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Interactive Audio Lesson
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Fourier Series Representation
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Today we will learn how to represent the initial conditions of the vibrating string using Fourier series. Specifically, we'll focus on how to calculate the coefficients A_n.
How do we obtain the coefficients A_n using our initial function f(x)?
Great question, Student_1! We calculate A_n using the integral formula: A_n = (2/L) ∫_0^L f(x) sin(nπx/L) dx. This essentially projects our function f(x) onto the sine basis functions.
What happens if f(x) contains frequencies outside the range of our basis functions?
If f(x) contains such frequencies, then our Fourier series may not converge correctly. It's important to ensure our function is periodic or defined appropriately within the interval.
Can you give us a quick recap of the A_n formula?
Certainly! A_n is calculated with (2/L) times the integral of f(x) multiplied by sin(nπx/L) over the interval from 0 to L. This is essential for linking back to our wave equation solution!
Determining Coefficient B_n
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Now, let’s shift our focus to the initial velocity of the vibrating string, represented by g(x). For this, we determine the coefficients B_n.
What’s the formula for B_n again?
B_n is found using B_n = (2/n^2πc) ∫_0^L g(x) sin(nπx/L) dx. This coefficient contributes to the time-dependent part of the solution.
Why do we divide by n^2πc in B_n?
That’s a great inquiry, Student_4! The n^2πc term normalizes the contribution of the initial velocity across different frequencies, ensuring each mode of vibration is weighted properly.
Can we apply this in real-world engineering problems?
Yes! Understanding these coefficients allows engineers to analyze how vibrations in structures will propagate, which is crucial in civil engineering.
Introduction & Overview
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Quick Overview
Standard
Using the initial conditions of a vibrating string represented by the wave equation, this section details the process of calculating the Fourier series coefficients, specifically focusing on the coefficients A_n and B_n, derived through integrals involving the initial shape and velocity of the string.
Detailed
Detailed Summary
In this section, we explore the determination of coefficients for the Fourier series solution of the vibrating string modeled by the wave equation. The coefficients A_n and B_n are essential in expressing the string's displacement over time as a series of sine functions.
Key Points:
- Fourier Series Representation: The initial condition for the displacement of the string, u(x, 0) = f(x), can be represented in terms of the sine functions of the Fourier series involving the coefficients A_n. The formula for A_n can be simplified as follows:
$$A_n = rac{2}{L} imes ext{∫}_0^L f(x) ext{sin}igg( rac{n ext{π}x}{L}igg) ext{dx}$$
- Velocity Initial Condition: The initial velocity ∂u/∂t(x,0) = g(x) determines the coefficients B_n, represented as:
$$B_n = rac{2}{n^2 ext{π}c} imes ext{∫}_0^L g(x) ext{sin}igg( rac{n ext{π}x}{L}igg) ext{dx}$$
- Overall Importance: The coefficients A_n and B_n are critical for constructing the solution u(x, t) and understanding how the string behaves over time under various initial conditions.
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Determining Coefficient A
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From the initial conditions:
X∞ (cid:16)nπx(cid:17) 2 Z L (cid:16)nπx(cid:17)
u(x,0)=f(x)= A sin ⇒A = f(x)sin dx
n L n L L
n=1 0
Detailed Explanation
This chunk describes how to determine the coefficients 'A' for the Fourier series expansion of the initial displacement of the string. We start with the initial condition given by 'u(x,0) = f(x)', which represents the shape of the vibrating string at time t=0. The formula for coefficient 'A' involves integrating 'f(x)sin(nπx/L)' over the length of the string from 0 to L. This integration helps capture how much of each sine component is present in the initial shape of the string.
Examples & Analogies
Imagine you are trying to recreate a melody on a musical instrument. The initial shape of the melody can be compared to the function 'f(x)', while the different notes or harmonics are like the sine functions. Just as you would find how much of each note is needed to reproduce the melody, here we are figuring out how much of each sine wave (harmonic) is needed to represent the initial position of the string.
Determining Coefficient B
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∂ ∂u t(x,t)(cid:12) (cid:12) (cid:12)
g(x)= X∞ n Lπc B nsin(cid:16)n Lπx(cid:17) ⇒B n = n2 πcZ L g(x)sin(cid:16)n Lπx(cid:17) dx
t=0 n=1 0
Detailed Explanation
This chunk addresses how to find the coefficients 'B', which represent the initial velocities of the points along the string. We utilize the initial velocity condition, '∂u/∂t(x,0) = g(x)', which describes how fast each point on the string is moving at time t=0. The formula for 'B' involves a similar integration as with 'A', this time integrating 'g(x)sin(nπx/L)' over the length of the string. This captures the initial velocity profile of the vibrating string.
Examples & Analogies
Think of a group of people on a trampoline. Each person represents a point on the string, and how high each person jumps initially represents their velocity. To understand how to launch everyone at the right speed to achieve a coordinated bounce, you would need to evaluate how high each person initially jumped. Just like measuring those jumps, we are determining how much of the initial velocity we need to capture for the vibrating string.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Fourier Series: A method to decompose functions into a sum of sine waves.
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Coefficient A_n: Represents the contributions of the initial displacement in the solution.
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Coefficient B_n: Represents the contributions of the initial velocity in the solution.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Using the initial displacement f(x) = x for a string of length L = 2, calculate A_n.
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Given g(x) = x^2, determine B_n for n=1, in the context of the series expansion for the vibrating string.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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For every n in the sine, A is the height, B is the line.
📖 Fascinating Stories
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Consider a string stretched tight, with A's revealing its initial height, while B's venture into velocity, mapping motion's reality.
🧠 Other Memory Gems
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Remember: Displacement (A) arrives first, Velocity (B) then bursts.
🎯 Super Acronyms
A = Away (displacement), B = Buzz (velocity)!
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Fourier Series
Definition:
A method to express a function as a sum of periodic components, used in analyzing waveforms.
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Term: Coefficient A_n
Definition:
The coefficients in the Fourier series that represent the initial displacement of the string.
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Term: Coefficient B_n
Definition:
The coefficients in the Fourier series that represent the initial velocity of the string.