Sound Waves
5.0 Application of doppler's effect in different situations
5.0 Application of doppler's effect in different situations
General formula for doppler's effect
By combining the above cases, a general formula is generated, $$f' = \left( {\frac{{{\text{Velocity of sound relative to the observer}}}}{{{\text{Velocity of sound relative to the source}}}}} \right)f$$ or $$f' = \left( {\frac{{{{\overrightarrow v }_{wo}}}}{{{{\overrightarrow v }_{ws}}}}} \right)f$$
| S. No. | Situation | Explanation | Apparent frequency |
|---|---|---|---|
| 1 | Source moves towards the statiodnary observer | 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 $\left( {{{\overrightarrow v }_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {\overrightarrow v _{ws}} = {\overrightarrow v _w} - {\overrightarrow v _s} \\ {\overrightarrow v _{ws}} = \left( {{v_w} - {v_s}} \right)\hat i\quad ...(i) \\\end{aligned} \end{equation} $$ Velocity of observer, ${{\vec v}_o} = 0$ Velocity of sound, ${\overrightarrow v _w} = {v_w}\widehat i$ Velocity of sound relative to observer $\left( {{{\overrightarrow v }_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {\overrightarrow v _{wo}} = {\overrightarrow v _w} - {\overrightarrow v _o} \\ {\overrightarrow v _{wo}} = {v_w}\hat i\quad ...(ii) \\\end{aligned} \end{equation} $$ Also, we know, $$\begin{equation} \begin{aligned} f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f \\ f' = \left( {\frac{{{v_w}}}{{{v_w} - {v_s}}}} \right)f \\\end{aligned} \end{equation} $$ | $$f' = \left( {\frac{{{v_w}}}{{{v_w} - {v_s}}}} \right)f$$ |
| 2 | Source moves away from the stationary observer | 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 $\left( {{{\overrightarrow v }_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {\overrightarrow v _{ws}} = {\overrightarrow v _w} - {\overrightarrow v _s} \\ {\overrightarrow v _{ws}} = \left( {{v_w} +{v_s}} \right)\hat i\quad ...(i) \\\end{aligned} \end{equation} $$ Velocity of observer, ${{\vec v}_o} = 0$ Velocity of sound, ${\overrightarrow v _w} = {v_w}\widehat i$ Velocity of sound relative to observer $\left( {{{\overrightarrow v }_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {\overrightarrow v _{wo}} = {\overrightarrow v _w} - {\overrightarrow v _o} \\ {\overrightarrow v _{wo}} = {v_w}\hat i\quad ...(ii) \\\end{aligned} \end{equation} $$ Also, we know, $$\begin{equation} \begin{aligned} f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f \\ f' = \left( {\frac{{{v_w}}}{{{v_w} + {v_s}}}} \right)f \\\end{aligned} \end{equation} $$ | $$f' = \left( {\frac{{{v_w}}}{{{v_w} + {v_s}}}} \right)f$$ |
| 3 | Observer moves towards the stationary source | Velocity of source, ${\overrightarrow v _s} = 0$ Velocity of sound, ${\overrightarrow v _w} = {v_w}\widehat i$ Velocity of sound relative to source $\left( {{{\overrightarrow v }_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{ws}} = {{\vec v}_w} - {{\vec v}_s} \\ {{\vec v}_{ws}} = {v_w}\hat i\quad ...(i) \\\end{aligned} \end{equation} $$ 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 $\left( {{{\overrightarrow v }_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = \left( {{v_w} + {v_o}} \right)\hat i\quad ...(ii) \\\end{aligned} \end{equation} $$ Also, we know, $$\begin{equation} \begin{aligned} f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f \\ f' = \left( {\frac{{{v_w} + {v_o}}}{{{v_w}}}} \right)f \\\end{aligned} \end{equation} $$ | $$f' = \left( {\frac{{{v_w} + {v_o}}}{{{v_w}}}} \right)f$$ |
| 4 | Observer moves away from the stationary source | Velocity of source, ${\overrightarrow v _s} = 0$ Velocity of sound, ${\overrightarrow v _w} = {v_w}\widehat i$ Velocity of sound relative to source $\left( {{{\overrightarrow v }_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{ws}} = {{\vec v}_w} - {{\vec v}_s} \\ {{\vec v}_{ws}} = {v_w}\hat i\quad ...(i) \\\end{aligned} \end{equation} $$ 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 $\left( {{{\overrightarrow v }_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = \left( {{v_w} - {v_o}} \right)\hat i\quad ...(ii) \\\end{aligned} \end{equation} $$ Also, we know, $$\begin{equation} \begin{aligned} f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f \\ f' = \left( {\frac{{{v_w} - {v_o}}}{{{v_w}}}} \right)f \\\end{aligned} \end{equation} $$ | $$f' = \left( {\frac{{{v_w} - {v_o}}}{{{v_w}}}} \right)f$$ |
| 5 | Source and observer both moves towards each other | 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 $\left( {{{\overrightarrow v }_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {\overrightarrow v _{ws}} = {\overrightarrow v _w} - {\overrightarrow v _s} \\ {\overrightarrow v _{ws}} = \left( {{v_w} - {v_s}} \right)\hat i\quad ...(i) \\\end{aligned} \end{equation} $$ 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 $\left( {{{\overrightarrow v }_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = \left( {{v_w} + {v_o}} \right)\hat i\quad ...(ii) \\\end{aligned} \end{equation} $$ Also, we know, $$\begin{equation} \begin{aligned} f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f \\ f' = \left( {\frac{{{v_w} + {v_o}}}{{{v_w} - {v_s}}}} \right)f \\\end{aligned} \end{equation} $$ | $$f' = \left( {\frac{{{v_w} + {v_o}}}{{{v_w} - {v_s}}}} \right)f$$ |
| 6 | Source and observer both move away from each other | 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 $\left( {{{\overrightarrow v }_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {\overrightarrow v _{ws}} = {\overrightarrow v _w} - {\overrightarrow v _s} \\ {\overrightarrow v _{ws}} = \left( {{v_w} +{v_s}} \right)\hat i\quad ...(i) \\\end{aligned} \end{equation} $$ 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 $\left( {{{\overrightarrow v }_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = \left( {{v_w} - {v_o}} \right)\hat i\quad ...(ii) \\\end{aligned} \end{equation} $$ Also, we know, $$\begin{equation} \begin{aligned} f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f \\ f' = \left( {\frac{{{v_w} - {v_o}}}{{{v_w} +{v_s}}}} \right)f \\\end{aligned} \end{equation} $$ | $$f' = \left( {\frac{{{v_w} - {v_o}}}{{{v_w} + {v_s}}}} \right)f$$ |
| 7 | Source moves towards the observer and observer moves away from the source | 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 $\left( {{{\overrightarrow v }_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {\overrightarrow v _{ws}} = {\overrightarrow v _w} - {\overrightarrow v _s} \\ {\overrightarrow v _{ws}} = \left( {{v_w} - {v_s}} \right)\hat i\quad ...(i) \\\end{aligned} \end{equation} $$ 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 $\left( {{{\overrightarrow v }_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = \left( {{v_w} - {v_o}} \right)\hat i\quad ...(ii) \\\end{aligned} \end{equation} $$ Also, we know, $$\begin{equation} \begin{aligned} f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f \\ f' = \left( {\frac{{{v_w} - {v_o}}}{{{v_w} - {v_s}}}} \right)f \\\end{aligned} \end{equation} $$ | $$f' = \left( {\frac{{{v_w} - {v_o}}}{{{v_w} - {v_s}}}} \right)f$$ |
| 8 | Source moves away from the observer and observer moves towards the source | 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 $\left( {{{\overrightarrow v }_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {\overrightarrow v _{ws}} = {\overrightarrow v _w} - {\overrightarrow v _s} \\ {\overrightarrow v _{ws}} = \left( {{v_w} + {v_s}} \right)\hat i\quad ...(i) \\\end{aligned} \end{equation} $$ 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 $\left( {{{\overrightarrow v }_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = \left( {{v_w} + {v_o}} \right)\hat i\quad ...(ii) \\\end{aligned} \end{equation} $$ Also, we know, $$\begin{equation} \begin{aligned} f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f \\ f' = \left( {\frac{{{v_w} + {v_o}}}{{{v_w} + {v_s}}}} \right)f \\\end{aligned} \end{equation} $$ | $$f' = \left( {\frac{{{v_w} + {v_o}}}{{{v_w} + {v_s}}}} \right)f$$ |
