10.5 - Standard Fourier Cosine and Sine Transform Pairs
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Fourier Cosine Transform Definition
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Let's begin with the Fourier Cosine Transform, commonly referred to as FCT. It is defined for functions defined on the interval [0,∞), represented mathematically as \( F_c(s) = \frac{2}{\pi} \int_0^{\infty} f(x) \cos(sx) dx \). This transform allows us to analyze functions in terms of their frequency components. Can anyone tell me why the cosine function is used here?
I think it's because cosine represents an even function that fits into our analysis for non-negative domains!
Exactly! Cosine functions are even and thus suitable for representing functions over semi-infinite domains. How do you think this could help in engineering applications?
Maybe it helps with problems like heat transfer where we deal with boundary conditions?
Right! FCT is widely used in engineering for such analyses. Let's remember the acronym FCT for 'Fourier Cosine Transform'.
Fourier Sine Transform Definition
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Now, let's move on to the Fourier Sine Transform, or FST. It is similar to FCT but employs sine functions and is represented as \( F_s(s) = \frac{2}{\pi} \int_0^{\infty} f(x) \sin(sx) dx \). Who can explain why we utilize sine functions here?
Since sine functions are odd, they're useful for domains where the function is constrained to zero at the boundary, like vibrations in a fixed string!
Excellent! FST is particularly effective in handling situations where the function must vanish at the boundary. Remember, we can think of FST as fitting perfectly with the sine wave properties!
Can this be applied to real-world problems like wave propagation?
Absolutely, wave propagation is a classic application of FST. Don't forget to associate FST with its utility in vibrations and oscillations.
Standard Transform Pairs
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Now that we have covered both transforms, let’s look at the standard pairs. For example, for the function \( f(x) = e^{-ax} \), the Fourier Cosine Transform results in \( F_c(s) = \frac{2a}{\pi(a^2+s^2)} \). Can someone derive the FST for this function?
Uh, maybe it could be similar since it’s an exponential decay, resulting in a similar form?
Good intuition! The Fourier Sine Transform of the same function gives us \( F_s(s) = \frac{2as}{\pi(a^2+s^2)} \). Notice how sine introduces the variable s in the numerator. How does this relate to application?
It shows how both transforms express different behaviors depending on the physical situation!
Exactly! Each transform uncovers unique insights based on the nature of the systems we analyze. Remember, practice these pairs as they are vital in engineering problems!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section elaborates on the Fourier cosine and sine transform pairs, presenting definitions and examples of common functions. These transforms are crucial for solving boundary value problems in civil engineering, particularly those related to heat transfer and structural analysis.
Detailed
In the study of transforms, the Fourier cosine and sine transforms play a vital role, especially in engineering contexts. The section delineates standard transform pairs for essential functions, describing how each pair maps a function in the spatial domain to the frequency domain. For example, the Fourier cosine transform of the exponential decay function \( e^{-ax} \) reveals properties that are pivotal for system analyses in heat conduction. Similarly, each sine transform pair is established, reinforcing the versatility of these transforms in applications ranging from structural deflection to wave propagation. Understanding these pairs not only aids in computational tasks but also supports effective problem-solving in engineering scenarios involving complex boundary conditions.
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Fourier Cosine Transform Pair
Chapter 1 of 2
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Chapter Content
Fourier Cosine Transform
Function f(x) Transform F (s)
e^{-ax}
2 ·
a
π
a²+s²
x
2 ·
s
π
s²
sin(ax)
2 ·
a
π
s²+a²
Detailed Explanation
This chunk outlines the Fourier Cosine Transform pairs for specific functions. The Fourier Cosine Transform (FCT) is particularly useful for functions defined over the semi-infinite domain. For example, for the function e^{-ax}, the transform is given as F(s) = (2/a) * (π / (a² + s²)), which indicates how the frequency spectrum of the function is represented in the cosine domain. Similarly, for x and sin(ax), we can find their corresponding transforms that help engineers solve problems involving vibrations and heat transfer.
Examples & Analogies
Imagine you are a sound engineer working with different sounds. Each sound can be represented in terms of its frequencies. The Fourier Cosine Transform operates similarly, where it decomposes complex sound waves (like e^{-ax}) into simpler components (cosine functions) that can be more easily analyzed or processed.
Fourier Sine Transform Pair
Chapter 2 of 2
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Chapter Content
Fourier Sine Transform
Function f(x) Transform F (s)
e^{-ax}
2 ·
s
π
a²+s²
x
2 ·
2a
π
(a²+s²)²
sin(ax)
2 ·
s
π
s²+a²
Detailed Explanation
Similar to the cosine transforms, the Fourier Sine Transform (FST) pairs for functions defined over the semi-infinite domain are listed here. For example, the transform of e^{-ax} is represented as F(s) = (2/s) * (π / (a² + s²)), helping to represent the function in the sine domain. Each pair elucidates how different functions relate to their sine transforms, streamlining calculations in various applications in engineering, especially where boundary conditions apply.
Examples & Analogies
Think of the FST as a tool that helps unlock the secrets of how waves behave. Just as a musician can adjust the pitch of notes played on an instrument to create special harmonics, the Fourier Sine Transform breaks apart complex waveforms into simpler parts (sine waves) that can reveal how those waves will behave over time, similar to waves in a pool when throwing stones.
Key Concepts
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Fourier Cosine Transform: A tool used for analyzing functions over the semi-infinite domain [0,∞) with cosine functions.
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Fourier Sine Transform: Used for functions vanishing at the boundary, applying sine functions for analysis.
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Standard Transform Pairs: Common functions and their Fourier transforms, crucial for practical engineering applications.
Examples & Applications
Fourier Cosine Transform of \( f(x) = e^{-ax} \) yields \( F_c(s) = \frac{2a}{\pi(a^2+s^2)} \).
Fourier Sine Transform of the same function gives \( F_s(s) = \frac{2as}{\pi(a^2+s^2)} \).
Memory Aids
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Rhymes
When the boundary's bright, think cosine's light; if it's zero at the edge, then sine's your pledge!
Stories
Imagine a beam with ends held tight, where energy flows without a fright. Cosine connects, sine sets free, simulating how structures bend with glee.
Memory Tools
COSINE - 'C' for Convert functions using Cosine, 'O' for Origin from zero, 'S' for Standard pairs we show, 'I' for Inverse when we need to follow, 'N' for Neat when calculating, 'E' for Engineering applications.
Acronyms
FCT - 'F' for Fourier, 'C' for Cosine, 'T' for Transform; remember this for clear function mappings.
Flash Cards
Glossary
- Fourier Cosine Transform (FCT)
A transform that converts a function defined on the interval [0,∞) into a frequency domain representation using cosine functions.
- Fourier Sine Transform (FST)
A transform that converts a function defined on the interval [0,∞) into a frequency domain representation using sine functions.
- Transform Pairs
A set of functions and their corresponding transformations that are commonly used in Fourier analysis.
- Boundary Value Problems
Mathematical problems involving differential equations where the solution is required to satisfy certain conditions at the boundaries of the domain.
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