# 相對論性多普勒效應

## 機制（一個簡單例子）

${\displaystyle t={\frac {\lambda }{c-v}}={\frac {1}{(1-v/c)f_{\mathrm {s} }}}}$

${\displaystyle t_{\mathrm {o} }={\frac {t}{\gamma }}={\frac {1}{\gamma (1-v/c)f_{\mathrm {s} }}}}$

${\displaystyle f_{\mathrm {o} }={\frac {1}{t_{\mathrm {o} }}}=\gamma (1-v/c)f_{\mathrm {s} }={\sqrt {\frac {1-v/c}{1+v/c}}}\,f_{\mathrm {s} }}$

## 通式

### 當運動沿著波動傳遞路線

${\displaystyle f_{\mathrm {o} }={\sqrt {\frac {1-v/c}{1+v/c}}}\,f_{\mathrm {s} }}$

${\displaystyle \lambda _{\mathrm {o} }={\sqrt {\frac {1+v/c}{1-v/c}}}\,\lambda _{\mathrm {s} }}$

${\displaystyle z+1={\frac {\lambda _{\mathrm {o} }}{\lambda _{\mathrm {s} }}}={\sqrt {\frac {1+v/c}{1-v/c}}}}$

${\displaystyle {\frac {\Delta f}{f}}\simeq -{\frac {v}{c}}\qquad {\frac {\Delta \lambda }{\lambda }}\simeq {\frac {v}{c}}\qquad z\simeq {\frac {v}{c}}}$

### 當運動沿著任意方向

${\displaystyle f_{\mathrm {o} }={\frac {f_{\mathrm {s} }}{\gamma \left(1-{\frac {v\cos \theta }{c}}\right)}}}$

${\displaystyle f_{\mathrm {o} }=\gamma \left(1-{\frac {v\cos \theta }{c}}\right)f_{\mathrm {s} }}$

${\displaystyle {\frac {\Delta f}{f}}\simeq -{\frac {v\cos \theta }{c}}}$