1-7 Energy and power of a traveling wave on a string

We want to relate F_y to the tension applied to the string.

This is the instantaneous power transmitted to the portion of the string to the right of point a. This formula is applicable to any wave on a string, sinusoidal or not.

Consider a wave moving from left to right as shown in the figure. Choose any point on the string and call it point a. The portion of the string to the left of point a applies a force F on the portion of the string to the right of point a. The x-component of F, F_x, does not do any work since its direction is always perpendicular to the motion of point a. All the work done by the force F is due to its y-component, F_y.  The rate at which the force F is providing energy to the portion of the string to the right of point a is equal to rate at which this force is doing work

Thus, the power transmitted form the left to the right of point a averaged over a time of one period is

For a sinusoidal wave, the instantaneous power transmitted at point a is

Since the string elements do not move along x-direction, from Newton's second law, F_x = t.