Answer
The proof is below.
Work Step by Step
Normally, the period would be how fast the wave approaches you, which is:
$T = \frac{\lambda}{v}$
However, now you are also moving at speed u in the direction of the wave, so we add this velocity to obtain:
$T' = \frac{\lambda}{v+u}$
We are now asked to show that this can be changed to be like equation 14.16. Thus, using the fact that the period is the inverse of the frequency and substitution, we find:
$f'= \frac{v+u}{\lambda}$
$f'= (\frac{v}{\lambda})(1+\frac{u}{v})$
$f'= (f)(1+\frac{u}{v})$
Thus, the proof is complete.
