Consider the uniform string with constant mass per unit length \(m\), constant tension \(T\), and length \(L\) shown in Figure 7.37. One end of the string is fixed, and the other end is attached to a spring with stiffness \(k\) that is constrained to move in the vertical direction. Find the frequency equation, the first three natural frequencies, and plot the first three mode shapes. Use the ratio \(k L / T=3\).

eeeee m, L, T y(x,t) Figure 7.37: String spring-constrained at one end and fixed at the other end.