11.2. One-Dimensional Wave Equation

Try these first

1. 1.

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

   Hint

   Recall the relationship between time and space derivatives.

2. 2.

   What does the variable $c$ represent in the wave equation?

   Hint

   Think about how fast the wave travels.

3. 3.

   What does $u(x, t)$ represent in the wave equation?

   • Velocity
   • Displacement
   • Frequency

   Hint

   Think about what the wave is doing at particular points.

4. 4.

   Is the one-dimensional wave equation applicable to sound and light waves?

   • True
   • False

   Hint

   Consider the types of waves you encounter daily.

5. 5.

   Solve the wave equation $\frac{\partial^{2}u}{\partial t^{2}} = \frac{\partial^{2}u}{\partial x^{2}}$ with fixed endpoints for initial conditions $u(x,0) = 0$ and $u_t(x,0) = 0$.

   Hint

   Consider how fixed endpoints affect the standing wave pattern.

6. 6.

   Derive the wave equation in the context of a stretched string under tension, detailing each step.

   Hint

   Think about the forces acting on a tiny piece of the string and how they relate.