Practice General Solution - 19.6 | 19. Modelling – Membrane, Two-Dimensional Wave Equation | Mathematics (Civil Engineering -1)
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General Solution

19.6 - General Solution

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Practice Questions

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Question 1 Easy

What is the general form of the solution for a vibrating membrane?

💡 Hint: Look for a summation involving cosine and sine functions.

Question 2 Easy

What role do the coefficients $A_{nm}$ and $B_{nm}$ play in the general solution?

💡 Hint: Think about how these coefficients are influenced by the initial state of the membrane.

4 more questions available

Interactive Quizzes

Quick quizzes to reinforce your learning

Question 1

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}

💡 Hint: Look for the summation involving sine functions.

Question 2

Natural frequency is represented by which symbol?

True
False

💡 Hint: Think about the terms we used to describe frequencies.

1 more question available

Challenge Problems

Push your limits with advanced challenges

Challenge 1 Hard

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.

💡 Hint: Remember the definition of Fourier coefficients from calculus!

Challenge 2 Hard

Discuss how varying the tension in the membrane affects the natural frequencies of the modes.

💡 Hint: Think about how the stiffness of a material influences vibration.

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