Concepts of Physics

IIT JEE Physics (1978 to 2018: 41 Years) Topic-wise Complete Solutions

Doppler Effect

This apparent change in the frequency of sound due to relative motion between the source and the observer is given by \begin{align} \nu=\frac{v+u_o}{v-u_s}\nu_0\nonumber \end{align} where, $v$ is the speed of sound in the medium, $u_0$ is the speed of the observer w.r.t. the medium, considered positive when it moves towards the source and negative when it moves away from the source, and $u_s$ is the speed of the source w.r.t. the medium, considered positive when it moves towards the observer and negative when it moves away from the observer.

Problems from IIT JEE

Problem (IIT JEE 2013): Two vehicles, each moving with speed $u$ on the same horizontal straight road, are approaching each other. Wind blows along the road with velocity $w$. One of these vehicles blows a whistle of frequency $f_1$. An observer in the other vehicle hears the frequency of the whistle to be $f_2$. The speed of sound in still air is $v$. The correct statement(s) is (are),

  1. If the wind blows from the observer to the source, $f_2 > f_1$.
  2. If the wind blows from the source to the observer, $f_2 > f_1$.
  3. If the wind blows from the observer to the source, $f_2 < f_1$.
  4. If the wind blows from the source to the observer, $f_2 < f_1$.

Solution: Doppler Effect Doppler's effect equation is, \begin{align} \label{yvb:eqn:1} f_2=\left[\frac{v+u_o}{v-u_s}\right]f_1. \end{align} Let the wind speed be $w$ and wind moves towards the source in case (i) and towards the observer in case (ii) (see figure). In the case (i), $u_o$, the speed of observer w.r.t. medium considered positive when it moves towards the source is $+(u-w)$ and $u_s$, the speed of source w.r.t. medium considered positive when it moves towards observer is $+(u+w)$. Substitute these values in first equation to get, \begin{align} f_2=\left[\frac{v+(u-w)}{v-(u+w)}\right]f_1, \nonumber \end{align} which is greater than $f_1$ (assuming $u>w$). In the case (ii), $u_o$ is $+(u+w)$ and $u_s$ is $+(u-w)$. Substitute these values in first equation to get, \begin{align} f_2=\left[\frac{v+(u+w)}{v-(u-w)}\right]f_1, \nonumber \end{align} which is again greater than $f_1$. The readers are encouraged to show that first equation can be generalized to, \begin{align} f_2=\left[\frac{v+w+u_o}{v+w-u_s}\right]f_1, \nonumber \end{align} where $w$ is positive if the wind blows in the direction of sound and negative if it blows opposite to the direction of sound. Thus, A and B are correct options.