(a) Consider the time-dependent wave function
\[D_{y}=f(x, t)= \begin{cases}b(x-c t) & \text { for } 01.0 \mathrm{~m},\end{cases}\]

where \(b=0.80\) and \(c=2.0 \mathrm{~m} / \mathrm{s}\). Plot the time-independent wave function for a few values of \(t\) to verify that the function corresponds to a wave traveling in the positive \(x\) direction at a speed of \(2.0 \mathrm{~m} / \mathrm{s}\).

(b) Let the shape of a wave at \(t=0\) be described by the function
\[f(x, 0)=\frac{a}{x^{2}+b}\]

If the wave travels in the negative \(x\) direction at a speed \(c\), what is the mathematical form of the time-dependent wave function \(f(x, t)\) ?