To solve part (b), let's carefully analyze the given equation:

The solution for $u(x,t)$ is assumed to be: $u(x,t) = e^{-ikx - i\omega t} + r e^{ikx - i\omega t}$

Here, $r$ is the unknown coefficient we need to find, representing the reflection coefficient.

To find $r$, follow these steps:

Step 1: Plug the solution into the governing equation

Substitute $u(x,t)$ into the differential equation (presumably a wave equation or similar). Since you haven't provided the governing equation, I'll assume a general wave equation: $\frac{\partial^2 u}{\partial t^2} = v^2 \frac{\partial^2 u}{\partial x^2},$ where $v = \frac{\omega}{k}$ (as instructed in part (c)).

Step 2: Differentiate $u(x,t)$

Differentiate $u(x,t)$ with respect to $t$ and $x$.

1. Time derivatives: $\frac{\partial u}{\partial t} = -i\omega \left(e^{-ikx - i\omega t} + r e^{ikx - i\omega t}\right),$ $\frac{\partial^2 u}{\partial t^2} = -\omega^2 \left(e^{-ikx - i\omega t} + r e^{ikx - i\omega t}\right).$

2. Space derivatives: $\frac{\partial u}{\partial x} = -ik e^{-ikx - i\omega t} + ik r e^{ikx - i\omega t},$ $\frac{\partial^2 u}{\partial x^2} = -k^2 e^{-ikx - i\omega t} - k^2 r e^{ikx - i\omega t}.$

Step 3: Plug into the wave equation

Substituting these into the wave equation: $-\omega^2 \left(e^{-ikx - i\omega t} + r e^{ikx - i\omega t}\right) = v^2 \left(-k^2 e^{-ikx - i\omega t} - k^2 r e^{ikx - i\omega t}\right).$

Using $v^2 = \frac{\omega^2}{k^2}$: $-\omega^2 \left(e^{-ikx - i\omega t} + r e^{ikx - i\omega t}\right) = \frac{\omega^2}{k^2} \left(-k^2 e^{-ikx - i\omega t} - k^2 r e^{ikx - i\omega t}\right).$

Simplify both sides: $-\omega^2 e^{-ikx - i\omega t} - \omega^2 r e^{ikx - i\omega t} = -\omega^2 e^{-ikx - i\omega t} - \omega^2 r e^{ikx - i\omega t}.$

Step 4: Solve for $r$

From the boundary conditions (typically continuity of the wave and its derivative at some point), you solve for $r$. Let me know if the problem provides any specific constraints or additional information!