(a) A bat locates insects by emitting ultrasonic “chirps” and then listening for echoes from
the bugs. Suppose a bat chirp has a frequency of 25 kHz. How fast would the bat have to
fly, and in what direction, for you to just barely be able to hear the chirp at 20 kHz?
(b) If this sound has an intensity of 2.0 × 10−3 W/m2
. Your eardrum is 6.0 mm in diameter.
How much energy will be transferred to your eardrum while listening to the bat for 1.0 min?
(a) The Doppler effect is used to determine the frequency shift of a sound wave relative to a moving observer. The equation for the Doppler effect is given by:
f' = (v + vo) / (v + vs) * f
Where:
f' = observed frequency
f = source frequency
v = speed of sound in air (assumed to be 343 m/s)
vo = observer's velocity
vs = source's velocity
To find the velocity of the bat required for you to hear the chirp at 20 kHz, we can rearrange the equation:
20 kHz = (343 m/s + vo) / (343 m/s + vs) * 25 kHz
Dividing both sides of the equation by 25 kHz:
0.8 = (343 m/s + vo) / (343 m/s + vs)
Rearranging the equation again:
vo = 0.8 * (343 m/s + vs) - 343 m/s
We can assume that the velocity of sound vs is 0, as the sound waves are propagating through still air. Therefore:
vo = 0.8 * (343 m/s) - 343 m/s
vo = -68.6 m/s
The bat would have to fly at a velocity of approximately 68.6 m/s in the opposite direction for you to barely hear the chirp at 20 kHz.
(b) The energy transferred to your eardrum can be calculated using the formula:
Energy = Power * Time
Power is given by the formula:
Power = Intensity * Area
The area of your eardrum can be calculated using the formula for the area of a circle:
Area = π * (d/2)^2
Where:
d = diameter of your eardrum (6.0 mm = 0.006 m)
Plugging in the values:
Area = π * (0.006 m/2)^2
Area = π * (0.006 m)^2 / 4
Area = 7.06 x 10^-5 m^2
Now we can calculate the power:
Power = Intensity * Area
Power = (2.0 x 10^-3 W/m^2) * (7.06 x 10^-5 m^2)
Power = 1.41 x 10^-7 W
Finally, we can calculate the energy transferred to your eardrum over 1.0 minute:
Energy = Power * Time
Energy = (1.41 x 10^-7 W) * (60 s)
Energy = 8.46 x 10^-6 J
So, approximately 8.46 x 10^-6 Joules of energy will be transferred to your eardrum while listening to the bat for 1.0 minute.