Consider an infinite number of masses \(m\) connected in a linear array to an infinite number of springs \(k\). In equilibrium the masses are separated by distance \(a\). Now allow small-amplitude transverse displacements of the masses, and take the limit as \(a \rightarrow 0\), with an infinite number of infinitesimal masses and an infinite number of infinitesimal spring-constants, so that the shape of the array as a function of time and space is given by \(\eta(t, x)\), where \(\eta\) is transverse to the direction of the array in equilibrium. Show that if the amplitude is very small, then \(\eta(t, x)\) obeys a linear wave equation, whose solutions can be traveling or standing transverse waves.