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
6.2 When the medium also moves with source and observer
2.2 Relation between pressure wave and density wave
6.2 When the medium also moves with source and observer
6.4 Questions
Before understanding the effect of motion of medium on doppler's effect, lets understand some basic terminologies.
As we know, $$\begin{equation} \begin{aligned} {\overrightarrow v _{wm}} = {\overrightarrow v _w} - {\overrightarrow v _m} \\ {\overrightarrow v _w} = {\overrightarrow v _{wm}} + {\overrightarrow v _m} \\\end{aligned} \end{equation} $$
where,
${\overrightarrow v _{w}} = $ Net velocity of sound
Note: ${\overrightarrow v _{w}} = $ includes the effect of actual velocity of sound and the velocity of the medium
${\overrightarrow v _{m}} = $ Velocity of medium
${\overrightarrow v _{wm}} = $ Velocity of sound relative to medium.
Note: ${\overrightarrow v _{wm}} $ is also known as the actual velocity of sound. In question always ${\overrightarrow v _{wm}} $ is given.
So, we have to calculate ${\overrightarrow v _{w}} $ and apply doppler effect.
6.2.1 When the medium moves towards the observer
Actual velcoity of sound, ${\overrightarrow v _{wm}} = v\widehat i$
Velocity of medium, ${\overrightarrow v _{wm}} = {v_m}\widehat i$
Net velocity of sound, ${\overrightarrow v _{w}} $
So, $$\begin{equation} \begin{aligned} {{\vec v}_{wm}} = {{\vec v}_w} - {{\vec v}_m} \\ {{\vec v}_w} = {{\vec v}_{wm}} + {{\vec v}_m} \\ {{\vec v}_w} = \left( {v + {v_m}} \right)\hat i\quad ...(i) \\\end{aligned} \end{equation} $$
Velocity of source, ${\overrightarrow v _s} = {v_s}\widehat i$
Velocity of sound, ${\overrightarrow v _w} = {v_w}\widehat i$
Velocity of sound relative to source $\left( {{{\overrightarrow v }_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {\overrightarrow v _{ws}} = {\overrightarrow v _w} - {\overrightarrow v _s} \\ {\overrightarrow v _{ws}} = \left( {{v_w} - {v_s}} \right)\hat i\quad ...(ii) \\\end{aligned} \end{equation} $$
Velocity of observer, ${{\vec v}_o} = - {v_o}\widehat i$
Velocity of sound, ${\overrightarrow v _w} = {v_w}\widehat i$
Velocity of sound relative to observer $\left( {{{\overrightarrow v }_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = \left( {{v_w} + {v_o}} \right)\hat i\quad ...(iii) \\\end{aligned} \end{equation} $$
Also, we know, $$f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f\quad ...(iv)$$ From equation $(ii),(iii)$ & $(iv)$ we get, $$f' = \left( {\frac{{{v_w} + {v_o}}}{{{v_w} - {v_s}}}} \right)f$$ or $$f' = \left( {\frac{{v + {v_m} + {v_o}}}{{v + {v_m} - {v_s}}}} \right)f$$
6.2.2 When the medium moves away from the observer
Actual velcoity of sound, ${\overrightarrow v _{wm}} = v\widehat i$
Velocity of medium, ${\overrightarrow v _{wm}} = -{v_m}\widehat i$
Net velocity of sound, ${\overrightarrow v _{w}} $
So, $$\begin{equation} \begin{aligned} {{\vec v}_{wm}} = {{\vec v}_w} - {{\vec v}_m} \\ {{\vec v}_w} = {{\vec v}_{wm}} + {{\vec v}_m} \\ {{\vec v}_w} = \left( {v - {v_m}} \right)\hat i\quad ...(i) \\\end{aligned} \end{equation} $$
Velocity of source, ${\overrightarrow v _s} = {v_s}\widehat i$
Velocity of sound, ${\overrightarrow v _w} = {v_w}\widehat i$
Velocity of sound relative to source $\left( {{{\overrightarrow v }_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {\overrightarrow v _{ws}} = {\overrightarrow v _w} - {\overrightarrow v _s} \\ {\overrightarrow v _{ws}} = \left( {{v_w} - {v_s}} \right)\hat i\quad ...(ii) \\\end{aligned} \end{equation} $$
Velocity of observer, ${{\vec v}_o} = - {v_o}\widehat i$
Velocity of sound, ${\overrightarrow v _w} = {v_w}\widehat i$
Velocity of sound relative to observer $\left( {{{\overrightarrow v }_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = \left( {{v_w} + {v_o}} \right)\hat i\quad ...(iii) \\\end{aligned} \end{equation} $$
Also, we know, $$f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f\quad ...(iv)$$ From equation $(ii),(iii)$ & $(iv)$ we get, $$f' = \left( {\frac{{{v_w} + {v_o}}}{{{v_w} - {v_s}}}} \right)f$$ or $$f' = \left( {\frac{{v - {v_m} + {v_o}}}{{v - {v_m} - {v_s}}}} \right)f$$
Note: If both the source and observer are at rest then there is no effect on frequency due to motion of air.
