Physics > Sound Waves > 6.0 Doppler's effect in two dimension
Sound Waves
1.0 Introduction
2.0 Displacement and pressure Waves
2.1 Relation between displacement wave and pressure wave
2.2 Relation between pressure wave and density wave
3.0 Speed of a longitudinal Wave
4.0 Doppler's Effect
5.0 Application of doppler's effect in different situations
6.0 Doppler's effect in two dimension
6.1 When Medium is at rest while source and observer is moving
6.2 When the medium also moves with source and observer
6.4 Questions
7.0 Characteristic of Sound waves
6.1 When Medium is at rest while source and observer is moving
2.2 Relation between pressure wave and density wave
6.2 When the medium also moves with source and observer
6.4 Questions
When the medium is at rest
Velocity of medium, ${\overrightarrow v _{m}}=0 $
Velocity of sound relative to medium, ${\overrightarrow v _{wm}} $
Net velocity of sound, ${\overrightarrow v _{w}} $
As we know, $${\overrightarrow v _w} = {\overrightarrow v _{wm}} + {\overrightarrow v _m}$$ So, $${\overrightarrow v _w} = {\overrightarrow v _{wm}}$$ That is why we never discussed the difference between ${\overrightarrow v _{wm}} $ & ${\overrightarrow v _{w}} $ in previous sections.
6.1.1 Source is moving & observer is at rest
Let a railway locomotive moving with velocity ${\overrightarrow v _s}$ which makes an angle $\theta $ with the observer at rest.
Velocity of source, ${\overrightarrow v _s} = {v_s}\widehat i$
Veloctiy of source along the line, ${v_{s'}} = {v_s}\cos \theta $
Velocity of sound $\left( {{v_w}} \right)$ along the line
Velocity of sound relative to the source along the line $\left( {{v_{ws'}}} \right)$, $$\begin{equation} \begin{aligned} {v_{ws'}} = {v_w} - {v_{s'}} \\ {v_{ws'}} = {v_w} - {v_s}\cos \theta \quad ...(i) \\\end{aligned} \end{equation} $$
Velocity of observer, ${\overrightarrow v _o} = 0 $
Veloctiy of observer along the line, ${v_{o'}} = 0 $
Velocity of sound $\left( {{v_w}} \right)$ along the line
Velocity of sound relative to the observer along the line $\left( {{v_{wo'}}} \right)$, $$\begin{equation} \begin{aligned} {v_{wo'}} = {v_w} - {v_{o'}} \\ {v_{wo'}} = {v_w} \quad ...(ii) \\\end{aligned} \end{equation} $$
Also, we know, $$f' = \left( {\frac{{{{\vec v}_{wo'}}}}{{{{\vec v}_{ws'}}}}} \right)f\quad ...(iii)$$ From equation $(i),(ii)$ & $(iii)$ we get, $$f' = \left( {\frac{{{v_w}}}{{{v_w} - {v_s}\cos \theta }}} \right)f$$
6.1.2 Both source and observer is moving
Let a railway locomotive moving with velocity ${\overrightarrow v _s}$ which makes an angle $\theta $ with the observer moving with velocity ${\overrightarrow v _o}$
Velocity of source, ${\overrightarrow v _s} = {v_s}\widehat i$
Veloctiy of source along the line, ${v_{s'}} = {v_s}\cos \theta $
Velocity of sound $\left( {{v_w}} \right)$ along the line
Velocity of sound relative to the source along the line $\left( {{v_{ws'}}} \right)$, $$\begin{equation} \begin{aligned} {v_{ws'}} = {v_w} - {v_{s'}} \\ {v_{ws'}} = {v_w} - {v_s}\cos \theta \quad ...(i) \\\end{aligned} \end{equation} $$
Velocity of observer, ${\overrightarrow v _o} = {v_o}\widehat i$
Veloctiy of observer along the line, ${v_{o'}} = {v_o}\cos \theta $
Velocity of sound $\left( {{v_w}} \right)$ along the line
Velocity of sound relative to the observer along the line $\left( {{v_{wo'}}} \right)$, $$\begin{equation} \begin{aligned} {v_{wo'}} = {v_w} - {v_{o'}} \\ {v_{wo'}} = {v_w} - {v_o}\cos \theta \quad ...(ii) \\\end{aligned} \end{equation} $$
Also, we know, $$f' = \left( {\frac{{{{\vec v}_{wo'}}}}{{{{\vec v}_{ws'}}}}} \right)f\quad ...(iii)$$ From equation $(i),(ii)$ & $(iii)$ we get, $$f' = \left( {\frac{{{v_w} - {v_o}\cos \theta }}{{{v_w} - {v_s}\cos \theta }}} \right)f$$
