We derived a general expression for waves \(y(t, x)\) on a long string, in terms of the initial displacement \(y(0, x) \equiv f(x)\) and velocity \(\partial y(0, x) / \partial t \equiv\) \(g(x)\). Suppose that the initial displacement is \(y(0, x)=f(x)\), where \(f(x)\) is some given function.

(a) What \(g(x)\) would be required, in terms of \(f(x)\), so that for any time \(t>0\), there is only a wave traveling to the right: \(y(t, x)=f(x-v t)\) ?

(b) Find this \(g(x)\) in the special case that \(f(x)\) is the Gaussian function \(f(x)=A e^{-x^{2} / b^{2}}\), where \(A\) and \(b\) are constants.