11.3. Introduction

Try these first

1. 1.

   What does the wave equation model?

   Hint

   Think of examples of waves.

2. 2.

   What is the standard form of the wave equation?

   Hint

   Recall the variables involved.

3. 3.

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

   • $$\\frac{\\partial^2 u}{\\partial t^2} = c^2 \\frac{\\partial^2 u}{\\partial x^2}$$
   • $$\\frac{\\partial u}{\\partial t} = c \\frac{\\partial^2 u}{\\partial x^2}$$
   • $$u = c^2 t^2 + x^2$$

   Hint

   Look for the equation that involves second derivatives.

4. 4.

   True or False: The wave equation can only be applied to sound waves.

   • True
   • False

   Hint

   Consider other examples of wave phenomena.

5. 5.

   Given a vibrating string fixed at both ends, write the general solution using Fourier series. Consider the fixed conditions at $u(0,t)=0$ and $u(L,t)=0$.

   Hint

   Think about how to incorporate boundary conditions into the Fourier expansion.

6. 6.

   Solve the one-dimensional wave equation with initial conditions $u(x,0) = sin(x)$ and $u_t(x,0) = 0$ on the interval $[0,\pi]$ with Dirichlet conditions.

   Hint

   Remember to apply the boundary conditions to derive the specific solution.