A bat flying toward a wall at a speed of 6.47 m/s emits an ultrasound wave with a frequency of 27.5 kHz. What frequency does the reflected wave have when it reaches the flying bat? (Assume that the speed of sound is v = 341 m/s.)
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To find the frequency of the reflected wave, we need to consider the Doppler effect. The Doppler effect describes the change in frequency or wavelength of a wave due to the relative motion between the source of the wave and the observer.
In this case, the bat is the source emitting the ultrasound wave, and it is moving towards the wall. When the wave reflects off the wall and reaches the bat, the bat is still moving towards the wall. Therefore, the observer (the bat) is moving towards the source (the wall).
The Doppler effect formula for frequency is given by:
f' = (v + vo) / (v + vs) * f
Where:
f' is the observed frequency (reflected wave frequency)
f is the emitted frequency (original wave frequency)
v is the speed of sound (341 m/s)
vo is the velocity of the observer (bat) relative to the medium (wall)
vs is the velocity of the source (wall) relative to the medium (bat)
In this case, since the bat is moving towards the wall, the velocity of the observer (vo) is positive, and the velocity of the source (vs) is negative.
Given:
f = 27.5 kHz = 27.5 x 10^3 Hz
v = 341 m/s
vo = 6.47 m/s
vs = -6.47 m/s
Substituting these values into the formula:
f' = (341 + 6.47) / (341 - (-6.47)) * 27.5 x 10^3
Simplifying:
f' = 347.47 / 347.47 * 27.5 x 10^3
f' = 27.5 x 10^3 Hz
Therefore, the frequency of the reflected wave, when it reaches the flying bat, is still 27.5 kHz or 27.5 x 10^3 Hz.