Final answer:
The two lowest frequencies at which dogs have an increased sensitivity due to the resonances of their ear canal are approximately 1700 Hz and 5100 Hz.
Step-by-step explanation:
To calculate the lowest frequencies at which dogs have an increased sensitivity in hearing due to the resonance of their ear canal, we can use the following information:
- The speed of sound in warm air (37.0°C) is approximately 350 m/s.
- A dog's ear canal is approximately 5.2 cm or 0.052 m long.
- The ear canal behaves like a tube closed at one end.
The fundamental frequency and the first overtone for a tube closed at one end can be found using the formula:
∑f = v/(4L), where
- ∑f is the fundamental frequency or first harmonic
- v is the speed of sound
- L is the length of the ear canal
The first overtone or second harmonic is three times the fundamental frequency.
Therefore, the lowest resonant frequency (fundamental frequency) for a dog's ear canal is:
f = v/(4L) = 350 m/s / (4 * 0.052 m) = 1682.69 Hz ≈ 1700 Hz
And the first overtone is:
first overtone = 3 * ∑f = 3 * 1682.69 Hz = 5048.08 Hz ≈ 5100 Hz
Thus, the two lowest frequencies at which dogs have an increased sensitivity are approximately 1700 Hz and 5100 Hz.