Chapter 42: Problem 18

Question: Microwave, which travel at the speed of light, are reflected from a distant airplane approaching the wave source. It is found that when the reflected waves are beat against the waves radiating from the source the beat frequency is $\nu_{\rm beat}~=~990~{\rm Hz}$. If the microwaves have wavelength $\lambda~=~0.12~{\rm m}$, what is the approach speed of the plane?

Solution: The approaching aircraft sees microwaves of frequency

$$\nu^\prime~=~\nu { \sqrt{1-{v^2\over c^2}}\over 1-{v\over c}}$$

which is the frequency of the microwaves reflected back toward the source. Therefore the source sees reflected microwaves of frequency

$$\nu^{\prime\prime}~=~\nu^\prime { \sqrt{1-{v^2\over c^2}}\over 1-... ...{1-{v^2\over c^2}\over (1-{v\over c})^2}~=~\nu {1+{v\over c}\over 1-{v\over c}}$$

The beat frequency is the difference $\nu_{\rm beat}~=~ \nu^{\prime\prime}-\nu$, which is

$$\nu_{\rm beat}~=~ \nu {2 {v\over c}\over 1-{v\over c}}$$

leading to

$${v\over c}~=~{ {\nu_{\rm beat}\over\nu} \over 2 + {\nu_{\rm beat}\over\nu} }~=~59~{\rm m/s}$$