14.3.2. Step 2: Solve the Simplified PDE

Try these first

1. 1.

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

   Hint

   Think about the relationship between time and space in wave propagation.

2. 2.

   Define D'Alembert's solution in relation to the wave equation.

   Hint

   Consider the two traveling waveforms.

3. 3.

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

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

   Hint

   Think of the waves moving in opposite directions.

4. 4.

   True or False: The wave equation is a first-order differential equation.

   • True
   • False

   Hint

   Consider the orders of derivatives in the equation.

5. 5.

   Given an initial displacement $u(x, 0) = e^{-x^2}$ and velocity $\frac{\partial u}{\partial t}(x,0) = 0$, find explicit forms for $f$ and $g$.

   Hint

   Integrate the given conditions step by step.

6. 6.

   Analyze the effect of changing wave speed $c$ on the solution form, considering $p(x,t) = A sin(kx - \omega t)$.

   Hint

   Use relationships between the wave parameters to detail this.