14.4. Applying Initial Conditions

Try these first

1. 1.

   What are the two initial conditions typically defined for wave problems?

   Hint

   Think about what describes the wave at the start.

2. 2.

   What does $u(x, 0) = \phi(x)$ signify?

   Hint

   What component of the wave might this represent?

3. 3.

   What are the initial conditions for the wave equation?

   • Initial displacement and initial acceleration
   • Initial displacement and initial velocity
   • Initial position and acceleration

   Hint

   Think about what you need to set the state of the wave.

4. 4.

   True or False: The solution for the wave equation can be determined without initial conditions.

   • True
   • False

   Hint

   Consider how we define the state of a wave.

5. 5.

   Given the wave equation with initial conditions $u(x, 0) = e^{-x}$ and $\frac{\partial u}{\partial t}(x, 0) = 2e^{-x}$, find the expressions for $f(x)$ and $g(x)$.

   Hint

   Think about how to apply the conditions to formulate the two functions.

6. 6.

   Apply D'Alembert's Solution for $u(x, 0) = cos(x)$ and $\psi(x) = \sin(x)$ and find the resultant wave function.

   Hint

   Use the properties of the cosine function when finding specific solutions.