20. Rectangular Membrane, Use of Double Fourier Series

Try these first

1. 1.

   What does the variable $u(x,y,t)$ represent in the wave equation?

   Hint

   Think about what is changing in the membrane.

2. 2.

   Define what boundary conditions are in the context of a rectangular membrane.

   Hint

   Consider how the edges of the membrane are held.

3. 3.

   What is the two-dimensional wave equation used to model?

   • The motion of a particle
   • The vibration of a membrane
   • The flow of liquid

   Hint

   Think about structures that can vibrate.

4. 4.

   True or False: The double Fourier series can be used to model forced vibrations?

   • True
   • False

   Hint

   Recall the different analysis conditions.

5. 5.

   Given a rectangular membrane of dimensions 2m x 3m and tension resulting in a wave speed of 10 m/s, calculate the first three fundamental frequencies using separation of variables.

   Hint

   Plug in the dimensions into your eigenvalue expressions.

6. 6.

   Design a small rectangular membrane structure to minimize vibrations caused by wind. Which Fourier coefficients would be most critical for ensuring stability?

   Hint

   Refer back to how lower frequencies relate to stronger vibrations.