Question

Horseshoe bats (genus Rhinolophus) emit sounds from their nostrils, then listen to the frequency of the sound reflected from their prey to determine the prey's speed. The horse shoe that gives the bat its name is a depression around the nostrils that acts like a focussing mirror so that the bat emits sound in a narrow beam like a flash light. A Rhinolophus is flying at a speed $v_{bat}$ and emits sound of frequency $f_{bat}$ the sound it hears reflected from an insect flying towards it has higher frequency $f_{ref}$ , speed of sound is $v_s$. Find the speed of the insect $v_{insect}$ :

• A. $v_s\left[\frac{f_{ref}(v_s-v_{bat})-f_{bat}(v_s+v_{bat})}{f_{ref}(v_s-v_{bat})+f_{bat}(v_s+v_{bat})}\right]$
• B. $v_s\left[\frac{f_{ref}(v_s+v_{bat})-f_{bat}(v_s-v_{bat})}{f_{ref}(v_s+v_{bat})+f_{bat}(v_s-v_{bat})}\right]$
• C. $v_s\left[\frac{f_{ref}(v_s+v_{bat})-f_{bat}(v_s-v_{bat})}{f_{ref}(v_s-v_{bat})+f_{bat}(v_s+v_{bat})}\right]$
• D. $v_s\left[\frac{f_{ref}(v_s+v_{bat})+f_{bat}(v_s-v_{bat})}{f_{ref}(v_s-v_{bat})-f_{bat}(v_s+v_{bat})}\right]$

Answer 1

The correct option is A $v_s\left[\frac{f_{ref}(v_s-v_{bat})-f_{bat}(v_s+v_{bat})}{f_{ref}(v_s-v_{bat})+f_{bat}(v_s+v_{bat})}\right]$

The sound first hits the insect. The insect observes some different frequency due to Doppler effect. The frequency of reflected sound is same as that of this frequency.

The bat finally observes this incident sound with a further different frequency.

Frequency observed by insect $=f_1=f_{bat}\left(\frac{v_s+v_{insect}}{v_s-v_{bat}}\right)$

Now this frequency is being reflected and being observed by bat.

Thus, $f_{ref}=f_1\left(\frac{v_s+v_{bat}}{v_s-v_{insect}}\right)=f_{bat}\left(\frac{v_s+v_{insect}}{v_s-v_{bat}}\right)\left(\frac{v_s+v_{bat}}{v_s-v_{insect}}\right)$

$⟹v_{insect}=v_s\left[\frac{f_{ref}(v_s-v_{bat})-f_{bat}(v_s+v_{bat})}{f_{ref}(v_s-v_{bat})+f_{bat}(v_s+v_{bat})}\right]$