One end of a rod of length \(L\) is held at \(x=0\) while the other end is stretched from \(x=L\) to \(x=(1+a) L\), where \(a\) is a constant. In this way an arbitrary point \(x\) in the rod is moved to \((1+a) x\). Then at time \(t=0\) the rod is released.

(a) What is the initial value of the displacement function \(\eta(0, x)\) ?

(b) Find \(\eta(t, x)\).

(c) Show that the velocity at the left end of the rod is either \(2 a v\) or \(-2 a v\), alternating between these vales with a time interval \(L / v\), where \(v\) is the wave velocity in the rod. You might want to use the doubling trick from the text.