\magnification=\magstep2

$$
x({\rm car}) \quad = \quad L + v t 
$$

$$
\eqalign{
x({\rm wave\ 1}) \quad &= \quad \phantom{2}cT + ct \cr
x({\rm wave\ 2}) \quad &= \quad 2cT + ct \cr
x({\rm wave\ 3}) \quad &= \quad 3cT + ct \cr
}
$$

$$
\eqalign{
{\rm Event\ 2:\ wave\ 2\ meets\ car\ } \quad x_2 &=   \qquad\qquad\qquad t_2 = \cr
{\rm Event\ 1:\ wave\ 1\ meets\ car\ } \quad x_1 &=   \qquad\qquad\qquad t_1 = \cr
}
$$

$$
T' \quad = \quad t'_1 - t'_2
$$


$$
{ {1}\over{f} } \quad = \quad T \quad = \quad t_1 - t_2 
$$

$$
T' \quad = \quad T \thinspace\thinspace 
       { {\sqrt{1 + \beta } } \over {\sqrt {1 - \beta } } }
$$



$$
\eqalign{
f' \quad &= \quad {{1}\over{T'}}  \quad = \quad {{1}\over{T}} \thinspace\thinspace 
       { {\sqrt{1 - \beta } } \over {\sqrt {1 + \beta } } } \cr
  \cr
f_{\rm obs}       &= \quad {f_{\rm source}} \thinspace\thinspace 
       { {\sqrt{1 - \beta } } \over {\sqrt {1 + \beta } } } \cr
}
$$

$$
f' \quad \simeq \quad f \Big( \qquad \qquad \qquad \Big) 
$$


$$
f' \quad \simeq \quad f \Big( 1 - {{v}\over{c}} \Big) 
$$



$$
\eqalign{
{\rm Event\ 2:\ wave\ 2\ meets\ car\ } \quad x'_2 &=   \cr
 \cr
\phantom{{\rm Event\ 2:\ wave\ 2\ meets\ car\ }} \quad t'_2 &=   \cr
 \cr
{\rm Event\ 1:\ wave\ 1\ meets\ car\ } \quad x'_1 &=   \cr
 \cr
\phantom{{\rm Event\ 1:\ wave\ 1\ meets\ car\ }} \quad t'_1 &=   \cr
}
$$

$$
{ {1}\over{f'} } \quad = \quad T' \quad = \quad t'_1 - t'_2 
$$



$$
\eqalign{
v_s \quad &= \quad f \lambda \cr
c \quad &= \quad f \lambda \cr
}
$$

$$
f \quad = \quad   { {1}\over{T} }
$$

$$
T \quad = \quad  { {1} \over {f} }
$$

$$
{(\Delta t)}_{\rm car} \quad = \quad {{1}\over{\gamma}} \thinspace {(\Delta t)}_{\rm Fred} 
$$

$$
{T}_{\rm car} \quad = \quad {{1}\over{\gamma}} \thinspace {T}_{\rm Fred} 
$$

$$
{f}_{\rm car} \quad = \quad {\gamma} \thinspace {f}_{\rm Fred} 
$$

$$
\eqalign{
{f}_{\rm source} \quad &= \quad {\gamma} \thinspace {f}_{\rm obs}  \cr
\cr
{f}_{\rm obs} \quad &= \quad {{1}\over{\gamma}} \thinspace {f}_{\rm source}  \cr
}
$$



\bye
