14. The One-Dimensional Wave Equation

Try these first

1. 1.

   What is the general form of D'Alembert's solution?

   Hint

   Recall the formula we discussed in class.

2. 2.

   Identify what u represents in the wave equation.

   Hint

   Think about the physical concept of a wave.

3. 3.

   Which of the following is the correct expression for D'Alembert's solution?

   • u(x,t) = f(x + ct) + g(x - ct)
   • u(x,t) = f(x - ct) + g(x - ct)
   • u(x,t) = f(x + ct) - g(x + ct)

   Hint

   Remember the forms discussed in the D'Alembert section.

4. 4.

   The wave equation represents what kind of partial differential equation?

   • True
   • False

   Hint

   Think about the characteristics of these equations.

5. 5.

   Solve ∂²u/∂t² = 9 ∂²u/∂x² with initial conditions u(x,0) = cos(x) and ∂u/∂t|_{t=0} = sin(2x). Derive the wave function.

   Hint

   Connect initial conditions with corresponding derivative forms.

6. 6.

   Discuss the implications of dispersive and non-dispersive media in relation to wave propagation.

   Hint

   Illustrate with examples from diverse media to solidify understanding.