4.1 Different case of Doppler's Effect

Source $S$ moves away from the observer $O$ which is at rest.

Velocity of source $ = {\vec v_s} = -{v_s}\widehat i$

Velocity of sound $ = {\vec v_w} = v\widehat i$

Velocity of sound relative to the source $ = {\vec v_{ws}}$

So, $$\begin{equation} \begin{aligned} {{\vec v}_{ws}} = {{\vec v}_w} - {{\vec v}_s} \\ {{\vec v}_{ws}} = v\hat i - \left( { - {v_s}\hat i} \right) \\ {{\vec v}_{ws}} = \left( {v + {v_s}} \right)\hat i \\\end{aligned} \end{equation} $$ Frequency emitted by the source $=f$

Due the motion of the source wavelength is modified.

So, wavelength of sound wave is, $$\begin{equation} \begin{aligned} \lambda ' = \frac{{{\text{velocity of sound relative to the source}}}}{f} \\ \lambda ' = \left( {\frac{{v + {v_s}}}{f}} \right)\quad ...(i) \\\end{aligned} \end{equation} $$

Velocity of observer $ = {\vec v_o} = 0$

Velocity of sound $ = {\vec v_w} = v\widehat i$

Velocity of sound relative to the observer $ = {\vec v_{wo}}$

So, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = v\hat i - 0 \\ {{\vec v}_{wo}} = v\hat i \\\end{aligned} \end{equation} $$ Let the frequency observed by observer be $f'$

Wavelength of sound wave is, $$\begin{equation} \begin{aligned} \lambda ' = \frac{{{\text{velocity of sound relative to the observer}}}}{{f'}} \\ f' = \left( {\frac{v}{{\lambda '}}} \right)\quad ...(ii) \\\end{aligned} \end{equation} $$ From equation $(i)$ and $(ii)$ we get, $$f' = \left( {\frac{v}{{v + {v_s}}}} \right)f$$

Observer $O$ moves towards the source $S$ which is at rest.

Velocity of source $ = {\vec v_s} = 0$

Velocity of sound $ = {\vec v_w} = v\widehat i$

Velocity of sound relative to the source $ = {\vec v_{ws}}$

So, $$\begin{equation} \begin{aligned} {{\vec v}_{ws}} = {{\vec v}_w} - 0 \\ {{\vec v}_{ws}} = v\hat i - 0 \\ {{\vec v}_{ws}} = v\hat i \\\end{aligned} \end{equation} $$ Frequency emitted by the source $=f$

Wavelength of the sound wave is, $$\begin{equation} \begin{aligned} \lambda = \frac{{{\text{velocity of sound relative to the source}}}}{f} \\ \lambda = \left( {\frac{v}{f}} \right)\quad ...(i) \\\end{aligned} \end{equation} $$

Velocity of observer $ = {{\vec v}_o} = - v\widehat i$

Velocity of sound $ = {\vec v_w} = v\widehat i$

Velocity of sound relative to the observer $ = {\vec v_{wo}}$

So, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = v\hat i - \left( { - {v_o}\widehat i} \right) \\ {{\vec v}_{wo}} = \left( {v + {v_o}} \right)\hat i \\\end{aligned} \end{equation} $$ Let the frequency observed by observer be $f'$

Wavelength of sound wave is, $$\begin{equation} \begin{aligned} \lambda = \frac{{{\text{velocity of sound relative to the observer}}}}{{f'}} \\ f' = \left( {\frac{{v + {v_o}}}{\lambda }} \right)\quad ...(ii) \\\end{aligned} \end{equation} $$ From equation $(i)$ and $(ii)$ we get, $$f' = \left( {\frac{{v + {v_o}}}{v}} \right)f$$

Observer $O$ moves away from the source $S$ which is at rest.

Velocity of source $ = {\vec v_s} = 0$

Velocity of sound $ = {\vec v_w} = v\widehat i$

Velocity of sound relative to the source $ = {\vec v_{ws}}$

So, $$\begin{equation} \begin{aligned} {{\vec v}_{ws}} = {{\vec v}_w} - 0 \\ {{\vec v}_{ws}} = v\hat i - 0 \\ {{\vec v}_{ws}} = v\hat i \\\end{aligned} \end{equation} $$ Frequency emitted by the source $=f$

Wavelength of the sound wave is, $$\begin{equation} \begin{aligned} \lambda = \frac{{{\text{velocity of sound relative to the source}}}}{f} \\ \lambda = \left( {\frac{v}{f}} \right)\quad ...(i) \\\end{aligned} \end{equation} $$

Velocity of observer $ = {{\vec v}_o} = v\widehat i$

Velocity of sound $ = {\vec v_w} = v\widehat i$

Velocity of sound relative to the observer $ = {\vec v_{wo}}$

So, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = v\hat i - {v_o}\widehat i \\ {{\vec v}_{wo}} = \left( {v - {v_o}} \right)\hat i \\\end{aligned} \end{equation} $$ Let the frequency observed by observer be $f'$

Wavelength of sound wave is, $$\begin{equation} \begin{aligned} \lambda = \frac{{{\text{velocity of sound relative to the observer}}}}{{f'}} \\ f' = \left( {\frac{{v - {v_o}}}{\lambda }} \right)\quad ...(ii) \\\end{aligned} \end{equation} $$ From equation $(i)$ and $(ii)$ we get, $$f' = \left( {\frac{{v - {v_o}}}{v}} \right)f$$