Dolphins emit clicks of sound for communication and echolocation. A marine biologist is monitoring a dolphin swimming in seawater where the speed of sound is 1522 m/s. When the dolphin is swimming directly away at 7.9 m/s, the marine biologist measures the number of clicks occurring per second to be at a frequency of 2336 Hz. What is the difference (in Hz) between this frequency and the number of clicks per second actually emitted by the dolphin?
To find the difference in frequency (in Hz) between the measured frequency and the actual frequency emitted by the dolphin, we need to account for the Doppler effect.
The Doppler effect is the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the source of the wave. In this case, the dolphin is moving away from the observer, causing a shift in the frequency of the clicks.
The formula to calculate the observed frequency (f') due to the Doppler effect is:
f' = (v + v_obs) / (v + v_sound) * f
Where:
f' is the observed frequency
v is the speed of sound in the medium (1522 m/s)
v_obs is the velocity of the observer (marine biologist) relative to the medium (7.9 m/s, as the dolphin is swimming away)
v_sound is the velocity of the source (dolphin) relative to the medium (unknown)
f is the emitted frequency
We can rearrange the formula to solve for the difference in frequency (Δf):
Δf = f - f'
Let's substitute the given values into the formula:
f' = (1522 + 7.9) / (1522) * 2336
f' ≈ 2342.5 Hz
Δf = 2336 Hz - 2342.5 Hz
Δf ≈ -6.5 Hz
Therefore, the difference in frequency between the measured value and the actual emitted frequency by the dolphin is approximately -6.5 Hz.