20.3. Solving the Spatial Equations

Try these first

1. 1.

   What is the standard form of the two-dimensional wave equation?

   Hint

   What do the symbols represent in this equation?

2. 2.

   Define boundary conditions in your own words.

   Hint

   Think about fixed membranes.

3. 3.

   What equation governs the displacement of a rectangular membrane?

   • Wave function
   • Wave equation
   • Fourier series

   Hint

   Recall the equation discussed.

4. 4.

   True or False: The boundary conditions for a fixed boundary membrane allow the edges to move freely.

   • True
   • False

   Hint

   Think about what 'fixed' implies.

5. 5.

   Given the dimensions of a membrane and an external force applied, derive the modified wave equation and discuss the effect of boundary conditions.

   Hint

   Think beyond the boundaries to how forces interact dynamically.

6. 6.

   Research and present a case study where Fourier series modeling prevented structural failure in a civil engineering project.

   Hint

   Look for examples in bridges or tall buildings.