9.2 - Derivation of the Fourier Integral
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Introduction to Fourier Integrals
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Today, we are diving into Fourier Integrals, which are crucial for analyzing non-periodic functions. Can anyone explain why we need Fourier Integrals instead of Fourier Series?
Because Fourier Series only work for periodic functions?
Correct! Fourier Series gives us a discrete representation for periodic functions, while Fourier Integrals let us handle continuous functions that are not periodic. Great start! Can someone give an example of a non-periodic function?
The temperature distribution along a rod after heating is a non-periodic function.
Exactly! It's all about analyzing complex systems in engineering. Let’s consider how we derive the integral form from the series.
Deriving the Fourier Integral
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To derive the Fourier Integral, we begin with the Fourier Series of a function defined on the interval. Who can remind us how a Fourier Series looks?
It’s a combination of sine and cosine terms with coefficients!
"Yes, we express functions as sums of these terms. As the interval expands to infinity, we transition from sums to integrals. The Fourier integral allows us to represent a function as.
Fourier Integral Representation
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Now we have the integral representation. Why do we need this for engineering specifically?
It helps solve problems with non-periodic conditions, like temperature changes over time.
Exactly! It’s vital for solving heat conduction problems in rods and structural analysis during earthquakes. Can you think of more applications?
What about vibrations in structures?
And also in soil mechanics, right?
Right again! This flexibility makes Fourier Integrals a powerful tool in civil engineering and physics.
Introduction & Overview
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Quick Overview
Standard
In this section, the derivation of the Fourier Integral is presented, detailing how piecewise continuous functions can be expressed as integrals of sine and cosine terms. This transition from Fourier Series to Fourier Integrals is essential for analyzing non-periodic functions in various engineering applications.
Detailed
Detailed Summary
In the derivation of the Fourier Integral, we start with a piecewise continuous function, f(x), defined on the entire real line (-∞, ∞), which meets specific criteria such as the Dirichlet conditions and absolute integrability. The Fourier Series allows us to express f(x) on a finite interval [-L, L] using an infinite sum of sine and cosine functions, represented as:
$$
f(x) = a_0 + rac{1}{2} rac{1}{L} igg( ext{sum of } a_n ext{cos}igg( rac{n ext{π}x}{L}igg) + b_n ext{sin}igg( rac{n ext{π}x}{L}igg) \
\ n = 1, 2, ... \ igg)$
As L approaches infinity, the discrete sums transform into continuous integrals. The Fourier Integral formula combines sines and cosines into:
$$
f(x) = rac{1}{ ext{π}} igg( ext{integral of } (A(ω) ext{cos}(ωx) + B(ω) ext{sin}(ωx)) dω \ ext{from } 0 ext{ to } ∞ \ igg)$$
with functions A(ω) and B(ω) defined as:
$$A(ω) = rac{1}{ ext{π}} ext{integral of } f(t) ext{cos}(ωt)dt \ ext{and} \ B(ω) = rac{1}{ ext{π}} ext{integral of } f(t) ext{sin}(ωt)dt$$
This representation simplifies the analysis of engineering problems with non-periodic boundary conditions and transient phenomena.
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Introduction to Fourier Integrals
Chapter 1 of 5
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Chapter Content
Let f(x) be a piecewise continuous function on (−∞,∞) that satisfies the Dirichlet conditions and is absolutely integrable, i.e.,
\[ \int_{-\infty}^{\infty} |f(x)|dx < \infty \]
Detailed Explanation
This portion introduces the function f(x) that will be analyzed. It specifies that f(x) is defined over the entire real line and must satisfy particular mathematical properties. The Dirichlet conditions ensure that the integral converges, which is essential for applying Fourier methods effectively. The condition of absolute integrability means that the area under the graph of |f(x)| is finite, allowing us to analyze f(x) with Fourier techniques.
Examples & Analogies
Imagine measuring the total amount of liquid in a lake, where the depth at various points represents the values of f(x). For us to consistently measure and analyze the liquid level, we need a finite amount (absolute integrability) ensuring that our measurements are meaningful and don't yield infinite volume.
From Fourier Series to Fourier Integral
Chapter 2 of 5
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Chapter Content
We begin with the Fourier series of f(x) defined on [−L,L]:
\[ f(x) = \sum_{n=0}^{\infty} \left( a_0 + \sum_{n=1}^{\infty} \left( \frac{a_n}{2} \cos\left( \frac{n\pi x}{L} \right) + b_n \sin\left( \frac{n\pi x}{L} \right) \right) \right) \]
Detailed Explanation
Here, the text presents the traditional Fourier series, which allows us to represent periodic functions as sums of sines and cosines. As L approaches infinity, we transition from a summation (from discrete harmonics) to an integral, illustrating how the discrete frequencies become continuous. This shift is essential for analyzing non-periodic functions, where we utilize Fourier integrals instead of series.
Examples & Analogies
Think of a choir where each singer represents a different frequency (harmonic) contributing to the overall sound (the function f(x)). As the choir grows larger (L approaches infinity), it transforms from distinct individual voices (summation) to a continuous blend of sound, analogous to smooth music played by an orchestra (integral).
The Fourier Integral Expression
Chapter 3 of 5
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Chapter Content
Let \( \omega = n\pi \). As L → ∞, ω → ω becomes a continuous variable. The sum becomes an integral:
\[ f(x) = \int_{0}^{\infty} [A(\omega)\cos(\omega x) + B(\omega)\sin(\omega x)] d\omega \]
Detailed Explanation
This transformation signifies that as the series expands infinitely, the behavior of f(x) can be replicated by integrating over the continuous spectrum of frequencies (ω). A(ω) and B(ω) represent the coefficients that dictate the contribution of cosine and sine waves, respectively, to the function f(x). This integral formulation is crucial for representing functions that are not limited to periodic behavior.
Examples & Analogies
Imagine creating a wide frequency radio that captures all the sounds in an environment rather than just one melody. Here, the radio picks up a continuous range of signals (integral), which corresponds to how the Fourier integral captures all frequency components of f(x), thereby allowing for a more comprehensive analysis.
Defining Fourier Coefficients
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Chapter Content
Where,
\[ A(\omega) = \frac{1}{\pi}\int_{-\infty}^{\infty} f(t)\cos(\omega t) dt, \quad B(\omega) = \frac{1}{\pi}\int_{-\infty}^{\infty} f(t)\sin(\omega t) dt \]
Detailed Explanation
This section defines how to compute the coefficients A(ω) and B(ω) for the Fourier integral. They are calculated using integrals of f(t) multiplied by cosine and sine functions, respectively. This reveals the contribution of each frequency to the overall function f(x) and allows us to reconstruct f(x) fully based on these coefficients.
Examples & Analogies
Think of A(ω) and B(ω) as the ingredients needed to bake a cake. You determine how much flour (cosine component) and sugar (sine component) to use based on a recipe (function f(t)). By understanding how each ingredient contributes to the cake, we can recreate it accurately—similar to how these coefficients allow for the accurate reconstruction of the original function.
Conclusion of the Derivation
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Chapter Content
This is the Fourier Integral Representation of f(x).
Detailed Explanation
This final statement encapsulates the result of our derivation, confirming that we have successfully formulated f(x) as a Fourier integral, enabling us to analyze and solve various engineering problems involving non-periodic functions.
Examples & Analogies
It's like finishing a complex puzzle. Throughout the process, each piece was examined and connected together (derivation) until the full picture emerged (Fourier Integral Representation), ready for practical applications in fields like engineering and physics.
Key Concepts
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Fourier Series: Representation of functions as sums of sine and cosine terms.
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Transition to Fourier Integral: Moving from discrete sums to continuous integrals.
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A and B coefficients: Essential components that represent the influence of sine and cosine terms in the integral representation.
Examples & Applications
The temperature distribution in a long concrete beam due to an instantaneous point source can be analyzed using Fourier integrals.
Vibration analysis of a beam exposed to non-periodic loads utilizes the Fourier Integral representation.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
In every Fourier realm, sines and cosines helm, for functions so wide, we take them in stride!
Stories
Imagine a long, hot metal rod. When heated at one end, its temperature disperses, like waves from a pebble tossed into a pond, spreading out in an endless cycle - just like Fourier Integrals.
Memory Tools
For Fourier, Just Remember: A = Amplitude, B = Basis functions, C = Continuous function conditions.
Acronyms
Fourier
F(e) = F(un) of Four functions for Flexible boundaries.
Flash Cards
Glossary
- Fourier Integral
A mathematical representation that allows a non-periodic function to be expressed as a continuous superposition of sine and cosine functions.
- Piecewise Continuous Function
A function that is continuous except at a finite number of points.
- Dirichlet Conditions
Conditions that need to be satisfied for Fourier series to converge, including absolute integrability.
- Absolute Integrable
A function f(x) is absolutely integrable if the integral of its absolute value is finite over the entire real line.
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