What happens to the total energy E of a vibrating string (see Exercise 4.4.9) R...

What happens to the total energy E of a vibrating string (see Exercise 4.4.9)

Reference Exercise 4.4.9:

From (4.4.1), derive conservation of energy for a vibrating string,

where the total energy E is the sum of the kinetic energy, defined by

and the potential energy, defined by

Reference 4.4.1: