19.6. General Solution

Try these first

1. 1.

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

   Hint

   Look for a summation involving cosine and sine functions.

2. 2.

   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.

3. 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}

   Hint

   Look for the summation involving sine functions.

4. 4.

   Natural frequency is represented by which symbol?

   • True
   • False

   Hint

   Think about the terms we used to describe frequencies.

5. 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.

   Hint

   Remember the definition of Fourier coefficients from calculus!

6. 6.

   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.