17.4. Method of Separation of Variables

Try these first

1. 1.

   What is the wave equation?

   Hint

   Think about waves in mediums like strings or air.

2. 2.

   Name one application of the separation of variables method.

   Hint

   Consider musical instruments.

3. 3.

   What does the separation of variables method help us solve?

   • A linear equation
   • A partial differential equation
   • An algebraic equation

   Hint

   Recall that the wave equation is a type of partial differential equation.

4. 4.

   Is the separation of variables applicable for boundary value problems?

   • True
   • False

   Hint

   Think about the initial and boundary conditions.

5. 5.

   Given a vibrating string of length L=2 meters, derive the first three mode shapes for the boundary conditions X(0) = 0 and X(2) = 0.

   Hint

   Recall how mode shapes derive from boundary conditions and eigenvalues.

6. 6.

   If T(t) has a cosine term, describe its role in the temporal behavior of the wave solution.

   Hint

   Think about how the wave oscillates based on initial conditions.