Practice Eigenvalues And Eigenfunctions (20.10) - Rectangular Membrane, Use of Double Fourier Series

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6 sample questions

Question 1.
Define eigenvalue and give an example.
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Think about the formula mentioned in class.
Question 2.
Explain the relationship between eigenvalues and eigenfunctions.
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Consider the definitions you learned.
3Question 3.
What is the formula for the eigenvalue α?
- α = nπ/a
- α = mπ/b
- α = n * m
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Look for the formula mentioned in class.
4Question 4.
True or False: Eigenfunctions describe the boundary conditions of the system.
- True
- False
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Think about what each term represents.
Question 5.
A rectangular membrane has a length of 2m and a width of 1m. Calculate the first three eigenvalues.
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Use the formulas for α and β.
Question 6.
If the initial displacement of a vibrating membrane is given by the function f(x,y) = sin(πx/2)sin(πy), identify the corresponding eigenvalue pairs.
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What values of n and m fit the given function?

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