19. Modelling – Membrane, Two-Dimensional Wave Equation

Try these first

1. 1.

   What is the two-dimensional wave equation?

   Hint

   Look for how displacement relates to wave speed.

2. 2.

   What does a membrane represent in this context?

   Hint

   Think of musical instruments, like drums.

3. 3.

   What is the main equation governing the motion of a vibrating membrane?

   • ∂²u/∂t² = 0
   • ∂²u/∂t² = c²∇²u
   • ∂u/∂x = 0

   Hint

   Focus on the relationship of tension and mass.

4. 4.

   True or False: A vibrating membrane's boundary conditions are not important.

   • True
   • False

   Hint

   Think about what happens at the edges.

5. 5.

   Consider a square membrane fixed at all edges. If it experiences a specific force at its center, determine how the initial conditions would change.

   Hint

   Visualize the force applied and its effect on the membrane.

6. 6.

   Given the wave speed equation, derive how changing the tension affects the natural frequencies of vibration.

   Hint

   Think about how more tension stretches the membrane.