11.4. Summary

Try these first

1. 1.

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

   Hint

   It involves second derivatives with respect to both time and space.

2. 2.

   What variables are represented in the wave equation?

   Hint

   Look for the terms that involve movement in time and space.

3. 3.

   What does the wave equation describe?

   • Thermodynamics
   • Wave propagation
   • Electromagnetism

   Hint

   Think about the physical processes involved in different media.

4. 4.

   True or False: D'Alembert's solution allows for wave shapes to change over time.

   • True
   • False

   Hint

   Recall how traveling waves behave.

5. 5.

   A string fixed at both ends vibrates according to the equation $\frac{\partial^2 u}{\partial t^2} = 16 \frac{\partial^2 u}{\partial x^2}$. If the initial conditions are $u(x,0) = 0$ and $\frac{\partial u}{\partial t}(x,0) = 10 \sin(\frac{\pi x}{L})$, find the solution for $u(x,t)$.

   Hint

   Apply separation of variables and use Fourier series expansions.

6. 6.

   Consider a wave traveling in a string modeled by the equation $\frac{\partial^2 u}{\partial t^2} = 25 \frac{\partial^2 u}{\partial x^2}$. If the free end at x = L has an amplitude of 5 cm, derive the expressions for displacement and motion at x over time.

   Hint

   Identify the wave parameters and relate them back to the wave equation.