6.4 Questions

2.1 Relation between displacement wave and pressure wave
2.2 Relation between pressure wave and density wave

4.1 Different case of Doppler's Effect

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

Question 3. A source of sound which emits a sound of frequency 600 $Hz$ is moving towards a wall with a velocity of $30\ m/s$. Two observers $A$ & $B$ moving with velocity $20\ m/s$ as shown in the figure. Velocity of sound in air is $330\ m/s$.

(a) Find the frequency of direct sound observed by observer $A$.

(b) Find the frequency of direct sound observed by observer $B$.

(c) Find the frequency of reflected sound observed by observer $A$.

(d) Find the frequency of reflected sound observed by observer $B$.

Velocity of sound, ${\vec v_w} = - 330\;\widehat i$

Velocity of source, ${\vec v_s} = 30\;\widehat i$

Velocity of observer $A$, ${\vec v_o} = - 20\;\widehat i$

Velocity of sound relative to observer $\left( {{{\vec v}_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = - 310\;\widehat i \\\end{aligned} \end{equation} $$

Velocity of sound relative to source $\left( {{{\vec v}_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{ws}} = {{\vec v}_w} - {{\vec v}_s} \\ {{\vec v}_{ws}} = - 360\;\widehat i \\\end{aligned} \end{equation} $$

Frequency of direct sound observed by observer $A$, $$\begin{equation} \begin{aligned} f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f = \left( {\frac{{ - 310}}{{ - 360}}} \right)(600) \\ f' = 516.67\;Hz \\\end{aligned} \end{equation} $$

Velocity of sound, ${\vec v_w} = 330\;\widehat i$

Velocity of source, ${\vec v_s} = 30\;\widehat i$

Velocity of observer $B$, ${\vec v_o} = 20\;\widehat i$

Velocity of sound relative to observer $\left( {{{\vec v}_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = 310\;\widehat i \\\end{aligned} \end{equation} $$

Velocity of sound relative to source $\left( {{{\vec v}_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{ws}} = {{\vec v}_w} - {{\vec v}_s} \\ {{\vec v}_{ws}} = 300\;\widehat i \\\end{aligned} \end{equation} $$

Frequency of direct sound observed by observer $A$, $$\begin{equation} \begin{aligned} f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f = \left( {\frac{{310}}{{ 300}}} \right)(600) \\ f' = 620\;Hz \\\end{aligned} \end{equation} $$

Velocity of sound, ${\vec v_w} = 330\;\widehat i$

Velocity of source, ${\vec v_s} = 30\;\widehat i$

Velocity of observer (wall), ${\vec v_o} = 0$

Velocity of sound relative to observer $\left( {{{\vec v}_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = 330\;\widehat i \\\end{aligned} \end{equation} $$

Velocity of sound relative to source $\left( {{{\vec v}_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{ws}} = {{\vec v}_w} - {{\vec v}_s} \\ {{\vec v}_{ws}} = 300\;\widehat i \\\end{aligned} \end{equation} $$

Frequency of sound at wall, $$\begin{equation} \begin{aligned} f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f = \left( {\frac{{ 330}}{{ 300}}} \right)(600) \\ f' = 660\;Hz \\\end{aligned} \end{equation} $$

Frequecy of sound refected by wall is 660 $Hz$ (Wall does not change the frequency).

Velocity of sound, ${\vec v_w} = -330\;\widehat i$

Velocity of source (wall), ${\vec v_s} = 0$

Velocity of observer $A$ (wall), ${{\vec v}_o} = - 20\;\hat i$

Velocity of sound relative to observer $\left( {{{\vec v}_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = - 310\;\hat i \\\end{aligned} \end{equation} $$

Velocity of sound relative to source $\left( {{{\vec v}_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{ws}} = {{\vec v}_w} - {{\vec v}_s} \\ {{\vec v}_{ws}} = -330\;\widehat i \\\end{aligned} \end{equation} $$

Frequency of sound at wall, $$\begin{equation} \begin{aligned} f'' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f = \left( {\frac{{ -310}}{{ -330}}} \right)(660) \\ f'' = 620\;Hz \\\end{aligned} \end{equation} $$

Velocity of sound, ${\vec v_w} = -330\;\widehat i$

Velocity of source (wall), ${\vec v_s} = 0$

Velocity of observer $B$ (wall), ${{\vec v}_o} = 20\;\hat i$

Velocity of sound relative to observer $\left( {{{\vec v}_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = - 350\;\hat i \\\end{aligned} \end{equation} $$

Velocity of sound relative to source $\left( {{{\vec v}_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{ws}} = {{\vec v}_w} - {{\vec v}_s} \\ {{\vec v}_{ws}} = -330\;\widehat i \\\end{aligned} \end{equation} $$

Frequency of sound at wall, $$\begin{equation} \begin{aligned} f'' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f = \left( {\frac{{ -350}}{{ -330}}} \right)(660) \\ f'' = 700\;Hz \\\end{aligned} \end{equation} $$

Question 4. A car approaching a crossing at a speed of $20m/s$ sounds a horn of frequency $500 Hz$ when $80m$ from the crossing. Speed of sound in air is $330 m/s$. What frequency is heard by an observer $60m$ from the crossing on the straight road which crosses car road at right angles?

Velocity of sound, ${\vec v_w} = \;{v_w}\;\hat i$

Velocity of source, ${\vec v_s} = {v_s}\cos \theta \;\hat i$

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

Velocity of sound relative to observer $\left( {{{\vec v}_{wo}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{wo}} = {{\vec v}_w} - {{\vec v}_o} \\ {{\vec v}_{wo}} = {v_w}\;\hat i \\\end{aligned} \end{equation} $$

Velocity of sound relative to source $\left( {{{\vec v}_{ws}}} \right)$, $$\begin{equation} \begin{aligned} {{\vec v}_{ws}} = {{\vec v}_w} - {{\vec v}_s} \\ {{\vec v}_{ws}} = \left( {{v_w} - {v_s}\cos \theta } \right)\;\hat i \\\end{aligned} \end{equation} $$

Frequency of sound observed by observer, $$\begin{equation} \begin{aligned} f' = \left( {\frac{{{{\vec v}_{wo}}}}{{{{\vec v}_{ws}}}}} \right)f \\ f' = \left( {\frac{{{v_w}}}{{{v_w} - {v_s}\cos \theta }}} \right)f \\ f' = \left( {\frac{{330}}{{330 - 20\left( {\frac{{80}}{{100}}} \right)}}} \right)(500) \\ f' = 525.47\;Hz \\\end{aligned} \end{equation} $$