Doppler Effect and It's Cases

Doppler Effect

The apparent change in frequency of sound wave due to the relative motion of source of sound of sound and observer is called Doppler’s effect. For example: You hear the high pitch of the siren of approaching ambulance and you notice dropping of pitch sudenly as ambulance passes you whic is dpppler effect. This phenomenon was first derived by Australian Scientist Doppler. So, it is Doppler’s effect.

Let ‘v’ be the velocity of sound ‘λ’ be the wavelength of sound wave and ‘f’ be the frequency.

$$\text {Then,} \: f = \frac {v}{\lambda } $$

Cases

1. When source of sound moves towards the Observer in rest

   When source of sound moves towards observer in rest, then wavelength of sound decreases. The apparent change in wavelength is given by
   \begin{align*} \lambda ‘ &= \frac {v – u_s}{f} \\ \text {If} \: ‘f’' \: \text { be the apparent change in frequency, Then} \\ f’ &= \frac {v}{\lambda ‘} = \frac {v}{(v –v_s)/f} \\ f’ &= \frac {v}{v – u_s} \times f \dots (i) \\ v &= \text {velocity of sound} \\ u_s &= \text {velocity of source} \\ \lambda ‘ &= \text {changed wavelength} \\ f &= \text {frequency of sound wave} \\ \text {Since,} \\ v>v –u_s \: i.e \: f’>f \\ \end{align*}So, frequency increases when source wave is towards the observer in rest.

2. When source of sound moves away from the Observer in rest

   When source of sound moves away from the observer in rest, the wavelength of sound wave. Therefore, apparent change in wavelength is given by
   \begin{align*} \lambda ‘ &= \frac {v +u_s}{f} \\ v &= \text {velocity of sources} \\ u_s &= \text {velocity of sound} \\ \lambda ‘ &= \text {changed wavelength} \\ f &= \text {frequency of sound wave} \\\text {If f’ be the apparent change in frequency.} \\ \text {Then,} \: fi &= \frac {v}{\lambda } = \frac {v}{(v +u_s)/f} \\ f’ &= \frac {v}{v +u_S} \times f \dots (ii) \\ \text {Since,} \\ v<v + u_s \: i.e \: f’<f \\ \end{align*}So, frequency decreases then source moves away from the observer in rest.

3. When observer moves towards the source in rest

   When observer towards the source in stationary then relative velocity of sound wave to the observer is v +u_o.
   \begin{align*} f’ &= \frac {\text {relative velocity of sound}}{\text {wavelength}} \\ &= \frac {v + u_0}{v/f} = \left ( \frac {v + u_0}{v} \right ) f \\ \therefore f’ &= \left ( \frac {v + u_0}{v} \right ) f \dots (iii) \\ v +u_0 > v \: i.e \: f’ >f \\ \end{align*} So, frequency increases when observer moves towards the source in rest.

4. When observer moves away from the source in rest

   When observer moves away from the source in rest then relative velocity of sound wave to the observer is v + u_0.
   \begin{align*} f’ &= \frac {\text {relative velocity of sound}}{\text {wavelength}} \\ &= \frac {v - u_0}{v/f} = \left ( \frac {v - u_0}{v} \right ) f \\ \therefore f’ &= \left ( \frac {v- u_0}{v} \right ) f \dots (iii) \\ \text {Since, } \\ v -u_0 < v \: i.e \: f’ <f \\ \end{align*} So, frequency decreases when observer moves away from the source in rest.

5. When source and observer moves towards each other

   When the source and observer are approaching towards each other with the velocity u_s and u_o respectively, then
   \begin{align*} \text {velocity of the waves relative to the observer,} v_r = v + u_0, \\ \text {and apparent wavelength,} \lambda ‘ = \frac {v – u_s}{f} \\ \text {Putting these values in equation} \: (v) \: \text {we get} \\ f’ &= \frac {v’}{\lambda ‘} = \frac {v + u_0}{(v – u_s)/f} = \frac {v + u_0}{v –v_s}f \\ \therefore f’ &= \frac {v + u_0}{v –v_s}f \dots (v)\\ \text {Since, } \\ v -u_0 < v + u_s \: i.e \: f’ >f \\ \end{align*}So, frequency increases when source and observer towards each other

6. When source and observer moves away from each other

   When the source and observer moves away from each other with the velocity u_s and u_o respectively, then
   \begin{align*} \text {velocity of the waves relative to the observer,} v_r = v - u_0, \\ \text {and apparent wavelength,} \lambda ‘ = \frac {v + u_s}{f} \\ \text {Putting these values in equation} \: (v) \: \text {we get} \\ f’ &= \frac {v’}{\lambda ‘} = \frac {v - u_0}{(v + u_s)/f} = \frac {v - u_0}{v +v_s}f \\ \therefore f’ &= \frac {v - u_0}{v + v_s}f \dots (vi)\\ \text {Since, } \\ v -u_0 < v + u_s \: i.e \: f’ < f \\ \end{align*}So, frequency decreases when source and observer moves away from each other .

7. When source leads the observer

   When the source and observer moves in same direction and the source is leads the observer, then
   \begin{align*} \text {velocity of the waves relative to the observer,} v_r = v + u_0, \\ \text {and apparent wavelength,} \lambda ‘ = \frac {v + u_s}{f} \\ \text {Putting these values in equation} \: (v) \: \text {we get} \\ f’ &= \frac {v’}{\lambda ‘} = \frac {v + u_0}{(v + u_s)/f} = \frac {v + u_0}{v +v_s}f \\ \therefore f’ &= \frac {v + u_0}{v + v_s}f \dots (vii)\\ \text {So, frequency will change depending on}\: u_o \: \text {and} \: u_s. \\ \end{align*}

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.