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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. (b) Explain why the note rises when the string is tightened.

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. (b) Explain why the note rises when the string is tightened.

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 1?

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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