Question

Obtain an expression for the apparent frequency of a note by an observer when
(a) source alone is in motion towards the observer.
(b) source alone is in motion away from the observer.
(c) observer alone is in motion towards the source.
(d) observer alone is in motion away from the source.
(e) when the source and observer are moving towards each other.​

Answer 1

The expression for apparent frequency of the sound heard by the observer is,

$\displaystyle\longrightarrow\sf{\nu'=\dfrac{v'}{\lambda'}\quad\quad\dots(1)}$

where $\displaystyle\sf{v'}$ and $\displaystyle\sf{\lambda'}$ are the apparent velocity and apparent wavelength of the sound respectively.

Since the apparent velocity $\displaystyle\sf{v'}$ is due to the motion of observer (apparent velocity is the velocity of sound heard by the listener),

$\displaystyle\longrightarrow\sf{v'=v\pm v_o}$

where $\displaystyle\sf{v}$ is the actual velocity of the sound in air and $\displaystyle\sf{v_o}$ is the velocity of observer, which has positive sign if it is opposite to that of sound and negative sign if it is in the same direction as that of the sound.

Since the apparent wavelength is due to the motion in source,

$\displaystyle\longrightarrow\sf{\lambda'=\dfrac{v\pm v_s}{\nu}}$

where $\displaystyle\sf{\nu}$ is the actual frequency of the sound, and $\displaystyle\sf{v_s}$ is the velocity of the source, which also has positive sign if it is opposite to that of sound and negative sign if it is in the same direction as that of the sound.

Hence (1) becomes,

$\displaystyle\longrightarrow\sf{\nu'=\dfrac{v\pm v_o}{\left(\dfrac{v\pm v_s}{\nu}\right)}}$

$\displaystyle\longrightarrow\sf{\underline{\underline{\nu'=\nu\left(\dfrac{v\pm v_o}{v\pm v_s}\right)}}}$

Case (a) :- If source alone is in motion towards the observer who is in rest,

$\displaystyle\sf{\boxed{\sf{S}}\longrightarrow\quad\quad \boxed{\sf{O}}}\\\\\sf{\quad v\longrightarrow}$

• Observer is in rest, therefore $\displaystyle\sf{v_o=0.}$

• Source moves in the same direction of the sound, therefore $\displaystyle\sf{v-v_s}$ is taken.

Hence the apparent frequency is,

$\displaystyle\longrightarrow\sf{\underline{\underline{\nu'=\nu\left(\dfrac{v}{v-v_s}\right)}}}$

Case (b) :- If source alone is in motion away from the observer who is in rest,

$\displaystyle\sf{\longleftarrow\boxed{\sf{S}}\quad\quad \boxed{\sf{O}}}\\\\\sf{\quad \quad \ v\longrightarrow}$

• Observer is in rest, therefore $\displaystyle\sf{v_L=0.}$

• Source moves opposite to the sound, therefore $\displaystyle\sf{v+v_s}$ is taken.

Hence the apparent frequency is,

$\displaystyle\longrightarrow\sf{\underline{\underline{\nu'=\nu\left(\dfrac{v}{v+v_s}\right)}}}$

Case (c) :- If observer alone is in motion towards the source which is in rest,

$\displaystyle\sf{\boxed{\sf{S}}\quad\quad\longleftarrow\boxed{\sf{O}}}\\\\\sf{\quad\!v\longrightarrow}$

• Source is in rest, therefore $\displaystyle\sf{v_s=0.}$

• Observer moves opposite to the sound, therefore $\displaystyle\sf{v+v_o}$ is taken.

Hence the apparent frequency is,

$\displaystyle\longrightarrow\sf{\underline{\underline{\nu'=\nu\left(\dfrac{v+v_o}{v}\right)}}}$

Case (d) :- If observer alone is in motion away from the source which is in rest,

$\displaystyle\sf{\boxed{\sf{S}}\quad\quad\boxed{\sf{O}}\longrightarrow}\\\\\sf{\quad\!v\longrightarrow}$

• Source is in rest, therefore $\displaystyle\sf{v_s=0.}$

• Observer moves in the same direction of the sound, therefore $\displaystyle\sf{v-v_o}$ is taken.

Hence the apparent frequency is,

$\displaystyle\longrightarrow\sf{\underline{\underline{\nu'=\nu\left(\dfrac{v-v_o}{v}\right)}}}$

Case (e) :- If both source and observer are moving towards each other,

$\displaystyle\sf{\boxed{\sf{S}}\longrightarrow\quad\quad\longleftarrow\boxed{\sf{O}}}\\\\\sf{\quad\!v\longrightarrow}$

• Source is moving in the same direction of the sound, therefore $\displaystyle\sf{v-v_s}$ is taken.

• Observer moves opposite to the sound, therefore $\displaystyle\sf{v+v_o}$ is taken.

Hence the apparent frequency is,

$\displaystyle\longrightarrow\sf{\underline{\underline{\nu'=\nu\left(\dfrac{v+v_o}{v-v_s}\right)}}}$

$\displaystyle\sf{\left[\ \boxed{\sf{S}}=source,\quad\boxed{\sf{O}}=observer,\quad v=velocity\ of\ sound\ \!\right]}$