Consider the fixed-fixed uniform string made of two sections shown in Figure 7.40. The left section of the string has a constant mass per unit length \(m\), and the right section of the string has the constant mass per unit length \(2 m\).

(a) From a free-body diagram and Newton's second law of motion, under what assumptions can the tension be considered constant?

(b) Assuming constant tension in both strings, find the frequency equation, the first three natural frequencies, and plot the first three mode shapes.

m x + y(x) (x2) 2m X2 Figure 7.40: Discontinuous string.