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1. (a) Use the Fourier expansion to explain why the note produced by a violin string rises sharply by one octave when the string is clamped exactly at its midpoint. Explain why the note rises when the string is tightened. (b) 2. Consider a metal rod (0 < x < 1), insulated along its sides but not at its ends, which is initially at temperature = 1. Suddenly both ends are plunged into a bath of temperature = 0. Write the differential equation, boundary conditions, and initial condition. Write the formula for the temperature u(x, t) at later times. In this problem, assume the infinite series expansion 3 π.χ. 1 3πx 1 + sin + sin 1 3 1 5 5πx 1 A quantum-mechanical particle on the line with an infinite potential out- 4 1 (sin T

1. (a) Use the Fourier expansion to explain why the note produced by a violin string rises sharply by one octave when the string is clamped exactly at its midpoint. Explain why the note rises when the string is tightened. (b) 2. Consider a metal rod (0 < x < 1), insulated along its sides but not at its ends, which is initially at temperature = 1. Suddenly both ends are plunged into a bath of temperature = 0. Write the differential equation, boundary conditions, and initial condition. Write the formula for the temperature u(x, t) at later times. In this problem, assume the infinite series expansion 3 π.χ. 1 3πx 1 + sin + sin 1 3 1 5 5πx 1 A quantum-mechanical particle on the line with an infinite potential out- 4 1 (sin T

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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[Second Order Equations] How do you solve question 2?

1. (a) Use the Fourier expansion to explain why the note produced by a violin string rises sharply by one octave when the string is clamped exactly at its midpoint. Explain why the note rises when the string is tightened. (b) 2. Consider a metal rod (0 < x < 1), insulated along its sides but not at its ends, which is initially at temperature = 1. Suddenly both ends are plunged into a bath of temperature = 0. Write the differential equation, boundary conditions, and initial condition. Write the formula for the temperature u(x, t) at later times. In this problem, assume the infinite series expansion 1 4 5. 6. П π.χ 1 3πx 1 sin + = sin + sin 1 3 1 5 5πx 1 3. A quantum-mechanical particle on the line with an infinite potential out- side the interval (0, 1) (“particle in a box") is given by Schrödinger's equation u, = iuxx on (0, 1) with Dirichlet conditions at the ends. Separate the variables and use (8) to find its representation as a series. 4. Consider waves in a resistant medium that satisfy the problem U₁₁ = c²uxx-ru₁ for 0 < x < 1 u= 0 at both ends u(x, 0) = (x) u₁(x, 0) = (x), where r is a constant, 0 < r < 2лc/l. Write down the series expansion of the solution. Do the same for 2лc/l <r < 4лc/l. Separate the variables for the equation tu₁ = Uxx +2u with the boundary conditions u(0, t) = u(π, t) = 0. Show that there are an infinite number of solutions that satisfy the initial condition u(x, 0) = 0. So uniqueness is false for this equation!
Transcribed Image Text:1. (a) Use the Fourier expansion to explain why the note produced by a violin string rises sharply by one octave when the string is clamped exactly at its midpoint. Explain why the note rises when the string is tightened. (b) 2. Consider a metal rod (0 < x < 1), insulated along its sides but not at its ends, which is initially at temperature = 1. Suddenly both ends are plunged into a bath of temperature = 0. Write the differential equation, boundary conditions, and initial condition. Write the formula for the temperature u(x, t) at later times. In this problem, assume the infinite series expansion 1 4 5. 6. П π.χ 1 3πx 1 sin + = sin + sin 1 3 1 5 5πx 1 3. A quantum-mechanical particle on the line with an infinite potential out- side the interval (0, 1) (“particle in a box") is given by Schrödinger's equation u, = iuxx on (0, 1) with Dirichlet conditions at the ends. Separate the variables and use (8) to find its representation as a series. 4. Consider waves in a resistant medium that satisfy the problem U₁₁ = c²uxx-ru₁ for 0 < x < 1 u= 0 at both ends u(x, 0) = (x) u₁(x, 0) = (x), where r is a constant, 0 < r < 2лc/l. Write down the series expansion of the solution. Do the same for 2лc/l <r < 4лc/l. Separate the variables for the equation tu₁ = Uxx +2u with the boundary conditions u(0, t) = u(π, t) = 0. Show that there are an infinite number of solutions that satisfy the initial condition u(x, 0) = 0. So uniqueness is false for this equation!
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