplitudes of the constant term, cosine terms, and sine terms, respectively, in the Fourier Series representation of a function. They are calculated using definite integrals of the function multiplied by cosine or sine functions over one fundamental period. The formulas provided are: a₀ = 2/T ∫f(t) dt a<0xE2><0x82><0x99>= 2/T ∫ cos(nx) f(t) dt b<0xE2><0x82><0x99> = 2/T ∫ sin(nx) f(t) dt

What is the significance of the indices 'm' and 'n' in the context of Fourier Series integrals?

The indices 'm' and 'n' represent the order of the harmonics (integer multiples of the fundamental frequency) in the cosine and sine terms. When evaluating the integrals for the Fourier coefficients, only terms where the indices of the integrating function and the component of the Fourier series are identical (m = n) and the trigonometric functions are the same (both sine or both cosine) result in non-zero values. This orthogonality property is crucial for isolating each coefficient.

What does the provided JavaScript simulation allow users to do?

The JavaScript simulation allows users to explore different function combinations (sines, cosines, and their products or sums) and visualize their integration over one fundamental period. Users can adjust parameters (a, b) and indices (m, n) using sliders and observe how these changes affect the function and its integral. The simulation helps understand which functions yield non-zero definite integrals over a period.

What is the relationship between the definite integral of sine and cosine functions (and their linear combinations or products) over a fundamental period and their Fourier coefficients?

The definite integral of sine and cosine functions (or their specific combinations) over a fundamental period is directly related to the Fourier coefficients. As highlighted in the "Experiments" section, the definite integral of simple sine and cosine functions with integer indices over the interval [0, 2Ï€] is typically zero. Non-zero integrals occur for "mixed" functions only when the sine or cosine terms have the same type and identical indices, which allows for the determination of the non-zero Fourier coefficients for those specific frequencies present in the function.

According to the experiments described, under what conditions is the definite integral of combinations of sine and cosine functions non-zero over a fundamental period?

The experiments demonstrate that the definite integral of combinations of sine and cosine functions over a fundamental period [0, 2Ï€] is generally zero, except when integrating the square of a sine or cosine term (implicitly suggested when "both terms are of the same type and have identical indices"). For example, integrating cos(nx) * cos(nx) or sin(nx) * sin(nx) will yield a non-zero result, which is fundamental to extracting the a<0xE2><0x82><0x99> and b<0xE2><0x82><0x99> coefficients.

What fundamental characteristics of sine and cosine functions are key to the results observed in the Fourier analysis?

The key characteristics of sine and cosine functions that underlie the results are their periodicity and their orthogonality over a fundamental period. Orthogonality means that the integral of the product of two different sine or cosine functions (or a sine and a cosine function) with integer multiples of the fundamental frequency over one period is zero. This property allows the Fourier Series to uniquely decompose a periodic function into its constituent sinusoidal components, as each coefficient can be independently determined through integration due to this orthogonality.

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