Editing Doppler effect (section)

==General==

In classical physics, where the speeds of the source and the receiver relative to the medium are lower than the speed of waves in the medium, the relationship between observed frequency <math>f</math> and emitted frequency <math>f_\text{0}</math> is given by:<ref name=halliday>{{cite book |last1=Walker |first1=Jearl |last2=Resnick |first2=Robert |authorlink2=Robert Resnick |last3=Halliday |first3=David |authorlink3=David Halliday (physicist) |title=Halliday & Resnick Fundamentals of Physics |date=2007 |publisher=Wiley |isbn=9781118233764 |oclc=436030602 |edition=8th}}</ref>
<math display="block">f = \left( \frac{c \pm v_\text{r}}{c \mp v_\text{s}} \right) f_0 </math>
where
*<math>c </math> is the propagation speed of waves in the medium;
*<math>v_\text{r} </math> is the speed of the receiver relative to the medium. In the formula, <math>v_\text{r} </math> is added to <math>c</math> if the receiver is moving towards the source, subtracted if the receiver is moving away from the source;
*<math>v_\text{s} </math> is the speed of the source relative to the medium. <math>v_\text{s} </math> is subtracted from <math>c</math> if the source is moving towards the receiver, added if the source is moving away from the receiver.

Note this relationship predicts that the frequency will decrease if either source or receiver is moving away from the other.

Equivalently, under the assumption that the source is either directly approaching or receding from the observer:
<math display="block">\frac{f}{v_{wr}} = \frac{f_0}{v_{ws}} = \frac{1}{\lambda}</math>
where 
*<math>v_{wr}</math> is the wave's speed relative to the receiver;
*<math>v_{ws}</math> is the wave's speed relative to the source;
*<math>\lambda</math> is the wavelength.

If the source approaches the observer at an angle (but still with a constant speed), the observed frequency that is first heard is higher than the object's emitted frequency. Thereafter, there is a [[monotonic]] decrease in the observed frequency as it gets closer to the observer, through equality when it is coming from a direction perpendicular to the relative motion (and was emitted at the point of closest approach; but when the wave is received, the source and observer will no longer be at their closest), and a continued monotonic decrease as it recedes from the observer. When the observer is very close to the path of the object, the transition from high to low frequency is very abrupt. When the observer is far from the path of the object, the transition from high to low frequency is gradual.

If the speeds <math>v_\text{s} </math> and <math>v_\text{r} \,</math> are small compared to the speed of the wave, the relationship between observed frequency <math>f</math> and emitted frequency <math>f_\text{0}</math> is approximately<ref name=halliday />
{|
|-
!Observed frequency!!Change in frequency
|-
|width=70%|{{center|<math>f = \left(1+\frac{\Delta v}{c}\right) f_0</math>}}|||{{center|<math>\Delta f = \frac{\Delta v}{c} f_0</math>}}
|}
where
*<math>\Delta f = f - f_0 </math>
*<math>\Delta v = -(v_\text{r} - v_\text{s}) </math> is the opposite of the relative speed of the receiver with respect to the source: it is positive when the source and the receiver are moving towards each other.

{{math proof|proof=
Given <math>f = \left( \frac{c + v_\text{r}}{c + v_\text{s}} \right) f_0</math>

we divide for <math>c</math>
<math display="block">f
= \left( \frac{1 + \frac{v_\text{r}} {c}} {1 + \frac{v_\text{s}} {c}} \right) f_0
= \left( 1 + \frac{v_\text{r}}{c} \right) \left( \frac{1}{1 + \frac{v_\text{s}} {c}} \right) f_0 </math>

Since <math>\frac{v_\text{s}}{c} \ll 1</math> we can substitute using the [[Taylor's series]] expansion of <math>\frac{1} {1 + x}</math> truncating all <math>x^2</math> and higher terms:
<math display="block"> \frac{1} {1 + \frac{v_\text{s}}{c}} \approx 1 - \frac{v_\text{s}}{c}</math>

When substituted in the last line, one gets:
<math display="block"> \left( 1 + \frac{v_\text{r}}{c} \right) \left( 1 - \frac{v_\text{s}}{c} \right) f_0 
= \left( 1 + \frac{v_\text{r}}{c} - \frac{v_\text{s}}{c} - \frac{v_\text{r} v_\text{s}}{c^2} \right) f_0 </math>

For small <math>v_\text{s} </math> and <math>v_\text{r}</math>, the last term <math> \frac{v_\text{r} v_\text{s}}{c^2} </math> becomes insignificant, hence:
<math display="block"> \left( 1 + \frac{v_\text{r} - v_\text{s}}{c}\right) f_0 </math>
}}

<gallery mode="packed" heights="250px">
File:Dopplereffectstationary.gif|Stationary sound source produces sound waves at a constant frequency {{math|''f''}}, and the wave-fronts propagate symmetrically away from the source at a constant speed c. The distance between wave-fronts is the wavelength. All observers will hear the same frequency, which will be equal to the actual frequency of the source where {{math|1=''f'' = ''f''{{sub|0}}}}.

File:Dopplereffectsourcemovingrightatmach0.7.gif|The same sound source is [[radiating]] sound waves at a constant frequency in the same medium. However, now the sound source is moving with a speed {{math|1=''υ''{{sub|s}} = 0.7 ''c''}}. Since the source is moving, the centre of each new [[wavefront]] is now slightly displaced to the right. As a result, the wave-fronts begin to bunch up on the right side (in front of) and spread further apart on the left side (behind) of the source. An observer in front of the source will hear a higher frequency {{math|1=''f'' = {{sfrac|''c'' + 0|''c'' – 0.7''c''}} ''f''{{sub|0}} = 3.33 ''f''{{sub|0}}}} and an observer behind the source will hear a lower frequency {{math|1=''f'' = {{sfrac|''c'' − 0|''c'' + 0.7''c''}} ''f''{{sub|0}} = 0.59 ''f''{{sub|0}}}}.

File:Dopplereffectsourcemovingrightatmach1.0.gif|Now the source is moving at the speed of sound in the medium ({{math|1=''υ''{{sub|s}} = ''c''}}). The wave fronts in front of the source are now all bunched up at the same point. As a result, an observer in front of the source will detect nothing until the source arrives and an observer behind the source will hear a lower frequency {{math|1=''f'' = {{sfrac|''c'' – 0|''c'' + ''c''}} ''f''{{sub|0}} = 0.5 ''f''{{sub|0}} }}.

File:Dopplereffectsourcemovingrightatmach1.4.gif|The sound source has now surpassed the speed of sound in the medium, and is traveling at 1.4 ''c''. Since the source is moving faster than the sound waves it creates, it actually leads the advancing wavefront. The sound source will pass by a stationary observer before the observer hears the sound. As a result, an observer in front of the source will detect nothing and an observer behind the source will hear a lower frequency {{math|1=''f'' = {{sfrac|''c'' – 0|''c'' + 1.4''c''}} ''f''{{sub|0}} = 0.42 ''f''{{sub|0}}}}.
</gallery>