Two oscillatory motions are given by \(x(t)=A \sin (m \omega t)\) and \(y(t)=A \sin (n \omega t+\phi)\), where \(m\) and \(n\) are positive integers. Consider displaying these motions on a single graph with \(x\) and \(y\) axes oriented perpendicular to each other. What restrictions are necessary on \(m, n\), and \(\phi\) if the trajectory is to be a closed curve? These curves are called Lissajous figures. Plot these curves for (a) \(m=n=1, \phi=0 ;(b) m=n=1, \phi=\pi / 4\); and (c) \(m=2, n=3, \phi=0\).