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

### Additional material - Derivation of average power

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.