14.5. Final Form of D'Alembert’s Solution (with initial conditions)

Try these first

1. 1.

   What is D'Alembert's solution?

   Hint

   Think about how waves behave with given initial states.

2. 2.

   What do the terms $\phi(x)$ and $\psi(x)$ represent?

   Hint

   What do we predict when we start with a wave?

3. 3.

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

   • u(x,t) = f(x + ct)
   • u(x,t) = f(x + ct) + g(x - ct)
   • u(x,t) = c²u(x,t)

   Hint

   Think about how waves can part ways.

4. 4.

   True or False: D'Alembert's solution applies only in the scenario of non-linear wave forms.

   • True
   • False

   Hint

   Recall the types of equations we discussed.

5. 5.

   Given an initial displacement of $u(x,0) = e^{-x^2}$ and an initial velocity $\partial u / \partial t(x,0) = e^{-x^2}$, derive the complete wave solution using D'Alembert's formula.

   Hint

   Recognize how the exponential function behaves when traveling across the wave.

6. 6.

   Analyze how increasing the wave speed $c$ affects the propagation of $u(x,0) = ext{sin}(x)$ and $\partial u / \partial t(x,0) = 0$.

   Hint

   Think about a race—how does speed make a difference?