Practice General Solution (19.6) - Modelling – Membrane, Two-Dimensional Wave Equation

Practice: General Solution

Lesson 19.6Modelling – Membrane, Two-Dimensional Wave Equation

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9 questions

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6 sample questions

Question 1.
What is the general form of the solution for a vibrating membrane?
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Look for a summation involving cosine and sine functions.
Question 2.
What role do the coefficients $A_{nm}$ and $B_{nm}$ play in the general solution?
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Think about how these coefficients are influenced by the initial state of the membrane.
3Question 3.
What is the general form of the solution for a membrane?
- A) u(x,y,t) = A cos(ωt) + B sin(ωt)
- B) u(x,y,t) = ∑_{n=1}^{∞} ∑_{m=1}^{∞} [A_{nm} cos(ω_{nm} t) + B_{nm} sin(ω_{nm} t)] sin(πnx/a) sin(πmy/b)
- C) u(x,y,t) = A sin(ωt)
- D) u(x,y,t) = B e^{kt}
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Look for the summation involving sine functions.
4Question 4.
Natural frequency is represented by which symbol?
- True
- False
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Think about the terms we used to describe frequencies.
Question 5.
Given a rectangular membrane fixed at its edges with length a and width b, derive the expressions for $A_{nm}$ and $B_{nm}$ based on specific initial conditions.
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Remember the definition of Fourier coefficients from calculus!
Question 6.
Discuss how varying the tension in the membrane affects the natural frequencies of the modes.
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Think about how the stiffness of a material influences vibration.

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