Chapter 16, Problem 16.69P

BIO Horseshoe bats (genus Rhinolophus) emit sounds from their nostrils and then listen to the frequency of the sound reflected from their prey to determine the prey’s speed. (The “horseshoe” that gives the bat its name is a depression around the nostrils that acts like a focusing mirror, so that the bat emits sound in a narrow beam like a flashlight.) A Rhinolophus flying at speed υ_bat emits sound of frequency f_bat; the sound it hears reflected from an insect flying toward it has a higher frequency f_refl. (a) Show that the speed of the insect is

${\upsilon }_{\text{in sect}}=\upsilon \left[\frac{f_{\text{refl}}\left(\upsilon -{\upsilon }_{\text{bat}}\right)-f_{\text{bat}}\left(\upsilon +{\upsilon }_{\text{bat}}\right)}{f_{\text{refl}}\left(\upsilon -{\upsilon }_{\text{bat}}\right)+f_{\text{bat}}\left(\upsilon +{\upsilon }_{\text{bat}}\right)}\right]$

where υ is the speed of sound. (b) If f_bat = 80.7 kHz, f_refl = 83.5 kHz. and υ_bat = 3.9 m/s, calculate the speed of the insect.