14.3. Derivation of D'Alembert’s Solution

Try these first

1. 1.

   Define the one-dimensional wave equation.

   Hint

   What does it describe?

2. 2.

   State what $u(x,t)$ represents in the wave equation.

   Hint

   Think about wave motion.

3. 3.

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

   • \\( \\frac{\\partial^2 u}{\\partial t^2} = c^2 \\frac{\\partial^2 u}{\\partial x^2} \\)
   • \\( u = f(x+t) + g(x-t) \\)
   • \\( u = c(x-t) \\)

   Hint

   Look for the equation that involves second derivatives.

4. 4.

   True or False: Initial conditions are not necessary to solve the wave equation.

   • True
   • False

   Hint

   Think about how we find specific solutions.

5. 5.

   Given the wave equation $\frac{\partial^2 u}{\partial t^2} = 9 \frac{\partial^2 u}{\partial x^2}$ and the initial conditions $u(x, 0) = e^{-x^2}$, ; \frac{\partial u}{\partial t}(x, 0) = 0 ), derive D'Alembert's solution and find $u(x,t)$.

   Hint

   How does the Gaussian shape of the initial condition influence the wave function?

6. 6.

   Create your own initial conditions based on real-life wave situations, derive the respective D'Alembert's form, and discuss the implications of your findings.

   Hint

   Variation in initial conditions will significantly alter wave behavior.